1. Introduction
To understand the holographic emergence of bulk dynamics in AdS/CFT [1–3], various approaches have been suggested [4–9]. In [9], after including the conformal transformation of the renormalization scale, it was shown that the bulk dynamics of a scalar field was highly constrained by the SO(2, d) conformal symmetry of the dual CFT scalar operators. Furthermore, the holographic emergence of the dynamics for the bulk gravity itself is governed by the generic duality relation between the boundary global symmetry and the bulk gauge symmetry. Applying it to the SO(2, d) conformal symmetry in CFTd, the bulk AdSd+1 gravity is reformulated as a SO(2, d) gauge theory in [10]. In this formalism, the pullback of the bulk Einstein equation on a co-dimension one hypersurface Σ can be naturally related to the covariant anomaly of the CFT SO(2, d) conservation law. Providing that the SO(2, d) conservation law is not anomalous for any local renormalization scale, the full bulk Einstein equation will be automatically implied.
In this paper, we study the formal mathematical structure of the holographic SO(2, d) anomaly systematically. In section 2.1 , we briefly review the bulk SO(2, d) covariant action [10] for AdS gravity. The covariant formula of anomalous boundary conservation law is revealed during the Noether procedure in section 2.2 . In section 3.1.1 , the bulk Chern–Simons like action is constructed by the homotopic integration. To resolve the issue for large gauge transformation, a topological term is imposed in section 3.1.2 . In section 3.2 , the boundary effective action is also constructed via the homotopic integration method. Then in section 3.3 , the descendent structure of the holographic SO(2, d) anomaly is established after obtaining the d + 2 form anomaly characteristic class. In section 4 , we summarize our results and discuss possible generalizations. The details of our derivations are summarized in the Appendices.
2. AdS gravity as SO(2, d) gauge theory
2.1. Bulk covariant action
By unifying the vielbein ${e}^{\hat{a}}$ and the SO(1, d) spin connection ${\omega }_{\,\,\hat{b}}^{\hat{a}}$ to the SO(2, d) gauge field ${A}_{\,\,\hat{\beta }}^{\hat{\alpha }}$, the Palatini's 1st order formalism for the AdSd+1 gravity was reformulated as a SO(2, d) gauge theory in [10]. The SO(2, d) indices $\hat{\alpha },\hat{\beta }=\hat{\bullet },\hat{0},\cdots \,{,}\hat{d}$ contain one more extra direction $\hat{\bullet }$ than the vielbein indices $\hat{a},\hat{b}=0,\cdots \,{,}d$. Thus naively we have 3 ) as the bulk covariant action since it is given by the summation of manifestly SO(2, d) covariant terms. Obviously, ${Y}^{\hat{\alpha }}$ is just an auxiliary field since it can be totally fixed via the SO(2, d) gauge transformation. The physical degrees of freedom remain same as the original Palatini action. A more systematical discussion can be found in appendix A .
$\begin{eqnarray}{A}_{\,\,\hat{\beta }}^{\hat{\alpha }}=\left(\begin{array}{cc}{A}_{\,\,\hat{\bullet }}^{\hat{\bullet }} & {A}_{\,\,\hat{b}}^{\hat{\bullet }}\\ {A}_{\,\,\hat{\bullet }}^{\hat{a}} & {A}_{\,\,\hat{b}}^{\hat{a}}\end{array}\right)=\left(\begin{array}{cc}0 & -{{\ell }}^{-1}{e}_{\hat{b}}\\ {{\ell }}^{-1}{e}^{\hat{a}} & {\omega }_{\,\,\hat{b}}^{\hat{a}}\end{array}\right).\end{eqnarray}$
To figure out the SO(2, d) → SO(1, d) reduction in the SO(2, d) covariant way, we further introduce the ruler field ${Y}^{\hat{\alpha }}$ which satisfies the constraint $\begin{eqnarray}{\eta }_{\hat{\alpha }\hat{\beta }}{Y}^{\hat{\alpha }}{Y}^{\hat{\beta }}=-{{\ell }}^{2}.\end{eqnarray}$
In terms of ${Y}^{\hat{\alpha }}$ and the curvature 2-form ${F}_{\,\,\hat{\beta }}^{\hat{\alpha }}\,={\rm{d}}{A}_{\,\,\hat{\beta }}^{\hat{\alpha }}+{A}_{\,\,\hat{\gamma }}^{\hat{\alpha }}\wedge {A}_{\,\,\hat{\beta }}^{\hat{\gamma }}$, the Palatini action of the AdSd+1 Einstein gravity is reformulated as $\begin{eqnarray}\begin{array}{rcl}{S}_{(d+1)}^{{\rm{cov}}} & = & {\displaystyle \int }_{M}{{ \mathcal L }}_{(d+1)}^{{\rm{cov}}}\\ & = & \frac{1}{2{\kappa }^{2}{\ell }\,(d-1)!}{\displaystyle \int }_{M}{\epsilon }_{\hat{\alpha }\hat{\beta }{\hat{\alpha }}_{1}\cdots {\hat{\alpha }}_{d}}\\ & & \times \left[{F}^{\hat{\alpha }\hat{\beta }}-\frac{2}{(d+1){{\ell }}^{2}}{\rm{D}}{Y}^{\hat{\alpha }}\wedge {\rm{D}}{Y}^{\hat{\beta }}\right]\\ & & \times \wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{1}}\wedge \cdots \wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{d-1}}{Y}^{{\hat{\alpha }}_{d}}.\end{array}\end{eqnarray}$
where D = d + A is the SO(2, d) gauge covariant derivative. We shall refer (In this formalism, the spacetime metric is given by the gauge invariant quadratic form B , it can be shown directly that the action (3 ) is equivalent to the original Einstein–Hilbert action by using equations (5 ) and (6 ).
$\begin{eqnarray}{g}_{MN}={\eta }_{\hat{\alpha }\hat{\beta }}{{\rm{D}}}_{M}{Y}^{\hat{\alpha }}{{\rm{D}}}_{N}{Y}^{\hat{\beta }}.\end{eqnarray}$
When the metric is not degenerate, $\{{Y}^{\hat{\alpha }},{{\rm{D}}}_{M}{Y}^{\hat{\alpha }}\}$ forms an intrinsic basis of the d + 2 dimensional space of the SO(2, d) vector representation. The spacetime geometric quantities can be recovered by expanding the SO(2, d) covariant quantities on the intrinsic basis. For example, the SO(2, d) field strength is expanded as $\begin{eqnarray}\begin{array}{r}{({F}_{{M}_{1}{M}_{2}})}^{\hat{\alpha }\hat{\beta }}=\left({{R}^{{N}_{1}{N}_{2}}}_{{M}_{1}{M}_{2}}+2{{\ell }}^{-2}{\delta }_{[{M}_{1}}^{{N}_{1}}{\delta }_{{M}_{2}]}^{{N}_{2}}\right)\\ \,\times {{\rm{D}}}_{{N}_{1}}{Y}^{[\hat{\alpha }}{{\rm{D}}}_{{N}_{2}}{Y}^{\hat{\beta }]}-4{{\ell }}^{-2}{{t}^{N}}_{[{M}_{1}{M}_{2}]}{Y}^{[\hat{\alpha }}{{\rm{D}}}_{N}{Y}^{\hat{\beta }]},\end{array}\end{eqnarray}$
where ${{t}^{N}}_{{M}_{1}{M}_{2}}$ is the torsion and ${{R}^{{N}_{1}{N}_{2}}}_{{M}_{1}{M}_{2}}$ is the Riemann curvature for the torsional connection. We also notice that $\begin{eqnarray}\begin{array}{l}\frac{1}{{\ell }\,p!(d+1-p)!}{\epsilon }_{{\hat{\alpha }}_{0}\cdots {\hat{\alpha }}_{p-1}{\hat{\alpha }}_{p}\cdots {\hat{\alpha }}_{d+1}}\\ \quad \times \,{\epsilon }^{{M}_{0}\cdots {M}_{p-1}{M}_{p}\cdots {M}_{d}}\\ \quad \times \,{{\rm{D}}}_{{M}_{p}}{Y}^{{\hat{\alpha }}_{p}}\cdots {{\rm{D}}}_{{M}_{d}}{Y}^{{\hat{\alpha }}_{d}}{Y}^{{\hat{\alpha }}_{d+1}}\\ \,={{\rm{D}}}^{[{M}_{0}}{Y}_{[{\hat{\alpha }}_{0}}\cdots {{\rm{D}}}^{{M}_{p-1}]}{Y}_{{\hat{\alpha }}_{p-1}]},\end{array}\end{eqnarray}$
where the spacetime indices are raised by gMN and $\begin{eqnarray}{\epsilon }^{01\cdots d}=\frac{1}{\sqrt{g}},g=-\det ({g}_{MN}).\end{eqnarray}$
As explained in detail in appendix The variation of the bulk covariant action on a manifold M with boundary Σ = ∂M is given by A , the bulk equations of motion for δA13 ), the bulk equations of motion for δY
$\begin{eqnarray}\begin{array}{rcl}\delta {S}_{(d+1)}^{{\rm{cov}}} & = & {\displaystyle \int }_{M}\left[\delta {A}^{\hat{\alpha }\hat{\beta }}\wedge {({{ \mathcal J }}_{(d+1)})}_{\hat{\alpha }\hat{\beta }}+\delta {Y}^{\hat{\alpha }}{({{ \mathcal J }}_{(d+1)})}_{\hat{\alpha }}\right]\\ & & +{\displaystyle \int }_{{\rm{\Sigma }}}\left[\delta {A}^{\hat{\alpha }\hat{\beta }}\wedge {({{ \mathcal K }}_{(d+1)}^{{\rm{cov}}})}_{\hat{\alpha }\hat{\beta }}+\delta {Y}^{\hat{\alpha }}{({{ \mathcal K }}_{(d+1)}^{{\rm{cov}}})}_{\hat{\alpha }}\right],\end{array}\end{eqnarray}$
where ${{ \mathcal J }}_{(d+1)}$ are the bulk off-shell currents $\begin{eqnarray}\begin{array}{rcl}{({{ \mathcal J }}_{(d+1)})}_{\hat{\alpha }\hat{\beta }} & = & \frac{\partial {{ \mathcal L }}_{(d+1)}^{{\rm{cov}}}}{\partial {A}^{\hat{\alpha }\hat{\beta }}}+{\rm{d}}\left(\frac{\partial {{ \mathcal L }}_{(d+1)}^{{\rm{cov}}}}{\partial ({\rm{d}}{A}^{\hat{\alpha }\hat{\beta }})}\right)\\ & = & \frac{{(-1)}^{d+1}{\ell }}{4{\kappa }^{2}(d-2)!}{\epsilon }_{\hat{\alpha }\hat{\beta }{\hat{\alpha }}_{1}\cdots {\hat{\alpha }}_{d}}\\ & & \times \,{F}^{{\hat{\alpha }}_{1}{\hat{\alpha }}_{2}}\wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{3}}\wedge \cdots \wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{d}},\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{rcl}{({{ \mathcal J }}_{(d+1)})}_{\hat{\alpha }} & = & \frac{\partial {{ \mathcal L }}_{(d+1)}^{{\rm{cov}}}}{\partial {Y}^{\hat{\alpha }}}-{\rm{d}}\left(\frac{\partial {{ \mathcal L }}_{(d+1)}^{{\rm{cov}}}}{\partial ({\rm{d}}{Y}^{\hat{\alpha }})}\right)\\ & = & \frac{{(-1)}^{d+1}}{2{\kappa }^{2}{\ell }\,(d-2)!}{\epsilon }_{\hat{\alpha }\hat{\beta }{\hat{\alpha }}_{1}\cdots {\hat{\alpha }}_{d}}\\ & & \times {Y}^{\hat{\beta }}{\rm{D}}({F}^{{\hat{\alpha }}_{1}{\hat{\alpha }}_{2}}\wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{2}}\wedge \cdots \wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{d}}),\end{array}\end{eqnarray}$
and ${{ \mathcal K }}_{(d+1)}^{{\rm{cov}}}$ are the bulk off-shell Noether potentials [11] $\begin{eqnarray}\begin{array}{rcl}{({{ \mathcal K }}_{(d+1)}^{{\rm{cov}}})}_{\hat{\alpha }\hat{\beta }} & = & \frac{\partial {{ \mathcal L }}_{(d+1)}^{{\rm{cov}}}}{\partial ({\rm{d}}{A}^{\hat{\alpha }\hat{\beta }})}=\frac{{(-1)}^{d+1}}{2{\kappa }^{2}{\ell }\,(d-1)!}\\ & & \times {\epsilon }_{\hat{\alpha }\hat{\beta }{\hat{\alpha }}_{1}\cdots {\hat{\alpha }}_{d}}{Y}^{{\hat{\alpha }}_{1}}{\rm{D}}{Y}^{{\hat{\alpha }}_{2}}\wedge \cdots \wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{d}},\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{rcl}{({{ \mathcal K }}_{(d+1)}^{{\rm{cov}}})}_{\hat{\alpha }} & = & \frac{\partial {{ \mathcal L }}_{(d+1)}^{{\rm{cov}}}}{\partial ({\rm{d}}{Y}^{\hat{\alpha }})}=\frac{{(-1)}^{d}}{2{\kappa }^{2}{\ell }\,(d-2)!}\\ & & \times \,{\epsilon }_{\hat{\alpha }\hat{\beta }{\hat{\alpha }}_{1}\cdots {\hat{\alpha }}_{d}}{Y}^{\hat{\beta }}\left[{F}^{{\hat{\alpha }}_{1}{\hat{\alpha }}_{2}}\wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{3}}\wedge \cdots \wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{d}}\right.\\ & & \left.-\frac{2}{{{\ell }}^{2}(d-1)}{\rm{D}}{Y}^{{\hat{\alpha }}_{1}}\wedge \cdots \wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{d}}\right].\end{array}\end{eqnarray}$
As analysed in [10] as well as appendix $\begin{eqnarray}\begin{array}{rcl}{({{ \mathcal J }}_{(d+1)})}_{\hat{\alpha }\hat{\beta }} & = & \frac{{(-1)}^{d+1}{\ell }}{4{\kappa }^{2}(d-2)!}{\epsilon }_{\hat{\alpha }\hat{\beta }{\hat{\alpha }}_{1}\cdots {\hat{\alpha }}_{d}}\\ & & \times \,{F}^{{\hat{\alpha }}_{1}{\hat{\alpha }}_{2}}\wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{3}}\wedge \cdots \wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{d}}=0\end{array}\end{eqnarray}$
are equivalent to the Einstein equation plus the torsion free condition. Providing equation ( $\begin{eqnarray}\begin{array}{l}{({{ \mathcal J }}_{(d+1)})}_{\hat{\alpha }}=\frac{{(-1)}^{d+1}}{2{\kappa }^{2}{\ell }\,(d-2)!}{\epsilon }_{\hat{\alpha }\hat{\beta }{\hat{\alpha }}_{1}\cdots {\hat{\alpha }}_{d}}\\ \quad \times \,{Y}^{\hat{\beta }}{\rm{D}}({F}^{{\hat{\alpha }}_{1}{\hat{\alpha }}_{2}}\wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{3}}\wedge \cdots \wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{d}})=0\end{array}\end{eqnarray}$
are automatically satisfied. Therefore, the introducing of ${Y}^{\hat{\alpha }}$ does not impose any additional constraints beyond the original Palatini equations.2.2. The off-shell conservation laws
Substituting the infinitesimal SO(2, d) gauge transformation 8 ), the gauge invariance of the action ${\delta }_{u}{S}_{(d+1)}^{{\rm{cov}}}=0$ implies the off-shell bulk conservation law 19 ) give rise to the covariant formalism of the CFT currents
$\begin{eqnarray}{\delta }_{u}{Y}^{\hat{\alpha }}={u}_{\,\,\hat{\beta }}^{\hat{\alpha }}{Y}^{\hat{\beta }},\qquad {\delta }_{u}{A}^{\hat{\alpha }\hat{\beta }}=-{\rm{D}}{u}^{\hat{\alpha }\hat{\beta }},\end{eqnarray}$
into the general variation ( $\begin{eqnarray}{\rm{D}}{({{ \mathcal J }}_{(d+1)})}_{\hat{\alpha }\hat{\beta }}+{Y}_{[\hat{\beta }}{({{ \mathcal J }}_{(d+1)})}_{\hat{\alpha }]}=0.\end{eqnarray}$
For the manifold with boundary Σ $\begin{eqnarray}{{\mathbb{Y}}}^{\hat{\alpha }}={f}_{{\rm{\Sigma }}}^{* }[{Y}^{\hat{\alpha }}],\quad {{\mathbb{A}}}^{\hat{\alpha }\hat{\beta }}={f}_{{\rm{\Sigma }}}^{* }[{A}^{\hat{\alpha }\hat{\beta }}],\end{eqnarray}$
the gauge invariance also implies the boundary off-shell anomalous conservation law $\begin{eqnarray}{\mathbb{D}}{({{ \mathcal J }}_{(d)}^{{\rm{cov}}})}_{\hat{\alpha }\hat{\beta }}+{{\mathbb{Y}}}_{[\hat{\beta }}{({{ \mathcal J }}_{(d)}^{{\rm{cov}}})}_{\hat{\alpha }]}={({{ \mathcal A }}_{(d)}^{{\rm{cov}}})}_{\hat{\alpha }\hat{\beta }},\end{eqnarray}$
where the boundary currents and anomaly are given by pullback of the bulk Noether potential and current on Σ respectively $\begin{eqnarray}\begin{array}{rcl}{({{ \mathcal J }}_{(d)}^{{\rm{cov}}})}_{\hat{\alpha }\hat{\beta }} & = & -{f}_{{\rm{\Sigma }}}^{* }[{({{ \mathcal K }}_{(d+1)}^{{\rm{cov}}})}_{\hat{\alpha }\hat{\beta }}],\\ {({{ \mathcal J }}_{(d)}^{{\rm{cov}}})}_{\hat{\alpha }} & = & -{f}_{{\rm{\Sigma }}}^{* }[{({{ \mathcal K }}_{(d+1)}^{{\rm{cov}}})}_{\hat{\alpha }}],\\ {({{ \mathcal A }}_{(d)}^{{\rm{cov}}})}_{\hat{\alpha }\hat{\beta }} & = & -{f}_{{\rm{\Sigma }}}^{* }[{({{ \mathcal J }}_{(d+1)})}_{\hat{\alpha }\hat{\beta }}].\end{array}\end{eqnarray}$
The same results were obtained in the Hamiltonian analysis in [10]. The boundary currents and anomaly are manifestly covariant under the local SO(2, d) transformation. From the dual CFT point of view, equation ( $\begin{eqnarray}\begin{array}{rcl}{({{ \mathcal J }}_{(d)}^{{\rm{cov}}})}_{\hat{\alpha }\hat{\beta }} & = & \left\langle \frac{\delta {S}_{{\rm{CFT}}}}{\delta {{\mathbb{A}}}^{\hat{\alpha }\hat{\beta }}}\right\rangle ,\\ {({{ \mathcal J }}_{(d)}^{{\rm{cov}}})}_{\hat{\alpha }} & = & \left\langle \frac{\delta {S}_{{\rm{CFT}}}}{\delta {{\mathbb{Y}}}^{\hat{\alpha }}}\right\rangle ,\end{array}\end{eqnarray}$
as well as the corresponding covariant SO(2, d) anomaly. When the pullback of the bulk Einstein equations on the hypersurface z = ζ(x) is satisfied, the covariant anomaly vanishes for the CFT with the local renormalization scale μ = ζ−1. Inversely, if we require that the CFT SO(2, d) covariant conservation law is not anomalous for any local renormalization scale, the full bulk Einstein equations are automatically implied.3. The descendent structure of SO(2, d) anomaly
3.1. Bulk consistent action
3.1.1. Chern–Simons like action
To describe the descendent structure of SO(2, d) anomaly, we need to introduce the homotopic quantities
$\begin{eqnarray}\begin{array}{rcl}{\rm{D}}(s)Y & = & {\rm{d}}Y+sAY,\\ F(s) & = & {\rm{d}}A(s)+A(s)\wedge A(s)\\ & = & sF-s(1-s)A\wedge A\end{array}\end{eqnarray}$
by the replacement $\begin{eqnarray}A\to A(s)=sA.\end{eqnarray}$
As in the case of chiral anomalies [12], the bulk Chern–Simons like action can be constructed in terms of the homotopic integration over the boundary covariant anomaly. That is $\begin{eqnarray}\begin{array}{rcl}{S}_{(d+1)}^{{\rm{CS}}} & = & {\displaystyle \int }_{M}{{ \mathcal L }}_{(d+1)}^{{\rm{CS}}}\\ & = & -{\displaystyle \int }_{M}{\displaystyle \int }_{0}^{1}{\rm{d}}s\,{A}^{\hat{\alpha }\hat{\beta }}\wedge {({{ \mathcal A }}_{(d)}^{{\rm{cov}}})}_{\hat{\alpha }\hat{\beta }}(A(s),Y)\\ & = & {\displaystyle \int }_{M}{\displaystyle \int }_{0}^{1}{\rm{d}}s\,{A}^{\hat{\alpha }\hat{\beta }}\wedge {({{ \mathcal J }}_{(d+1)})}_{\hat{\alpha }\hat{\beta }}(A(s),Y)\\ & = & \frac{{(-1)}^{d+1}{\ell }}{4{\kappa }^{2}(d-2)!}{\displaystyle \int }_{M}{\displaystyle \int }_{0}^{1}{\rm{d}}s\,{\epsilon }_{\hat{\alpha }\hat{\beta }{\hat{\alpha }}_{1}\cdots {\hat{\alpha }}_{d}}\\ & & \times \,{A}^{\hat{\alpha }\hat{\beta }}\wedge {F}^{{\hat{\alpha }}_{1}{\hat{\alpha }}_{2}}(s)\wedge {\rm{D}}(s){Y}^{{\hat{\alpha }}_{3}}\wedge \cdots \wedge {\rm{D}}(s){Y}^{{\hat{\alpha }}_{d}}.\end{array}\end{eqnarray}$
The variation of ${S}_{(d+1)}^{{\rm{CS}}}$ is given by
$\begin{eqnarray}\begin{array}{rcl}\delta {S}_{(d+1)}^{{\rm{CS}}} & = & {\displaystyle \int }_{M}\left[\delta {A}^{\hat{\alpha }\hat{\beta }}\wedge {({{ \mathcal J }}_{(d+1)})}_{\hat{\alpha }\hat{\beta }}+\delta {Y}^{\hat{\alpha }}{({{ \mathcal J }}_{(d+1)})}_{\hat{\alpha }}\right]\\ & & +{\displaystyle \int }_{{\rm{\Sigma }}}\left[\delta {A}^{\hat{\alpha }\hat{\beta }}\wedge {({{ \mathcal K }}_{(d+1)}^{{\rm{CS}}})}_{\hat{\alpha }\hat{\beta }}+\delta {Y}^{\hat{\alpha }}{({{ \mathcal K }}_{(d+1)}^{{\rm{CS}}})}_{\hat{\alpha }}\right],\end{array}\end{eqnarray}$
where the Noether potentials are $\begin{eqnarray}\begin{array}{rcl}{({{ \mathcal K }}_{(d+1)}^{{\rm{CS}}})}_{\hat{\alpha }\hat{\beta }} & = & \frac{{(-1)}^{d+1}{\ell }}{4{\kappa }^{2}(d-2)!}{\displaystyle \int }_{0}^{1}{\rm{d}}s\,{\epsilon }_{\hat{\alpha }\hat{\beta }{\hat{\alpha }}_{1}\cdots {\hat{\alpha }}_{d}}\\ & & \times \,{A}^{{\hat{\alpha }}_{1}{\hat{\alpha }}_{2}}(s)\wedge {\rm{D}}(s){Y}^{{\hat{\alpha }}_{3}}\wedge \cdots \wedge {\rm{D}}(s){Y}^{{\hat{\alpha }}_{d}},\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{rcl}{({{ \mathcal K }}_{(d+1)}^{{\rm{CS}}})}_{\hat{\alpha }} & = & \frac{{(-1)}^{d}}{2{\kappa }^{2}{\ell }\,(d-2)!}{\epsilon }_{\hat{\alpha }\hat{\beta }{\hat{\alpha }}_{1}\cdots {\hat{\alpha }}_{d}}\\ & & \times \,{Y}^{\hat{\beta }}\left[{F}^{{\hat{\alpha }}_{1}{\hat{\alpha }}_{2}}\wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{3}}\wedge \cdots \wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{d}}\right.\\ & & \left.-{\int }_{0}^{1}{\rm{d}}s\,{\rm{D}}(s)\left({A}^{{\hat{\alpha }}_{1}{\hat{\alpha }}_{2}}\wedge {\rm{D}}(s){Y}^{{\hat{\alpha }}_{3}}\wedge \cdots \wedge {\rm{D}}(s){Y}^{{\hat{\alpha }}_{d}}\right)\right].\end{array}\end{eqnarray}$
We notice that $\delta {S}_{(d+1)}^{{\rm{CS}}}$ gives rise to the same bulk terms as $\delta {S}_{(d+1)}^{{\rm{cov}}}$. Thus, these two Lagrangians differ only by a closed term, and both of them can be viewed as the SO(2, d) uplifting of the bulk Einstein gravity. For d = 2, ${S}_{(d+1)}^{{\rm{CS}}}$ comes back to the SO(2, 2) = SL(2, R) × SL(2, R) Chern–Simons gauge theory [13].As in the chiral anomaly case, the Chern–Simons like action (23 ) is invariant under the perturbative gauge transformations, but not invariant under the large gauge transformations. Let us consider the finite SO(2, d) gauge transformation
$\begin{eqnarray}\begin{array}{rcl}Y\to \tilde{Y} & = & UY,\\ A\to \tilde{A} & = & UA{U}^{-1}-{\rm{d}}U{U}^{-1}\\ & = & U(A-\hat{A}){U}^{-1},\end{array}\end{eqnarray}$
where we denote $\hat{A}={U}^{-1}{\rm{d}}U$. The corresponding homotopic quantities are $\begin{eqnarray}\begin{array}{rcl}\hat{A}(\hat{s}) & = & \hat{s}\hat{A}=\hat{s}{U}^{-1}{\rm{d}}U,\\ \hat{{\rm{D}}}(\hat{s})Y & = & {\rm{d}}Y+\hat{s}\hat{A}Y={\rm{d}}Y+s{U}^{-1}{\rm{d}}UY,\\ \hat{F}(\hat{s}) & = & {\rm{d}}\hat{A}(\hat{s})+\hat{A}(\hat{s})\wedge \hat{A}(\hat{s})\\ & = & -\hat{s}(1-\hat{s})\hat{A}\wedge \hat{A}=\hat{F}(1-\hat{s}),\\ A(s,\hat{s}) & = & sA+\hat{s}\hat{A},\\ {\rm{D}}(s,\hat{s})Y & = & {\rm{d}}Y+sAY+\hat{s}\hat{A}Y,\\ F(s,\hat{s}) & = & {\rm{d}}A(s,\hat{s})+A(s,\hat{s})\wedge A(s,\hat{s})\\ & = & F(s)+\hat{F}(\hat{s})+s\hat{s}(A\wedge \hat{A}+\hat{A}\wedge A).\end{array}\end{eqnarray}$
In terms of these homotopic quantities, the finite gauge transformation of the ${{ \mathcal L }}_{(d+1)}^{{\rm{CS}}}[A,Y]$ can be expressed as $\begin{eqnarray}\begin{array}{l}\frac{4{\kappa }^{2}(d-2)!}{{(-1)}^{d+1}{\ell }}\left({{ \mathcal L }}_{(d+1)}^{{\rm{CS}}}[\tilde{A},\tilde{Y}]-{{ \mathcal L }}_{(d+1)}^{{\rm{CS}}}[A,Y]\right)\\ \quad =-{\displaystyle \int }_{0}^{1}{\rm{d}}\hat{s}\,{\epsilon }_{\hat{\alpha }\hat{\beta }{\hat{\alpha }}_{1}\cdots {\hat{\alpha }}_{d}}{\hat{A}}^{\hat{\alpha }\hat{\beta }}\wedge {\hat{F}}^{{\hat{\alpha }}_{1}{\hat{\alpha }}_{2}}(\hat{s})\\ \quad \wedge \hat{{\rm{D}}}(\hat{s}){Y}^{{\hat{\alpha }}_{3}}\wedge \cdots \wedge \hat{{\rm{D}}}(\hat{s}){Y}^{{\hat{\alpha }}_{d}}\\ \qquad \,\,+{\rm{d}}\left[{\displaystyle \int }_{0}^{1}{\rm{d}}s{\displaystyle \int }_{0}^{1-s}{\rm{d}}\hat{s}\,{\epsilon }_{\hat{\alpha }\hat{\beta }{\hat{\alpha }}_{1}\cdots {\hat{\alpha }}_{d}}{\hat{A}}^{\hat{\alpha }\hat{\beta }}\wedge {A}^{{\hat{\alpha }}_{1}{\hat{\alpha }}_{2}}\right.\\ \qquad \left.\wedge {\rm{D}}(s,\hat{s}){Y}^{{\hat{\alpha }}_{3}}\wedge \cdots \wedge {\rm{D}}(s,\hat{s}){Y}^{{\hat{\alpha }}_{d}}\right].\end{array}\end{eqnarray}$
Obviously, the bulk term vanishes for infinitesimal gauge transformations. It means that the perturbative gauge invariance is unbroken up to the boundary term. However, for large gauge transformations, the bulk term is no longer zero and the full SO(2, d) invariance is broken.3.1.2. Topological term and bulk consistent action
Due to the existence of the ruler field, one can always introduce an A-independent term as 30 ) is just a boundary term
$\begin{eqnarray}\begin{array}{rcl}{S}_{(d+1)}^{{\rm{top}}} & = & {\displaystyle \int }_{M}{{ \mathcal L }}_{(d+1)}^{{\rm{top}}}=\frac{d}{{\kappa }^{2}(d+1)!{{\ell }}^{3}}\\ & & \times \,{\displaystyle \int }_{M}{\epsilon }_{{\hat{\alpha }}_{0}\cdots {\hat{\alpha }}_{d+1}}{\rm{d}}{Y}^{{\hat{\alpha }}_{0}}\wedge \cdots \wedge {\rm{d}}{Y}^{{\hat{\alpha }}_{d}}{Y}^{{\hat{\alpha }}_{d+1}}.\end{array}\end{eqnarray}$
It is a topological term since ${{ \mathcal L }}_{(d+1)}^{{\rm{top}}}$ is closed $\begin{eqnarray}\begin{array}{l}{\rm{d}}({\epsilon }_{{\hat{\alpha }}_{0}\cdots {\hat{\alpha }}_{d+1}}{\rm{d}}{Y}^{{\hat{\alpha }}_{0}}\wedge \cdots \wedge {\rm{d}}{Y}^{{\hat{\alpha }}_{d}}{Y}^{{\hat{\alpha }}_{d+1}})\\ \quad ={(-1)}^{d+1}{\epsilon }_{{\hat{\alpha }}_{0}\cdots {\hat{\alpha }}_{d+1}}{\rm{d}}{Y}^{{\hat{\alpha }}_{0}}\wedge \cdots \wedge {\rm{d}}{Y}^{{\hat{\alpha }}_{d+1}}=0.\end{array}\end{eqnarray}$
Due to the topological nature, the variation of equation ( $\begin{eqnarray}\begin{array}{rcl}\delta {S}_{(d+1)}^{{\rm{top}}} & = & \frac{{(-1)}^{d}}{{\kappa }^{2}{{\ell }}^{3}(d-1)!}{\displaystyle \int }_{{\rm{\Sigma }}}{\epsilon }_{{\hat{\alpha }}_{1}\cdots {\hat{\alpha }}_{d+2}}\\ & & \times \,\delta {Y}^{{\hat{\alpha }}_{1}}{Y}^{{\hat{\alpha }}_{2}}{\rm{d}}{Y}^{{\hat{\alpha }}_{3}}\wedge \cdots \wedge {\rm{d}}{Y}^{{\hat{\alpha }}_{d+2}}.\end{array}\end{eqnarray}$
Especially, ${S}_{(d+1)}^{{\rm{top}}}$ is invariant under the bulk perturbative SO(2, d) gauge transformations.Under the finite SO(2, d) gauge transformations, ${{ \mathcal L }}_{(d+1)}^{{\rm{top}}}$ gives rise to 33 ) is exactly same as the one appeared in equation (29 ). Therefore, the large gauge transformation of the Chern–Simons like action can be compensated by adding the topological term
$\begin{eqnarray}\begin{array}{r}\frac{4{\kappa }^{2}(d-2)!}{{(-1)}^{d+1}{\ell }}\left({{ \mathcal L }}_{(d+1)}^{{\rm{top}}}[\tilde{Y}]-{{ \mathcal L }}_{(d+1)}^{{\rm{top}}}[Y]\right)\\ \quad =-{\displaystyle \int }_{0}^{1}{\rm{d}}\hat{s}\,{\epsilon }_{{\hat{\alpha }}_{0}\cdots {\hat{\alpha }}_{d+1}}{\hat{A}}^{{\hat{\alpha }}_{0}{\hat{\alpha }}_{1}}\wedge {\hat{F}}^{{\hat{\alpha }}_{2}{\hat{\alpha }}_{3}}(\hat{s})\\ \qquad \wedge \hat{{\rm{D}}}(\hat{s}){Y}^{{\hat{\alpha }}_{4}}\wedge \cdots \wedge \hat{{\rm{D}}}(\hat{s}){Y}^{{\hat{\alpha }}_{d+1}}\\ \qquad +\frac{2{(-1)}^{d}}{(d-1){{\ell }}^{2}}{\rm{d}}\left[{\displaystyle \int }_{0}^{1}{\rm{d}}\hat{s}\,{\epsilon }_{{\hat{\alpha }}_{0}\cdots {\hat{\alpha }}_{d+1}}{\hat{A}}^{{\hat{\alpha }}_{0}{\hat{\alpha }}_{1}}\right.\\ \qquad \left.\wedge \hat{{\rm{D}}}(\hat{s}){Y}^{{\hat{\alpha }}_{2}}\wedge \cdots \wedge \hat{{\rm{D}}}(\hat{s}){Y}^{{\hat{\alpha }}_{d}}{Y}^{{\hat{\alpha }}_{d+1}}\right].\end{array}\end{eqnarray}$
The bulk term in equation ( $\begin{eqnarray}{S}_{(d+1)}^{{\rm{con}}}={\int }_{M}{{ \mathcal L }}_{(d+1)}^{{\rm{con}}}={\int }_{M}[{{ \mathcal L }}_{(d+1)}^{{\rm{CS}}}-{{ \mathcal L }}_{(d+1)}^{{\rm{top}}}].\end{eqnarray}$
Up to the boundary terms, this bulk consistent action ${S}_{(d+1)}^{{\rm{con}}}$ is invariant under both perturbative and large gauge transformations.Since $\delta {S}_{(d+1)}^{{\rm{top}}}$ is just a boundary term, the variation of the consistent action gives rise to the same bulk term as in $\delta {S}_{(d+1)}^{{\rm{CS}}}$ and $\delta {S}_{(d+1)}^{{\rm{cov}}}$. We have
$\begin{eqnarray}\begin{array}{rcl}\delta {S}_{(d+1)}^{{\rm{con}}} & = & {\displaystyle \int }_{M}\left[\delta {A}^{\hat{\alpha }\hat{\beta }}\wedge {({{ \mathcal J }}_{(d+1)})}_{\hat{\alpha }\hat{\beta }}+\delta {Y}^{\hat{\alpha }}{({{ \mathcal J }}_{(d+1)})}_{\hat{\alpha }}\right]\\ & & +{\displaystyle \int }_{{\rm{\Sigma }}}\left[\delta {A}^{\hat{\alpha }\hat{\beta }}\wedge {({{ \mathcal K }}_{(d+1)}^{{\rm{con}}})}_{\hat{\alpha }\hat{\beta }}+\delta {Y}^{\hat{\alpha }}{({{ \mathcal K }}_{(d+1)}^{{\rm{con}}})}_{\hat{\alpha }}\right],\end{array}\end{eqnarray}$
where ${({{ \mathcal K }}_{(d+1)}^{{\rm{con}}})}_{\hat{\alpha }\hat{\beta }}={({{ \mathcal K }}_{(d+1)}^{{\rm{CS}}})}_{\hat{\alpha }\hat{\beta }}$ since ${S}_{(d+1)}^{{\rm{top}}}$ is A-independent and only ${({{ \mathcal K }}_{(d+1)})}_{\hat{\alpha }}$ is modified by the topological term $\begin{eqnarray}\begin{array}{rcl}{({{ \mathcal K }}_{(d+1)}^{{\rm{con}}})}_{\hat{\alpha }} & = & {({{ \mathcal K }}_{(d+1)}^{{\rm{CS}}})}_{\hat{\alpha }}-\frac{{(-1)}^{d}}{{\kappa }^{2}{{\ell }}^{3}(d-1)!}\\ & & \times \,{\epsilon }_{\hat{\alpha }\hat{\beta }{\hat{\alpha }}_{1}\cdots {\hat{\alpha }}_{d}}{Y}^{\hat{\beta }}{\rm{d}}{Y}^{{\hat{\alpha }}_{1}}\wedge \cdots \wedge {\rm{d}}{Y}^{{\hat{\alpha }}_{d}}.\end{array}\end{eqnarray}$
The pullback of the consistent Noether potentials gives rise to the Bardeen–Zumino polynomial [14] $\begin{eqnarray}\begin{array}{rcl}{({{ \mathcal P }}_{(d)})}_{\hat{\alpha }\hat{\beta }} & = & -{f}_{{\rm{\Sigma }}}^{* }[{({{ \mathcal K }}_{(d+1)}^{{\rm{con}}})}_{\hat{\alpha }\hat{\beta }}],\\ {({{ \mathcal P }}_{(d)})}_{\hat{\alpha }} & = & -{f}_{{\rm{\Sigma }}}^{* }[{({{ \mathcal K }}_{(d+1)}^{{\rm{con}}})}_{\hat{\alpha }}],\end{array}\end{eqnarray}$
which are basically the differences between the boundary covariant and consistent currents.3.2. Boundary effective action
Similar to equation (23 ), one can also construct the boundary relative effective action by the homotopic integration over the boundary current [15]. That is, 29 ) and (33 ), we find equation (39 ) is explicitly the boundary term produced by the gauge transformation of the bulk consistent action ${S}_{(d+1)}^{{\rm{con}}}$. Thus we have the expected finite descendent relation 39 ) gives rise to the consistent anomaly
$\begin{eqnarray}\begin{array}{rcl}{W}_{(d)} & = & {\displaystyle \int }_{{\rm{\Sigma }}}{{ \mathcal L }}_{(d)}^{{\rm{eff}}}={\displaystyle \int }_{{\rm{\Sigma }}}{\displaystyle \int }_{0}^{1}{\rm{d}}s\,{A}^{\hat{\alpha }\hat{\beta }}\wedge {({{ \mathcal J }}_{(d)}^{{\rm{cov}}})}_{\hat{\alpha }\hat{\beta }}(A(s),Y)\\ & = & -\frac{1}{2{\kappa }^{2}{\ell }(d-1)!}{\displaystyle \int }_{{\rm{\Sigma }}}{\displaystyle \int }_{0}^{1}{\rm{d}}s\,{\epsilon }_{\hat{\alpha }\hat{\beta }{\hat{\alpha }}_{1}\cdots {\hat{\alpha }}_{d}}\\ & & \times {A}^{\hat{\alpha }\hat{\beta }}\wedge {\rm{D}}(s){Y}^{{\hat{\alpha }}_{1}}\wedge \cdots \wedge {\rm{D}}(s){Y}^{{\hat{\alpha }}_{d-1}}{Y}^{{\hat{\alpha }}_{d}}.\end{array}\end{eqnarray}$
Under the finite SO(2, d) gauge transformations, W(d) transforms as $\begin{eqnarray}\begin{array}{l}{W}_{(d)}[\tilde{A},\tilde{Y}]-{W}_{(d)}[A,Y]\\ \qquad =\frac{{(-1)}^{d+1}{\ell }}{4{\kappa }^{2}(d-2)!}{\displaystyle \int }_{{\rm{\Sigma }}}{\displaystyle \int }_{0}^{1}{\rm{d}}\hat{s}\,{\epsilon }_{\hat{\alpha }\hat{\beta }{\hat{\alpha }}_{1}\cdots {\hat{\alpha }}_{d}}\\ \qquad \times \,\left[{\displaystyle \int }_{0}^{1-\hat{s}}{\rm{d}}s\,{\hat{A}}^{\hat{\alpha }\hat{\beta }}\wedge {A}^{{\hat{\alpha }}_{1}{\hat{\alpha }}_{2}}\right.\\ \qquad \wedge {\rm{D}}(s,\hat{s}){Y}^{{\hat{\alpha }}_{3}}\wedge \cdots \wedge {\rm{D}}(s,\hat{s}){Y}^{{\hat{\alpha }}_{d}}\\ \qquad \left.+\frac{2}{(d-1){{\ell }}^{2}}{\hat{A}}^{\hat{\alpha }\hat{\beta }}{Y}^{{\hat{\alpha }}_{1}}\wedge \hat{{\rm{D}}}(\hat{s}){Y}^{{\hat{\alpha }}_{2}}\wedge \cdots \wedge \hat{{\rm{D}}}(\hat{s}){Y}^{{\hat{\alpha }}_{d}}\right].\end{array}\end{eqnarray}$
Comparing with equations ( $\begin{eqnarray}{{\rm{\Delta }}}_{U}{{ \mathcal L }}_{(d+1)}^{{\rm{con}}}[A,Y]={\rm{d}}({{\rm{\Delta }}}_{U}{{ \mathcal L }}_{(d)}^{{\rm{eff}}}[A,Y]).\end{eqnarray}$
At the infinitesimal limit, equation ( $\begin{eqnarray}\begin{array}{l}{\displaystyle \int }_{{\rm{\Sigma }}}{u}^{\hat{\alpha }\hat{\beta }}{({{ \mathcal A }}_{(d)}^{{\rm{con}}})}_{\hat{\alpha }\hat{\beta }}=\delta {W}_{(d)}[A,Y]\\ =\frac{{(-1)}^{d}{\ell }}{4{\kappa }^{2}\,(d-2)!}{\displaystyle \int }_{{\rm{\Sigma }}}{u}^{\hat{\alpha }\hat{\beta }}{\epsilon }_{\hat{\alpha }\hat{\beta }{\hat{\alpha }}_{1}\cdots {\hat{\alpha }}_{d}}\\ \,\times \,\left[{\displaystyle \int }_{0}^{1}{\rm{d}}s\,(1-s)\,{\rm{d}}\left({A}^{{\hat{\alpha }}_{1}{\hat{\alpha }}_{2}}\wedge {\rm{D}}(s){Y}^{{\hat{\alpha }}_{3}}\wedge \cdots \wedge {\rm{D}}(s){Y}^{{\hat{\alpha }}_{d}}\right)\right.\\ \,\left.+\frac{2}{(d-1){{\ell }}^{2}}{\rm{d}}{Y}^{{\hat{\alpha }}_{1}}\wedge \cdots \wedge {\rm{d}}{Y}^{{\hat{\alpha }}_{d}}\right],\end{array}\end{eqnarray}$
where the total derivative terms have been subtracted since ∂Σ = ∂∂M = 0. The boundary consistent anomaly satisfies that the infinitesimal descendent relation $\begin{eqnarray}{\delta }_{u}{I}_{(d+1)}^{{\rm{con}}}={\rm{d}}{I}_{(d)}^{{\rm{con}}},\end{eqnarray}$
where we denote that $\begin{eqnarray}\begin{array}{rcl}{I}_{(d)}^{{\rm{con}}} & = & {u}^{\hat{\alpha }\hat{\beta }}{({{ \mathcal A }}_{(d)}^{{\rm{con}}})}_{\hat{\alpha }\hat{\beta }},\\ {I}_{(d+1)}^{{\rm{con}}} & = & {{ \mathcal L }}_{(d+1)}^{{\rm{con}}}.\end{array}\end{eqnarray}$
The corresponding consistent currents are
$\begin{eqnarray}\begin{array}{l}{({{ \mathcal J }}_{(d)}^{{\rm{con}}})}_{\hat{\alpha }\hat{\beta }}=\frac{\delta {W}_{(d)}}{\delta {A}^{\hat{\alpha }\hat{\beta }}}=\frac{{(-1)}^{d}}{2{\kappa }^{2}{\ell }(d-1)!}{\epsilon }_{\hat{\alpha }\hat{\beta }{\hat{\alpha }}_{1}\cdots {\hat{\alpha }}_{d}}\\ \,\times \,\left[{Y}^{{\hat{\alpha }}_{1}}{\rm{D}}{Y}^{{\hat{\alpha }}_{2}}\wedge \cdots \wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{d}}-\frac{{{\ell }}^{2}(d-1)}{2}\right.\\ \,\left.\times \,{\displaystyle \int }_{0}^{1}{\rm{d}}s\,{A}^{{\hat{\alpha }}_{1}{\hat{\alpha }}_{2}}(s)\wedge {\rm{D}}(s){Y}^{{\hat{\alpha }}_{3}}\wedge \cdots \wedge {\rm{D}}(s){Y}^{{\hat{\alpha }}_{d}}\right],\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{l}{({{ \mathcal J }}_{(d)}^{{\rm{con}}})}_{\hat{\alpha }}=\frac{\delta {W}_{(d)}}{\delta {Y}^{\hat{\alpha }}}=\frac{{(-1)}^{d}}{2{\kappa }^{2}{\ell }(d-1)!}{\epsilon }_{\hat{\alpha }\hat{\beta }{\hat{\alpha }}_{1}\cdots {\hat{\alpha }}_{d}}\\ \times \,{Y}^{\hat{\beta }}\left[\frac{2}{{{\ell }}^{2}}\,({\rm{D}}{Y}^{{\hat{\alpha }}_{2}}\wedge \cdots \wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{d}}-{\rm{d}}{Y}^{{\hat{\alpha }}_{2}}\wedge \cdots \wedge {\rm{d}}{Y}^{{\hat{\alpha }}_{d}})\right.\\ \left.-(d-1){\displaystyle \int }_{0}^{1}{\rm{d}}s\,{\rm{D}}(s)[{A}^{{\hat{\alpha }}_{1}{\hat{\alpha }}_{2}}\wedge {\rm{D}}(s){Y}^{{\hat{\alpha }}_{3}}\wedge \cdots \wedge {\rm{D}}(s){Y}^{{\hat{\alpha }}_{d}}]\right].\end{array}\end{eqnarray}$
One can further verify that the consistent currents satisfy the off-shell consistent conservation law $\begin{eqnarray}{\rm{D}}{({{ \mathcal J }}_{(d)}^{{\rm{con}}})}_{\hat{\alpha }\hat{\beta }}+{Y}_{[\hat{\beta }}{({{ \mathcal J }}_{(d)}^{{\rm{con}}})}_{\hat{\alpha }]}={({{ \mathcal A }}_{(d)}^{{\rm{con}}})}_{\hat{\alpha }\hat{\beta }},\end{eqnarray}$
as well as the relations [14] $\begin{eqnarray}\begin{array}{rcl}{({{ \mathcal J }}_{(d)}^{{\rm{con}}})}_{\hat{\alpha }\hat{\beta }} & = & {({{ \mathcal J }}_{(d)}^{{\rm{cov}}})}_{\hat{\alpha }\hat{\beta }}-{({{ \mathcal P }}_{(d)})}_{\hat{\alpha }\hat{\beta }},\\ {({{ \mathcal J }}_{(d)}^{{\rm{con}}})}_{\hat{\alpha }} & = & {({{ \mathcal J }}_{(d)}^{{\rm{cov}}})}_{\hat{\alpha }}-{({{ \mathcal P }}_{(d)})}_{\hat{\alpha }}.\end{array}\end{eqnarray}$
Since the Bardeen–Zumino polynomial does not contribute to the homotopic integration $\begin{eqnarray}\begin{array}{l}{A}^{\hat{\alpha }\hat{\beta }}\wedge {({{ \mathcal P }}_{(d)})}_{\hat{\alpha }\hat{\beta }}(sA,Y)\propto {\displaystyle \int }_{0}^{1}{\rm{d}}\tilde{s}\,{\epsilon }_{\hat{\alpha }\hat{\beta }{\hat{\alpha }}_{1}\cdots {\hat{\alpha }}_{d}}{A}^{\hat{\alpha }\hat{\beta }}\\ \qquad \wedge {A}^{{\hat{\alpha }}_{1}{\hat{\alpha }}_{2}}(\tilde{s)}\wedge {\rm{D}}(s{\tilde{s)Y}}^{{\hat{\alpha }}_{3}}\wedge \cdots \wedge {\rm{D}}(s{\tilde{s)Y}}^{{\hat{\alpha }}_{d}}=0,\end{array}\end{eqnarray}$
the effective action can also be constructed in terms of the homotopic integration over the consistent current [15] $\begin{eqnarray}\begin{array}{rcl}{W}_{(d)} & = & {\displaystyle \int }_{{\rm{\Sigma }}}{\displaystyle \int }_{0}^{1}{\rm{d}}s\,{A}^{\hat{\alpha }\hat{\beta }}\wedge {({{ \mathcal J }}_{(d)}^{{\rm{cov}}})}_{\hat{\alpha }\hat{\beta }}(A(s),Y)\\ & = & {\displaystyle \int }_{{\rm{\Sigma }}}{\displaystyle \int }_{0}^{1}{\rm{d}}s\,{A}^{\hat{\alpha }\hat{\beta }}\wedge {({{ \mathcal J }}_{(d)}^{{\rm{con}}})}_{\hat{\alpha }\hat{\beta }}(A(s),Y).\end{array}\end{eqnarray}$
3.3. Characteristic class
The characteristic class of the SO(2, d) anomaly is given by the exterior derivative of the bulk consistent Lagrangian C , the exterior derivative of the bulk covariant Lagrangian gives rise to the same result
$\begin{eqnarray}\begin{array}{rcl}{I}_{(d+2)} & = & {\rm{d}}{{ \mathcal L }}_{(d+1)}^{{\rm{con}}}=\frac{{(-1)}^{d+1}{\ell }}{8{\kappa }^{2}(d-2)!}{\epsilon }_{\hat{\alpha }\hat{\beta }{\hat{\alpha }}_{1}\cdots {\hat{\alpha }}_{d}}\\ & & \times \,{F}^{\hat{\alpha }\hat{\beta }}\wedge {F}^{{\hat{\alpha }}_{1}{\hat{\alpha }}_{2}}\wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{3}}\wedge \cdots \wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{d}}.\end{array}\end{eqnarray}$
As shown in [10] as well as appendix $\begin{eqnarray}{I}_{(d+2)}={\rm{d}}{{ \mathcal L }}_{(d+1)}^{{\rm{cov}}}.\end{eqnarray}$
In fact, the bulk consistent action is just the summation of the bulk covariant action and the boundary effective action $\begin{eqnarray}\begin{array}{rcl}{{ \mathcal L }}_{(d+1)}^{{\rm{cov}}} & = & {{ \mathcal L }}_{(d+1)}^{{\rm{con}}}-{\rm{d}}{{ \mathcal L }}_{(d)}^{{\rm{eff}}}\\ & = & {{ \mathcal L }}_{(d+1)}^{{\rm{CS}}}-{{ \mathcal L }}_{(d+1)}^{{\rm{top}}}-{\rm{d}}{{ \mathcal L }}_{(d)}^{{\rm{eff}}}.\end{array}\end{eqnarray}$
Under the finite gauge transformations, the bulk and boundary terms of ${{\rm{\Delta }}}_{U}{S}_{(d+1)}^{{\rm{CS}}}$ are compensated respectively by the transformations of topological term and the boundary effective action. This is consistent with the fact that ${{ \mathcal L }}_{(d+1)}^{{\rm{cov}}}$ is manifestly SO(2, d) gauge invariant.As in the usual treatment of chiral anomalies, one can also establish the descendent structure 40 ) is absent in this approach.
$\begin{eqnarray}\begin{array}{rcl}{I}_{(d+2)} & = & {\rm{d}}{{ \mathcal L }}_{(d+1)}^{{\rm{CS}}}\\ {\delta }_{u}{I}_{(d+1)}^{{\rm{CS}}} & = & {\rm{d}}{I}_{(d)}^{{\rm{CS}}},\end{array}\end{eqnarray}$
directly by the bulk Chern–Simons term $\begin{eqnarray}{I}_{(d+1)}^{{\rm{CS}}}={{ \mathcal L }}_{(d+1)}^{{\rm{CS}}},{I}_{(d)}^{{\rm{CS}}}={u}^{\hat{\alpha }\hat{\beta }}{({{ \mathcal A }}_{(d)}^{{\rm{CS}}})}_{\hat{\alpha }\hat{\beta }}.\end{eqnarray}$
The corresponding anomaly $\begin{eqnarray}\begin{array}{l}{({{ \mathcal A }}_{(d)}^{{\rm{CS}}})}_{\hat{\alpha }\hat{\beta }}=\frac{{(-1)}^{d}{\ell }}{4{\kappa }^{2}\,(d-2)!}{\epsilon }_{\hat{\alpha }\hat{\beta }{\hat{\alpha }}_{1}\cdots {\hat{\alpha }}_{d}}\\ \times {\displaystyle \int }_{0}^{1}{\rm{d}}s\,(1-s)\,{\rm{d}}\left({A}^{{\hat{\alpha }}_{1}{\hat{\alpha }}_{2}}\wedge {\rm{D}}(s){Y}^{{\hat{\alpha }}_{3}}\wedge \cdots \wedge {\rm{D}}(s){Y}^{{\hat{\alpha }}_{d}}\right)\end{array}\end{eqnarray}$
appears from the infinitesimal gauge transformation of the topologically improved effective action $\begin{eqnarray}\hat{W}={W}_{(d)}+{S}_{(d+1)}^{{\rm{top}}}.\end{eqnarray}$
Due to the existence of the bulk term in ${{\rm{\Delta }}}_{U}{S}_{(d+1)}^{{\rm{CS}}}$ and ${{\rm{\Delta }}}_{U}{S}_{(d+1)}^{{\rm{top}}}$, the Chern–Simons descendent relation is correct only for the perturbative gauge transformations. The finite descendent relation like equation (4. Summary
In this paper, we establish the descendent structure of the holographic SO(2, d) anomaly. Due to the existence of the ruler field, one can write down the explicit form of the boundary covariant and consistent currents. The bulk Chern–Simons like action and the boundary effective action are constructed in terms of the homotopic integration method. To compensate the large gauge transformation of the Chern–Simons like action, we introduce a gauge field independent topological term which is constructed only by the ruler field. It is curious to discuss the effect of the topological term in the story of cobordism invariance [16, 17].
In the Einstein gravity, the SO(2, d) anomaly is governed by the characteristic class 59 ).
$\begin{eqnarray}\begin{array}{rcl}{I}_{(d+2)} & = & \frac{{(-1)}^{d+1}{\ell }}{8{\kappa }^{2}(d-2)!}{\epsilon }_{{\hat{\alpha }}_{0}\cdots {\hat{\alpha }}_{d+1}}{F}^{{\hat{\alpha }}_{0}{\hat{\alpha }}_{1}}\\ & & \wedge {F}^{{\hat{\alpha }}_{2}{\hat{\alpha }}_{3}}\wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{4}}\wedge \cdots \wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{d+1}}.\end{array}\end{eqnarray}$
It is natural to consider the generalization of this formula in various cases. For example, one can investigate its supersymmetric generalization based on the local OSp(1∣4; R) covariant formalism [18] of N = 1 AdS4 supergravity. On the other hand, the most general formula of the characteristic class in the pure bosonic case could be $\begin{eqnarray}\begin{array}{rcl}{I}_{(d+2)} & = & \displaystyle \sum _{i=1}^{\unicode{x0230A}\frac{d}{2}\unicode{x0230B}+1}{a}_{i}{\epsilon }_{{\hat{\alpha }}_{0}\cdots {\hat{\alpha }}_{d+1}}\\ & & \times \,{F}^{{\hat{\alpha }}_{0}{\hat{\alpha }}_{1}}\wedge \cdots \wedge {F}^{{\hat{\alpha }}_{2i-2}{\hat{\alpha }}_{2i-1}}\wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{2i}}\\ & & \wedge \cdots \wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{d+1}}.\end{array}\end{eqnarray}$
Now the corresponding bulk theory will be the SO(2, d) gauge theory uplifting of the Lovelock gravity [19] for which the bulk covariant action is in the form $\begin{eqnarray}\begin{array}{rcl}{S}_{(d+1)}^{{\rm{cov}}} & = & {\displaystyle \int }_{M}\displaystyle \sum _{i=0}^{\unicode{x0230A}\frac{d+1}{2}\unicode{x0230B}}{\lambda }_{i}{\epsilon }_{{\hat{\alpha }}_{0}\cdots {\hat{\alpha }}_{d+1}}\\ & & \times \,{F}^{{\hat{\alpha }}_{0}{\hat{\alpha }}_{1}}\wedge \cdots \wedge {F}^{{\hat{\alpha }}_{2i-2}{\hat{\alpha }}_{2i-1}}\wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{2i}}\\ & & \wedge \cdots \wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{d}}{Y}^{{\hat{\alpha }}_{d+1}}.\end{array}\end{eqnarray}$
In [20], it is suggested that the string low energy effective action could always be recast as Lovelock type of theory to all orders in ${\alpha }^{{\prime} }$. Therefore, by considering the gauge field and ruler field associated with the stringy gauge symmetries, it is possible to recast the string low energy effective theory as a gauge theory in the form of equation (Appendix A SO(2, d) uplifting of Einstein gravity
A.1. Vielbein formalism of Einstein gravity
In terms of the vielbein formalism, the (D = d + 1)-dimensional Einstein–Hilbert action with negative cosmological constant A.2 ), we get the Einstein equation
$\begin{eqnarray}{\rm{\Lambda }}=-\frac{(D-1)(D-2)}{2{{\ell }}^{2}}\end{eqnarray}$
can be re-written as $\begin{eqnarray}\begin{array}{c}S[{{e}}^{a}]=\frac{1}{2{\kappa }^{2}\,(D-2)!}\displaystyle \int {\epsilon }_{{a}_{1}\cdots {a}_{D}}\\ \,\times \left[{{\rm{\Theta }}}^{{a}_{1}{a}_{2}}+\frac{(D-2)}{D\,{{\ell }}^{2}}{{e}}^{{a}_{1}}\wedge {{e}}^{{a}_{2}}\right]\wedge {{e}}^{{a}_{3}}\wedge \cdots \wedge {{e}}^{{a}_{D}}.\end{array}\end{eqnarray}$
Our convention for the unit total antisymmetric tensor ${\epsilon }_{{a}_{1}\cdots {a}_{D}}$ of the SO(1, d) vielbein indices ai = 0, ⋯ , d is $\begin{eqnarray}{\epsilon }_{0\cdots d}=1.\end{eqnarray}$
In the usual 2nd order understanding of Einstein gravity, the fundamental dynamical field is just the vielbein 1-form ${{e}}^{a}={{e}}_{\,\,M}^{a}{\rm{d}}{x}^{M}$cc where {xM} are the bulk spacetime coordinates. The curvature 2-form $\begin{eqnarray}\begin{array}{rcl}{{\rm{\Theta }}}_{\,\,b}^{a} & = & {\rm{d}}{\omega }_{\,\,b}^{a}+{\omega }_{\,\,c}^{a}\wedge {\omega }_{\,\,b}^{c}\\ & = & \frac{1}{2}{R}_{\,\,bMN}^{a}{\rm{d}}{x}^{M}\wedge {\rm{d}}{x}^{N},\end{array}\end{eqnarray}$
is the field strength of the spin connection 1-form ωab = −ωba which is just the gauge field of the local SO(1, d) group. Providing the torsion free condition $\begin{eqnarray}{{D}}^{(\omega )}{{e}}^{a}={\rm{d}}{{e}}^{a}+{\omega }_{\,\,b}^{a}\wedge {{e}}^{b}=0,\end{eqnarray}$
the spin connection is totally fixed by the vielbein. Taking the variation of the action ($\begin{eqnarray}\left({{\rm{\Theta }}}^{[{a}_{1}{a}_{2}}+{{\ell }}^{-2}{{e}}^{[{a}_{1}}\wedge {{e}}^{{a}_{2}}\right)\wedge {{e}}^{{a}_{3}}\wedge \cdots \wedge {{e}}^{{a}_{D-1}]}=0.\end{eqnarray}$
In the Palatini's 1st order understanding of gravity, both the vielbein and the spin connection are regarded as independent variables A.9 ) is equivalent to the torsion free condition (A.5 ). Therefore, the Palatini action is equivalent to the original AdS gravity classically.
$\begin{eqnarray}\begin{array}{r}S[{{e}}^{a},{\omega }_{\,\,b}^{a}]=\frac{1}{2{\kappa }^{2}\,(D-2)!}\displaystyle \int {\epsilon }_{{a}_{1}\cdots {a}_{D}}\\ \times \,\left[{{\rm{\Theta }}}^{{a}_{1}{a}_{2}}+\frac{(D-2)}{D\,{{\ell }}^{2}}{{e}}^{{a}_{1}}\wedge {{e}}^{{a}_{2}}\right]\wedge {{e}}^{{a}_{3}}\wedge \cdots \wedge {{e}}^{{a}_{D}}.\end{array}\end{eqnarray}$
The corresponding EOM's from the variation of ea and ωab are respectively $\begin{eqnarray}\left({{\rm{\Theta }}}^{[{a}_{1}{a}_{2}}+{{\ell }}^{-2}{{e}}^{[{a}_{1}}\wedge {{e}}^{{a}_{2}}\right)\wedge {{e}}^{{a}_{3}}\wedge \cdots \wedge {{e}}^{{a}_{D-1}]}=0,\end{eqnarray}$
$\begin{eqnarray}{{\rm{D}}}^{(\omega )}{{e}}^{[{a}_{1}}\wedge {{e}}^{{a}_{2}}\wedge \cdots \wedge {{e}}^{{a}_{D-2}]}=0.\end{eqnarray}$
Providing that the vielbein ea is not degenerate, the second equation (A.2. From SO(1, d) to SO(2, d)
The Palatini's only have the SO(1, d) gauge symmetry which comes from the ambiguities in choosing the vielbein. As explained in the main text, there is a natural way to unify the vielbein ea and the SO(1, d) spin connection ωab as different components of the SO(2, d) gauge field ${A}^{\hat{\alpha }\hat{\beta }}$. That is A.10 ). After the gauge choice, ${Y}^{\hat{\alpha }}$ is totally fixed and the residual gauge symmetry becomes SO(1, d). Thus the introducing of the ruler field does not introduce any additional physical degree of freedom. In the language of the fiber bundle, the ruler field is just a section ${y}^{\hat{\alpha }}={Y}^{\hat{\alpha }}(x)$ of the AdS bundle E
$\begin{eqnarray}{A}^{\hat{a}\hat{b}}={\omega }^{ab},\,\,\,{A}^{\hat{a}\hat{\bullet }}={{\ell }}^{-1}{e}^{a}=-{A}^{\hat{\bullet }\hat{a}},\end{eqnarray}$
where we use $\hat{\bullet }$ to denote the additional indices of SO(2, d) vector space and $\hat{\alpha },\hat{\beta }$ are the vector indices of SO(2, d) group. Then the corresponding field strength is related to the curvature as well as the torsion $\begin{eqnarray}\begin{array}{rcl}{F}^{ab} & = & {\rm{d}}{A}^{ab}+{A}_{\,\,\hat{\gamma }}^{a}\wedge {A}^{\hat{\gamma }b}\\ & = & {\rm{d}}{\omega }^{ab}+{{\omega }^{a}}_{c}\wedge {\omega }^{cb}+{{\ell }}^{-2}{e}^{a}\wedge {e}^{b}\\ & = & {{\rm{\Theta }}}^{ab}+{{\ell }}^{-2}{e}^{a}\wedge {e}^{b},\\ {F}^{a\hat{\bullet }} & = & {\rm{d}}{A}^{a\hat{\bullet }}+{A}_{\,\,\hat{\gamma }}^{a}\wedge {A}^{\hat{\gamma }\hat{\bullet }}\\ & = & {{\ell }}^{-1}({\rm{d}}{e}^{a}+{{\omega }^{a}}_{c}\wedge {e}^{c})={{\ell }}^{-1}\,{{\rm{D}}}^{(\omega )}{e}^{a}.\end{array}\end{eqnarray}$
From the SO(2, d) point of view, the choice of the $\hat{\bullet }$ direction itself is only invariant under the SO(1, d) subgroup perpendicular to the $\hat{\bullet }$ direction. By a generic SO(2, d) rotation, the choice of the $\hat{\bullet }$ direction shall also be changed covariant. To exhibit the choice of the $\hat{\bullet }$ direction in a SO(2, d) covariant way, we introduce an auxiliary field ${Y}^{\hat{\alpha }}$ which is in the vector representation of SO(2, d). Since the ${Y}^{\hat{\alpha }}$ field is imposed only for pointing out the $\hat{\bullet }$ direction covariantly, its norm is not relevant in the SO(1, d) → SO(2, d) uplifting. Thus we also need to impose the SO(2, d) invariant constraint $\begin{eqnarray}{Y}^{\hat{\alpha }}{Y}_{\hat{\alpha }}=-{{\ell }}^{2},\end{eqnarray}$
which fixes the norm of ${Y}^{\hat{\alpha }}$. Such an auxiliary field is named as ruler field in [10]. Obviously, the formula of the ruler field ${Y}^{\hat{\alpha }}$ can be totally fixed by the SO(2, d) gauge choice, e.g. the Einstein gauge $\begin{eqnarray}{Y}_{\hat{a}}=0,\,\,\,\,{Y}_{\hat{\bullet }}={\ell },\end{eqnarray}$
which recovers our initial idea of SO(2, d) uplifting ($\begin{eqnarray}\begin{array}{ccccc}{{\rm{AdS}}}_{d+1} & \longrightarrow & E & \mathop{\longrightarrow }\limits^{\pi } & {M}_{D}\\ \{{y}^{\hat{\alpha }}\} & \longrightarrow & \{{y}^{\hat{\alpha }},{x}^{M}\} & \mathop{\longrightarrow }\limits^{\pi } & \{{x}^{M}\}\end{array}\end{eqnarray}$
upon the D = d + 1 dimensional base manifold MD with coordinates {xM}. The fiber space is the pure AdS space $\begin{eqnarray}{{\rm{AdS}}}_{d+1}=\frac{SO(2,d)}{SO(1,d)}\end{eqnarray}$
which is described by the redundant embedding coordinates ${y}^{\hat{\alpha }}$ with the constraint $\begin{eqnarray}{\eta }_{\hat{\alpha }\hat{\beta }}{y}^{\hat{\alpha }}{y}^{\hat{\beta }}=-{{\ell }}^{2}.\end{eqnarray}$
In the Einstein gauge (A.13 ), we have A.8 ) and (A.9 ) can be nicely unified in a SO(2, d) covariant way A.13 ), we have A.13 ), it recovers the 1st order formalism (A.7 ) of Einstein gravity.
$\begin{eqnarray}\begin{array}{rcl}{\rm{D}}{Y}^{\hat{a}} & = & {\rm{d}}{Y}^{\hat{a}}+{A}_{\,\,\hat{\beta }}^{\hat{a}}{Y}^{\hat{\beta }}={\ell }{A}^{\hat{a}\hat{\bullet }}={e}^{a},\\ {\rm{D}}{Y}^{\hat{\bullet }} & = & {\rm{d}}{Y}^{\hat{\bullet }}+{A}_{\,\,\hat{\beta }}^{\hat{\bullet }}{Y}^{\hat{\beta }}=0,\\ {\rm{D}}{\rm{D}}{Y}^{\hat{a}} & = & {F}_{\,\,\hat{\beta }}^{\hat{a}}{Y}^{\hat{\beta }}={\ell }{F}^{\hat{a}\hat{\bullet }}={{\rm{D}}}^{(\omega )}{e}^{a},\\ {\rm{D}}{\rm{D}}{Y}^{\hat{\bullet }} & = & {F}_{\,\,\hat{b}}^{\hat{\bullet }}{Y}^{\hat{b}}=0,\end{array}\end{eqnarray}$
where D = d + A is the SO(2, d) covariant derivative. Now the Palatini EOMs ($\begin{eqnarray}{F}^{[{\hat{\alpha }}_{1}{\hat{\alpha }}_{2}}\wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{3}}\wedge \cdots \wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{d}]}=0.\end{eqnarray}$
One can further realize the uplifting at the action level. In the Einstein gauge ($\begin{eqnarray}\begin{array}{l}\left({{\rm{\Theta }}}^{[{a}_{1}{a}_{2}}+{{\ell }}^{-2}{e}^{[{a}_{1}}\wedge {e}^{{a}_{2}}\right)\wedge {e}^{{a}_{3}}\wedge \cdots \wedge {e}^{{a}_{D}]}\\ \quad =(D+1)\,{{\ell }}^{-1}{F}^{[{\hat{\alpha }}_{1}{\hat{\alpha }}_{2}}\wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{3}}\wedge \cdots \wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{D}}{Y}^{{\hat{\alpha }}_{D+1}]},\\ {e}^{[{a}_{1}}\wedge {e}^{{a}_{2}}\wedge {e}^{{a}_{3}}\wedge \cdots \wedge {e}^{{a}_{D}]}\\ \quad =(D+1){{\ell }}^{-1}{\rm{D}}{Y}^{[{\hat{\alpha }}_{1}}\wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{2}}\wedge \cdots \wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{D}}{Y}^{{\hat{\alpha }}_{D+1}]}.\end{array}\end{eqnarray}$
It suggests that the following manifestly SO(2, d) gauge invariant action is $\begin{eqnarray}\begin{array}{rcl}{S}^{{\rm{cov}}}[Y,A] & = & \frac{1}{2{\kappa }^{2}{\ell }\,(D-2)!}{\displaystyle \int }_{M}{\epsilon }_{{\hat{\alpha }}_{1}\cdots {\hat{\alpha }}_{D+1}}\\ & & \times \,\left[{F}^{{\hat{\alpha }}_{1}{\hat{\alpha }}_{2}}-\frac{2}{D{{\ell }}^{2}}{\rm{D}}{Y}^{{\hat{\alpha }}_{1}}\wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{2}}\right]\wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{3}}\\ & & \wedge \cdots \wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{D}}{Y}^{{\hat{\alpha }}_{D+1}}\end{array}\end{eqnarray}$
where ${\epsilon }_{{\hat{\alpha }}_{1}\cdots {\hat{\alpha }}_{D+1}}$ is the unit total antisymmetric tensor of the SO(2, d) indices ${\hat{\alpha }}_{i}=\hat{\bullet },\hat{0},\cdots \,{,}\hat{d}.$ In the Einstein gauge (A.3. Equations of motion
Now let us check that the equations of motion for (A.20 ) is just the unified Palatini equations of motion (A.18 ). We notice that A.25 ) backwardly, we have reformulated the $\delta {Y}^{\hat{\alpha }}{V}_{\hat{\alpha }}$ term such that
$\begin{eqnarray*}\begin{array}{l}\delta {\displaystyle \int }_{M}{\epsilon }_{{\hat{\alpha }}_{1}\cdots {\hat{\alpha }}_{D+1}}{F}^{{\hat{\alpha }}_{1}{\hat{\alpha }}_{2}}\wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{3}}\wedge \cdots \wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{D}}{Y}^{{\hat{\alpha }}_{D+1}}\\ \,\,={\displaystyle \int }_{M}\left\{{\epsilon }_{{\hat{\alpha }}_{1}\cdots {\hat{\alpha }}_{D+1}}{\rm{D}}\delta {A}^{{\hat{\alpha }}_{1}{\hat{\alpha }}_{2}}\wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{3}}\wedge \cdots \wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{D}}{Y}^{{\hat{\alpha }}_{D+1}}\right.\\ \quad +(D-2){\epsilon }_{{\hat{\alpha }}_{1}\cdots {\hat{\alpha }}_{D+1}}{F}^{{\hat{\alpha }}_{1}{\hat{\alpha }}_{2}}\wedge ({\rm{D}}\delta {Y}^{{\hat{\alpha }}_{3}}+\delta {A}_{\,\,\hat{\beta }}^{{\hat{\alpha }}_{3}}{Y}^{\hat{\beta }})\\ \quad \wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{4}}\wedge \cdots \wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{D}}{Y}^{{\hat{\alpha }}_{D+1}}\\ \quad \left.+{\epsilon }_{{\hat{\alpha }}_{1}\cdots {\hat{\alpha }}_{D+1}}{F}^{{\hat{\alpha }}_{1}{\hat{\alpha }}_{2}}\wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{3}}\wedge \cdots \wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{D}}\delta {Y}^{{\hat{\alpha }}_{D+1}}\right\}\\ \,\,={\displaystyle \int }_{M}\left\{(D-2){\epsilon }_{{\hat{\alpha }}_{1}\cdots {\hat{\alpha }}_{D+1}}\delta {A}^{{\hat{\alpha }}_{1}{\hat{\alpha }}_{2}}\wedge {F}_{\,\,\hat{\beta }}^{{\hat{\alpha }}_{3}}{Y}^{\hat{\beta }}\right.\\ \quad \wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{4}}\wedge \cdots \wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{D}}{Y}^{{\hat{\alpha }}_{D+1}}\\ \quad +{(-1)}^{D}{\epsilon }_{{\hat{\alpha }}_{1}\cdots {\hat{\alpha }}_{D+1}}\delta {A}^{{\hat{\alpha }}_{1}{\hat{\alpha }}_{2}}\wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{3}}\end{array}\end{eqnarray*}$
$\begin{eqnarray}\begin{array}{l}\quad \wedge \cdots \wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{D}}\wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{D+1}}\\ \quad -(D-2)(D-3){\epsilon }_{{\hat{\alpha }}_{1}\cdots {\hat{\alpha }}_{D+1}}\delta {Y}^{{\hat{\alpha }}_{1}}\\ \quad \times {F}^{{\hat{\alpha }}_{2}{\hat{\alpha }}_{3}}\wedge {F}_{\,\,\hat{\beta }}^{{\hat{\alpha }}_{4}}{Y}^{\hat{\beta }}\wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{5}}\wedge \cdots \wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{D}}{Y}^{{\hat{\alpha }}_{D+1}}\\ \quad +{(-1)}^{D}(D-2){\epsilon }_{{\hat{\alpha }}_{1}\cdots {\hat{\alpha }}_{D+1}}\delta {Y}^{{\hat{\alpha }}_{1}}{F}^{{\hat{\alpha }}_{2}{\hat{\alpha }}_{3}}\\ \quad \wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{4}}\wedge \cdots \wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{D}}\wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{D+1}}\\ \quad +(D-2){\epsilon }_{{\hat{\alpha }}_{1}\cdots {\hat{\alpha }}_{D+1}}\delta {A}_{\,\,\hat{\beta }}^{{\hat{\alpha }}_{1}}{Y}^{\hat{\beta }}\\ \quad \wedge {F}^{{\hat{\alpha }}_{2}{\hat{\alpha }}_{3}}\wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{4}}\wedge \cdots \wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{D}}{Y}^{{\hat{\alpha }}_{D+1}}\\ \quad \left.+{\epsilon }_{{\hat{\alpha }}_{1}\cdots {\hat{\alpha }}_{D+1}}{F}^{{\hat{\alpha }}_{1}{\hat{\alpha }}_{2}}\wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{3}}\wedge \cdots \wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{D}}\delta {Y}^{{\hat{\alpha }}_{D+1}}\right\},\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{l}\delta {\displaystyle \int }_{M}{\epsilon }_{{\hat{\alpha }}_{1}\cdots {\hat{\alpha }}_{D+1}}{\rm{D}}{Y}^{{\hat{\alpha }}_{1}}\wedge \cdots \wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{D}}{Y}^{{\hat{\alpha }}_{D+1}}\\ \,\,={\displaystyle \int }_{M}\left\{D{\epsilon }_{{\hat{\alpha }}_{1}\cdots {\hat{\alpha }}_{D+1}}({\rm{D}}\delta {Y}^{{\hat{\alpha }}_{1}}+\delta {A}_{\,\,\hat{\beta }}^{{\hat{\alpha }}_{1}}{Y}^{\hat{\beta }})\right.\\ \quad \wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{2}}\wedge \cdots \wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{D}}{Y}^{{\hat{\alpha }}_{D+1}}\\ \quad \left.+{\epsilon }_{{\hat{\alpha }}_{1}\cdots {\hat{\alpha }}_{D+1}}{\rm{D}}{Y}^{{\hat{\alpha }}_{1}}\wedge \cdots \wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{D}}\delta {Y}^{{\hat{\alpha }}_{D+1}}\right\}\\ \,\,={\displaystyle \int }_{M}\left\{-D(D-1){\epsilon }_{{\hat{\alpha }}_{1}\cdots {\hat{\alpha }}_{D+1}}\delta {Y}^{{\hat{\alpha }}_{1}}{F}_{\,\,\hat{\beta }}^{{\hat{\alpha }}_{2}}{Y}^{\hat{\beta }}\right.\\ \quad \wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{3}}\wedge \cdots \wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{D}}{Y}^{{\hat{\alpha }}_{D+1}}\\ \quad +{(-1)}^{D}D\,{\epsilon }_{{\hat{\alpha }}_{1}\cdots {\hat{\alpha }}_{D+1}}\delta {Y}^{{\hat{\alpha }}_{1}}\wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{2}}\\ \quad \wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{3}}\wedge \cdots \wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{D}}\wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{D+1}}\\ \quad +D{\epsilon }_{{\hat{\alpha }}_{1}\cdots {\hat{\alpha }}_{D+1}}\delta {A}_{\,\,\hat{\beta }}^{{\hat{\alpha }}_{1}}{Y}^{\hat{\beta }}\wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{2}}\wedge \cdots \wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{D}}{Y}^{{\hat{\alpha }}_{D+1}}\\ \quad \left.+{\epsilon }_{{\hat{\alpha }}_{1}\cdots {\hat{\alpha }}_{D+1}}{\rm{D}}{Y}^{{\hat{\alpha }}_{1}}\wedge \cdots \wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{D}}\delta {Y}^{{\hat{\alpha }}_{D+1}}\right\}.\end{array}\end{eqnarray}$
At the 3rd steps, we have used the integration by parts as well as $\begin{eqnarray}{\rm{D}}{\rm{D}}Y=FY,\qquad {\rm{D}}F=0.\end{eqnarray}$
By using the constraint ${Y}^{\hat{\beta }}{Y}_{\hat{\beta }}=-{{\ell }}^{2}$ as well as the facts $\begin{eqnarray}{\rm{D}}{Y}^{\hat{\beta }}{Y}_{\hat{\beta }}=0,\quad \delta {Y}^{\hat{\beta }}{Y}_{\hat{\beta }}=0,\end{eqnarray}$
which are implied by the constraint, we can get the resummation identities5 $\begin{eqnarray}\begin{array}{l}{\epsilon }_{{\hat{\alpha }}_{1}\cdots {\hat{\alpha }}_{D+1}}\delta {A}_{\,\,\hat{\beta }}^{{\hat{\alpha }}_{1}}{Y}^{\hat{\beta }}\wedge {F}^{{\hat{\alpha }}_{2}{\hat{\alpha }}_{3}}\\ \quad \wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{4}}\wedge \cdots \wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{D}}{Y}^{{\hat{\alpha }}_{D+1}}\\ \,\,=-{\epsilon }_{{\hat{\alpha }}_{1}\cdots {\hat{\alpha }}_{D+1}}\delta {A}^{{\hat{\alpha }}_{1}{\hat{\alpha }}_{2}}\wedge {F}_{\,\,\hat{\beta }}^{{\hat{\alpha }}_{3}}{Y}^{\hat{\beta }}\\ \quad \wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{4}}\wedge \cdots \wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{D}}{Y}^{{\hat{\alpha }}_{D+1}}\\ \quad +\frac{{(-1)}^{D}{{\ell }}^{2}}{2}{\epsilon }_{{\hat{\alpha }}_{1}\cdots {\hat{\alpha }}_{D+1}}\delta {A}^{{\hat{\alpha }}_{1}{\hat{\alpha }}_{2}}\\ \quad \wedge {F}^{{\hat{\alpha }}_{3}{\hat{\alpha }}_{4}}\wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{5}}\wedge \cdots \wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{D+1}},\,\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{c}{\epsilon }_{{\hat{\alpha }}_{1}\cdots {\hat{\alpha }}_{D+1}}\delta {Y}^{{\hat{\alpha }}_{1}}{F}^{{\hat{\alpha }}_{2}{\hat{\alpha }}_{3}}\wedge {F}_{\,\,\hat{\beta }}^{{\hat{\alpha }}_{4}}{Y}^{\hat{\beta }}\\ \quad \wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{5}}\wedge \cdots \wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{D}}{Y}^{{\hat{\alpha }}_{D+1}}\\ \quad =\frac{{(-1)}^{D+1}{{\ell }}^{2}}{4}{\epsilon }_{{\hat{\alpha }}_{1}\cdots {\hat{\alpha }}_{D+1}}\delta {Y}^{{\hat{\alpha }}_{1}}{F}^{{\hat{\alpha }}_{2}{\hat{\alpha }}_{3}}\\ \quad \wedge {F}^{{\hat{\alpha }}_{4}{\hat{\alpha }}_{5}}\wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{6}}\wedge \cdots \wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{D+1}},\,\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{l}{\epsilon }_{{\hat{\alpha }}_{1}\cdots {\hat{\alpha }}_{D+1}}\delta {A}_{\,\,\hat{\beta }}^{{\hat{\alpha }}_{1}}{Y}^{\hat{\beta }}\wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{2}}\wedge \cdots \wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{D}}{Y}^{{\hat{\alpha }}_{D+1}}\,\\ \,=\frac{{(-1)}^{D}{{\ell }}^{2}}{2}{\epsilon }_{{\hat{\alpha }}_{1}\cdots {\hat{\alpha }}_{D+1}}\delta {A}^{{\hat{\alpha }}_{1}{\hat{\alpha }}_{2}}\wedge {F}^{{\hat{\alpha }}_{3}{\hat{\alpha }}_{4}}\\ \quad \wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{5}}\wedge \cdots \wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{D+1}},\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{l}{\epsilon }_{{\hat{\alpha }}_{1}\cdots {\hat{\alpha }}_{D+1}}\delta {Y}^{{\hat{\alpha }}_{1}}{F}_{\,\,\hat{\beta }}^{{\hat{\alpha }}_{2}}{Y}^{\hat{\beta }}\wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{3}}\wedge \cdots \wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{D}}{Y}^{{\hat{\alpha }}_{D+1}}\\ =\frac{{(-1)}^{D+1}{{\ell }}^{2}}{2}{\epsilon }_{{\hat{\alpha }}_{1}\cdots {\hat{\alpha }}_{D+1}}\delta {Y}^{{\hat{\alpha }}_{1}}{F}^{{\hat{\alpha }}_{2}{\hat{\alpha }}_{3}}\wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{4}}\\ \quad \wedge \cdots \wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{D+1}}.\end{array}\end{eqnarray}$
Thus $\begin{eqnarray}\begin{array}{l}\delta {\displaystyle \int }_{M}{\epsilon }_{{\hat{\alpha }}_{1}\cdots {\hat{\alpha }}_{D+1}}{F}^{{\hat{\alpha }}_{1}{\hat{\alpha }}_{2}}\wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{3}}\wedge \cdots \wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{D}}{Y}^{{\hat{\alpha }}_{D+1}}\\ ={\displaystyle \int }_{M}\left\{{(-1)}^{D}{\epsilon }_{{\hat{\alpha }}_{1}\cdots {\hat{\alpha }}_{D+1}}\delta {A}^{{\hat{\alpha }}_{1}{\hat{\alpha }}_{2}}\wedge \left[\frac{{{\ell }}^{2}(D-2)}{2}{F}^{{\hat{\alpha }}_{3}{\hat{\alpha }}_{4}}\right.\right.\\ \quad \left.+{\rm{D}}{Y}^{{\hat{\alpha }}_{3}}\wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{4}}\right]\wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{5}}\wedge \cdots \wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{D+1}}\\ \quad +{(-1)}^{D}{\epsilon }_{{\hat{\alpha }}_{1}\cdots {\hat{\alpha }}_{D+1}}\delta {Y}^{{\hat{\alpha }}_{1}}\left[\frac{(D-2)(D-3){{\ell }}^{2}}{4}{F}^{{\hat{\alpha }}_{2}{\hat{\alpha }}_{3}}\right.\\ \quad \wedge {F}^{{\hat{\alpha }}_{4}{\hat{\alpha }}_{5}}\wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{6}}\wedge \cdots \wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{D+1}}\\ \quad \left.\left.+(D-1){F}^{{\hat{\alpha }}_{2}{\hat{\alpha }}_{3}}\wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{4}}\wedge \cdots \wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{D+1}}\right]\right\},\,\\ \end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{l}\delta {\displaystyle \int }_{M}{\epsilon }_{{\hat{\alpha }}_{1}\cdots {\hat{\alpha }}_{D+1}}{\rm{D}}{Y}^{{\hat{\alpha }}_{1}}\wedge \cdots \wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{D}}{Y}^{{\hat{\alpha }}_{D+1}}\\ ={\displaystyle \int }_{M}\left\{\frac{{(-1)}^{D}{{\ell }}^{2}D}{2}{\epsilon }_{{\hat{\alpha }}_{1}\cdots {\hat{\alpha }}_{D+1}}\delta {A}^{{\hat{\alpha }}_{1}{\hat{\alpha }}_{2}}\right.\\ \quad \wedge {F}^{{\hat{\alpha }}_{3}{\hat{\alpha }}_{4}}\wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{5}}\wedge \cdots \wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{D+1}}\\ \quad +{(-1)}^{D}\,{\epsilon }_{{\hat{\alpha }}_{1}\cdots {\hat{\alpha }}_{D+1}}\delta {Y}^{{\hat{\alpha }}_{1}}\left[\frac{D(D-1){{\ell }}^{2}}{2}{F}^{{\hat{\alpha }}_{2}{\hat{\alpha }}_{3}}\right.\\ \quad \left.\left.\wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{4}}\wedge \cdots \wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{D+1}}+(D+1){\rm{D}}{Y}^{{\hat{\alpha }}_{2}}\wedge \cdots \wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{D+1}}\right]\right\}\\ ={\displaystyle \int }_{M}\left\{\frac{{(-1)}^{D}{{\ell }}^{2}D}{2}{\epsilon }_{{\hat{\alpha }}_{1}\cdots {\hat{\alpha }}_{D+1}}\delta {A}^{{\hat{\alpha }}_{1}{\hat{\alpha }}_{2}}\wedge {F}^{{\hat{\alpha }}_{3}{\hat{\alpha }}_{4}}\right.\\ \wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{5}}\wedge \cdots \wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{D+1}}+\frac{{(-1)}^{D}D(D-1){{\ell }}^{2}}{2}\\ \,\,\left.\times \,{\epsilon }_{{\hat{\alpha }}_{1}\cdots {\hat{\alpha }}_{D+1}}\delta {Y}^{{\hat{\alpha }}_{1}}{F}^{{\hat{\alpha }}_{2}{\hat{\alpha }}_{3}}\wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{4}}\wedge \cdots \wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{D+1}}\right\},\end{array}\end{eqnarray}$
where we have used that $\delta {Y}^{[{\hat{\alpha }}_{1}}{\rm{D}}{Y}^{{\hat{\alpha }}_{2}}\wedge \cdots \wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{D+1}]}=0$ since there are only D linear independent directions which are orthogonal with ${Y}^{\hat{\alpha }}$. In total, we have $\begin{eqnarray}\begin{array}{l}2{\kappa }^{2}{\ell }\,(D-2)!\,\delta {S}^{{\rm{cov}}}[Y,A]\\ =\delta {\displaystyle \int }_{M}{\epsilon }_{{\hat{\alpha }}_{1}\cdots {\hat{\alpha }}_{D+1}}\left[{F}^{{\hat{\alpha }}_{1}{\hat{\alpha }}_{2}}-\frac{2}{D{{\ell }}^{2}}{\rm{D}}{Y}^{{\hat{\alpha }}_{1}}\wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{2}}\right]\\ \wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{3}}\wedge \cdots \wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{D}}{Y}^{{\hat{\alpha }}_{D+1}}\\ ={\displaystyle \int }_{M}\left\{\frac{{(-1)}^{D}{{\ell }}^{2}(D-2)}{2}{\epsilon }_{{\hat{\alpha }}_{1}\cdots {\hat{\alpha }}_{D+1}}\delta {A}^{{\hat{\alpha }}_{1}{\hat{\alpha }}_{2}}\right.\\ \wedge {F}^{{\hat{\alpha }}_{3}{\hat{\alpha }}_{4}}\wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{5}}\wedge \cdots \wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{D+1}}\\ +\frac{{(-1)}^{D}(D-2)(D-3){{\ell }}^{2}}{4}{\epsilon }_{{\hat{\alpha }}_{1}\cdots {\hat{\alpha }}_{D+1}}\delta {Y}^{{\hat{\alpha }}_{1}}\\ \left.\times \,{F}^{{\hat{\alpha }}_{2}{\hat{\alpha }}_{3}}\wedge {F}^{{\hat{\alpha }}_{4}{\hat{\alpha }}_{5}}\wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{6}}\wedge \cdots \wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{D+1}}\right\}\\ ={\displaystyle \int }_{M}\left\{\frac{{(-1)}^{D}{{\ell }}^{2}(D-2)}{2}{\epsilon }_{{\hat{\alpha }}_{1}\cdots {\hat{\alpha }}_{D+1}}\delta {A}^{{\hat{\alpha }}_{1}{\hat{\alpha }}_{2}}\right.\\ \wedge {F}^{{\hat{\alpha }}_{3}{\hat{\alpha }}_{4}}\wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{5}}\wedge \cdots \wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{D+1}}\\ -(D-2)(D-3){\epsilon }_{{\hat{\alpha }}_{1}\cdots {\hat{\alpha }}_{D+1}}\delta {Y}^{{\hat{\alpha }}_{1}}{F}^{{\hat{\alpha }}_{2}{\hat{\alpha }}_{3}}\\ \left.\wedge {F}_{\,\,\hat{\beta }}^{{\hat{\alpha }}_{4}}{Y}^{\hat{\beta }}\wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{5}}\wedge \cdots \wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{D}}{Y}^{{\hat{\alpha }}_{D+1}}\right\}.\end{array}\end{eqnarray}$
At the last step, by using the resummation identity ($\begin{eqnarray*}{V}_{\hat{\alpha }}{Y}^{\hat{\alpha }}=0\,\iff \,{V}^{\hat{\alpha }}=({\delta }_{\hat{\beta }}^{\hat{\alpha }}+{{\ell }}^{-2}{Y}^{\hat{\alpha }}{Y}_{\hat{\beta }}){V}^{\hat{\beta }}.\end{eqnarray*}$
In summary, the bulk equations of motion for δA are A.18 ). On the other hand, since the components of $\delta {Y}^{\hat{\alpha }}$ are constrained by the linear relation
$\begin{eqnarray}{\epsilon }_{\hat{\alpha }\hat{\beta }{\hat{\alpha }}_{1}\cdots {\hat{\alpha }}_{d}}{F}^{{\hat{\alpha }}_{1}{\hat{\alpha }}_{2}}\wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{3}}\wedge \cdots \wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{d}}=0,\end{eqnarray}$
which are exactly the unified Palatini equations ($\begin{eqnarray}{Y}_{\hat{\alpha }}\delta {Y}^{\hat{\alpha }}=0,\end{eqnarray}$
the correct EOM of $\delta {Y}^{\hat{\alpha }}$ should be given by the ${V}_{\hat{\alpha }}$ satisfying ${V}_{\hat{\alpha }}{Y}^{\hat{\alpha }}=0$. That is $\begin{eqnarray}\begin{array}{l}{\epsilon }_{\hat{\alpha }{\hat{\alpha }}_{1}\cdots {\hat{\alpha }}_{d+1}}{F}^{{\hat{\alpha }}_{1}{\hat{\alpha }}_{2}}\wedge {F}_{\,\,\hat{\beta }}^{{\hat{\alpha }}_{3}}{Y}^{\hat{\beta }}\wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{4}}\\ \quad \wedge \cdots \wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{d}}{Y}^{{\hat{\alpha }}_{d+1}}=0.\end{array}\end{eqnarray}$
Since $\begin{eqnarray}\begin{array}{l}{\epsilon }_{\hat{\alpha }\hat{\beta }{\hat{\alpha }}_{1}\cdots {\hat{\alpha }}_{d}}{Y}^{\hat{\beta }}{\rm{D}}({F}^{{\hat{\alpha }}_{1}{\hat{\alpha }}_{2}}\wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{3}}\wedge \cdots \wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{d}})\\ \,\,={(-1)}^{d}(d-2){\epsilon }_{\hat{\alpha }{\hat{\alpha }}_{1}\cdots {\hat{\alpha }}_{d+1}}{F}^{{\hat{\alpha }}_{1}{\hat{\alpha }}_{2}}\wedge {F}_{\,\,\hat{\beta }}^{{\hat{\alpha }}_{3}}{Y}^{\hat{\beta }}\\ \quad \wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{4}}\wedge \cdots \wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{d}}{Y}^{{\hat{\alpha }}_{d+1}},\end{array}\end{eqnarray}$
the $\delta {Y}^{\hat{\alpha }}$ EOM is automatically satisfied if the EOM for δA is satisfied. Therefore, the introducing of ${Y}^{\hat{\alpha }}$ does not impose any additional constraints beyond the original Palatini equations of motion.A.4. Proof of resummation identities
Let us take
$\begin{eqnarray}\begin{array}{l}{\epsilon }_{{\hat{\alpha }}_{1}\cdots {\hat{\alpha }}_{D+1}}\delta {A}_{\,\,\hat{\beta }}^{{\hat{\alpha }}_{1}}{Y}^{\hat{\beta }}\wedge {F}^{{\hat{\alpha }}_{2}{\hat{\alpha }}_{3}}\wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{4}}\\ \quad \wedge \cdots \wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{D}}{Y}^{{\hat{\alpha }}_{D+1}}\\ \,\,=-{\epsilon }_{{\hat{\alpha }}_{1}\cdots {\hat{\alpha }}_{D+1}}\delta {A}^{{\hat{\alpha }}_{1}{\hat{\alpha }}_{2}}\wedge {F}_{\,\,\hat{\beta }}^{{\hat{\alpha }}_{3}}{Y}^{\hat{\beta }}\\ \quad \wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{4}}\wedge \cdots \wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{D}}{Y}^{{\hat{\alpha }}_{D+1}}\\ \quad +\frac{{(-1)}^{D}{{\ell }}^{2}}{2}{\epsilon }_{{\hat{\alpha }}_{1}\cdots {\hat{\alpha }}_{D+1}}\delta {A}^{{\hat{\alpha }}_{1}{\hat{\alpha }}_{2}}\wedge {F}^{{\hat{\alpha }}_{3}{\hat{\alpha }}_{4}}\\ \quad \wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{5}}\wedge \cdots \wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{D+1}}\end{array}\end{eqnarray}$
as a example to illustrate the proof of the resummation identities. To avoid confusion of the notations in the middle steps, let us recover the notation of summations which is ignored in the Einstein summation convention. We find $\begin{eqnarray*}\begin{array}{l}\displaystyle \sum _{{\hat{\alpha }}_{1},\cdots \,{,}{\hat{\alpha }}_{D+1},\hat{\beta }}{\epsilon }_{{\hat{\alpha }}_{1}\cdots {\hat{\alpha }}_{D+1}}\delta {A}_{\,\,\hat{\beta }}^{{\hat{\alpha }}_{1}}{Y}^{\hat{\beta }}\wedge {F}^{{\hat{\alpha }}_{2}{\hat{\alpha }}_{3}}\\ \quad \wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{4}}\wedge \cdots \wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{D}}{Y}^{{\hat{\alpha }}_{D+1}}\\ =\displaystyle \sum _{{\hat{\alpha }}_{1},\cdots \,{,}{\hat{\alpha }}_{D+1}}{\epsilon }_{{\hat{\alpha }}_{1}\cdots {\hat{\alpha }}_{D+1}}\displaystyle \sum _{\hat{\beta }}\delta {A}_{\,\,\hat{\beta }}^{{\hat{\alpha }}_{1}}{Y}^{\hat{\beta }}\wedge {F}^{{\hat{\alpha }}_{2}{\hat{\alpha }}_{3}}\\ \quad \wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{4}}\wedge \cdots \wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{D}}{Y}^{{\hat{\alpha }}_{D+1}}\\ =\displaystyle \sum _{{\hat{\alpha }}_{1},\cdots \,{,}{\hat{\alpha }}_{D+1}}{\epsilon }_{{\hat{\alpha }}_{1}\cdots {\hat{\alpha }}_{D+1}}\delta {A}_{\,\,{\hat{\alpha }}_{1}}^{{\hat{\alpha }}_{1}}{Y}^{{\hat{\alpha }}_{1}}\wedge {F}^{{\hat{\alpha }}_{2}{\hat{\alpha }}_{3}}\end{array}\end{eqnarray*}$
$\begin{eqnarray*}\begin{array}{l}\quad \wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{4}}\wedge \cdots \wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{D}}{Y}^{{\hat{\alpha }}_{D+1}}\\ \quad +\displaystyle \sum _{{\hat{\alpha }}_{1},\cdots \,{,}{\hat{\alpha }}_{D+1}}{\epsilon }_{{\hat{\alpha }}_{1}\cdots {\hat{\alpha }}_{D+1}}\delta {A}_{\,\,{\hat{\alpha }}_{2}}^{{\hat{\alpha }}_{1}}{Y}^{{\hat{\alpha }}_{2}}\wedge {F}^{{\hat{\alpha }}_{2}{\hat{\alpha }}_{3}}\\ \wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{4}}\wedge \cdots \wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{D}}{Y}^{{\hat{\alpha }}_{D+1}}\\ \quad +\displaystyle \sum _{{\hat{\alpha }}_{1},\cdots \,{,}{\hat{\alpha }}_{D+1}}{\epsilon }_{{\hat{\alpha }}_{1}\cdots {\hat{\alpha }}_{D+1}}\delta {A}_{\,\,{\hat{\alpha }}_{3}}^{{\hat{\alpha }}_{1}}{Y}^{{\hat{\alpha }}_{3}}\wedge {F}^{{\hat{\alpha }}_{2}{\hat{\alpha }}_{3}}\\ \quad \wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{4}}\wedge \cdots \wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{D}}{Y}^{{\hat{\alpha }}_{D+1}}\\ \quad +\displaystyle \sum _{{\hat{\alpha }}_{1},\cdots \,{,}{\hat{\alpha }}_{D+1}}{\epsilon }_{{\hat{\alpha }}_{1}\cdots {\hat{\alpha }}_{D+1}}\delta {A}_{\,\,{\hat{\alpha }}_{4}}^{{\hat{\alpha }}_{1}}{Y}^{{\hat{\alpha }}_{4}}\wedge {F}^{{\hat{\alpha }}_{2}{\hat{\alpha }}_{3}}\,\end{array}\end{eqnarray*}$
$\begin{eqnarray*}\begin{array}{l}\quad \wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{4}}\wedge \cdots \wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{D}}{Y}^{{\hat{\alpha }}_{D+1}}\\ \quad +\cdots +\displaystyle \sum _{{\hat{\alpha }}_{1},\cdots \,{,}{\hat{\alpha }}_{D+1}}{\epsilon }_{{\hat{\alpha }}_{1}\cdots {\hat{\alpha }}_{D+1}}\delta {A}_{\,\,{\hat{\alpha }}_{D}}^{{\hat{\alpha }}_{1}}{Y}^{{\hat{\alpha }}_{D}}\wedge {F}^{{\hat{\alpha }}_{2}{\hat{\alpha }}_{3}}\\ \quad \wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{4}}\wedge \cdots \wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{D}}{Y}^{{\hat{\alpha }}_{D+1}}\\ \quad +\displaystyle \sum _{{\hat{\alpha }}_{1},\cdots \,{,}{\hat{\alpha }}_{D+1}}{\epsilon }_{{\hat{\alpha }}_{1}\cdots {\hat{\alpha }}_{D+1}}\delta {A}_{\,\,{\hat{\alpha }}_{D+1}}^{{\hat{\alpha }}_{1}}{Y}^{{\hat{\alpha }}_{D+1}}\wedge {F}^{{\hat{\alpha }}_{2}{\hat{\alpha }}_{3}}\\ \quad \wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{4}}\wedge \cdots \wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{D}}{Y}^{{\hat{\alpha }}_{D+1}}\\ \,\,=0+2\displaystyle \sum _{{\hat{\alpha }}_{1},\cdots \,{,}{\hat{\alpha }}_{D+1}}{\epsilon }_{{\hat{\alpha }}_{1}\cdots {\hat{\alpha }}_{D+1}}\delta {A}_{\,\,{\hat{\alpha }}_{2}}^{{\hat{\alpha }}_{1}}{Y}^{{\hat{\alpha }}_{2}}\\ \quad \wedge {F}^{{\hat{\alpha }}_{2}{\hat{\alpha }}_{3}}\wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{4}}\wedge \cdots \wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{D}}{Y}^{{\hat{\alpha }}_{D+1}}\,\end{array}\end{eqnarray*}$
$\begin{eqnarray}\begin{array}{l}\quad +(D-3)\displaystyle \sum _{{\hat{\alpha }}_{1},\cdots \,{,}{\hat{\alpha }}_{D+1}}{\epsilon }_{{\hat{\alpha }}_{1}\cdots {\hat{\alpha }}_{D+1}}\delta {A}_{\,\,{\hat{\alpha }}_{4}}^{{\hat{\alpha }}_{1}}{Y}^{{\hat{\alpha }}_{4}}\\ \quad \wedge {F}^{{\hat{\alpha }}_{2}{\hat{\alpha }}_{3}}\wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{4}}\wedge \cdots \wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{D}}{Y}^{{\hat{\alpha }}_{D+1}}\\ \quad +\displaystyle \sum _{{\hat{\alpha }}_{1},\cdots \,{,}{\hat{\alpha }}_{D+1}}{\epsilon }_{{\hat{\alpha }}_{1}\cdots {\hat{\alpha }}_{D+1}}\delta {A}_{\,\,{\hat{\alpha }}_{D+1}}^{{\hat{\alpha }}_{1}}{Y}^{{\hat{\alpha }}_{D+1}}\\ \wedge {F}^{{\hat{\alpha }}_{2}{\hat{\alpha }}_{3}}\wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{4}}\wedge \cdots \wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{D}}{Y}^{{\hat{\alpha }}_{D+1}}.\,\end{array}\end{eqnarray}$
At the 3rd step, we have used the fact that in our present computation $\begin{eqnarray}{\epsilon }_{{\hat{\alpha }}_{1}\cdots {\hat{\alpha }}_{D+1}}\displaystyle \sum _{\hat{\beta }=\hat{\bullet },\hat{0},\cdots \,{,}\hat{d}}\,\iff \,{\epsilon }_{{\hat{\alpha }}_{1}\cdots {\hat{\alpha }}_{D+1}}\displaystyle \sum _{\hat{\beta }={\hat{\alpha }}_{1},\cdots \,{,}{\hat{\alpha }}_{D+1}}.\end{eqnarray}$
Since ${\epsilon }_{{\hat{\alpha }}_{1}\cdots {\hat{\alpha }}_{D+1}}$ is non-zero only when ${\hat{\alpha }}_{i}\ne {\hat{\alpha }}_{j}\,(\forall i\ne j)$. At the 4th step, the terms with same structures are collected together.On the other hand, we similarly notice that
$\begin{eqnarray*}\begin{array}{l}2\displaystyle \sum _{{\hat{\alpha }}_{1},\cdots \,{,}{\hat{\alpha }}_{D+1}}{\epsilon }_{{\hat{\alpha }}_{1}\cdots {\hat{\alpha }}_{D+1}}\delta {A}^{{\hat{\alpha }}_{1}{\hat{\alpha }}_{2}}{Y}_{{\hat{\alpha }}_{2}}\wedge {F}^{{\hat{\alpha }}_{2}{\hat{\alpha }}_{3}}\\ \quad \wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{4}}\wedge \cdots \wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{D}}{Y}^{{\hat{\alpha }}_{D+1}}\\ \quad =\displaystyle \sum _{{\hat{\alpha }}_{1},\cdots \,{,}{\hat{\alpha }}_{D+1}}{\epsilon }_{{\hat{\alpha }}_{1}\cdots {\hat{\alpha }}_{D+1}}\displaystyle \sum _{\hat{\beta }}\delta {A}^{{\hat{\alpha }}_{1}{\hat{\alpha }}_{2}}{Y}_{\hat{\beta }}\\ \quad \wedge {F}^{\hat{\beta }{\hat{\alpha }}_{3}}\wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{4}}\wedge \cdots \wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{D}}{Y}^{{\hat{\alpha }}_{D+1}}\\ \quad -\displaystyle \sum _{{\hat{\alpha }}_{1},\cdots \,{,}{\hat{\alpha }}_{D+1}}{\epsilon }_{{\hat{\alpha }}_{1}\cdots {\hat{\alpha }}_{D+1}}\delta {A}^{{\hat{\alpha }}_{1}{\hat{\alpha }}_{2}}{Y}_{{\hat{\alpha }}_{3}}\wedge {F}^{{\hat{\alpha }}_{3}{\hat{\alpha }}_{3}}\,\end{array}\end{eqnarray*}$
$\begin{eqnarray*}\begin{array}{l}\quad \wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{4}}\wedge \cdots \wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{D}}{Y}^{{\hat{\alpha }}_{D+1}}\\ \quad -(D-3)\displaystyle \sum _{{\hat{\alpha }}_{1},\cdots \,{,}{\hat{\alpha }}_{D+1}}{\epsilon }_{{\hat{\alpha }}_{1}\cdots {\hat{\alpha }}_{D+1}}\delta {A}^{{\hat{\alpha }}_{1}{\hat{\alpha }}_{2}}{Y}_{{\hat{\alpha }}_{4}}\wedge {F}^{{\hat{\alpha }}_{4}{\hat{\alpha }}_{3}}\\ \quad \wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{4}}\wedge \cdots \wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{D}}{Y}^{{\hat{\alpha }}_{D+1}}\\ \quad -\displaystyle \sum _{{\hat{\alpha }}_{1},\cdots \,{,}{\hat{\alpha }}_{D+1}}{\epsilon }_{{\hat{\alpha }}_{1}\cdots {\hat{\alpha }}_{D+1}}\delta {A}^{{\hat{\alpha }}_{1}{\hat{\alpha }}_{2}}{Y}_{{\hat{\alpha }}_{D+1}}\wedge {F}^{{\hat{\alpha }}_{D+1}{\hat{\alpha }}_{3}}\\ \quad \wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{4}}\wedge \cdots \wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{D}}{Y}^{{\hat{\alpha }}_{D+1}}\\ \quad =\displaystyle \sum _{{\hat{\alpha }}_{1},\cdots \,{,}{\hat{\alpha }}_{D+1}}{\epsilon }_{{\hat{\alpha }}_{1}\cdots {\hat{\alpha }}_{D+1}}\displaystyle \sum _{\hat{\beta }}\delta {A}^{{\hat{\alpha }}_{1}{\hat{\alpha }}_{2}}{Y}_{\hat{\beta }}\wedge {F}^{\hat{\beta }{\hat{\alpha }}_{3}}\,\end{array}\end{eqnarray*}$
$\begin{eqnarray}\begin{array}{l}\quad \wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{4}}\wedge \cdots \wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{D}}{Y}^{{\hat{\alpha }}_{D+1}}-0\\ \quad -(D-3)\displaystyle \sum _{{\hat{\alpha }}_{1},\cdots \,{,}{\hat{\alpha }}_{D+1}}{\epsilon }_{{\hat{\alpha }}_{1}\cdots {\hat{\alpha }}_{D+1}}\delta {A}^{{\hat{\alpha }}_{1}{\hat{\alpha }}_{2}}{Y}_{{\hat{\alpha }}_{4}}\wedge {F}^{{\hat{\alpha }}_{4}{\hat{\alpha }}_{3}}\\ \quad \wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{4}}\wedge \cdots \wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{D}}{Y}^{{\hat{\alpha }}_{D+1}}\\ \quad -\displaystyle \sum _{{\hat{\alpha }}_{1},\cdots \,{,}{\hat{\alpha }}_{D+1}}{\epsilon }_{{\hat{\alpha }}_{1}\cdots {\hat{\alpha }}_{D+1}}\delta {A}^{{\hat{\alpha }}_{1}{\hat{\alpha }}_{2}}{Y}_{{\hat{\alpha }}_{D+1}}\wedge {F}^{{\hat{\alpha }}_{D+1}{\hat{\alpha }}_{3}}\\ \quad \wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{4}}\wedge \cdots \wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{D}}{Y}^{{\hat{\alpha }}_{D+1}},\end{array}\end{eqnarray}$
$\begin{eqnarray*}\begin{array}{l}2\displaystyle \sum _{{\hat{\alpha }}_{1},\cdots \,{,}{\hat{\alpha }}_{D+1}}{\epsilon }_{{\hat{\alpha }}_{1}\cdots {\hat{\alpha }}_{D+1}}\delta {A}^{{\hat{\alpha }}_{1}{\hat{\alpha }}_{4}}{Y}_{{\hat{\alpha }}_{4}}\wedge {F}^{{\hat{\alpha }}_{2}{\hat{\alpha }}_{3}}\\ \quad \wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{4}}\wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{5}}\wedge \cdots \wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{D}}{Y}^{{\hat{\alpha }}_{D+1}}\\ \quad =\displaystyle \sum _{{\hat{\alpha }}_{1},\cdots \,{,}{\hat{\alpha }}_{D+1}}{\epsilon }_{{\hat{\alpha }}_{1}\cdots {\hat{\alpha }}_{D+1}}\displaystyle \sum _{\hat{\beta }}\delta {A}^{{\hat{\alpha }}_{1}{\hat{\alpha }}_{4}}{Y}_{\hat{\beta }}\wedge {F}^{{\hat{\alpha }}_{2}{\hat{\alpha }}_{3}}\\ \quad \wedge {\rm{D}}{Y}^{\hat{\beta }}\wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{5}}\wedge \cdots \wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{D}}{Y}^{{\hat{\alpha }}_{D+1}}\\ \quad -2\displaystyle \sum _{{\hat{\alpha }}_{1},\cdots \,{,}{\hat{\alpha }}_{D+1}}{\epsilon }_{{\hat{\alpha }}_{1}\cdots {\hat{\alpha }}_{D+1}}\delta {A}^{{\hat{\alpha }}_{1}{\hat{\alpha }}_{4}}{Y}_{{\hat{\alpha }}_{2}}\wedge {F}^{{\hat{\alpha }}_{2}{\hat{\alpha }}_{3}}\,\end{array}\end{eqnarray*}$
$\begin{eqnarray*}\begin{array}{l}\quad \wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{2}}\wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{5}}\wedge \cdots \wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{D}}{Y}^{{\hat{\alpha }}_{D+1}}\\ \quad -(D-4)\displaystyle \sum _{{\hat{\alpha }}_{1},\cdots \,{,}{\hat{\alpha }}_{D+1}}{\epsilon }_{{\hat{\alpha }}_{1}\cdots {\hat{\alpha }}_{D+1}}\delta {A}^{{\hat{\alpha }}_{1}{\hat{\alpha }}_{4}}{Y}_{{\hat{\alpha }}_{5}}\wedge {F}^{{\hat{\alpha }}_{2}{\hat{\alpha }}_{3}}\\ \quad \wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{5}}\wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{5}}\wedge \cdots \wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{D}}{Y}^{{\hat{\alpha }}_{D+1}}\\ \quad -\displaystyle \sum _{{\hat{\alpha }}_{1},\cdots \,{,}{\hat{\alpha }}_{D+1}}{\epsilon }_{{\hat{\alpha }}_{1}\cdots {\hat{\alpha }}_{D+1}}\delta {A}^{{\hat{\alpha }}_{1}{\hat{\alpha }}_{4}}{Y}_{{\hat{\alpha }}_{D+1}}\wedge {F}^{{\hat{\alpha }}_{2}{\hat{\alpha }}_{3}}\\ \quad \wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{D+1}}\wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{5}}\wedge \cdots \wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{D}}{Y}^{{\hat{\alpha }}_{D+1}}\,\end{array}\end{eqnarray*}$
$\begin{eqnarray}\begin{array}{l}\quad =\displaystyle \sum _{{\hat{\alpha }}_{1},\cdots \,{,}{\hat{\alpha }}_{D+1}}{\epsilon }_{{\hat{\alpha }}_{1}\cdots {\hat{\alpha }}_{D+1}}\displaystyle \sum _{\hat{\beta }}\delta {A}^{{\hat{\alpha }}_{1}{\hat{\alpha }}_{4}}{Y}_{\hat{\beta }}\wedge {F}^{{\hat{\alpha }}_{2}{\hat{\alpha }}_{3}}\\ \quad \wedge {\rm{D}}{Y}^{\hat{\beta }}\wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{5}}\wedge \cdots \wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{D}}{Y}^{{\hat{\alpha }}_{D+1}}\\ -2\displaystyle \sum _{{\hat{\alpha }}_{1},\cdots \,{,}{\hat{\alpha }}_{D+1}}{\epsilon }_{{\hat{\alpha }}_{1}\cdots {\hat{\alpha }}_{D+1}}\delta {A}^{{\hat{\alpha }}_{1}{\hat{\alpha }}_{4}}{Y}_{{\hat{\alpha }}_{2}}\wedge {F}^{{\hat{\alpha }}_{2}{\hat{\alpha }}_{3}}\\ \quad \wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{2}}\wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{5}}\wedge \cdots \wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{D}}{Y}^{{\hat{\alpha }}_{D+1}}-0\\ \quad -\displaystyle \sum _{{\hat{\alpha }}_{1},\cdots \,{,}{\hat{\alpha }}_{D+1}}{\epsilon }_{{\hat{\alpha }}_{1}\cdots {\hat{\alpha }}_{D+1}}\delta {A}^{{\hat{\alpha }}_{1}{\hat{\alpha }}_{4}}{Y}_{{\hat{\alpha }}_{D+1}}\wedge {F}^{{\hat{\alpha }}_{2}{\hat{\alpha }}_{3}}\\ \quad \wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{D+1}}\wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{5}}\wedge \cdots \wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{D}}{Y}^{{\hat{\alpha }}_{D+1}},\,\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{l}2\displaystyle \sum _{{\hat{\alpha }}_{1},\cdots \,{,}{\hat{\alpha }}_{D+1}}{\epsilon }_{{\hat{\alpha }}_{1}\cdots {\hat{\alpha }}_{D+1}}\delta {A}^{{\hat{\alpha }}_{1}{\hat{\alpha }}_{D+1}}{Y}_{{\hat{\alpha }}_{D+1}}\wedge {F}^{{\hat{\alpha }}_{2}{\hat{\alpha }}_{3}}\\ \quad \wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{4}}\wedge \cdots \wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{D}}{Y}^{{\hat{\alpha }}_{D+1}}\\ =\displaystyle \sum _{{\hat{\alpha }}_{1},\cdots \,{,}{\hat{\alpha }}_{D+1}}{\epsilon }_{{\hat{\alpha }}_{1}\cdots {\hat{\alpha }}_{D+1}}\displaystyle \sum _{\hat{\beta }}\delta {A}^{{\hat{\alpha }}_{1}{\hat{\alpha }}_{D+1}}{Y}_{\hat{\beta }}\wedge {F}^{{\hat{\alpha }}_{2}{\hat{\alpha }}_{3}}\\ \quad \wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{4}}\wedge \cdots \wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{D}}{Y}^{\hat{\beta }}\\ -2\displaystyle \sum _{{\hat{\alpha }}_{1},\cdots \,{,}{\hat{\alpha }}_{D+1}}{\epsilon }_{{\hat{\alpha }}_{1}\cdots {\hat{\alpha }}_{D+1}}\delta {A}^{{\hat{\alpha }}_{1}{\hat{\alpha }}_{D+1}}{Y}_{{\hat{\alpha }}_{2}}\wedge {F}^{{\hat{\alpha }}_{2}{\hat{\alpha }}_{3}}\\ \quad \wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{4}}\wedge \cdots \wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{D}}{Y}^{{\hat{\alpha }}_{2}}\\ -(D-3)\displaystyle \sum _{{\hat{\alpha }}_{1},\cdots \,{,}{\hat{\alpha }}_{D+1}}{\epsilon }_{{\hat{\alpha }}_{1}\cdots {\hat{\alpha }}_{D+1}}\delta {A}^{{\hat{\alpha }}_{1}{\hat{\alpha }}_{D+1}}{Y}_{{\hat{\alpha }}_{4}}\wedge {F}^{{\hat{\alpha }}_{2}{\hat{\alpha }}_{3}}\\ \quad \wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{4}}\wedge \cdots \wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{D}}{Y}^{{\hat{\alpha }}_{4}}.\end{array}\end{eqnarray}$
It follows that $\begin{eqnarray*}\begin{array}{l}\displaystyle \sum _{{\hat{\alpha }}_{1},\cdots \,{,}{\hat{\alpha }}_{D+1},\hat{\beta }}{\epsilon }_{{\hat{\alpha }}_{1}\cdots {\hat{\alpha }}_{D+1}}\delta {A}_{\,\,\hat{\beta }}^{{\hat{\alpha }}_{1}}{Y}^{\hat{\beta }}\wedge {F}^{{\hat{\alpha }}_{2}{\hat{\alpha }}_{3}}\\ \quad \wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{4}}\wedge \cdots \wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{D}}{Y}^{{\hat{\alpha }}_{D+1}}\\ =2\displaystyle \sum _{{\hat{\alpha }}_{1},\cdots \,{,}{\hat{\alpha }}_{D+1}}{\epsilon }_{{\hat{\alpha }}_{1}\cdots {\hat{\alpha }}_{D+1}}\delta {A}_{\,\,{\hat{\alpha }}_{2}}^{{\hat{\alpha }}_{1}}{Y}^{{\hat{\alpha }}_{2}}\wedge {F}^{{\hat{\alpha }}_{2}{\hat{\alpha }}_{3}}\\ \quad \wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{4}}\wedge \cdots \wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{D}}{Y}^{{\hat{\alpha }}_{D+1}}\\ +(D-3)\displaystyle \sum _{{\hat{\alpha }}_{1},\cdots \,{,}{\hat{\alpha }}_{D+1}}{\epsilon }_{{\hat{\alpha }}_{1}\cdots {\hat{\alpha }}_{D+1}}\delta {A}_{\,\,{\hat{\alpha }}_{4}}^{{\hat{\alpha }}_{1}}{Y}^{{\hat{\alpha }}_{4}}\wedge {F}^{{\hat{\alpha }}_{2}{\hat{\alpha }}_{3}}\,\end{array}\end{eqnarray*}$
$\begin{eqnarray*}\begin{array}{l}\quad \wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{4}}\wedge \cdots \wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{D}}{Y}^{{\hat{\alpha }}_{D+1}}\\ \quad +\displaystyle \sum _{{\hat{\alpha }}_{1},\cdots \,{,}{\hat{\alpha }}_{D+1}}{\epsilon }_{{\hat{\alpha }}_{1}\cdots {\hat{\alpha }}_{D+1}}\delta {A}_{\,\,{\hat{\alpha }}_{D+1}}^{{\hat{\alpha }}_{1}}{Y}^{{\hat{\alpha }}_{D+1}}\wedge {F}^{{\hat{\alpha }}_{2}{\hat{\alpha }}_{3}}\\ \quad \wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{4}}\wedge \cdots \wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{D}}{Y}^{{\hat{\alpha }}_{D+1}}\\ \quad =\displaystyle \sum _{{\hat{\alpha }}_{1},\cdots \,{,}{\hat{\alpha }}_{D+1}}{\epsilon }_{{\hat{\alpha }}_{1}\cdots {\hat{\alpha }}_{D+1}}\displaystyle \sum _{\hat{\beta }}\delta {A}^{{\hat{\alpha }}_{1}{\hat{\alpha }}_{2}}{Y}_{\hat{\beta }}\wedge {F}^{\hat{\beta }{\hat{\alpha }}_{3}}\\ \quad \wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{4}}\wedge \cdots \wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{D}}{Y}^{{\hat{\alpha }}_{D+1}}\\ \quad -(D-3)\displaystyle \sum _{{\hat{\alpha }}_{1},\cdots \,{,}{\hat{\alpha }}_{D+1}}{\epsilon }_{{\hat{\alpha }}_{1}\cdots {\hat{\alpha }}_{D+1}}\delta {A}^{{\hat{\alpha }}_{1}{\hat{\alpha }}_{2}}{Y}_{{\hat{\alpha }}_{4}}\wedge {F}^{{\hat{\alpha }}_{4}{\hat{\alpha }}_{3}}\end{array}\end{eqnarray*}$
$\begin{eqnarray*}\begin{array}{l}\quad \wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{4}}\wedge \cdots \wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{D}}{Y}^{{\hat{\alpha }}_{D+1}}\\ \quad -\displaystyle \sum _{{\hat{\alpha }}_{1},\cdots \,{,}{\hat{\alpha }}_{D+1}}{\epsilon }_{{\hat{\alpha }}_{1}\cdots {\hat{\alpha }}_{D+1}}\delta {A}^{{\hat{\alpha }}_{1}{\hat{\alpha }}_{2}}{Y}_{{\hat{\alpha }}_{D+1}}\wedge {F}^{{\hat{\alpha }}_{D+1}{\hat{\alpha }}_{3}}\\ \quad \wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{4}}\wedge \cdots \wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{D}}{Y}^{{\hat{\alpha }}_{D+1}}\\ \quad +\frac{D-3}{2}\displaystyle \sum _{{\hat{\alpha }}_{1},\cdots \,{,}{\hat{\alpha }}_{D+1}}{\epsilon }_{{\hat{\alpha }}_{1}\cdots {\hat{\alpha }}_{D+1}}\displaystyle \sum _{\hat{\beta }}\delta {A}^{{\hat{\alpha }}_{1}{\hat{\alpha }}_{4}}{Y}_{\hat{\beta }}\wedge {F}^{{\hat{\alpha }}_{2}{\hat{\alpha }}_{3}}\\ \quad \wedge {\rm{D}}{Y}^{\hat{\beta }}\wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{5}}\wedge \cdots \wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{D}}{Y}^{{\hat{\alpha }}_{D+1}}\\ \quad -(D-3)\displaystyle \sum _{{\hat{\alpha }}_{1},\cdots \,{,}{\hat{\alpha }}_{D+1}}{\epsilon }_{{\hat{\alpha }}_{1}\cdots {\hat{\alpha }}_{D+1}}\delta {A}^{{\hat{\alpha }}_{1}{\hat{\alpha }}_{4}}{Y}_{{\hat{\alpha }}_{2}}\wedge {F}^{{\hat{\alpha }}_{2}{\hat{\alpha }}_{3}}\\ \quad \wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{2}}\wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{5}}\wedge \cdots \wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{D}}{Y}^{{\hat{\alpha }}_{D+1}}\end{array}\end{eqnarray*}$
$\begin{eqnarray*}\begin{array}{l}\quad -\frac{D-3}{2}\displaystyle \sum _{{\hat{\alpha }}_{1},\cdots \,{,}{\hat{\alpha }}_{D+1}}{\epsilon }_{{\hat{\alpha }}_{1}\cdots {\hat{\alpha }}_{D+1}}\delta {A}^{{\hat{\alpha }}_{1}{\hat{\alpha }}_{4}}{Y}_{{\hat{\alpha }}_{D+1}}\wedge {F}^{{\hat{\alpha }}_{2}{\hat{\alpha }}_{3}}\\ \quad \wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{D+1}}\wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{5}}\wedge \cdots \wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{D}}{Y}^{{\hat{\alpha }}_{D+1}}\\ \quad +\frac{1}{2}\displaystyle \sum _{{\hat{\alpha }}_{1},\cdots \,{,}{\hat{\alpha }}_{D+1}}{\epsilon }_{{\hat{\alpha }}_{1}\cdots {\hat{\alpha }}_{D+1}}\displaystyle \sum _{\hat{\beta }}\delta {A}^{{\hat{\alpha }}_{1}{\hat{\alpha }}_{D+1}}{Y}_{\hat{\beta }}\wedge {F}^{{\hat{\alpha }}_{2}{\hat{\alpha }}_{3}}\\ \quad \wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{4}}\wedge \cdots \wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{D}}{Y}^{\hat{\beta }}\\ \quad -\displaystyle \sum _{{\hat{\alpha }}_{1},\cdots \,{,}{\hat{\alpha }}_{D+1}}{\epsilon }_{{\hat{\alpha }}_{1}\cdots {\hat{\alpha }}_{D+1}}\delta {A}^{{\hat{\alpha }}_{1}{\hat{\alpha }}_{D+1}}{Y}_{{\hat{\alpha }}_{2}}\wedge {F}^{{\hat{\alpha }}_{2}{\hat{\alpha }}_{3}}\end{array}\end{eqnarray*}$
$\begin{eqnarray*}\begin{array}{l}\quad \wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{4}}\wedge \cdots \wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{D}}{Y}^{{\hat{\alpha }}_{2}}\\ \quad -\frac{D-3}{2}\displaystyle \sum _{{\hat{\alpha }}_{1},\cdots \,{,}{\hat{\alpha }}_{D+1}}{\epsilon }_{{\hat{\alpha }}_{1}\cdots {\hat{\alpha }}_{D+1}}\delta {A}^{{\hat{\alpha }}_{1}{\hat{\alpha }}_{D+1}}{Y}_{{\hat{\alpha }}_{4}}\\ \quad \wedge {F}^{{\hat{\alpha }}_{2}{\hat{\alpha }}_{3}}\wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{4}}\wedge \cdots \wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{D}}{Y}^{{\hat{\alpha }}_{4}}\\ \quad =\displaystyle \sum _{{\hat{\alpha }}_{1},\cdots \,{,}{\hat{\alpha }}_{D+1}}{\epsilon }_{{\hat{\alpha }}_{1}\cdots {\hat{\alpha }}_{D+1}}\displaystyle \sum _{\hat{\beta }}\delta {A}^{{\hat{\alpha }}_{1}{\hat{\alpha }}_{2}}{Y}_{\hat{\beta }}\\ \quad \wedge {F}^{\hat{\beta }{\hat{\alpha }}_{3}}\wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{4}}\wedge \cdots \wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{D}}{Y}^{{\hat{\alpha }}_{D+1}}\\ +\frac{D-3}{2}\displaystyle \sum _{{\hat{\alpha }}_{1},\cdots \,{,}{\hat{\alpha }}_{D+1}}{\epsilon }_{{\hat{\alpha }}_{1}\cdots {\hat{\alpha }}_{D+1}}\displaystyle \sum _{\hat{\beta }}\delta {A}^{{\hat{\alpha }}_{1}{\hat{\alpha }}_{4}}{Y}_{\hat{\beta }}\end{array}\end{eqnarray*}$
$\begin{eqnarray*}\begin{array}{l}\quad \wedge {F}^{{\hat{\alpha }}_{2}{\hat{\alpha }}_{3}}\wedge {\rm{D}}{Y}^{\hat{\beta }}\wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{5}}\wedge \cdots \wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{D}}{Y}^{{\hat{\alpha }}_{D+1}}\\ \quad +\frac{1}{2}\displaystyle \sum _{{\hat{\alpha }}_{1},\cdots \,{,}{\hat{\alpha }}_{D+1}}{\epsilon }_{{\hat{\alpha }}_{1}\cdots {\hat{\alpha }}_{D+1}}\displaystyle \sum _{\hat{\beta }}\delta {A}^{{\hat{\alpha }}_{1}{\hat{\alpha }}_{D+1}}{Y}_{\hat{\beta }}\\ \quad \wedge {F}^{{\hat{\alpha }}_{2}{\hat{\alpha }}_{3}}\wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{4}}\wedge \cdots \wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{D}}{Y}^{\hat{\beta }}\\ \quad -(D-3)\displaystyle \sum _{{\hat{\alpha }}_{1},\cdots \,{,}{\hat{\alpha }}_{D+1}}{\epsilon }_{{\hat{\alpha }}_{1}\cdots {\hat{\alpha }}_{D+1}}\left[\delta {A}^{{\hat{\alpha }}_{1}{\hat{\alpha }}_{4}}{Y}_{{\hat{\alpha }}_{2}}\wedge {F}^{{\hat{\alpha }}_{2}{\hat{\alpha }}_{3}}\right.\\ \quad \left.\wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{2}}\wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{5}}\wedge \cdots \wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{D}}{Y}^{{\hat{\alpha }}_{D+1}}+({\hat{\alpha }}_{4}\leftrightarrow {\hat{\alpha }}_{2})\right]\\ \quad -\frac{D-3}{2}\displaystyle \sum _{{\hat{\alpha }}_{1},\cdots \,{,}{\hat{\alpha }}_{D+1}}{\epsilon }_{{\hat{\alpha }}_{1}\cdots {\hat{\alpha }}_{D+1}}\left[\delta {A}^{{\hat{\alpha }}_{1}{\hat{\alpha }}_{4}}{Y}_{{\hat{\alpha }}_{D+1}}\right.\end{array}\end{eqnarray*}$
$\begin{eqnarray*}\begin{array}{l}\quad \wedge {F}^{{\hat{\alpha }}_{2}{\hat{\alpha }}_{3}}\wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{D+1}}\wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{5}}\wedge \cdots \wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{D}}{Y}^{{\hat{\alpha }}_{D+1}}\\ \quad \left.+({\hat{\alpha }}_{4}\leftrightarrow {\hat{\alpha }}_{d+1})\right]\\ \quad -\displaystyle \sum _{{\hat{\alpha }}_{1},\cdots \,{,}{\hat{\alpha }}_{D+1}}{\epsilon }_{{\hat{\alpha }}_{1}\cdots {\hat{\alpha }}_{D+1}}\left[\delta {A}^{{\hat{\alpha }}_{1}{\hat{\alpha }}_{D+1}}{Y}_{{\hat{\alpha }}_{2}}\wedge {F}^{{\hat{\alpha }}_{2}{\hat{\alpha }}_{3}}\right.\\ \quad \left.\wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{4}}\wedge \cdots \wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{D}}{Y}^{{\hat{\alpha }}_{2}}+({\hat{\alpha }}_{2}\leftrightarrow {\hat{\alpha }}_{d+1})\right]\\ \,=\displaystyle \sum _{{\hat{\alpha }}_{1},\cdots \,{,}{\hat{\alpha }}_{D+1}}{\epsilon }_{{\hat{\alpha }}_{1}\cdots {\hat{\alpha }}_{D+1}}\displaystyle \sum _{\hat{\beta }}\delta {A}^{{\hat{\alpha }}_{1}{\hat{\alpha }}_{2}}{Y}_{\hat{\beta }}\wedge {F}^{\hat{\beta }{\hat{\alpha }}_{3}}\end{array}\end{eqnarray*}$
$\begin{eqnarray}\begin{array}{l}\quad \wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{4}}\wedge \cdots \wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{D}}{Y}^{{\hat{\alpha }}_{D+1}}\\ \quad +\frac{D-3}{2}\displaystyle \sum _{{\hat{\alpha }}_{1},\cdots \,{,}{\hat{\alpha }}_{D+1}}{\epsilon }_{{\hat{\alpha }}_{1}\cdots {\hat{\alpha }}_{D+1}}\displaystyle \sum _{\hat{\beta }}\delta {A}^{{\hat{\alpha }}_{1}{\hat{\alpha }}_{4}}{Y}_{\hat{\beta }}\\ \quad \wedge {F}^{{\hat{\alpha }}_{2}{\hat{\alpha }}_{3}}\wedge {\rm{D}}{Y}^{\hat{\beta }}\wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{5}}\wedge \cdots \wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{D}}{Y}^{{\hat{\alpha }}_{D+1}}\\ +\frac{1}{2}\displaystyle \sum _{{\hat{\alpha }}_{1},\cdots \,{,}{\hat{\alpha }}_{D+1}}{\epsilon }_{{\hat{\alpha }}_{1}\cdots {\hat{\alpha }}_{D+1}}\displaystyle \sum _{\hat{\beta }}\delta {A}^{{\hat{\alpha }}_{1}{\hat{\alpha }}_{D+1}}{Y}_{\hat{\beta }}\\ \quad \wedge {F}^{{\hat{\alpha }}_{2}{\hat{\alpha }}_{3}}\wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{4}}\wedge \cdots \wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{D}}{Y}^{\hat{\beta }},\end{array}\end{eqnarray}$
where all the terms containing [ ⋯ ] vanish since ${\epsilon }_{{\hat{\alpha }}_{1}\cdots {\hat{\alpha }}_{D+1}}$ is totally antisymmetric. In summary, we have $\begin{eqnarray}\begin{array}{l}\delta {A}_{\,\,\hat{\beta }}^{{\hat{\alpha }}_{1}}{Y}^{\hat{\beta }}\wedge {F}^{{\hat{\alpha }}_{2}{\hat{\alpha }}_{3}}\wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{4}}\wedge \cdots \wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{D}}{Y}^{{\hat{\alpha }}_{D+1}}\\ ={\epsilon }_{{\hat{\alpha }}_{1}\cdots {\hat{\alpha }}_{D+1}}\delta {A}^{{\hat{\alpha }}_{1}{\hat{\alpha }}_{2}}{Y}_{\hat{\beta }}\wedge {F}^{\hat{\beta }{\hat{\alpha }}_{3}}\\ \quad \wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{4}}\wedge \cdots \wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{D}}{Y}^{{\hat{\alpha }}_{D+1}}\\ \quad +\frac{D-3}{2}{\epsilon }_{{\hat{\alpha }}_{1}\cdots {\hat{\alpha }}_{D+1}}\delta {A}^{{\hat{\alpha }}_{1}{\hat{\alpha }}_{4}}{Y}_{\hat{\beta }}\wedge {F}^{{\hat{\alpha }}_{2}{\hat{\alpha }}_{3}}\wedge {\rm{D}}{Y}^{\hat{\beta }}\\ \quad \wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{5}}\wedge \cdots \wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{D}}{Y}^{{\hat{\alpha }}_{D+1}}\\ \quad +\frac{1}{2}{\epsilon }_{{\hat{\alpha }}_{1}\cdots {\hat{\alpha }}_{D+1}}\delta {A}^{{\hat{\alpha }}_{1}{\hat{\alpha }}_{D+1}}{Y}_{\hat{\beta }}\wedge {F}^{{\hat{\alpha }}_{2}{\hat{\alpha }}_{3}}\wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{4}}\\ \quad \wedge \cdots \wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{D}}{Y}^{\hat{\beta }}\\ ={\epsilon }_{{\hat{\alpha }}_{1}\cdots {\hat{\alpha }}_{D+1}}\delta {A}^{{\hat{\alpha }}_{1}{\hat{\alpha }}_{2}}{Y}_{\hat{\beta }}\wedge {F}^{\hat{\beta }{\hat{\alpha }}_{3}}\wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{4}}\\ \quad \wedge \cdots \wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{D}}{Y}^{{\hat{\alpha }}_{D+1}}+0\\ \quad -\frac{{{\ell }}^{2}}{2}{\epsilon }_{{\hat{\alpha }}_{1}\cdots {\hat{\alpha }}_{D+1}}\delta {A}^{{\hat{\alpha }}_{1}{\hat{\alpha }}_{D+1}}\wedge {F}^{{\hat{\alpha }}_{2}{\hat{\alpha }}_{3}}\\ \quad \wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{4}}\wedge \cdots \wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{D}}\\ =-{\epsilon }_{{\hat{\alpha }}_{1}\cdots {\hat{\alpha }}_{D+1}}\delta {A}^{{\hat{\alpha }}_{1}{\hat{\alpha }}_{2}}\wedge {F}_{\,\,\hat{\beta }}^{{\hat{\alpha }}_{3}}{Y}^{\hat{\beta }}\wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{4}}\\ \quad \wedge \cdots \wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{D}}{Y}^{{\hat{\alpha }}_{D+1}}\\ \quad +\frac{{(-1)}^{D}{{\ell }}^{2}}{2}{\epsilon }_{{\hat{\alpha }}_{1}\cdots {\hat{\alpha }}_{D+1}}\delta {A}^{{\hat{\alpha }}_{1}{\hat{\alpha }}_{2}}\wedge \wedge {F}^{{\hat{\alpha }}_{3}{\hat{\alpha }}_{4}}\\ \quad \wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{5}}\wedge \cdots \wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{D+1}},\end{array}\end{eqnarray}$
where we have used that ${Y}^{\hat{\beta }}{Y}_{\hat{\beta }}=-{{\ell }}^{2}$ and ${\rm{D}}{Y}^{\hat{\beta }}{Y}_{\hat{\beta }}=0$.Appendix B Intrinsic expansion
In the SO(2, d) gauge theory formalism, the spacetime metric is given by
$\begin{eqnarray}{g}_{MN}={{\rm{D}}}_{M}{Y}^{\hat{\alpha }}{{\rm{D}}}_{N}{Y}_{\hat{\alpha }}.\end{eqnarray}$
When the metric gMN is not degenerate, together with $\begin{eqnarray}{Y}^{\hat{\alpha }}{Y}_{\hat{\alpha }}=-{{\ell }}^{2},\quad {Y}^{\hat{\alpha }}{{\rm{D}}}_{M}{Y}_{\hat{\alpha }}=0,\end{eqnarray}$
it is noticed that the (d + 2)–vectors $\{{Y}^{\hat{\alpha }},{{\rm{D}}}_{M}{Y}^{\hat{\alpha }}\}$ naturally forms an intrinsic basis of the (d + 2)–dimensional space of the SO(2, d) vector representation. The completion relation is obviously $\begin{eqnarray}{{\rm{D}}}_{M}{Y}^{\hat{\alpha }}{{\rm{D}}}^{M}{Y}_{\hat{\beta }}-{{\ell }}^{-2}{Y}^{\hat{\alpha }}{Y}_{\hat{\beta }}={\delta }_{\hat{\beta }}^{\hat{\alpha }},\end{eqnarray}$
where the spacetime indices are rising by gMN.To clarify the relation between the SO(2, d) gauge theory notations and the usual SO(2, d) invariant notations of gravity, let us expand the quantities with SO(2, d) indices by the intrinsic basis. We notice that
$\begin{eqnarray}\begin{array}{l}{Y}^{\hat{\alpha }}{{\rm{D}}}_{N}{{\rm{D}}}_{M}{Y}_{\hat{\alpha }}=-{g}_{MN},\\ {{\rm{D}}}_{N}{Y}^{\hat{\alpha }}{{\rm{D}}}_{({M}_{2}}{{\rm{D}}}_{{M}_{1})}{Y}_{\hat{\alpha }}=\frac{1}{2}\left[{{\rm{D}}}_{{M}_{2}}({{\rm{D}}}_{N}{Y}^{\hat{\alpha }}{{\rm{D}}}_{{M}_{1}}{Y}_{\hat{\alpha }})\right.\\ \qquad +{{\rm{D}}}_{{M}_{1}}({{\rm{D}}}_{N}{Y}^{\hat{\alpha }}{{\rm{D}}}_{{M}_{2}}{Y}_{\hat{\alpha }})\\ \qquad \left.-{{\rm{D}}}_{N}({{\rm{D}}}_{{M}_{1}}{Y}^{\hat{\alpha }}{{\rm{D}}}_{{M}_{2}}{Y}_{\hat{\alpha }})\right]\\ \qquad +{{\rm{D}}}_{[N}{{\rm{D}}}_{{M}_{1}]}{Y}^{\hat{\alpha }}{{\rm{D}}}_{{M}_{2}}{Y}_{\hat{\alpha }}\\ \qquad +{{\rm{D}}}_{{M}_{1}}{Y}^{\hat{\alpha }}{{\rm{D}}}_{[N}{{\rm{D}}}_{{M}_{2}]}{Y}_{\hat{\alpha }}\\ \quad =\frac{1}{2}\left({\partial }_{{M}_{2}}{g}_{N{M}_{1}}+{\partial }_{{M}_{1}}{g}_{N{M}_{2}}-{\partial }_{N}{g}_{{M}_{1}{M}_{2}}\right)\\ \qquad +{({F}_{N({M}_{1}})}^{\hat{\alpha }\hat{\beta }}{{\rm{D}}}_{{M}_{2})}{Y}_{\hat{\alpha }}\,{Y}_{\hat{\beta }},\\ {{\rm{D}}}_{N}{Y}^{\hat{\alpha }}{{\rm{D}}}_{[{M}_{2}}{{\rm{D}}}_{{M}_{1}]}{Y}_{\hat{\alpha }}=\frac{1}{2}{({F}_{{M}_{2}{M}_{1}})}^{\hat{\alpha }\hat{\beta }}{Y}_{\hat{\beta }}{{\rm{D}}}_{N}{Y}_{\hat{\alpha }}.\end{array}\end{eqnarray}$
Thus we have the expansion of DDY as $\begin{eqnarray}{{\rm{D}}}_{{M}_{2}}{{\rm{D}}}_{{M}_{1}}{Y}^{\hat{\alpha }}={{\ell }}^{-2}{g}_{{M}_{1}{M}_{2}}{Y}^{\hat{\alpha }}+{{\rm{\Gamma }}}_{\,\,{M}_{1}{M}_{2}}^{N}{{\rm{D}}}_{N}{Y}^{\hat{\alpha }},\end{eqnarray}$
which suggests the following spacetime connection ${{\rm{\Gamma }}}_{\,\,NP}^{M}$ with torsion ${t}_{\,\,NP}^{M}$ $\begin{eqnarray}\begin{array}{rcl}{{\rm{\Gamma }}}_{\,\,NP}^{M} & = & {\hat{{\rm{\Gamma }}}}_{\,\,NP}^{M}+{t}_{\,\,NP}^{M},\\ {\hat{{\rm{\Gamma }}}}_{M{M}_{1}{M}_{2}} & = & \frac{1}{2}\left({\partial }_{{M}_{2}}{g}_{N{M}_{1}}+{\partial }_{{M}_{1}}{g}_{N{M}_{2}}\right.\left.-{\partial }_{N}{g}_{{M}_{1}{M}_{2}}\right),\\ {t}_{N{M}_{1}{M}_{2}} & = & \frac{1}{2}\left[{({F}_{N{M}_{1}})}^{{\hat{\beta }}_{1}{\hat{\beta }}_{2}}{{\rm{D}}}_{{M}_{2}}{Y}_{{\hat{\beta }}_{1}}\right.\\ & & +{({F}_{N{M}_{2}})}^{{\hat{\beta }}_{1}{\hat{\beta }}_{2}}{{\rm{D}}}_{{M}_{1}}{Y}_{{\hat{\beta }}_{1}}\\ & & \left.-{({F}_{{M}_{1}{M}_{2}})}^{{\hat{\beta }}_{1}{\hat{\beta }}_{2}}{{\rm{D}}}_{N}{Y}_{{\hat{\beta }}_{1}}\right]{Y}_{{\hat{\beta }}_{2}}.\end{array}\end{eqnarray}$
The torsion free condition can be expressed covariantly as $\begin{eqnarray}{{\rm{D}}}_{[{M}_{1}}{{\rm{D}}}_{{M}_{2}]}{Y}^{\hat{\alpha }}=\frac{1}{2}{({F}_{{M}_{1}{M}_{2}})}_{\,\,\hat{\beta }}^{\hat{\alpha }}{Y}^{\hat{\beta }}=0.\end{eqnarray}$
Now we can define the torsional spacetime covariant derivative ∇ = ∂ + Γ, as well as the torsional total covariant derivative ${ \mathcal D }=\partial +{\rm{\Gamma }}+A={\rm{\nabla }}+A$. Similarly, the spacetime covariant derivative without torsion is $\hat{{\rm{\nabla }}}=\partial +\hat{{\rm{\Gamma }}}$, and the total covariant derivatives without torsion is $\hat{{ \mathcal D }}=\partial +\hat{{\rm{\Gamma }}}+A=\hat{{\rm{\nabla }}}+A$. In terms of $\hat{{ \mathcal D }}$ and ${ \mathcal D }$, we have $\begin{eqnarray}{\hat{{ \mathcal D }}}_{{M}_{2}}{\hat{{ \mathcal D }}}_{{M}_{1}}{Y}^{\hat{\alpha }}={{\ell }}^{-2}{g}_{{M}_{1}{M}_{2}}{Y}^{\hat{\alpha }}+{{t}^{N}}_{{M}_{1}{M}_{2}}{{\rm{D}}}_{N}{Y}^{\hat{\alpha }},\end{eqnarray}$
$\begin{eqnarray}{{ \mathcal D }}_{{M}_{2}}{{ \mathcal D }}_{{M}_{1}}{Y}^{\hat{\alpha }}={{\ell }}^{-2}{g}_{{M}_{1}{M}_{2}}{Y}^{\hat{\alpha }}.\end{eqnarray}$
Now, by using (B.5 ), one can further get the intrinsic expansion of higher derivative terms such as ${{\rm{D}}}_{{M}_{1}}{{\rm{D}}}_{{M}_{2}}{{\rm{D}}}_{{M}_{3}}{Y}^{\hat{\alpha }}$. In terms of the results for ${{\rm{D}}}_{[{M}_{1}}{{\rm{D}}}_{{M}_{2}]}{{\rm{D}}}_{{M}_{3}}{Y}^{\hat{\alpha }}$ and ${{\rm{D}}}_{[{M}_{1}}{{\rm{D}}}_{{M}_{2}]}{Y}^{\hat{\alpha }}$, we find that the intrinsic expansion of the SO(2, d) field strength is
$\begin{eqnarray}\begin{array}{l}{({F}_{{M}_{1}{M}_{2}})}^{\hat{\alpha }\hat{\beta }}=\left({{R}^{{N}_{1}{N}_{2}}}_{{M}_{1}{M}_{2}}+2{{\ell }}^{-2}{\delta }_{[{M}_{1}}^{{N}_{1}}{\delta }_{{M}_{2}]}^{{N}_{2}}\right)\\ \quad \times {{\rm{D}}}_{{N}_{1}}{Y}^{[\hat{\alpha }}{{\rm{D}}}_{{N}_{2}}{Y}^{\hat{\beta }]}-4{{\ell }}^{-2}{{t}^{N}}_{[{M}_{1}{M}_{2}]}{Y}^{[\hat{\alpha }}{{\rm{D}}}_{N}{Y}^{\hat{\beta }]}\\ \,=\left({\hat{{R}^{{N}_{1}{N}_{2}}}}_{{M}_{1}{M}_{2}}+2{\hat{{\rm{\nabla }}}}_{[{M}_{1}}{t}_{\,\,{M}_{2}]}^{{N}_{1}{N}_{2}}\right.\\ \quad \left.+2{t}_{\,\,N[{M}_{1}}^{{N}_{1}}{t}_{\,\,{M}_{2}]}^{N{N}_{2}}+2{{\ell }}^{-2}{\delta }_{[{M}_{1}}^{{N}_{1}}{\delta }_{{M}_{2}]}^{{N}_{2}}\right)\\ \quad \times \,{{\rm{D}}}_{{N}_{1}}{Y}^{[\hat{\alpha }}{{\rm{D}}}_{{N}_{2}}{Y}^{\hat{\beta }]}\\ \quad -4{{\ell }}^{-1}{{t}^{N}}_{[{M}_{1}{M}_{2}]}{Y}^{[\hat{\alpha }}{{\rm{D}}}_{N}{Y}^{\hat{\beta }]},\end{array}\end{eqnarray}$
where ${{R}^{{N}_{1}{N}_{2}}}_{{M}_{1}{M}_{2}}$ and ${\hat{{R}^{{N}_{1}{N}_{2}}}}_{{M}_{1}{M}_{2}}$ are respectively the curvature tensors for ∇ and $\hat{{\rm{\nabla }}}$. Furthermore, expanding the SO(2, d) Bianchi identity on the intrinsic basis, we get $\begin{eqnarray}\begin{array}{rcl}0 & = & {{\rm{D}}}_{[{M}_{3}}{({F}_{{M}_{1}{M}_{2}]})}^{\hat{\alpha }\hat{\beta }}\\ & = & \left({\hat{{\rm{\nabla }}}}_{[{M}_{3}}{\hat{R}}_{\,\,{M}_{1}{M}_{2}]}^{{N}_{1}{N}_{2}}-{\hat{R}}_{\,\,[{M}_{2}{M}_{3}{M}_{1}]}^{N}{t}_{\,\,N}^{{N}_{1}{N}_{2}}\right)\\ & & \times \,{{\rm{D}}}_{{N}_{1}}{Y}^{[\hat{\alpha }}{{\rm{D}}}_{{N}_{2}}{Y}^{\hat{\beta }]}-2{{\ell }}^{-2}{\hat{R}}_{\,\,[{M}_{3}{M}_{1}{M}_{2}]}^{N}\\ & & \times \,{Y}^{[\hat{\alpha }}{{\rm{D}}}_{N}{Y}^{\hat{\beta }]}.\end{array}\end{eqnarray}$
It is equivalent to the two Bianchi identities for the usual Riemann curvature.B.1. Recovering Einstein–Hilbert action
By using the intrinsic expansion, one can directly recover the Einstein–Hilbert action without taking any gauge choice.
We notice that B.14 ) as well as the expansion (B.10 ), we find that
$\begin{eqnarray}\begin{array}{l}{\epsilon }_{{\hat{\alpha }}_{1}\cdots {\hat{\alpha }}_{D+1}}{{\rm{D}}}_{{M}_{1}}{Y}^{{\hat{\alpha }}_{1}}\cdots {{\rm{D}}}_{{M}_{D}}{Y}^{{\hat{\alpha }}_{D}}{Y}^{{\hat{\alpha }}_{D+1}}={\ell }{g}^{\frac{1}{2}}{\varepsilon }_{{M}_{1}\cdots {M}_{D}},\end{array}\end{eqnarray}$
where $\begin{eqnarray}{\varepsilon }_{01\cdots d}=-1,\quad g=-\det ({g}_{MN}).\end{eqnarray}$
By using the resummation technique, one can further show that $\begin{eqnarray}\begin{array}{l}\frac{1}{{\ell }\,p!(d+1-p)!}{\epsilon }_{{\hat{\alpha }}_{0}\cdots {\hat{\alpha }}_{p-1}{\hat{\alpha }}_{p}\cdots {\hat{\alpha }}_{d+1}}{\epsilon }^{{M}_{0}\cdots {M}_{p-1}{M}_{p}\cdots {M}_{d}}\\ \,\,\times \,{{\rm{D}}}_{{M}_{p}}{Y}^{{\hat{\alpha }}_{p}}\cdots {{\rm{D}}}_{{M}_{d}}{Y}^{{\hat{\alpha }}_{d}}{Y}^{{\hat{\alpha }}_{d+1}}\,=\,{{\rm{D}}}^{[{M}_{0}}{Y}_{[{\hat{\alpha }}_{0}}\cdots {{\rm{D}}}^{{M}_{p-1}]}{Y}_{{\hat{\alpha }}_{p-1}]},\,\,\,\,\end{array}\end{eqnarray}$
where $\begin{eqnarray}{\epsilon }^{01\cdots d}=\frac{1}{\sqrt{g}}.\end{eqnarray}$
By using ($\begin{eqnarray}\begin{array}{l}\displaystyle \int {\epsilon }_{{\hat{\alpha }}_{1}\cdots {\hat{\alpha }}_{D+1}}{F}^{{\hat{\alpha }}_{1}{\hat{\alpha }}_{2}}\wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{3}}\wedge \cdots \wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{D}}{Y}^{{\hat{\alpha }}_{D+1}},\\ =\frac{1}{2}\,{\ell }\,2!\,(D-2)!\\ \quad \times \displaystyle \int {{\rm{d}}}^{D}x\,\sqrt{g}{({F}_{{M}_{1}{M}_{2}})}^{{\hat{\alpha }}_{1}{\hat{\alpha }}_{2}}{{\rm{D}}}^{[{M}_{1}}{Y}_{[{\hat{\alpha }}_{1}}{{\rm{D}}}^{{M}_{2}]}{Y}_{{\hat{\alpha }}_{2}]}\\ ={\ell }\,(D-2)!\displaystyle \int {{\rm{d}}}^{D}x\,\sqrt{g}\left[R+{{\ell }}^{-2}D(D-1)\right],\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{l}\displaystyle \int {\epsilon }_{{\hat{\alpha }}_{1}\cdots {\hat{\alpha }}_{D+1}}{\rm{D}}{Y}^{{\hat{\alpha }}_{1}}\wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{2}}\wedge \cdots \wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{D}}{Y}^{{\hat{\alpha }}_{D+1}}\\ ={\ell }\,D!\displaystyle \int {{\rm{d}}}^{D}x\,\sqrt{g}.\end{array}\end{eqnarray}$
It follows that $\begin{eqnarray}\begin{array}{l}{S}^{{\rm{cov}}}[Y,A]=\frac{1}{2{\kappa }^{2}{\ell }\,(D-2)!}{\displaystyle \int }_{M}{\epsilon }_{{\hat{\alpha }}_{1}\cdots {\hat{\alpha }}_{D+1}}\\ \quad \times \,\left[{F}^{{\hat{\alpha }}_{1}{\hat{\alpha }}_{2}}-\frac{2}{D{{\ell }}^{2}}{\rm{D}}{Y}^{{\hat{\alpha }}_{1}}\wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{2}}\right]\\ \quad \wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{3}}\wedge \cdots \wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{D}}{Y}^{{\hat{\alpha }}_{D+1}}\\ =\frac{1}{2{\kappa }^{2}}\displaystyle \int {{\rm{d}}}^{D}x\,\sqrt{g}\left[R+{{\ell }}^{-2}(D-2)(D-1)\right],\end{array}\end{eqnarray}$
which is exactly the Einstein–Hilbert action with ${\rm{\Lambda }}\,=-\frac{(D-1)(D-2)}{2{{\ell }}^{2}}$.Appendix C Structure of SO(2, d) anomaly
C.1. Generic structure of the anomaly
In the field theory, a classical symmetry which keeps the action invariant might be anomalous at the quantum level. The reason is that the path integral measure might not be invariant under the symmetry transformations [15]. If the symmetry is described by a Lie group ${ \mathcal G }$ with background gauge field A, the classical conservation law
$\begin{eqnarray}{\rm{D}}J=0\end{eqnarray}$
will get modified in presence of the anomaly. The anomalous conservation law $\begin{eqnarray}\langle {\rm{D}}J\rangle ={ \mathcal A }\end{eqnarray}$
has an extra source term ${ \mathcal A }$ which comes from the Jacobian of the matter field path integral measure under the infinitesimal gauge transformation.The explicit forms of the anomaly depend on the regularization procedures [12, 14, 15]. By integrating out the matter fields, one will get the effective action W[A] of the theory
$\begin{eqnarray}{{\rm{e}}}^{{\rm{i}}W[A]}=\int [{\mathfrak{D}}\phi ]{{\rm{e}}}^{{\rm{i}}S[\phi ;A]}.\end{eqnarray}$
In presence of the anomaly, the effective action W[A] is not gauge invariant and its gauge transformation $\begin{eqnarray}{{ \mathcal A }}^{{\rm{con}}}\sim {\delta }_{{ \mathcal G }}W[A]\end{eqnarray}$
will gives rise to the so-called the consistent anomaly ${{ \mathcal A }}^{{\rm{con}}}$. By construction, we have $\begin{eqnarray}{\rm{D}}{J}^{{\rm{con}}}={{ \mathcal A }}^{{\rm{con}}},\end{eqnarray}$
where the consistent current is simply $\begin{eqnarray}{J}^{{\rm{con}}}=\frac{\delta W[A]}{\delta A}.\end{eqnarray}$
When the Lie group ${ \mathcal G }$ is non-abelian, the consistent anomaly itself is not a covariant tensor under the gauge transformation. Alternatively, one can adjust the regularization procedures such that ⟨J⟩ and thus ⟨DJ⟩ are gauge covariant. This procedure gives rise to the covariant current Jcov and covariant anomaly ${{ \mathcal A }}^{{\rm{cov}}}$.By using the BRST formalism, one can establish a generic descendent description [12, 14, 15] of the non-abelian anomaly
$\begin{eqnarray}\begin{array}{rcl}{I}_{(d+2)} & = & {\rm{d}}{I}_{(d+1)}\\ {\delta }_{u}{I}_{(d+1)} & = & {\rm{d}}{I}_{(d)}\\ & \cdots \end{array}\end{eqnarray}$
where ${I}_{(d)}={{ \mathcal A }}^{{\rm{con}}}$ is the consistent anomaly. Now the anomaly in the d-dimensional QFT is uniquely governed by a gauge invariant closed (d + 2)-form, i.e. the characteristic class I(d+2). Given I(d+2), one can locally have a (d + 1)-form I(d+1) for which the exterior derivative gives rise to I(d+2). Under the infinitesimal gauge transformation, I(d+1) is invariant up to a boundary term which is exactly the anomaly ${I}_{(d)}={{ \mathcal A }}^{{\rm{con}}}$.In terms of the descendent formula, one can easily decide the explicit form of the consistent anomaly ${{ \mathcal A }}^{{\rm{con}}}$. However, in many cases, the explicit form of the current J would be non-local and is usually unknown. In the generic discussion of the anomaly [12, 14, 15], it is proved that the difference between the consistent and covariant currents is a local polynomial of the background gauge field, and it can be decided systematically. This polynomial is called the Bardeen–Zumino polynomial ${{ \mathcal P }}^{{\rm{BZ}}}$. Therefore, one can always get the explicit form of the covariant anomaly by adding ${\rm{D}}{{ \mathcal P }}^{{\rm{BZ}}}$ to ${{ \mathcal A }}^{{\rm{con}}}$. Given the covariant anomaly, the term I(d+1) can be obtained by the homotopic integration
$\begin{eqnarray}{I}_{(d+1)}={\int }_{0}^{1}{\rm{d}}s\,A\wedge {{ \mathcal A }}^{{\rm{cov}}}(A(s)).\end{eqnarray}$
Similarly, the regularized effective action can be expressed as the homotopic integration $\begin{eqnarray}{W}_{(d)}={\int }_{{\rm{\Sigma }}}{\int }_{0}^{1}{\rm{d}}s\,A\wedge { \mathcal J }(A(s)),\end{eqnarray}$
where one can use either the consistent current or covariant current since the homotopic integration of the Bardeen–Zumino polynomial ${{ \mathcal P }}^{{\rm{BZ}}}$ is always zero.The simplest example is the d = 2 chiral anomaly. In this case, the characteristic class is
$\begin{eqnarray}{I}_{4}={\rm{tr}}(F\wedge F)\end{eqnarray}$
and I(3) is just the d = 3 Chern–Simons term $\begin{eqnarray}\begin{array}{rcl}{I}_{3} & = & {\rm{tr}}\left(A\wedge {\rm{d}}A+\frac{2}{3}A\wedge A\wedge A\right)\\ & = & 2{\displaystyle \int }_{0}^{1}{\rm{d}}s\,{\rm{tr}}\left[A\wedge F(s)\right].\end{array}\end{eqnarray}$
The consistent and covariant anomalies are respectively $\begin{eqnarray}{{ \mathcal A }}^{{\rm{con}}}={\rm{d}}A,\quad {{ \mathcal A }}^{{\rm{cov}}}=2F,\end{eqnarray}$
which are consistent with $\begin{eqnarray}{{ \mathcal P }}^{{\rm{BZ}}}=A.\end{eqnarray}$
C.2. The bulk covariant action
The bulk covariant action on (D = d + 1)–dimensional manifold M is given by
$\begin{eqnarray}\begin{array}{l}2{\kappa }^{2}{\ell }\,(d-1)!{S}_{(d+1)}^{{\rm{cov}}}=2{\kappa }^{2}{\ell }\,(d-1)!{\displaystyle \int }_{M}{{ \mathcal L }}_{(d+1)}^{{\rm{cov}}}\\ ={\displaystyle \int }_{M}{\epsilon }_{\hat{\alpha }\hat{\beta }{\hat{\alpha }}_{1}\cdots {\hat{\alpha }}_{d}}\left[{F}^{\hat{\alpha }\hat{\beta }}-\frac{2}{(d+1){{\ell }}^{2}}{\rm{D}}{Y}^{\hat{\alpha }}\wedge {\rm{D}}{Y}^{\hat{\beta }}\right]\\ \quad \wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{1}}\wedge \cdots \wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{d-1}}{Y}^{{\hat{\alpha }}_{d}}.\end{array}\end{eqnarray}$
C.2.1. ${\rm{d}}{{ \mathcal L }}_{(d+1)}^{{\rm{cov}}}={I}_{(d+2)}$
By using the resummations
$\begin{eqnarray}\begin{array}{l}{\epsilon }_{\hat{\alpha }\hat{\beta }{\hat{\alpha }}_{1}\cdots {\hat{\alpha }}_{d}}{F}^{\hat{\alpha }\hat{\beta }}\wedge {F}^{{\hat{\alpha }}_{1}\hat{\gamma }}{Y}_{\hat{\gamma }}\wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{2}}\\ \quad \wedge \cdots \wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{d-1}}{Y}^{{\hat{\alpha }}_{d}}\\ \quad =-\frac{{(-1)}^{d}{{\ell }}^{2}}{4}{\epsilon }_{\hat{\alpha }\hat{\beta }{\hat{\alpha }}_{1}\cdots {\hat{\alpha }}_{d}}{F}^{\hat{\alpha }\hat{\beta }}\wedge {F}^{{\hat{\alpha }}_{1}{\hat{\alpha }}_{2}}\\ \quad \wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{3}}\wedge \cdots \wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{d}},\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{l}{\epsilon }_{\hat{\alpha }\hat{\beta }{\hat{\alpha }}_{1}\cdots {\hat{\alpha }}_{d}}{F}^{\hat{\alpha }\hat{\gamma }}{Y}_{\hat{\gamma }}\wedge {\rm{D}}{Y}^{\hat{\beta }}\\ \wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{1}}\wedge \cdots \wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{d-1}}{Y}^{{\hat{\alpha }}_{d}}\\ \quad =-\frac{{(-1)}^{d}{{\ell }}^{2}}{2}{\epsilon }_{\hat{\alpha }\hat{\beta }{\hat{\alpha }}_{1}\cdots {\hat{\alpha }}_{d}}{F}^{\hat{\alpha }\hat{\beta }}\wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{1}}\\ \quad \wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{2}}\wedge \cdots \wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{d}},\end{array}\end{eqnarray}$
we find that the exterior derivative of ${{ \mathcal L }}_{(d+1)}^{{\rm{cov}}}$ is given by $\begin{eqnarray}\begin{array}{l}2{\kappa }^{2}{\ell }\,(d-1)!{\rm{d}}{{ \mathcal L }}_{(d+1)}^{{\rm{cov}}}\\ ={\rm{d}}\left\{{\epsilon }_{\hat{\alpha }\hat{\beta }{\hat{\alpha }}_{1}\cdots {\hat{\alpha }}_{d}}\left[{F}^{\hat{\alpha }\hat{\beta }}-\frac{2}{(d+1){{\ell }}^{2}}{\rm{D}}{Y}^{\hat{\alpha }}\wedge {\rm{D}}{Y}^{\hat{\beta }}\right]\right.\\ \quad \left.\wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{1}}\wedge \cdots \wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{d-1}}{Y}^{{\hat{\alpha }}_{d}}\right\}\\ =(d-1){\epsilon }_{\hat{\alpha }\hat{\beta }{\hat{\alpha }}_{1}\cdots {\hat{\alpha }}_{d}}{F}^{\hat{\alpha }\hat{\beta }}\wedge {F}_{\,\,\hat{\gamma }}^{{\hat{\alpha }}_{1}}{Y}^{\hat{\gamma }}\\ \quad \wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{2}}\wedge \cdots \wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{d-1}}{Y}^{{\hat{\alpha }}_{d}}\\ -{(-1)}^{d}{\epsilon }_{\hat{\alpha }\hat{\beta }{\hat{\alpha }}_{1}\cdots {\hat{\alpha }}_{d}}{F}^{\hat{\alpha }\hat{\beta }}\wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{1}}\\ \quad \wedge \cdots \wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{d-1}}\wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{d}}\\ \quad -2{{\ell }}^{-2}{\epsilon }_{\hat{\alpha }\hat{\beta }{\hat{\alpha }}_{1}\cdots {\hat{\alpha }}_{d}}{F}_{\,\,\hat{\gamma }}^{\hat{\alpha }}{Y}^{\hat{\gamma }}\wedge {\rm{D}}{Y}^{\hat{\beta }}\\ \quad \wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{1}}\wedge \cdots \wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{d-1}}{Y}^{{\hat{\alpha }}_{d}}\\ =-\frac{{(-1)}^{d}(d-1){{\ell }}^{2}}{4}{\epsilon }_{\hat{\alpha }\hat{\beta }{\hat{\alpha }}_{1}\cdots \cdots {\hat{\alpha }}_{d}}{F}^{\hat{\alpha }\hat{\beta }}\\ \quad \wedge {F}^{{\hat{\alpha }}_{1}{\hat{\alpha }}_{2}}\wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{3}}\wedge \cdots \wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{d}}\\ -{(-1)}^{d}{\epsilon }_{\hat{\alpha }\hat{\beta }{\hat{\alpha }}_{1}\cdots {\hat{\alpha }}_{d}}{F}^{\hat{\alpha }\hat{\beta }}\wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{1}}\\ \quad \wedge \cdots \wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{d-1}}\wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{d}}\\ +{(-1)}^{d}{\epsilon }_{\hat{\alpha }\hat{\beta }{\hat{\alpha }}_{1}\cdots {\hat{\alpha }}_{d}}{F}^{\hat{\alpha }\hat{\beta }}\wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{1}}\\ \quad \wedge \cdots \wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{d-1}}\wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{d}}\\ =\frac{(d-1){{\ell }}^{2}}{4{(-1)}^{d+1}}{\epsilon }_{\hat{\alpha }\hat{\beta }{\hat{\alpha }}_{1}\cdots \cdots {\hat{\alpha }}_{d}}{F}^{\hat{\alpha }\hat{\beta }}\\ \quad \wedge {F}^{{\hat{\alpha }}_{1}{\hat{\alpha }}_{2}}\wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{3}}\wedge \cdots \wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{d}}.\end{array}\end{eqnarray}$
Thus the characteristic class of the anomaly is $\begin{eqnarray}\begin{array}{rcl}{I}_{(d+2)} & = & \frac{{(-1)}^{d+1}{\ell }}{8{\kappa }^{2}\,(d-2)!}{\epsilon }_{\hat{\alpha }\hat{\beta }{\hat{\alpha }}_{1}\cdots \cdots {\hat{\alpha }}_{d}}{F}^{\hat{\alpha }\hat{\beta }}\wedge {F}^{{\hat{\alpha }}_{1}{\hat{\alpha }}_{2}}\\ & & \wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{3}}\wedge \cdots \wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{d}}.\end{array}\end{eqnarray}$
C.2.2. Bulk off-shell currents and Noether potentials
In computing $\delta {S}_{(d+1)}^{{\rm{cov}}}$ in appendix A.3 , the boundary terms are omitted. To get the Noether potentials, we need to included the boundary terms on Σ = ∂M. By using the resummations 9 ), (10 ), (11 ), (12 ) for ${({{ \mathcal J }}_{(d+1)})}_{\hat{\alpha }\hat{\beta }},{({{ \mathcal J }}_{(d+1)})}_{\hat{\alpha }},{({{ \mathcal K }}_{(d+1)}^{{\rm{cov}}})}_{\hat{\alpha }\hat{\beta }},{({{ \mathcal K }}_{(d+1)}^{{\rm{cov}}})}_{\hat{\alpha }}$in the main text. Especially, we notice that
$\begin{eqnarray}\begin{array}{l}{\epsilon }_{\hat{\alpha }\hat{\beta }{\hat{\alpha }}_{1}\cdots {\hat{\alpha }}_{d}}\delta {A}^{\hat{\alpha }\hat{\gamma }}{Y}_{\hat{\gamma }}{Y}^{\hat{\beta }}\wedge {F}^{{\hat{\alpha }}_{1}{\hat{\alpha }}_{2}}\\ \wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{3}}\wedge \cdots \wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{d}}\\ =-\frac{{{\ell }}^{2}}{2}{\epsilon }_{\hat{\alpha }\hat{\beta }{\hat{\alpha }}_{1}\cdots {\hat{\alpha }}_{d}}\delta {A}^{\hat{\alpha }\hat{\beta }}\wedge {F}^{{\hat{\alpha }}_{1}{\hat{\alpha }}_{2}}\\ \wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{3}}\wedge \cdots \wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{d}}\\ -{\epsilon }_{\hat{\alpha }\hat{\beta }{\hat{\alpha }}_{1}\cdots {\hat{\alpha }}_{d}}\delta {A}^{\hat{\alpha }\hat{\beta }}\wedge {(FY)}^{{\hat{\alpha }}_{1}}{Y}^{{\hat{\alpha }}_{2}}\\ \wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{3}}\wedge \cdots \wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{d}},\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{l}{\epsilon }_{\hat{\alpha }\hat{\beta }{\hat{\alpha }}_{1}\cdots {\hat{\alpha }}_{d}}\delta {A}^{\hat{\alpha }\hat{\gamma }}{Y}_{\hat{\gamma }}{Y}^{\hat{\beta }}\wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{1}}\wedge \cdots \wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{d}}\\ =-\frac{{{\ell }}^{2}}{2}{\epsilon }_{\hat{\alpha }\hat{\beta }{\hat{\alpha }}_{1}\cdots {\hat{\alpha }}_{d}}\delta {A}^{\hat{\alpha }\hat{\beta }}\wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{1}}\wedge \cdots \wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{d}},\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{l}-{{\ell }}^{2}{\epsilon }_{\hat{\alpha }\hat{\beta }{\hat{\alpha }}_{1}\cdots {\hat{\alpha }}_{d}}\delta {Y}^{\hat{\alpha }}{F}^{\hat{\beta }{\hat{\alpha }}_{1}}\wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{2}}\wedge \cdots \wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{d}}\\ =-2{\epsilon }_{\hat{\alpha }\hat{\beta }{\hat{\alpha }}_{1}\cdots {\hat{\alpha }}_{d}}\delta {Y}^{\hat{\alpha }}{Y}^{\hat{\beta }}{(FY)}^{{\hat{\alpha }}_{1}}\wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{2}}\wedge \cdots \wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{d}},\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{l}-{{\ell }}^{2}{\epsilon }_{\hat{\alpha }\hat{\beta }{\hat{\alpha }}_{1}\cdots {\hat{\alpha }}_{d}}\delta {Y}^{\hat{\alpha }}{\rm{D}}{Y}^{\hat{\beta }}\wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{1}}\wedge \cdots \wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{d}}\\ ={\epsilon }_{\hat{\alpha }\hat{\beta }{\hat{\alpha }}_{1}\cdots {\hat{\alpha }}_{d}}\delta {Y}^{\hat{\alpha }}{\rm{D}}{Y}^{\hat{\beta }}\wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{1}}\wedge \cdots \wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{d}}{Y}^{\hat{\gamma }}{Y}_{\hat{\gamma }}=0,\end{array}\end{eqnarray}$
we get $\begin{eqnarray*}\begin{array}{l}\delta {\displaystyle \int }_{M}{\epsilon }_{\hat{\alpha }\hat{\beta }{\hat{\alpha }}_{1}\cdots {\hat{\alpha }}_{d}}\left[{F}^{\hat{\alpha }\hat{\beta }}-\frac{2}{(d+1){{\ell }}^{2}}{\rm{D}}{Y}^{\hat{\alpha }}\wedge {\rm{D}}{Y}^{\hat{\beta }}\right]\\ \wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{1}}\wedge \cdots \wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{d-1}}{Y}^{{\hat{\alpha }}_{d}}\\ ={\displaystyle \int }_{M}{\epsilon }_{\hat{\alpha }\hat{\beta }{\hat{\alpha }}_{1}\cdots {\hat{\alpha }}_{d}}\left\{{\rm{D}}\delta {A}^{\hat{\alpha }\hat{\beta }}\wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{1}}\wedge \cdots \wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{d-1}}{Y}^{{\hat{\alpha }}_{d}}\right.\\ \qquad +(d-1){F}^{\hat{\alpha }\hat{\beta }}\wedge {({\rm{D}}\delta Y+\delta AY)}^{{\hat{\alpha }}_{1}}\\ \qquad \wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{2}}\wedge \cdots \wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{d-1}}{Y}^{{\hat{\alpha }}_{d}}\\ +{F}^{\hat{\alpha }\hat{\beta }}\wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{1}}\wedge \cdots \wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{d-1}}\delta {Y}^{{\hat{\alpha }}_{d}}\end{array}\,\end{eqnarray*}$
$\begin{eqnarray*}\begin{array}{l}-\frac{2}{{{\ell }}^{2}}{({\rm{D}}\delta Y+\delta AY)}^{\hat{\alpha }}\wedge {\rm{D}}{Y}^{\hat{\beta }}\\ \wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{1}}\wedge \cdots \wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{d-1}}{Y}^{{\hat{\alpha }}_{d}}\\ -\frac{2}{(d+1){{\ell }}^{2}}{\rm{D}}{Y}^{\hat{\alpha }}\wedge {\rm{D}}{Y}^{\hat{\beta }}\wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{1}}\\ \left.\wedge \cdots \wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{d-1}}\delta {Y}^{{\hat{\alpha }}_{d}}\right\}\\ ={\displaystyle \int }_{M}{\epsilon }_{\hat{\alpha }\hat{\beta }{\hat{\alpha }}_{1}\cdots {\hat{\alpha }}_{d}}\left\{\delta {A}^{\hat{\alpha }\hat{\beta }}\wedge \left[{\rm{D}}({\rm{D}}{Y}^{{\hat{\alpha }}_{1}}\wedge \cdots \wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{d-1}}){Y}^{{\hat{\alpha }}_{d}}\right.\right.\\ \left.-{(-1)}^{d}{\rm{D}}{Y}^{{\hat{\alpha }}_{1}}\wedge \cdots \wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{d}}\right]\\ +(d-1){(\delta AY)}^{{\hat{\alpha }}_{1}}\wedge {F}^{\hat{\alpha }\hat{\beta }}\wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{2}}\end{array}\end{eqnarray*}$
$\begin{eqnarray*}\begin{array}{l}\wedge \cdots \wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{d-1}}{Y}^{{\hat{\alpha }}_{d}}\\ -\frac{2}{{{\ell }}^{2}}{(\delta AY)}^{\hat{\alpha }}\wedge {\rm{D}}{Y}^{\hat{\beta }}\wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{1}}\\ \wedge \cdots \wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{d-1}}{Y}^{{\hat{\alpha }}_{d}}\\ -(d-1)\delta {Y}^{{\hat{\alpha }}_{1}}\left[{\rm{D}}({F}^{\hat{\alpha }\hat{\beta }}\wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{2}}\wedge \cdots \wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{d-1}})\right.\\ \left.\times \,{Y}^{{\hat{\alpha }}_{d}}+{(-1)}^{d}{F}^{\hat{\alpha }\hat{\beta }}\wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{2}}\wedge \cdots \wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{d}}\right]\end{array}\,\end{eqnarray*}$
$\begin{eqnarray*}\begin{array}{l}+\frac{2}{{{\ell }}^{2}}\delta {Y}^{\hat{\alpha }}\left[{\rm{D}}({\rm{D}}{Y}^{\hat{\beta }}\wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{1}}\wedge \cdots \wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{d-1}})\right.\\ \left.\times \,{Y}^{{\hat{\alpha }}_{d}}+{(-1)}^{d}{\rm{D}}{Y}^{\hat{\beta }}\wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{1}}\wedge \cdots \wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{d}}\right]\\ +\delta {Y}^{{\hat{\alpha }}_{d}}{F}^{\hat{\alpha }\hat{\beta }}\wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{1}}\wedge \cdots \wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{d-1}}\\ -\frac{2}{(d+1){{\ell }}^{2}}\delta {Y}^{{\hat{\alpha }}_{d}}{\rm{D}}{Y}^{\hat{\alpha }}\wedge {\rm{D}}{Y}^{\hat{\beta }}\\ \left.\wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{1}}\wedge \cdots \wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{d-1}}\right\}\end{array}\,\end{eqnarray*}$
$\begin{eqnarray*}\begin{array}{l}+{\displaystyle \int }_{{\rm{\Sigma }}}{\epsilon }_{\hat{\alpha }\hat{\beta }{\hat{\alpha }}_{1}\cdots {\hat{\alpha }}_{d}}\left\{\delta {A}^{\hat{\alpha }\hat{\beta }}\wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{1}}\right.\\ \wedge \cdots \wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{d-1}}{Y}^{{\hat{\alpha }}_{d}}+(d-1)\delta {Y}^{{\hat{\alpha }}_{1}}{F}^{\hat{\alpha }\hat{\beta }}\\ \wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{2}}\wedge \cdots \wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{d-1}}{Y}^{{\hat{\alpha }}_{d}}\\ \left.-\frac{2}{{{\ell }}^{2}}\delta {Y}^{\hat{\alpha }}{\rm{D}}{Y}^{\hat{\beta }}\wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{1}}\wedge \cdots \wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{d-1}}{Y}^{{\hat{\alpha }}_{d}}\right\}\end{array}\,\end{eqnarray*}$
$\begin{eqnarray*}\begin{array}{l}\,={(-1)}^{d+1}(d-1){\displaystyle \int }_{M}{\epsilon }_{\hat{\alpha }\hat{\beta }{\hat{\alpha }}_{1}\cdots {\hat{\alpha }}_{d}}\\ \times \left\{\frac{{{\ell }}^{2}}{2}\delta {A}^{\hat{\alpha }\hat{\beta }}\wedge {F}^{{\hat{\alpha }}_{1}{\hat{\alpha }}_{2}}\wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{3}}\wedge \cdots \wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{d}}\right.\\ \left.+\delta {Y}^{\hat{\alpha }}{Y}^{\hat{\beta }}{\rm{D}}({F}^{{\hat{\alpha }}_{1}{\hat{\alpha }}_{2}}\wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{2}}\wedge \cdots \wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{d}})\right\}\\ +{\displaystyle \int }_{{\rm{\Sigma }}}{\epsilon }_{\hat{\alpha }\hat{\beta }{\hat{\alpha }}_{1}\cdots {\hat{\alpha }}_{d}}\left\{\delta {A}^{\hat{\alpha }\hat{\beta }}\wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{1}}\right.\end{array}\,\end{eqnarray*}$
$\begin{eqnarray}\begin{array}{l}\wedge \cdots \wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{d-1}}{Y}^{{\hat{\alpha }}_{d}}+{(-1)}^{d}\delta {Y}^{\hat{\alpha }}{Y}^{\hat{\beta }}\\ \times \left[(d-1){F}^{{\hat{\alpha }}_{1}{\hat{\alpha }}_{2}}\wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{3}}\wedge \cdots \wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{d}}\right.\\ \left.\left.-\frac{2}{{{\ell }}^{2}}{\rm{D}}{Y}^{{\hat{\alpha }}_{1}}\wedge \cdots \wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{d}}\right]\right\}.\end{array}\end{eqnarray}$
It implies the expressions ($\begin{eqnarray}{({{ \mathcal J }}_{(d+1)})}_{\hat{\alpha }}{Y}^{\hat{\alpha }}=0,\quad {({{ \mathcal K }}_{(d+1)}^{{\rm{cov}}})}_{\hat{\alpha }}{Y}^{\hat{\alpha }}=0,\end{eqnarray}$
which are the correct formula under the constraint ${Y}_{\hat{\alpha }}\delta {Y}^{\hat{\alpha }}=0$.C.2.3. SO(2, d) conservation law from ${S}_{(d+1)}^{{\rm{cov}}}$
Under the SO(2, d) gauge transformation
$\begin{eqnarray}{Y}^{\hat{\alpha }}\to {\tilde{Y}}^{\hat{\alpha }}={U}_{\,\,\hat{\beta }}^{\hat{\alpha }}{Y}^{\hat{\beta }},A\to \tilde{A}=UA{U}^{-1}-{\rm{d}}U{U}^{-1},\end{eqnarray}$
this bulk covariant action is manifestly invariant without any boundary tails $\begin{eqnarray}\begin{array}{rcl}{{\rm{\Delta }}}_{U}{{ \mathcal L }}_{(d+1)}^{{\rm{cov}}}(A,Y) & = & {{ \mathcal L }}_{(d+1)}^{{\rm{cov}}}(\tilde{A},\tilde{Y)}\\ & & -{{ \mathcal L }}_{(d+1)}^{{\rm{cov}}}(A,Y)=0.\end{array}\end{eqnarray}$
Substituting the infinitesimal gauge transformation C.30 ) works for any M and Σ, the boundary anomalous conservation law must also hold without the pullback
$\begin{eqnarray}\begin{array}{rcl}{\delta }_{u}{Y}^{\hat{\alpha }} & = & {u}_{\,\,\hat{\beta }}^{\hat{\alpha }}{Y}^{\hat{\beta }},\\ {\delta }_{u}{A}^{\hat{\alpha }\hat{\beta }} & = & -{\rm{D}}{u}^{\hat{\alpha }\hat{\beta }},\end{array}\end{eqnarray}$
to the variation of ${S}_{(d+1)}^{{\rm{cov}}}$, we get $\begin{eqnarray}\begin{array}{l}0={\delta }_{u}{S}_{(d+1)}^{{\rm{cov}}}(A,Y)={\delta }_{u}{\displaystyle \int }_{M}{{ \mathcal L }}_{(d+1)}^{{\rm{cov}}}(A,Y,F,{\rm{D}}Y)\\ ={\displaystyle \int }_{M}\left[-{\rm{D}}{u}^{\hat{\alpha }\hat{\beta }}\wedge {({{ \mathcal J }}_{(d+1)}^{{\rm{cov}}})}_{\hat{\alpha }\hat{\beta }}+{(uY)}^{\hat{\alpha }}{({{ \mathcal J }}_{(d+1)}^{{\rm{cov}}})}_{\hat{\alpha }}\right]\\ +{\displaystyle \int }_{{\rm{\Sigma }}}\left[-{\rm{D}}{u}^{\hat{\alpha }\hat{\beta }}\wedge {({{ \mathcal K }}_{(d+1)}^{{\rm{cov}}})}_{\hat{\alpha }\hat{\beta }}+{(uY)}^{\hat{\alpha }}{({{ \mathcal K }}_{(d+1)}^{{\rm{cov}}})}_{\hat{\alpha }}\right]\\ ={\displaystyle \int }_{M}{u}^{\hat{\alpha }\hat{\beta }}\left[{\rm{D}}{({{ \mathcal J }}_{(d+1)}^{{\rm{cov}}})}_{\hat{\alpha }\hat{\beta }}+{Y}_{[\hat{\beta }}{({{ \mathcal J }}_{(d+1)}^{{\rm{cov}}})}_{\hat{\alpha }]}\right]\\ +{\displaystyle \int }_{{\rm{\Sigma }}}{u}^{\hat{\alpha }\hat{\beta }}\left[-{({{ \mathcal J }}_{(d+1)}^{{\rm{cov}}})}_{\hat{\alpha }\hat{\beta }}+{\rm{D}}{({{ \mathcal K }}_{(d+1)}^{{\rm{cov}}})}_{\hat{\alpha }\hat{\beta }}\right.\\ \left.+{Y}_{[\hat{\beta }}{({{ \mathcal K }}_{(d+1)}^{{\rm{cov}}})}_{\hat{\alpha }]}\right]-{\displaystyle \int }_{\partial {\rm{\Sigma }}}{u}^{\hat{\alpha }\hat{\beta }}{({{ \mathcal K }}_{(d+1)}^{{\rm{cov}}})}_{\hat{\alpha }\hat{\beta }}.\end{array}\end{eqnarray}$
Fixing M, the arbitrariness of u implies the following off-shell bulk conservation law $\begin{eqnarray}{\rm{D}}{({{ \mathcal J }}_{(d+1)}^{{\rm{cov}}})}_{\hat{\alpha }\hat{\beta }}+{Y}_{[\hat{\beta }}{({{ \mathcal J }}_{(d+1)}^{{\rm{cov}}})}_{\hat{\alpha }]}=0,\end{eqnarray}$
as well as the boundary anomalous conservation law at the off-shell level $\begin{eqnarray}{\mathbb{D}}{({{ \mathcal J }}_{(d)}^{{\rm{cov}}})}_{\hat{\alpha }\hat{\beta }}+{{\mathbb{Y}}}_{[\hat{\beta }}{({{ \mathcal J }}_{(d)}^{{\rm{cov}}})}_{\hat{\alpha }]}={({{ \mathcal A }}_{(d)}^{{\rm{cov}}})}_{\hat{\alpha }\hat{\beta }},\end{eqnarray}$
where ${({{ \mathcal A }}_{(d)}^{{\rm{cov}}})}_{\hat{\alpha }\hat{\beta }}$ is the boundary covariant anomaly on Σ$\begin{eqnarray}\begin{array}{l}{({{ \mathcal A }}_{(d)}^{{\rm{cov}}})}_{\hat{\alpha }\hat{\beta }}({\rm{\Sigma }})={f}_{{\rm{\Sigma }}}^{* }[{({{ \mathcal J }}_{(d+1)}^{{\rm{cov}}})}_{\hat{\alpha }\hat{\beta }}]\\ =\frac{{(-1)}^{d+1}{\ell }}{4{\kappa }^{2}(d-2)!}{\epsilon }_{\hat{\alpha }\hat{\beta }{\hat{\alpha }}_{1}\cdots {\hat{\alpha }}_{d}}{{\mathbb{F}}}^{{\hat{\alpha }}_{1}{\hat{\alpha }}_{2}}\\ \quad \wedge {\mathbb{D}}{{\mathbb{Y}}}^{{\hat{\alpha }}_{3}}\wedge \cdots \wedge {\mathbb{D}}{{\mathbb{Y}}}^{{\hat{\alpha }}_{d}}.\end{array}\end{eqnarray}$
Since ($\begin{eqnarray}{\rm{D}}{({{ \mathcal K }}_{(d+1)}^{{\rm{cov}}})}_{\hat{\alpha }\hat{\beta }}+{Y}_{[\hat{\beta }}{({{ \mathcal K }}_{(d+1)}^{{\rm{cov}}})}_{\hat{\alpha }]}={({{ \mathcal J }}_{(d+1)}^{{\rm{cov}}})}_{\hat{\alpha }\hat{\beta }}.\end{eqnarray}$
By using the resummations $\begin{eqnarray}\begin{array}{l}{Y}_{[\hat{\beta }}{\epsilon }_{\hat{\alpha }]{\hat{\alpha }}_{0}{\hat{\alpha }}_{1}\cdots {\hat{\alpha }}_{d}}{Y}^{{\hat{\alpha }}_{0}}{\rm{D}}({F}^{{\hat{\alpha }}_{1}{\hat{\alpha }}_{2}}\wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{3}}\wedge \cdots \wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{d}})\\ =(d-2){Y}_{{\hat{\beta }}_{0}}{\delta }_{[\hat{\beta }}^{{\hat{\beta }}_{0}}{\delta }_{\hat{\alpha }]}^{{\hat{\alpha }}_{D}}{\epsilon }_{{\hat{\alpha }}_{D}{\hat{\alpha }}_{0}{\hat{\alpha }}_{1}\cdots {\hat{\alpha }}_{d}}{Y}^{{\hat{\alpha }}_{0}}\\ \times \,{F}^{{\hat{\alpha }}_{1}{\hat{\alpha }}_{2}}\wedge {(FY)}^{{\hat{\alpha }}_{3}}\wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{4}}\wedge \cdots \wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{d}}\\ =-{Y}_{[\hat{\beta }}{\epsilon }_{\hat{\alpha }]{\hat{\alpha }}_{0}{\hat{\alpha }}_{1}\cdots {\hat{\alpha }}_{d}}{Y}^{{\hat{\alpha }}_{0}}\\ \times {\rm{D}}({F}^{{\hat{\alpha }}_{1}{\hat{\alpha }}_{2}}\wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{3}}\wedge \cdots \wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{d}})\\ -{{\ell }}^{2}{\epsilon }_{\hat{\alpha }\hat{\beta }{\hat{\alpha }}_{1}\cdots {\hat{\alpha }}_{d}}{\rm{D}}({F}^{{\hat{\alpha }}_{1}{\hat{\alpha }}_{2}}\wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{3}}\wedge \cdots \wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{d}}),\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{l}{Y}_{[\hat{\beta }}{\epsilon }_{\hat{\alpha }]{\hat{\alpha }}_{0}{\hat{\alpha }}_{1}\cdots {\hat{\alpha }}_{d}}{Y}^{{\hat{\alpha }}_{0}}{F}^{{\hat{\alpha }}_{1}{\hat{\alpha }}_{2}}\\ \quad \wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{3}}\wedge \cdots \wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{d}}\\ ={Y}_{{\hat{\beta }}_{0}}{\delta }_{[\hat{\beta }}^{{\hat{\beta }}_{0}}{\delta }_{\hat{\alpha }]}^{{\hat{\alpha }}_{D}}{\epsilon }_{{\hat{\alpha }}_{D}{\hat{\alpha }}_{0}{\hat{\alpha }}_{1}\cdots {\hat{\alpha }}_{d}}{Y}^{{\hat{\alpha }}_{0}}\\ \quad \times \,{F}^{{\hat{\alpha }}_{1}{\hat{\alpha }}_{2}}\wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{3}}\wedge \cdots \wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{d}}\\ =-{Y}_{[\hat{\beta }}{\epsilon }_{\hat{\alpha }]{\hat{\alpha }}_{0}{\hat{\alpha }}_{1}\cdots {\hat{\alpha }}_{d}}{Y}^{{\hat{\alpha }}_{0}}{F}^{{\hat{\alpha }}_{1}{\hat{\alpha }}_{2}}\wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{3}}\\ \quad \wedge \cdots \wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{d}}-{{\ell }}^{2}{\epsilon }_{\hat{\alpha }\hat{\beta }{\hat{\alpha }}_{1}\cdots {\hat{\alpha }}_{d}}{F}^{{\hat{\alpha }}_{1}{\hat{\alpha }}_{2}}\wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{3}}\\ \quad \wedge \cdots \wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{d}}+2{\epsilon }_{\hat{\alpha }\hat{\beta }{\hat{\alpha }}_{1}{\hat{\alpha }}_{2}{\hat{\alpha }}_{3}\cdots {\hat{\alpha }}_{d}}{Y}^{{\hat{\alpha }}_{1}}\\ \quad \times {(FY)}^{{\hat{\alpha }}_{2}}\wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{3}}\wedge \cdots \wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{d}},\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{l}{Y}_{[\hat{\beta }}{\epsilon }_{\hat{\alpha }]{\hat{\alpha }}_{0}{\hat{\alpha }}_{1}\cdots {\hat{\alpha }}_{d}}{Y}^{{\hat{\alpha }}_{0}}{\rm{D}}{Y}^{{\hat{\alpha }}_{1}}\wedge \cdots \wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{d}}\\ ={Y}_{{\hat{\beta }}_{0}}{\delta }_{[\hat{\beta }}^{{\hat{\beta }}_{0}}{\delta }_{\hat{\alpha }]}^{{\hat{\alpha }}_{D}}{\epsilon }_{{\hat{\alpha }}_{D}{\hat{\alpha }}_{0}{\hat{\alpha }}_{1}\cdots {\hat{\alpha }}_{d}}{Y}^{{\hat{\alpha }}_{0}}{\rm{D}}{Y}^{{\hat{\alpha }}_{1}}\wedge \cdots \wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{d}}\\ =-{Y}_{[\hat{\beta }}{\epsilon }_{\hat{\alpha }]{\hat{\alpha }}_{0}{\hat{\alpha }}_{1}\cdots {\hat{\alpha }}_{d}}{Y}^{{\hat{\alpha }}_{0}}{\rm{D}}{Y}^{{\hat{\alpha }}_{1}}\wedge \cdots \wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{d}}\\ \,-{{\ell }}^{2}{\epsilon }_{\hat{\alpha }\hat{\beta }{\hat{\alpha }}_{1}\cdots {\hat{\alpha }}_{d}}{\rm{D}}{Y}^{{\hat{\alpha }}_{1}}\wedge \cdots \wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{d}}.\end{array}\end{eqnarray}$
One can verify that the off-shell conservation laws are indeed identities $\begin{eqnarray}\begin{array}{l}{\rm{D}}{({{ \mathcal J }}_{(d+1)}^{{\rm{cov}}})}_{\hat{\alpha }\hat{\beta }}+{Y}_{[\hat{\beta }}{({{ \mathcal J }}_{(d+1)}^{{\rm{cov}}})}_{\hat{\alpha }]}\\ =\frac{{(-1)}^{d+1}{\ell }}{4{\kappa }^{2}(d-2)!}\left[{\rm{D}}({\epsilon }_{\hat{\alpha }\hat{\beta }{\hat{\alpha }}_{1}\cdots {\hat{\alpha }}_{d}}{F}^{{\hat{\alpha }}_{1}{\hat{\alpha }}_{2}}\right.\\ \quad \wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{3}}\wedge \cdots \wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{d}})\\ \quad +\frac{2}{{{\ell }}^{2}}{Y}_{[\hat{\beta }}{\epsilon }_{\hat{\alpha }]{\hat{\alpha }}_{0}{\hat{\alpha }}_{1}\cdots {\hat{\alpha }}_{d}}{Y}^{{\hat{\alpha }}_{0}}\\ \quad \left.\times \,{\rm{D}}({F}^{{\hat{\alpha }}_{1}{\hat{\alpha }}_{2}}\wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{3}}\wedge \cdots \wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{d}})\right]\\ =\frac{{(-1)}^{d+1}{\ell }}{4{\kappa }^{2}(d-2)!}\left[{\epsilon }_{\hat{\alpha }\hat{\beta }{\hat{\alpha }}_{1}\cdots {\hat{\alpha }}_{d}}\right.\\ \quad \times {\rm{D}}\left({F}^{{\hat{\alpha }}_{1}{\hat{\alpha }}_{2}}\wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{3}}\wedge \cdots \wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{d}}\right)\\ \quad -{\epsilon }_{\hat{\alpha }\hat{\beta }{\hat{\alpha }}_{1}\cdots {\hat{\alpha }}_{d}}{\rm{D}}\left({F}^{{\hat{\alpha }}_{1}{\hat{\alpha }}_{2}}\wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{3}}\right.\\ \quad \left.\left.\wedge \cdots \wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{d}}\right)\right]\\ =0,\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{l}{\rm{D}}{({{ \mathcal K }}_{(d+1)}^{{\rm{cov}}})}_{\hat{\alpha }\hat{\beta }}+{Y}_{[\hat{\beta }}{({{ \mathcal K }}_{(d+1)}^{{\rm{cov}}})}_{\hat{\alpha }]}\\ =\frac{{(-1)}^{d}}{2{\kappa }^{2}{\ell }\,(d-1)!}\left\{-{\rm{D}}({\epsilon }_{\hat{\alpha }\hat{\beta }{\hat{\alpha }}_{1}\cdots {\hat{\alpha }}_{d}}{Y}^{{\hat{\alpha }}_{1}}\right.\\ \,\times \,{\rm{D}}{Y}^{{\hat{\alpha }}_{2}}\wedge \cdots \wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{d}})+{Y}_{[\hat{\beta }}{\epsilon }_{\hat{\alpha }]{\hat{\alpha }}_{0}{\hat{\alpha }}_{1}\cdots {\hat{\alpha }}_{d}}\\ \,\times \,{Y}^{{\hat{\alpha }}_{0}}\left[(d-1){F}^{{\hat{\alpha }}_{1}{\hat{\alpha }}_{2}}\wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{3}}\right.\\ \,\,\left.\left.\wedge \cdots \wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{d}}-\frac{2}{{{\ell }}^{2}}{\rm{D}}{Y}^{{\hat{\alpha }}_{1}}\wedge \cdots \wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{d}}\right]\right\}\\ =\frac{{(-1)}^{d}}{2{\kappa }^{2}{\ell }\,(d-1)!}\left\{-{\epsilon }_{\hat{\alpha }\hat{\beta }{\hat{\alpha }}_{1}\cdots {\hat{\alpha }}_{d}}\right.\\ \quad \times \,\left[{\rm{D}}{Y}^{{\hat{\alpha }}_{1}}\wedge \cdots \wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{d}}+(d-1){Y}^{{\hat{\alpha }}_{1}}\right.\\ \quad \left.\times {(FY)}^{{\hat{\alpha }}_{2}}\wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{3}}\wedge \cdots \wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{d}}\right]\\ +{\epsilon }_{\hat{\alpha }\hat{\beta }{\hat{\alpha }}_{1}\cdots {\hat{\alpha }}_{d}}\left[(d-1){Y}^{{\hat{\alpha }}_{1}}{(FY)}^{{\hat{\alpha }}_{2}}\right.\\ \quad \wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{3}}\wedge \cdots \wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{d}}-\frac{{{\ell }}^{2}(d-1)}{2}\\ \quad \times \,{F}^{{\hat{\alpha }}_{1}{\hat{\alpha }}_{2}}\wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{3}}\wedge \cdots \wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{d}}\\ \quad \left.\left.+{\rm{D}}{Y}^{{\hat{\alpha }}_{1}}\wedge \cdots \wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{d}}\right]\right\}\\ =\frac{{(-1)}^{d+1}{\ell }}{4{\kappa }^{2}\,(d-2)!}{\epsilon }_{\hat{\alpha }\hat{\beta }{\hat{\alpha }}_{1}\cdots {\hat{\alpha }}_{d}}{F}^{{\hat{\alpha }}_{1}{\hat{\alpha }}_{2}}\\ \quad \wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{3}}\wedge \cdots \wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{d}}={({{ \mathcal J }}_{(d+1)}^{{\rm{cov}}})}_{\hat{\alpha }\hat{\beta }}.\end{array}\end{eqnarray}$
C.3. The bulk consistent action
To construct the bulk consistent action, let us start from the following bulk CS action
$\begin{eqnarray}\begin{array}{l}\frac{4{\kappa }^{2}(d-2)!}{{(-1)}^{d+1}{\ell }}{S}_{(d+1)}^{{\rm{CS}}}[A,Y]\\ \,=\frac{4{\kappa }^{2}(d-2)!}{{(-1)}^{d+1}{\ell }}{\displaystyle \int }_{M}{{ \mathcal L }}_{(d+1)}^{{\rm{CS}}}\\ \,=\frac{4{\kappa }^{2}(d-2)!}{{(-1)}^{d+1}{\ell }}{\displaystyle \int }_{M}{\displaystyle \int }_{0}^{1}{\rm{d}}s\,{A}^{\hat{\alpha }\hat{\beta }}\\ \quad \wedge {({{ \mathcal A }}_{(d)}^{{\rm{cov}}})}_{\hat{\alpha }\hat{\beta }}(A(s),Y)\\ ={\displaystyle \int }_{M}{\displaystyle \int }_{0}^{1}{\rm{d}}s\,{\epsilon }_{\hat{\alpha }\hat{\beta }{\hat{\alpha }}_{1}\cdots {\hat{\alpha }}_{d}}{A}^{\hat{\alpha }\hat{\beta }}\wedge {F}^{{\hat{\alpha }}_{1}{\hat{\alpha }}_{2}}(s)\\ \quad \wedge {\rm{D}}(s){Y}^{{\hat{\alpha }}_{3}}\wedge \cdots \wedge {\rm{D}}(s){Y}^{{\hat{\alpha }}_{d}},\end{array}\end{eqnarray}$
where the homotopy notations are $\begin{eqnarray}\begin{array}{rcl}A(s) & = & sA,\\ {\rm{D}}(s)Y & = & {\rm{d}}Y+sAY,\\ F(s) & = & {\rm{d}}A(s)+A(s)\wedge A(s)=s{\rm{d}}A+{s}^{2}A\wedge A\\ & = & sF-s(1-s)A\wedge A\end{array}\end{eqnarray}$
and $\begin{eqnarray}\begin{array}{l}{({{ \mathcal A }}_{(d)}^{{\rm{cov}}})}_{\hat{\alpha }\hat{\beta }}={({{ \mathcal J }}_{(d+1)}^{{\rm{cov}}})}_{\hat{\alpha }\hat{\beta }}\\ =\frac{{(-1)}^{d+1}{\ell }}{4{\kappa }^{2}(d-2)!}{\epsilon }_{\hat{\alpha }\hat{\beta }{\hat{\alpha }}_{1}\cdots {\hat{\alpha }}_{d}}{F}^{{\hat{\alpha }}_{1}{\hat{\alpha }}_{2}}\\ \quad \wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{3}}\wedge \cdots \wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{d}}.\end{array}\end{eqnarray}$
C.3.1. ${\rm{d}}{{ \mathcal L }}_{(d+1)}^{{\rm{CS}}}={I}_{(d+2)}$
We notice that
$\begin{eqnarray}\begin{array}{l}{\rm{d}}[{\epsilon }_{\hat{\alpha }\hat{\beta }{\hat{\alpha }}_{1}\cdots {\hat{\alpha }}_{d}}{A}^{\hat{\alpha }\hat{\beta }}(s)\wedge {F}^{{\hat{\alpha }}_{1}{\hat{\alpha }}_{2}}(s)\\ \quad \wedge {\rm{D}}(s){Y}^{{\hat{\alpha }}_{3}}\wedge \cdots \wedge {\rm{D}}(s){Y}^{{\hat{\alpha }}_{d}}]\\ ={\epsilon }_{\hat{\alpha }\hat{\beta }{\hat{\alpha }}_{1}\cdots {\hat{\alpha }}_{d}}{\rm{D}}(s)[{A}^{\hat{\alpha }\hat{\beta }}(s)\wedge {F}^{{\hat{\alpha }}_{1}{\hat{\alpha }}_{2}}(s)\\ \quad \wedge {\rm{D}}(s){Y}^{{\hat{\alpha }}_{3}}\wedge \cdots \wedge {\rm{D}}(s){Y}^{{\hat{\alpha }}_{d}}]\\ ={\epsilon }_{\hat{\alpha }\hat{\beta }{\hat{\alpha }}_{1}\cdots {\hat{\alpha }}_{d}}{[F(s)+A(s)\wedge A(s)]}^{\hat{\alpha }\hat{\beta }}\\ \quad \wedge {F}^{{\hat{\alpha }}_{1}{\hat{\alpha }}_{2}}(s)\wedge {\rm{D}}(s){Y}^{{\hat{\alpha }}_{3}}\wedge \cdots \wedge {\rm{D}}(s){Y}^{{\hat{\alpha }}_{d}}\\ -(d-2){\epsilon }_{\hat{\alpha }\hat{\beta }{\hat{\alpha }}_{1}\cdots {\hat{\alpha }}_{d}}{A}^{\hat{\alpha }\hat{\beta }}(s)\wedge {F}^{{\hat{\alpha }}_{1}{\hat{\alpha }}_{2}}(s)\\ \quad \wedge {[F(s)Y]}^{{\hat{\alpha }}_{3}}\wedge \cdots \wedge {\rm{D}}(s){Y}^{{\hat{\alpha }}_{d}}\\ ={\epsilon }_{\hat{\alpha }\hat{\beta }{\hat{\alpha }}_{1}\cdots {\hat{\alpha }}_{d}}{[F(s)+A(s)\wedge A(s)]}^{\hat{\alpha }\hat{\beta }}\\ \quad \wedge {F}^{{\hat{\alpha }}_{1}{\hat{\alpha }}_{2}}(s)\wedge {\rm{D}}(s){Y}^{{\hat{\alpha }}_{3}}\wedge \cdots \wedge {\rm{D}}(s){Y}^{{\hat{\alpha }}_{d}}\\ +\frac{d-2}{2}{\epsilon }_{\hat{\alpha }\hat{\beta }{\hat{\alpha }}_{1}\cdots {\hat{\alpha }}_{d}}{F}^{\hat{\alpha }\hat{\beta }}(s)\wedge {F}^{{\hat{\alpha }}_{1}{\hat{\alpha }}_{2}}(s)\\ \quad \wedge {[A(s)Y]}^{{\hat{\alpha }}_{3}}\wedge {\rm{D}}(s){Y}^{{\hat{\alpha }}_{4}}\wedge \cdots \wedge {\rm{D}}(s){Y}^{{\hat{\alpha }}_{d}}\\ ={\epsilon }_{\hat{\alpha }\hat{\beta }{\hat{\alpha }}_{1}\cdots {\hat{\alpha }}_{d}}s{\partial }_{s}{F}^{\hat{\alpha }\hat{\beta }}(s)\wedge {F}^{{\hat{\alpha }}_{1}{\hat{\alpha }}_{2}}(s)\\ \quad \wedge {\rm{D}}(s){Y}^{{\hat{\alpha }}_{3}}\wedge \cdots \wedge {\rm{D}}(s){Y}^{{\hat{\alpha }}_{d}}\\ \quad +\frac{d-2}{2}{\epsilon }_{\hat{\alpha }\hat{\beta }{\hat{\alpha }}_{1}\cdots {\hat{\alpha }}_{d}}{F}^{\hat{\alpha }\hat{\beta }}(s)\wedge {F}^{{\hat{\alpha }}_{1}{\hat{\alpha }}_{2}}(s)\\ \quad \wedge s{\partial }_{s}{[{\rm{D}}(s)Y]}^{{\hat{\alpha }}_{3}}\wedge {\rm{D}}(s){Y}^{{\hat{\alpha }}_{4}}\wedge \cdots \wedge {\rm{D}}(s){Y}^{{\hat{\alpha }}_{d}}\\ =\frac{1}{2}s{\partial }_{s}[{\epsilon }_{\hat{\alpha }\hat{\beta }{\hat{\alpha }}_{1}\cdots {\hat{\alpha }}_{d}}{F}^{\hat{\alpha }\hat{\beta }}(s)\wedge {F}^{{\hat{\alpha }}_{1}{\hat{\alpha }}_{2}}(s)\\ \quad \wedge {\rm{D}}(s){Y}^{{\hat{\alpha }}_{3}}\wedge \cdots \wedge {\rm{D}}(s){Y}^{{\hat{\alpha }}_{d}}],\end{array}\end{eqnarray}$
where we have used that $\begin{eqnarray}\begin{array}{rcl}{\rm{D}}(s){A}^{{\hat{\alpha }}_{1}{\hat{\alpha }}_{2}}(s) & = & {[F(s)+A(s)\wedge A(s)]}^{{\hat{\alpha }}_{1}{\hat{\alpha }}_{2}}\\ & = & s{\partial }_{s}{F}^{{\hat{\alpha }}_{1}{\hat{\alpha }}_{2}}(s),\\ s{\partial }_{s}A(s) & = & A(s),\end{array}\end{eqnarray}$
as well as the resummation $\begin{eqnarray}\begin{array}{l}{\epsilon }_{\hat{\alpha }\hat{\beta }{\hat{\alpha }}_{1}\cdots {\hat{\alpha }}_{d}}{A}^{\hat{\alpha }\hat{\beta }}(s)\wedge {F}^{{\hat{\alpha }}_{1}{\hat{\alpha }}_{2}}(s)\wedge {F}^{{\hat{\alpha }}_{3}\hat{\gamma }}(s)\\ \quad \times \,{Y}_{\hat{\gamma }}\wedge {\rm{D}}(s){Y}^{{\hat{\alpha }}_{4}}\wedge \cdots \wedge {\rm{D}}(s){Y}^{{\hat{\alpha }}_{d}}\\ =-\frac{1}{2}{\epsilon }_{\hat{\alpha }\hat{\beta }{\hat{\alpha }}_{1}\cdots {\hat{\alpha }}_{d}}{F}^{\hat{\alpha }\hat{\beta }}(s)\wedge {F}^{{\hat{\alpha }}_{1}{\hat{\alpha }}_{2}}(s)\\ \quad \wedge {[A(s)Y]}^{{\hat{\alpha }}_{3}}\wedge {\rm{D}}(s){Y}^{{\hat{\alpha }}_{4}}\wedge \cdots \wedge {\rm{D}}(s){Y}^{{\hat{\alpha }}_{d}}.\end{array}\end{eqnarray}$
Thus the exterior derivative ${\rm{d}}{{ \mathcal L }}_{(d+1)}^{{\rm{CS}}}$ is the same as ${\rm{d}}{{ \mathcal L }}_{(d+1)}^{{\rm{cov}}}$ $\begin{eqnarray}\begin{array}{l}\frac{4{\kappa }^{2}(d-2)!}{{(-1)}^{d+1}{\ell }}{\rm{d}}{{ \mathcal L }}_{(d+1)}^{{\rm{con}}}\\ ={\rm{d}}\left[{\displaystyle \int }_{0}^{1}\frac{{\rm{d}}s}{s}\,{\epsilon }_{\hat{\alpha }\hat{\beta }{\hat{\alpha }}_{1}\cdots {\hat{\alpha }}_{d}}{A}^{\hat{\alpha }\hat{\beta }}(s)\right.\\ \left.\wedge {F}^{{\hat{\alpha }}_{1}{\hat{\alpha }}_{2}}(s)\wedge {\rm{D}}(s){Y}^{{\hat{\alpha }}_{3}}\wedge \cdots \wedge {\rm{D}}(s){Y}^{{\hat{\alpha }}_{d}}\right]\\ =\frac{1}{2}{\displaystyle \int }_{0}^{1}{\rm{d}}s\,{\partial }_{s}[{\epsilon }_{\hat{\alpha }\hat{\beta }{\hat{\alpha }}_{1}\cdots {\hat{\alpha }}_{d}}{F}^{\hat{\alpha }\hat{\beta }}(s)\\ \quad \wedge {F}^{{\hat{\alpha }}_{1}{\hat{\alpha }}_{2}}(s)\wedge {\rm{D}}(s){Y}^{{\hat{\alpha }}_{3}}\quad \wedge \cdots \wedge {\rm{D}}(s){Y}^{{\hat{\alpha }}_{d}}]\\ =\frac{1}{2}{\epsilon }_{\hat{\alpha }\hat{\beta }{\hat{\alpha }}_{1}\cdots {\hat{\alpha }}_{d}}{F}^{\hat{\alpha }\hat{\beta }}(s)\wedge {F}^{{\hat{\alpha }}_{1}{\hat{\alpha }}_{2}}(s)\\ \quad \wedge {\rm{D}}(s){Y}^{{\hat{\alpha }}_{3}}\wedge \cdots \wedge {\rm{D}}(s){Y}^{{\hat{\alpha }}_{d}}{\left|\right.}_{s=0}^{s=1}\\ =\frac{1}{2}{\epsilon }_{\hat{\alpha }\hat{\beta }{\hat{\alpha }}_{1}\cdots {\hat{\alpha }}_{d}}{F}^{\hat{\alpha }\hat{\beta }}\wedge {F}^{{\hat{\alpha }}_{1}{\hat{\alpha }}_{2}}\\ \quad \wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{3}}\wedge \cdots \wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{d}}\\ =\frac{4{\kappa }^{2}(d-2)!}{{(-1)}^{d+1}{\ell }}{I}_{(d+2)}\\ =\frac{4{\kappa }^{2}(d-2)!}{{(-1)}^{d+1}{\ell }}{\rm{d}}{{ \mathcal L }}_{(d+1)}^{{\rm{cov}}}.\end{array}\end{eqnarray}$
It means that the two Lagrangians ${{ \mathcal L }}_{(d+1)}^{{\rm{con}}}$ and ${{ \mathcal L }}_{(d+1)}^{{\rm{cov}}}$ differ only by a closed from.C.3.2. SO(2, d) gauge transformation of ${{ \mathcal L }}_{(d+1)}^{{\rm{CS}}}$
Denoting
$\begin{eqnarray}\hat{A}={U}^{-1}{\rm{d}}U,\quad \hat{A}(s)=s\hat{A}=s{U}^{-1}{\rm{d}}U,\end{eqnarray}$
the finite gauge transformation can be expressed as $\begin{eqnarray}\begin{array}{rcl}{Y}^{\hat{\alpha }}\to {\tilde{Y}}^{\hat{\alpha }} & = & {U}_{\,\,\hat{\beta }}^{\hat{\alpha }}{Y}^{\hat{\beta }},\\ A\to \tilde{A} & = & UA{U}^{-1}-{\rm{d}}U{U}^{-1}\\ & = & U(A-\hat{A}){U}^{-1}.\end{array}\end{eqnarray}$
Thus we have $\begin{eqnarray}\begin{array}{rcl}A(s)\to \tilde{A(s)} & = & s\tilde{A}=sUA{U}^{-1}-s{U}^{-1}{\rm{d}}U\\ & = & U(A(s)-s{U}^{-1}{\rm{d}}U){U}^{-1}\\ & = & U[A(s)-\hat{A}(s)]{U}^{-1}.\end{array}\end{eqnarray}$
It follows that $\begin{eqnarray}\begin{array}{rcl}\tilde{F(s)} & = & {\rm{d}}\tilde{A(s)}+\tilde{A(s)}\wedge \tilde{A(s)}\\ & = & s{\rm{d}}\tilde{A}+{s}^{2}\tilde{A}\wedge \tilde{A}\\ & = & s\tilde{F}-s(1-s)\tilde{A}\wedge \tilde{A}\\ & = & sU[F-(1-s)(A\wedge A-\hat{A}\wedge A\\ & & -A\wedge \hat{A}+\hat{A}\wedge \hat{A})]{U}^{-1}\\ & = & U[F(s)+s(1-s)(\hat{A}\wedge A+A\wedge \hat{A})\\ & & +\hat{F}(1-s)]{U}^{-1},\\ \tilde{{\rm{D}}(s)}\tilde{Y} & = & U{\rm{d}}Y+{\rm{d}}UY+UA(s)-s{\rm{d}}UY\\ & = & U[{\rm{d}}Y+sAY+(1-s)\hat{A}Y]\\ & = & U[{\rm{D}}(s)Y+(1-s)\hat{A}Y]\\ & = & U[\hat{{\rm{D}}}(1-s)Y+sAY],\end{array}\end{eqnarray}$
where we have used that $\begin{eqnarray}\begin{array}{rcl}\hat{{\rm{D}}}(s)Y & = & {\rm{d}}Y+s\hat{A}Y={\rm{d}}Y+s{U}^{-1}{\rm{d}}UY,\\ \hat{F}(s) & = & {\rm{d}}\hat{A}(s)+\hat{A}(s)\wedge \hat{A}(s)\\ & = & {\rm{d}}(s{U}^{-1}{\rm{d}}U)+{s}^{2}{U}^{-1}{\rm{d}}U\wedge {U}^{-1}{\rm{d}}U\\ & = & s(1-s){\rm{d}}({U}^{-1}{\rm{d}}U)\\ & = & -s(1-s)\hat{A}\wedge \hat{A}=\hat{F}(1-s).\end{array}\end{eqnarray}$
Thus $\begin{eqnarray*}\begin{array}{l}\frac{4{\kappa }^{2}(d-2)!}{{(-1)}^{d+1}{\ell }}{{ \mathcal L }}_{(d+1)}^{{\rm{CS}}}[\tilde{A},\tilde{Y}]\\ ={\displaystyle \int }_{0}^{1}{\rm{d}}s\,{\epsilon }_{\hat{\alpha }\hat{\beta }{\hat{\alpha }}_{1}\cdots {\hat{\alpha }}_{d}}{\tilde{A}}^{\hat{\alpha }\hat{\beta }}\wedge {\tilde{F}}^{{\hat{\alpha }}_{1}{\hat{\alpha }}_{2}}(s)\\ \quad \wedge \tilde{{\rm{D}}(s)}{\tilde{Y}}^{{\hat{\alpha }}_{3}}\wedge \cdots \wedge \tilde{{\rm{D}}(s)}{\tilde{Y}}^{{\hat{\alpha }}_{d}}\\ ={\displaystyle \int }_{0}^{1}{\rm{d}}s\,{\epsilon }_{\hat{\alpha }\hat{\beta }{\hat{\alpha }}_{1}\cdots {\hat{\alpha }}_{d}}{(A-\hat{A})}^{\hat{\alpha }\hat{\beta }}\wedge [F(s)\\ \quad +s(1-s)(\hat{A}\wedge A+A\wedge \hat{A})+\hat{F}(1-s){]}^{{\hat{\alpha }}_{1}{\hat{\alpha }}_{2}}\\ \quad \wedge {[{\rm{D}}(s)Y+(1-s)\hat{A}Y]}^{{\hat{\alpha }}_{3}}\\ \quad \wedge \cdots \wedge {[{\rm{D}}(s)Y+(1-s)\hat{A}Y]}^{{\hat{\alpha }}_{d}}\end{array}\end{eqnarray*}$
$\begin{eqnarray*}\begin{array}{l}={\displaystyle \int }_{0}^{1}{\rm{d}}s\,{\epsilon }_{\hat{\alpha }\hat{\beta }{\hat{\alpha }}_{1}\cdots {\hat{\alpha }}_{d}}{(A-\hat{A})}^{\hat{\alpha }\hat{\beta }}\wedge F{(s,1-s)}^{{\hat{\alpha }}_{1}{\hat{\alpha }}_{2}}\\ \quad \wedge {\rm{D}}(s,1-s){Y}^{{\hat{\alpha }}_{3}}\wedge \cdots \wedge {\rm{D}}(s,1-s){Y}^{{\hat{\alpha }}_{d}}\\ ={\displaystyle \int }_{0}^{1}{\rm{d}}s\,{\epsilon }_{\hat{\alpha }\hat{\beta }{\hat{\alpha }}_{1}\cdots {\hat{\alpha }}_{d}}{A}^{\hat{\alpha }\hat{\beta }}\wedge F{(s,1-s)}^{{\hat{\alpha }}_{1}{\hat{\alpha }}_{2}}\\ \quad \wedge {\rm{D}}(s,1-s){Y}^{{\hat{\alpha }}_{3}}\wedge \cdots \wedge {\rm{D}}(s,1-s){Y}^{{\hat{\alpha }}_{d}}\\ -{\displaystyle \int }_{0}^{1}{\rm{d}}\hat{s}\,{\epsilon }_{\hat{\alpha }\hat{\beta }{\hat{\alpha }}_{1}\cdots {\hat{\alpha }}_{d}}{\hat{A}}^{\hat{\alpha }\hat{\beta }}\wedge F{(1-\hat{s},\hat{s})}^{{\hat{\alpha }}_{1}{\hat{\alpha }}_{2}}\\ \quad \wedge {\rm{D}}(1-\hat{s},\hat{s}){Y}^{{\hat{\alpha }}_{3}}\wedge \cdots \wedge {\rm{D}}(1-\hat{s},\hat{s}){Y}^{{\hat{\alpha }}_{d}}\\ ={\displaystyle \int }_{0}^{1}{\rm{d}}s\,{\epsilon }_{\hat{\alpha }\hat{\beta }{\hat{\alpha }}_{1}\cdots {\hat{\alpha }}_{d}}{A}^{\hat{\alpha }\hat{\beta }}\wedge {F}^{{\hat{\alpha }}_{1}{\hat{\alpha }}_{2}}(s,0)\end{array}\end{eqnarray*}$
$\begin{eqnarray*}\begin{array}{l}\quad \wedge {\rm{D}}(s,0){Y}^{{\hat{\alpha }}_{3}}\wedge \cdots \wedge {\rm{D}}(s,0){Y}^{{\hat{\alpha }}_{d}}\\ +{\displaystyle \int }_{0}^{1}{\rm{d}}s{\displaystyle \int }_{0}^{1-s}{\rm{d}}\hat{s}\,{\partial }_{\hat{s}}\left[{\epsilon }_{\hat{\alpha }\hat{\beta }{\hat{\alpha }}_{1}\cdots {\hat{\alpha }}_{d}}{A}^{\hat{\alpha }\hat{\beta }}\right.\\ \quad \left.\wedge {F}^{{\hat{\alpha }}_{1}{\hat{\alpha }}_{2}}(s,\hat{s})\wedge {\rm{D}}(s,\hat{s}){Y}^{{\hat{\alpha }}_{3}}\wedge \cdots \wedge {\rm{D}}(s,\hat{s}){Y}^{{\hat{\alpha }}_{d}}\right]\\ -{\displaystyle \int }_{0}^{1}{\rm{d}}\hat{s}\,{\epsilon }_{\hat{\alpha }\hat{\beta }{\hat{\alpha }}_{1}\cdots {\hat{\alpha }}_{d}}{\hat{A}}^{\hat{\alpha }\hat{\beta }}\wedge {F}^{{\hat{\alpha }}_{1}{\hat{\alpha }}_{2}}(0,\hat{s})\\ \quad \wedge {\rm{D}}(0,\hat{s}){Y}^{{\hat{\alpha }}_{3}}\wedge \cdots \wedge {\rm{D}}(0,\hat{s}){Y}^{{\hat{\alpha }}_{d}}\\ -{\displaystyle \int }_{0}^{1}{\rm{d}}\hat{s}{\displaystyle \int }_{0}^{1-\hat{s}}{\rm{d}}s\,{\partial }_{s}\left[{\epsilon }_{\hat{\alpha }\hat{\beta }{\hat{\alpha }}_{1}\cdots {\hat{\alpha }}_{d}}{\hat{A}}^{\hat{\alpha }\hat{\beta }}\right.\end{array}\end{eqnarray*}$
$\begin{eqnarray*}\begin{array}{l}\quad \left.\wedge {F}^{{\hat{\alpha }}_{1}{\hat{\alpha }}_{2}}(s,\hat{s})\wedge {\rm{D}}(s,\hat{s}){Y}^{{\hat{\alpha }}_{3}}\wedge \cdots \wedge {\rm{D}}(s,\hat{s}){Y}^{{\hat{\alpha }}_{d}}\right]\\ ={\displaystyle \int }_{0}^{1}{\rm{d}}s\,{\epsilon }_{\hat{\alpha }\hat{\beta }{\hat{\alpha }}_{1}\cdots {\hat{\alpha }}_{d}}{A}^{\hat{\alpha }\hat{\beta }}\wedge {F}^{{\hat{\alpha }}_{1}{\hat{\alpha }}_{2}}(s)\\ \quad \wedge {\rm{D}}(s){Y}^{{\hat{\alpha }}_{3}}\wedge \cdots \wedge {\rm{D}}(s){Y}^{{\hat{\alpha }}_{d}}\\ -{\displaystyle \int }_{0}^{1}{\rm{d}}\hat{s}\,{\epsilon }_{\hat{\alpha }\hat{\beta }{\hat{\alpha }}_{1}\cdots {\hat{\alpha }}_{d}}{\hat{A}}^{\hat{\alpha }\hat{\beta }}\wedge {\hat{F}}^{{\hat{\alpha }}_{1}{\hat{\alpha }}_{2}}(\hat{s})\\ \quad \wedge \hat{{\rm{D}}}(\hat{s}){Y}^{{\hat{\alpha }}_{3}}\wedge \cdots \wedge \hat{{\rm{D}}}(\hat{s}){Y}^{{\hat{\alpha }}_{d}}\\ +{\displaystyle \int }_{0}^{1}{\rm{d}}s{\displaystyle \int }_{0}^{1-s}{\rm{d}}\hat{s}\,{\epsilon }_{\hat{\alpha }\hat{\beta }{\hat{\alpha }}_{1}\cdots {\hat{\alpha }}_{d}}\left\{{\partial }_{\hat{s}}\right.\end{array}\end{eqnarray*}$
$\begin{eqnarray*}\begin{array}{l}\quad \times \,\left[{A}^{\hat{\alpha }\hat{\beta }}\wedge {F}^{{\hat{\alpha }}_{1}{\hat{\alpha }}_{2}}(s,\hat{s})\right.\\ \quad \left.\wedge {\rm{D}}(s,\hat{s}){Y}^{{\hat{\alpha }}_{3}}\wedge \cdots \wedge {\rm{D}}(s,\hat{s}){Y}^{{\hat{\alpha }}_{d}}\right]\\ \quad -{\partial }_{s}\left[{\hat{A}}^{\hat{\alpha }\hat{\beta }}\wedge {F}^{{\hat{\alpha }}_{1}{\hat{\alpha }}_{2}}(s,\hat{s})\right.\\ \quad \left.\left.\wedge {\rm{D}}(s,\hat{s}){Y}^{{\hat{\alpha }}_{3}}\wedge \cdots \wedge {\rm{D}}(s,\hat{s}){Y}^{{\hat{\alpha }}_{d}}\right]\right\}\\ ={\displaystyle \int }_{0}^{1}{\rm{d}}s\,{\epsilon }_{\hat{\alpha }\hat{\beta }{\hat{\alpha }}_{1}\cdots {\hat{\alpha }}_{d}}{A}^{\hat{\alpha }\hat{\beta }}\wedge {F}^{{\hat{\alpha }}_{1}{\hat{\alpha }}_{2}}(s)\\ \quad \wedge {\rm{D}}(s){Y}^{{\hat{\alpha }}_{3}}\wedge \cdots \wedge {\rm{D}}(s){Y}^{{\hat{\alpha }}_{d}}\\ \quad -{\displaystyle \int }_{0}^{1}{\rm{d}}\hat{s}\,{\epsilon }_{\hat{\alpha }\hat{\beta }{\hat{\alpha }}_{1}\cdots {\hat{\alpha }}_{d}}{\hat{A}}^{\hat{\alpha }\hat{\beta }}\wedge {\hat{F}}^{{\hat{\alpha }}_{1}{\hat{\alpha }}_{2}}(\hat{s})\end{array}\,\end{eqnarray*}$
$\begin{eqnarray*}\begin{array}{l}\quad \wedge \hat{{\rm{D}}}(\hat{s}){Y}^{{\hat{\alpha }}_{3}}\wedge \cdots \wedge \hat{{\rm{D}}}(\hat{s}){Y}^{{\hat{\alpha }}_{d}}\\ \quad +{\displaystyle \int }_{0}^{1}{\rm{d}}s{\displaystyle \int }_{0}^{1-s}{\rm{d}}\hat{s}\,{\epsilon }_{\hat{\alpha }\hat{\beta }{\hat{\alpha }}_{1}\cdots {\hat{\alpha }}_{d}}\left\{{A}^{\hat{\alpha }\hat{\beta }}\right.\\ \quad \wedge {\rm{D}}(s,\hat{s}){\hat{A}}^{{\hat{\alpha }}_{1}{\hat{\alpha }}_{2}}\wedge {\rm{D}}(s,\hat{s}){Y}^{{\hat{\alpha }}_{3}}\wedge \cdots \wedge {\rm{D}}(s,\hat{s}){Y}^{{\hat{\alpha }}_{d}}\\ \quad +(d-2){A}^{\hat{\alpha }\hat{\beta }}\wedge {F}^{{\hat{\alpha }}_{1}{\hat{\alpha }}_{2}}(s,\hat{s})\wedge {(\hat{A}Y)}^{{\hat{\alpha }}_{3}}\\ \quad \wedge {\rm{D}}(s,\hat{s}){Y}^{{\hat{\alpha }}_{4}}\wedge \cdots \wedge {\rm{D}}(s,\hat{s}){Y}^{{\hat{\alpha }}_{d}}\\ \quad -{\hat{A}}^{\hat{\alpha }\hat{\beta }}\wedge {\rm{D}}(s,\hat{s}){A}^{{\hat{\alpha }}_{1}{\hat{\alpha }}_{2}}\end{array}\end{eqnarray*}$
$\begin{eqnarray*}\begin{array}{l}\quad \wedge {\rm{D}}(s,\hat{s}){Y}^{{\hat{\alpha }}_{3}}\wedge \cdots \wedge {\rm{D}}(s,\hat{s}){Y}^{{\hat{\alpha }}_{d}}\\ \quad -(d-2){\hat{A}}^{\hat{\alpha }\hat{\beta }}\wedge {F}^{{\hat{\alpha }}_{1}{\hat{\alpha }}_{2}}(s,\hat{s})\wedge {(AY)}^{{\hat{\alpha }}_{3}}\\ \quad \left.\wedge {\rm{D}}(s,\hat{s}){Y}^{{\hat{\alpha }}_{4}}\wedge \cdots \wedge {\rm{D}}(s,\hat{s}){Y}^{{\hat{\alpha }}_{d}}\right\}\\ ={\displaystyle \int }_{0}^{1}{\rm{d}}s\,{\epsilon }_{\hat{\alpha }\hat{\beta }{\hat{\alpha }}_{1}\cdots {\hat{\alpha }}_{d}}{A}^{\hat{\alpha }\hat{\beta }}\wedge {F}^{{\hat{\alpha }}_{1}{\hat{\alpha }}_{2}}(s)\\ \quad \wedge {\rm{D}}(s){Y}^{{\hat{\alpha }}_{3}}\wedge \cdots \wedge {\rm{D}}(s){Y}^{{\hat{\alpha }}_{d}}\\ \quad -{\displaystyle \int }_{0}^{1}{\rm{d}}\hat{s}\,{\epsilon }_{\hat{\alpha }\hat{\beta }{\hat{\alpha }}_{1}\cdots {\hat{\alpha }}_{d}}{\hat{A}}^{\hat{\alpha }\hat{\beta }}\wedge {\hat{F}}^{{\hat{\alpha }}_{1}{\hat{\alpha }}_{2}}(\hat{s})\end{array}\end{eqnarray*}$
$\begin{eqnarray*}\begin{array}{l}\quad \wedge \hat{{\rm{D}}}(\hat{s}){Y}^{{\hat{\alpha }}_{3}}\wedge \cdots \wedge \hat{{\rm{D}}}(\hat{s}){Y}^{{\hat{\alpha }}_{d}}\\ \quad +{\displaystyle \int }_{0}^{1}{\rm{d}}s{\displaystyle \int }_{0}^{1-s}{\rm{d}}\hat{s}\,{\epsilon }_{\hat{\alpha }\hat{\beta }{\hat{\alpha }}_{1}\cdots {\hat{\alpha }}_{d}}\\ \quad \times \,\left\{{\rm{D}}(s,\hat{s}){\hat{A}}^{\hat{\alpha }\hat{\beta }}\wedge {A}^{{\hat{\alpha }}_{1}{\hat{\alpha }}_{2}}\wedge {\rm{D}}(s,\hat{s}){Y}^{{\hat{\alpha }}_{3}}\right.\\ \quad \wedge \cdots \wedge {\rm{D}}(s,\hat{s}){Y}^{{\hat{\alpha }}_{d}}-{\hat{A}}^{\hat{\alpha }\hat{\beta }}\wedge {\rm{D}}(s,\hat{s}){A}^{{\hat{\alpha }}_{1}{\hat{\alpha }}_{2}}\\ \quad \wedge {\rm{D}}(s,\hat{s}){Y}^{{\hat{\alpha }}_{3}}\wedge \cdots \wedge {\rm{D}}(s,\hat{s}){Y}^{{\hat{\alpha }}_{d}}\\ \quad +(d-2){\hat{A}}^{\hat{\alpha }\hat{\beta }}\wedge {A}^{{\hat{\alpha }}_{1}{\hat{\alpha }}_{2}}\wedge {F}^{{\hat{\alpha }}_{3}\hat{\gamma }}(s,\hat{s}){Y}_{\hat{\gamma }}\end{array}\end{eqnarray*}$
$\begin{eqnarray*}\begin{array}{l}\quad \left.\wedge {\rm{D}}(s,\hat{s}){Y}^{{\hat{\alpha }}_{4}}\wedge \cdots \wedge {\rm{D}}(s,\hat{s}){Y}^{{\hat{\alpha }}_{d}}\right\}\\ ={\displaystyle \int }_{0}^{1}{\rm{d}}s\,{\epsilon }_{\hat{\alpha }\hat{\beta }{\hat{\alpha }}_{1}\cdots {\hat{\alpha }}_{d}}{A}^{\hat{\alpha }\hat{\beta }}\\ \quad \wedge {F}^{{\hat{\alpha }}_{1}{\hat{\alpha }}_{2}}(s)\wedge {\rm{D}}(s){Y}^{{\hat{\alpha }}_{3}}\wedge \cdots \wedge {\rm{D}}(s){Y}^{{\hat{\alpha }}_{d}}\\ \quad -{\displaystyle \int }_{0}^{1}{\rm{d}}\hat{s}\,{\epsilon }_{\hat{\alpha }\hat{\beta }{\hat{\alpha }}_{1}\cdots {\hat{\alpha }}_{d}}{\hat{A}}^{\hat{\alpha }\hat{\beta }}\\ \quad \wedge {\hat{F}}^{{\hat{\alpha }}_{1}{\hat{\alpha }}_{2}}(\hat{s})\wedge \hat{{\rm{D}}}(\hat{s}){Y}^{{\hat{\alpha }}_{3}}\wedge \cdots \wedge \hat{{\rm{D}}}(\hat{s}){Y}^{{\hat{\alpha }}_{d}}\\ \quad +{\displaystyle \int }_{0}^{1}{\rm{d}}s{\displaystyle \int }_{0}^{1-s}{\rm{d}}\hat{s}\,{\epsilon }_{\hat{\alpha }\hat{\beta }{\hat{\alpha }}_{1}\cdots {\hat{\alpha }}_{d}}\\ \quad \times \,{\rm{D}}(s,\hat{s})[{\hat{A}}^{\hat{\alpha }\hat{\beta }}\wedge {A}^{{\hat{\alpha }}_{1}{\hat{\alpha }}_{2}}\end{array}\end{eqnarray*}$
$\begin{eqnarray}\begin{array}{l}\quad \wedge {\rm{D}}(s,\hat{s}){Y}^{{\hat{\alpha }}_{3}}\wedge \cdots \wedge {\rm{D}}(s,\hat{s}){Y}^{{\hat{\alpha }}_{d}}]\\ =\frac{4{\kappa }^{2}(d-2)!}{{(-1)}^{d+1}{\ell }}{{ \mathcal L }}_{(d+1)}^{{\rm{CS}}}[A,Y]\\ \quad -{\displaystyle \int }_{0}^{1}{\rm{d}}\hat{s}\,{\epsilon }_{\hat{\alpha }\hat{\beta }{\hat{\alpha }}_{1}\cdots {\hat{\alpha }}_{d}}{\hat{A}}^{\hat{\alpha }\hat{\beta }}\wedge {\hat{F}}^{{\hat{\alpha }}_{1}{\hat{\alpha }}_{2}}(\hat{s})\\ \quad \wedge \hat{{\rm{D}}}(\hat{s}){Y}^{{\hat{\alpha }}_{3}}\wedge \cdots \wedge \hat{{\rm{D}}}(\hat{s}){Y}^{{\hat{\alpha }}_{d}}\\ \quad +{\displaystyle \int }_{0}^{1}{\rm{d}}s{\displaystyle \int }_{0}^{1-s}{\rm{d}}\hat{s}\,{\rm{d}}[{\epsilon }_{\hat{\alpha }\hat{\beta }{\hat{\alpha }}_{1}\cdots {\hat{\alpha }}_{d}}{\hat{A}}^{\hat{\alpha }\hat{\beta }}\wedge {A}^{{\hat{\alpha }}_{1}{\hat{\alpha }}_{2}}\\ \quad \wedge {\rm{D}}(s,\hat{s}){Y}^{{\hat{\alpha }}_{3}}\wedge \cdots \wedge {\rm{D}}(s,\hat{s}){Y}^{{\hat{\alpha }}_{d}}],\end{array}\end{eqnarray}$
where we have used the double homotopic notation $\begin{eqnarray}\begin{array}{rcl}A(s,\hat{s}) & = & sA+\hat{s}\hat{A},\\ {\rm{D}}(s,\hat{s})Y & = & {\rm{d}}Y+sAY+\hat{s}\hat{A}Y\\ & = & {\rm{D}}(s)Y+\hat{s}\hat{A}Y\\ & = & \hat{{\rm{D}}}(\hat{s})Y+sAY,\\ F(s,\hat{s}) & = & {\rm{d}}A(s,\hat{s})+A(s,\hat{s})\wedge A(s,\hat{s})\\ & = & s{\rm{d}}A+\hat{s}{\rm{d}}\hat{A}+(sA+\hat{s}\hat{A})\wedge (sA+\hat{s}\hat{A})\\ & = & F(s)+\hat{F}(\hat{s})+s\hat{s}(A\wedge \hat{A}+\hat{A}\wedge A),\\ {\rm{D}}(s,\hat{s})(tA+\hat{t}\hat{A}) & = & {\rm{d}}(tA+\hat{t}\hat{A})+(sA+\hat{s}\hat{A})\\ & & \wedge (tA+\hat{t}\hat{A})+(tA+\hat{t}\hat{A})\wedge (sA+\hat{s}\hat{A})\\ & = & (t{\partial }_{s}+\hat{t}{\partial }_{\hat{s}})F(s,\hat{s}),\\ A(t,\hat{t}) & = & tA+\hat{t}\hat{A}\\ & = & (t{\partial }_{s}+\hat{t}{\partial }_{\hat{s}})A(s,\hat{s}),\end{array}\end{eqnarray}$
as well as the resummation $\begin{eqnarray}\begin{array}{l}{\epsilon }_{\hat{\alpha }\hat{\beta }{\hat{\alpha }}_{1}\cdots {\hat{\alpha }}_{d}}{\hat{A}}^{\hat{\alpha }\hat{\beta }}\wedge {A}^{{\hat{\alpha }}_{1}{\hat{\alpha }}_{2}}\wedge {F}^{{\hat{\alpha }}_{3}\hat{\gamma }}(s,\hat{s}){Y}_{\hat{\gamma }}\\ \quad \wedge {\rm{D}}(s,\hat{s}){Y}^{{\hat{\alpha }}_{4}}\wedge \cdots \wedge {\rm{D}}(s,\hat{s}){Y}^{{\hat{\alpha }}_{d}}\\ ={\epsilon }_{\hat{\alpha }\hat{\beta }{\hat{\alpha }}_{1}\cdots {\hat{\alpha }}_{d}}{A}^{\hat{\alpha }\hat{\beta }}\wedge {F}^{{\hat{\alpha }}_{1}{\hat{\alpha }}_{2}}(s,\hat{s})\wedge {(\hat{A}Y)}^{{\hat{\alpha }}_{3}}\\ \quad \wedge {\rm{D}}(s,\hat{s}){Y}^{{\hat{\alpha }}_{4}}\wedge \cdots \wedge {\rm{D}}(s,\hat{s}){Y}^{{\hat{\alpha }}_{d}}\\ \quad -{\epsilon }_{\hat{\alpha }\hat{\beta }{\hat{\alpha }}_{1}\cdots {\hat{\alpha }}_{d}}{\hat{A}}^{\hat{\alpha }\hat{\beta }}\wedge {F}^{{\hat{\alpha }}_{1}{\hat{\alpha }}_{2}}(s,\hat{s})\wedge {(AY)}^{{\hat{\alpha }}_{1}}\\ \quad \wedge {\rm{D}}(s,\hat{s}){Y}^{{\hat{\alpha }}_{4}}\wedge \cdots \wedge {\rm{D}}(s,\hat{s}){Y}^{{\hat{\alpha }}_{d}}.\end{array}\end{eqnarray}$
C.3.3. SO(2, d) gauge transformation of ${{ \mathcal L }}_{(d+1)}^{{\rm{top}}}$
Under the finite gauge transformation
$\begin{eqnarray}{Y}^{\hat{\alpha }}\to {\tilde{Y}}^{\hat{\alpha }}={U}_{\,\,\hat{\beta }}^{\hat{\alpha }}{Y}^{\hat{\beta }},\end{eqnarray}$
the topological term transforms as $\begin{eqnarray*}\begin{array}{l}\frac{{\kappa }^{2}{{\ell }}^{3}(d+1)!}{d}{{ \mathcal L }}_{(d+1)}^{{\rm{top}}}[\tilde{Y}]\\ ={\epsilon }_{{\hat{\alpha }}_{0}\cdots {\hat{\alpha }}_{d+1}}{\rm{d}}{\tilde{Y}}^{{\hat{\alpha }}_{0}}\\ \quad \wedge \cdots \wedge {\rm{d}}{\tilde{Y}}^{{\hat{\alpha }}_{d}}{\tilde{Y}}^{{\hat{\alpha }}_{d+1}}\\ ={\epsilon }_{{\hat{\alpha }}_{0}\cdots {\hat{\alpha }}_{d+1}}{\rm{d}}({U}_{\,\,{\hat{\beta }}_{0}}^{{\hat{\alpha }}_{0}}{Y}^{{\hat{\beta }}_{0}})\\ \quad \wedge \cdots \wedge {\rm{d}}({U}_{\,\,{\hat{\beta }}_{d}}^{{\hat{\alpha }}_{d}}{Y}^{{\hat{\beta }}_{d}}){U}_{\,\,{\hat{\beta }}_{d+1}}^{{\hat{\alpha }}_{d+1}}{Y}^{{\hat{\beta }}_{d+1}}\\ ={\epsilon }_{{\hat{\alpha }}_{0}\cdots {\hat{\alpha }}_{d+1}}({\rm{d}}{U}_{\,\,{\hat{\beta }}_{0}}^{{\hat{\alpha }}_{0}}{Y}^{{\hat{\beta }}_{0}}+{U}_{\,\,{\hat{\beta }}_{0}}^{{\hat{\alpha }}_{0}}{\rm{d}}{Y}^{{\hat{\beta }}_{0}})\end{array}\,\end{eqnarray*}$
$\begin{eqnarray*}\begin{array}{l}\quad \wedge \cdots \wedge ({\rm{d}}{U}_{\,\,{\hat{\beta }}_{d}}^{{\hat{\alpha }}_{d}}{Y}^{{\hat{\beta }}_{d}}+{U}_{\,\,{\hat{\beta }}_{d}}^{{\hat{\alpha }}_{d}}{\rm{d}}{Y}^{{\hat{\beta }}_{d}})\\ \quad \times {U}_{\,\,{\hat{\beta }}_{d+1}}^{{\hat{\alpha }}_{d+1}}{Y}^{{\hat{\beta }}_{d+1}}\\ =\displaystyle \sum _{n=0}^{d+1}\frac{(d+1)!}{n!(d+1-n)!}{\epsilon }_{{\hat{\alpha }}_{0}\cdots {\hat{\alpha }}_{d+1}}{\rm{d}}{U}_{\,\,{\hat{\beta }}_{0}}^{{\hat{\alpha }}_{0}}{Y}^{{\hat{\beta }}_{0}}\\ \wedge \cdots \wedge {\rm{d}}{U}_{\,\,{\hat{\beta }}_{n-1}}^{{\hat{\alpha }}_{n-1}}{Y}^{{\hat{\beta }}_{n-1}}\wedge {U}_{\,\,{\hat{\beta }}_{n}}^{{\hat{\alpha }}_{n}}\end{array}\,\end{eqnarray*}$
$\begin{eqnarray*}\begin{array}{l}\quad \times \,{\rm{d}}{Y}^{{\hat{\beta }}_{n}}\wedge \cdots \wedge {U}_{\,\,{\hat{\beta }}_{d}}^{{\hat{\alpha }}_{d}}{\rm{d}}{Y}^{{\hat{\beta }}_{d}}{U}_{\,\,{\hat{\beta }}_{d+1}}^{{\hat{\alpha }}_{d+1}}{Y}^{{\hat{\beta }}_{d+1}}\\ =\displaystyle \sum _{n=0}^{d+1}\frac{(d+1)!}{n!(d+1-n)!}{\epsilon }_{{\hat{\beta }}_{0}\cdots {\hat{\beta }}_{d+1}}{({U}^{-1})}_{\,\,{\hat{\alpha }}_{0}}^{{\hat{\beta }}_{0}}\\ \quad \times \,{\rm{d}}{U}_{\,\,{\hat{\gamma }}_{0}}^{{\hat{\alpha }}_{0}}{Y}^{{\hat{\gamma }}_{0}}\wedge \cdots \wedge {({U}^{-1})}_{\,\,{\hat{\alpha }}_{n-1}}^{{\hat{\beta }}_{n-1}}\end{array}\,\end{eqnarray*}$
$\begin{eqnarray*}\begin{array}{l}\quad \times \,{\rm{d}}{U}_{\,\,{\hat{\gamma }}_{n-1}}^{{\hat{\alpha }}_{n-1}}{Y}^{{\hat{\gamma }}_{n-1}}\\ \quad \wedge {\rm{d}}{Y}^{{\hat{\beta }}_{n}}\wedge \cdots \wedge {\rm{d}}{Y}^{{\hat{\beta }}_{d}}{Y}^{{\hat{\beta }}_{d+1}}\\ =\displaystyle \sum _{n=0}^{d+1}\frac{(d+1)!}{n!(d+1-n)!}{\epsilon }_{{\hat{\alpha }}_{0}\cdots {\hat{\alpha }}_{d+1}}{({U}^{-1}{\rm{d}}UY)}^{{\hat{\alpha }}_{0}}\\ \quad \wedge \cdots \wedge {({U}^{-1}{\rm{d}}UY)}^{{\hat{\alpha }}_{n-1}}\wedge {\rm{d}}{Y}^{{\hat{\alpha }}_{n}}\\ \quad \wedge \cdots \wedge {\rm{d}}{Y}^{{\hat{\alpha }}_{d}}{Y}^{{\hat{\alpha }}_{d+1}}\\ ={\epsilon }_{{\hat{\alpha }}_{0}\cdots {\hat{\alpha }}_{d+1}}{({\rm{d}}Y+{U}^{-1}{\rm{d}}UY)}^{{\hat{\alpha }}_{0}}\end{array}\,\end{eqnarray*}$
$\begin{eqnarray*}\begin{array}{l}\quad \wedge \cdots \wedge {({\rm{d}}Y+{U}^{-1}{\rm{d}}UY)}^{{\hat{\alpha }}_{d}}{Y}^{{\hat{\alpha }}_{d+1}}\\ ={\epsilon }_{{\hat{\alpha }}_{0}\cdots {\hat{\alpha }}_{d+1}}\hat{{\rm{D}}}{Y}^{{\hat{\alpha }}_{0}}\wedge \cdots \wedge \hat{{\rm{D}}}{Y}^{{\hat{\alpha }}_{d}}{Y}^{{\hat{\alpha }}_{d+1}}\\ ={\epsilon }_{{\hat{\alpha }}_{0}\cdots {\hat{\alpha }}_{d+1}}{\rm{d}}{Y}^{{\hat{\alpha }}_{0}}\wedge \cdots \wedge {\rm{d}}{Y}^{{\hat{\alpha }}_{d}}{Y}^{{\hat{\alpha }}_{d+1}}\\ \quad +{\displaystyle \int }_{0}^{1}{\rm{d}}s\,{\partial }_{s}[{\epsilon }_{{\hat{\alpha }}_{0}\cdots {\hat{\alpha }}_{d+1}}\hat{{\rm{D}}}(s){Y}^{{\hat{\alpha }}_{0}}\wedge \cdots \wedge \hat{{\rm{D}}}(s){Y}^{{\hat{\alpha }}_{d}}{Y}^{{\hat{\alpha }}_{d+1}}]\\ ={\epsilon }_{{\hat{\alpha }}_{0}\cdots {\hat{\alpha }}_{d+1}}{\rm{d}}{Y}^{{\hat{\alpha }}_{0}}\wedge \cdots \wedge {\rm{d}}{Y}^{{\hat{\alpha }}_{d}}{Y}^{{\hat{\alpha }}_{d+1}}\end{array}\end{eqnarray*}$
$\begin{eqnarray*}\begin{array}{l}\quad +(d+1){\displaystyle \int }_{0}^{1}{\rm{d}}s\,{\epsilon }_{{\hat{\alpha }}_{0}\cdots {\hat{\alpha }}_{d+1}}{(\hat{A}Y)}^{{\hat{\alpha }}_{0}}\wedge \hat{{\rm{D}}}(s){Y}^{{\hat{\alpha }}_{1}}\\ \quad \wedge \cdots \wedge \hat{{\rm{D}}}(s){Y}^{{\hat{\alpha }}_{d}}{Y}^{{\hat{\alpha }}_{d+1}}\\ ={\epsilon }_{{\hat{\alpha }}_{0}\cdots {\hat{\alpha }}_{d+1}}{\rm{d}}{Y}^{{\hat{\alpha }}_{0}}\wedge \cdots \wedge {\rm{d}}{Y}^{{\hat{\alpha }}_{d}}{Y}^{{\hat{\alpha }}_{d+1}}\\ \quad -\frac{{(-1)}^{d}(d+1){{\ell }}^{2}}{2}{\displaystyle \int }_{0}^{1}{\rm{d}}s\,{\epsilon }_{{\hat{\alpha }}_{0}\cdots {\hat{\alpha }}_{d+1}}{\hat{A}}^{{\hat{\alpha }}_{0}{\hat{\alpha }}_{1}}\end{array}\end{eqnarray*}$
$\begin{eqnarray*}\begin{array}{l}\quad \wedge \hat{{\rm{D}}}(s){Y}^{{\hat{\alpha }}_{2}}\wedge \cdots \wedge \hat{{\rm{D}}}(s){Y}^{{\hat{\alpha }}_{d+1}}\\ ={\epsilon }_{{\hat{\alpha }}_{0}\cdots {\hat{\alpha }}_{d+1}}{\rm{d}}{Y}^{{\hat{\alpha }}_{0}}\wedge \cdots \wedge {\rm{d}}{Y}^{{\hat{\alpha }}_{d}}{Y}^{{\hat{\alpha }}_{d+1}}\\ \quad -\frac{(d+1){{\ell }}^{2}}{2}{\rm{d}}\left[{\displaystyle \int }_{0}^{1}{\rm{d}}s\,{\epsilon }_{{\hat{\alpha }}_{0}\cdots {\hat{\alpha }}_{d+1}}{\hat{A}}^{{\hat{\alpha }}_{0}{\hat{\alpha }}_{1}}\right.\\ \quad \left.\wedge \hat{{\rm{D}}}(s){Y}^{{\hat{\alpha }}_{2}}\wedge \cdots \wedge \hat{{\rm{D}}}(s){Y}^{{\hat{\alpha }}_{d}}{Y}^{{\hat{\alpha }}_{d+1}}\right]\end{array}\end{eqnarray*}$
$\begin{eqnarray*}\begin{array}{l}\quad +\frac{(d+1){{\ell }}^{2}}{2}{\displaystyle \int }_{0}^{1}{\rm{d}}s\,{\epsilon }_{{\hat{\alpha }}_{0}\cdots {\hat{\alpha }}_{d+1}}{\partial }_{s}\hat{F}{(s)}^{{\hat{\alpha }}_{0}{\hat{\alpha }}_{1}}\\ \quad \wedge \hat{{\rm{D}}}(s){Y}^{{\hat{\alpha }}_{2}}\wedge \cdots \wedge \hat{{\rm{D}}}(s){Y}^{{\hat{\alpha }}_{d}}{Y}^{{\hat{\alpha }}_{d+1}}\\ \quad -\frac{({d}^{2}-1){{\ell }}^{2}}{2}{\displaystyle \int }_{0}^{1}{\rm{d}}s\,{\epsilon }_{{\hat{\alpha }}_{0}\cdots {\hat{\alpha }}_{d+1}}{\hat{A}}^{{\hat{\alpha }}_{0}{\hat{\alpha }}_{1}}\\ \quad \wedge {[\hat{F}(s)Y]}^{{\hat{\alpha }}_{2}}\wedge \hat{{\rm{D}}}(s){Y}^{{\hat{\alpha }}_{3}}\wedge \cdots \wedge \hat{{\rm{D}}}(s){Y}^{{\hat{\alpha }}_{d}}{Y}^{{\hat{\alpha }}_{d+1}}\\ ={\epsilon }_{{\hat{\alpha }}_{0}\cdots {\hat{\alpha }}_{d+1}}{\rm{d}}{Y}^{{\hat{\alpha }}_{0}}\wedge \cdots \wedge {\rm{d}}{Y}^{{\hat{\alpha }}_{d}}{Y}^{{\hat{\alpha }}_{d+1}}\end{array}\end{eqnarray*}$
$\begin{eqnarray*}\begin{array}{l}\quad -\frac{(d+1){{\ell }}^{2}}{2}{\rm{d}}\left[{\displaystyle \int }_{0}^{1}{\rm{d}}s\,{\epsilon }_{{\hat{\alpha }}_{0}\cdots {\hat{\alpha }}_{d+1}}{\hat{A}}^{{\hat{\alpha }}_{0}{\hat{\alpha }}_{1}}\right.\\ \quad \left.\wedge \hat{{\rm{D}}}(s){Y}^{{\hat{\alpha }}_{2}}\wedge \cdots \wedge \hat{{\rm{D}}}(s){Y}^{{\hat{\alpha }}_{d}}{Y}^{{\hat{\alpha }}_{d+1}}\right]\\ \quad +\frac{(d+1){{\ell }}^{2}}{2}{\displaystyle \int }_{0}^{1}{\rm{d}}s\,{\epsilon }_{{\hat{\alpha }}_{0}\cdots {\hat{\alpha }}_{d+1}}{\partial }_{s}\\ \quad \times \,\left[\hat{F}{(s)}^{{\hat{\alpha }}_{0}{\hat{\alpha }}_{1}}\wedge \hat{{\rm{D}}}(s){Y}^{{\hat{\alpha }}_{2}}\wedge \cdots \wedge \hat{{\rm{D}}}(s){Y}^{{\hat{\alpha }}_{d}}{Y}^{{\hat{\alpha }}_{d+1}}\right]\\ \quad -\frac{({d}^{2}-1){{\ell }}^{2}}{2}{\displaystyle \int }_{0}^{1}{\rm{d}}s\,{\epsilon }_{{\hat{\alpha }}_{0}\cdots {\hat{\alpha }}_{d+1}}\hat{F}{(s)}^{{\hat{\alpha }}_{0}{\hat{\alpha }}_{1}}\\ \quad \wedge {(\hat{A}Y)}^{{\hat{\alpha }}_{2}}\wedge \hat{{\rm{D}}}(s){Y}^{{\hat{\alpha }}_{3}}\wedge \cdots \wedge \hat{{\rm{D}}}(s){Y}^{{\hat{\alpha }}_{d}}{Y}^{{\hat{\alpha }}_{d+1}}\end{array}\end{eqnarray*}$
$\begin{eqnarray*}\begin{array}{l}\quad -\frac{({d}^{2}-1){{\ell }}^{2}}{2}{\displaystyle \int }_{0}^{1}{\rm{d}}s\,{\epsilon }_{{\hat{\alpha }}_{0}\cdots {\hat{\alpha }}_{d+1}}{\hat{A}}^{{\hat{\alpha }}_{0}{\hat{\alpha }}_{1}}\\ \quad \wedge {[\hat{F}(s)Y]}^{{\hat{\alpha }}_{2}}\wedge \hat{{\rm{D}}}(s){Y}^{{\hat{\alpha }}_{3}}\wedge \cdots \wedge \hat{{\rm{D}}}(s){Y}^{{\hat{\alpha }}_{d}}{Y}^{{\hat{\alpha }}_{d+1}}\\ ={\epsilon }_{{\hat{\alpha }}_{0}\cdots {\hat{\alpha }}_{d+1}}{\rm{d}}{Y}^{{\hat{\alpha }}_{0}}\wedge \cdots \wedge {\rm{d}}{Y}^{{\hat{\alpha }}_{d}}{Y}^{{\hat{\alpha }}_{d+1}}\end{array}\end{eqnarray*}$
$\begin{eqnarray}\begin{array}{l}\quad -\frac{(d+1){{\ell }}^{2}}{2}{\rm{d}}\left[{\displaystyle \int }_{0}^{1}{\rm{d}}s\,{\epsilon }_{{\hat{\alpha }}_{0}\cdots {\hat{\alpha }}_{d+1}}{\hat{A}}^{{\hat{\alpha }}_{0}{\hat{\alpha }}_{1}}\right.\\ \quad \left.\wedge \hat{{\rm{D}}}(s){Y}^{{\hat{\alpha }}_{2}}\wedge \cdots \wedge \hat{{\rm{D}}}(s){Y}^{{\hat{\alpha }}_{d}}{Y}^{{\hat{\alpha }}_{d+1}}\right]\\ \quad +\frac{{(-1)}^{d}({d}^{2}-1){{\ell }}^{4}}{4}{\displaystyle \int }_{0}^{1}{\rm{d}}s\,{\epsilon }_{{\hat{\alpha }}_{0}\cdots {\hat{\alpha }}_{d+1}}{\hat{A}}^{{\hat{\alpha }}_{0}{\hat{\alpha }}_{1}}\\ \quad \wedge {\hat{F}}^{{\hat{\alpha }}_{2}{\hat{\alpha }}_{3}}(s)\wedge \hat{{\rm{D}}}(s)\\ \quad \times \,{Y}^{{\hat{\alpha }}_{4}}\wedge \cdots \wedge \hat{{\rm{D}}}(s){Y}^{{\hat{\alpha }}_{d+1}},\end{array}\end{eqnarray}$
where we have used that $\begin{eqnarray}\begin{array}{rcl}\hat{F(s)} & = & {\rm{d}}\hat{A(s)}+\hat{A(s)}\wedge \hat{A(s)}\\ & = & {\rm{d}}(s{U}^{-1}{\rm{d}}U)+(s{U}^{-1}{\rm{d}}U)\wedge (s{U}^{-1}{\rm{d}}U)\\ & = & -s{U}^{-1}{\rm{d}}U{U}^{-1}\wedge {\rm{d}}U+{s}^{2}{U}^{-1}{\rm{d}}U\wedge {U}^{-1}{\rm{d}}U\\ & = & s(s-1){U}^{-1}{\rm{d}}U\wedge {U}^{-1}{\rm{d}}U\\ & = & s(1-s){\rm{d}}({U}^{-1}{\rm{d}}U),\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{l}{\epsilon }_{{\hat{\beta }}_{0}\cdots {\hat{\beta }}_{d+1}}{({U}^{-1})}_{\,\,{\hat{\alpha }}_{0}}^{{\hat{\beta }}_{0}}\cdots {({U}^{-1})}_{\,\,{\hat{\alpha }}_{n}}^{{\hat{\beta }}_{n}}\\ ={\epsilon }_{{\hat{\alpha }}_{0}\cdots {\hat{\alpha }}_{d+1}}{U}_{\,\,{\hat{\beta }}_{n+1}}^{{\hat{\alpha }}_{n+1}}\cdots {U}_{\,\,{\hat{\beta }}_{d+1}}^{{\hat{\alpha }}_{d+1}},\end{array}\end{eqnarray}$
as well as the resummations $\begin{eqnarray}\begin{array}{l}{\epsilon }_{{\hat{\alpha }}_{0}\cdots {\hat{\alpha }}_{d+1}}{\hat{A}}^{{\hat{\alpha }}_{0}\hat{\beta }}{Y}_{\hat{\beta }}\wedge \hat{{\rm{D}}}(s){Y}^{{\hat{\alpha }}_{1}}\\ \quad \wedge \cdots \wedge \hat{{\rm{D}}}(s){Y}^{{\hat{\alpha }}_{d}}{Y}^{{\hat{\alpha }}_{d+1}}\\ =-\frac{{(-1)}^{d}{{\ell }}^{2}}{2}{\epsilon }_{{\hat{\alpha }}_{0}\cdots {\hat{\alpha }}_{d+1}}{\hat{A}}^{{\hat{\alpha }}_{0}{\hat{\alpha }}_{1}}\\ \quad \wedge \hat{{\rm{D}}}(s){Y}^{{\hat{\alpha }}_{2}}\wedge \cdots \wedge \hat{{\rm{D}}}(s){Y}^{{\hat{\alpha }}_{d+1}},\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{l}-{{\ell }}^{2}{\epsilon }_{{\hat{\alpha }}_{0}\cdots {\hat{\alpha }}_{d+1}}{\hat{A}}^{{\hat{\alpha }}_{0}{\hat{\alpha }}_{1}}\wedge {\hat{F}}^{{\hat{\alpha }}_{2}{\hat{\alpha }}_{3}}(s)\\ \quad \wedge \hat{{\rm{D}}}(s){Y}^{{\hat{\alpha }}_{4}}\wedge \cdots \wedge \hat{{\rm{D}}}(s){Y}^{{\hat{\alpha }}_{d+1}}\\ =2{(-1)}^{d}{\epsilon }_{{\hat{\alpha }}_{0}\cdots {\hat{\alpha }}_{d+1}}{\hat{F}}^{{\hat{\alpha }}_{0}{\hat{\alpha }}_{1}}(s)\wedge {(\hat{A}Y)}^{{\hat{\alpha }}_{2}}\\ \quad \wedge \hat{{\rm{D}}}(s){Y}^{{\hat{\alpha }}_{3}}\wedge \cdots \wedge \hat{{\rm{D}}}(s){Y}^{{\hat{\alpha }}_{d}}{Y}^{{\hat{\alpha }}_{d+1}}\\ +2{(-1)}^{d}{\epsilon }_{{\hat{\alpha }}_{0}\cdots {\hat{\alpha }}_{d+1}}{\hat{A}}^{{\hat{\alpha }}_{0}{\hat{\alpha }}_{1}}\wedge {[\hat{F}(s)Y]}^{{\hat{\alpha }}_{2}}\\ \quad \wedge \hat{{\rm{D}}}(s){Y}^{{\hat{\alpha }}_{3}}\wedge \cdots \wedge \hat{{\rm{D}}}(s){Y}^{{\hat{\alpha }}_{d}}{Y}^{{\hat{\alpha }}_{d+1}}.\end{array}\end{eqnarray}$
Together with ${{ \mathcal L }}_{(d+1)}^{{\rm{top}}}$, the bulk consistent action
$\begin{eqnarray}\begin{array}{l}2{\kappa }^{2}{\ell }(d-1)!{S}_{(d+1)}^{{\rm{con}}}\\ =2{\kappa }^{2}{\ell }(d-1)!({S}_{(d+1)}^{{\rm{CS}}}-{S}_{(d+1)}^{{\rm{top}}})\\ =\frac{{(-1)}^{d+1}(d-1){{\ell }}^{2}}{2}{\displaystyle \int }_{M}{\epsilon }_{{\hat{\alpha }}_{0}{\hat{\alpha }}_{1}\cdots {\hat{\alpha }}_{d+1}}\\ \quad \times \,\left[{\displaystyle \int }_{0}^{1}{\rm{d}}s\,{A}^{{\hat{\alpha }}_{0}{\hat{\alpha }}_{1}}\wedge {F}^{{\hat{\alpha }}_{2}{\hat{\alpha }}_{3}}(s)\right.\\ \quad \wedge {\rm{D}}(s){Y}^{{\hat{\alpha }}_{4}}\wedge \cdots \wedge {\rm{D}}(s){Y}^{{\hat{\alpha }}_{d+1}}\\ \left.+\frac{4{(-1)}^{d}}{({d}^{2}-1){{\ell }}^{4}}{\rm{d}}{Y}^{{\hat{\alpha }}_{0}}\wedge \cdots \wedge {\rm{d}}{Y}^{{\hat{\alpha }}_{d}}{Y}^{{\hat{\alpha }}_{d+1}}\right],\end{array}\end{eqnarray}$
transforms as $\begin{eqnarray}\begin{array}{l}\frac{4{\kappa }^{2}(d-2)!}{{(-1)}^{d+1}{\ell }}{S}_{(d+1)}^{{\rm{con}}}[\tilde{A},\tilde{Y}]\\ -\frac{4{\kappa }^{2}(d-2)!}{{(-1)}^{d+1}{\ell }}{S}_{(d+1)}^{{\rm{con}}}[A,Y]\\ ={\displaystyle \int }_{{\rm{\Sigma }}}{\displaystyle \int }_{0}^{1}{\rm{d}}s\,{\epsilon }_{\hat{\alpha }\hat{\beta }{\hat{\alpha }}_{1}\cdots {\hat{\alpha }}_{d}}{\hat{A}}^{\hat{\alpha }\hat{\beta }}\\ \quad \wedge \left[{\displaystyle \int }_{0}^{1-s}{\rm{d}}\hat{s}\,{A}^{{\hat{\alpha }}_{1}{\hat{\alpha }}_{2}}\wedge {\rm{D}}(s,\hat{s}){Y}^{{\hat{\alpha }}_{3}}\right.\\ \quad \wedge \cdots \wedge {\rm{D}}(s,\hat{s}){Y}^{{\hat{\alpha }}_{d}}\\ \quad +\frac{2}{(d-1){{\ell }}^{2}}{Y}^{{\hat{\alpha }}_{1}}\hat{{\rm{D}}}(s){Y}^{{\hat{\alpha }}_{2}}\\ \quad \left.\wedge \cdots \wedge \hat{{\rm{D}}}(s){Y}^{{\hat{\alpha }}_{d}}\right].\end{array}\end{eqnarray}$
The large gauge transforms in ${S}_{(d+1)}^{{\rm{CS}}}$ and ${S}_{(d+1)}^{{\rm{top}}}$ cancel each other, and thus ${S}_{(d+1)}^{{\rm{con}}}$ is gauge invariant up to boundary terms.C.3.4. The variation of ${S}_{(d+1)}^{{\rm{con}}}$
The variation of ${S}_{(d+1)}^{{\rm{CS}}}$ is given by
$\begin{eqnarray*}\begin{array}{l}\delta {\displaystyle \int }_{M}{\displaystyle \int }_{0}^{1}{\rm{d}}s\,{\epsilon }_{\hat{\alpha }\hat{\beta }{\hat{\alpha }}_{1}\cdots {\hat{\alpha }}_{d}}{A}^{\hat{\alpha }\hat{\beta }}\\ \quad \wedge {F}^{{\hat{\alpha }}_{1}{\hat{\alpha }}_{2}}(s)\wedge {\rm{D}}(s){Y}^{{\hat{\alpha }}_{3}}\wedge \cdots \wedge {\rm{D}}(s){Y}^{{\hat{\alpha }}_{d}}\\ ={\displaystyle \int }_{M}{\displaystyle \int }_{0}^{1}{\rm{d}}s\,{\epsilon }_{\hat{\alpha }\hat{\beta }{\hat{\alpha }}_{1}\cdots {\hat{\alpha }}_{d}}\left\{\delta {A}^{\hat{\alpha }\hat{\beta }}\right.\\ \quad \wedge {F}^{{\hat{\alpha }}_{1}{\hat{\alpha }}_{2}}(s)\wedge {\rm{D}}(s){Y}^{{\hat{\alpha }}_{3}}\wedge \cdots \wedge {\rm{D}}(s){Y}^{{\hat{\alpha }}_{d}}\\ \quad +\delta {A}^{\hat{\alpha }\hat{\beta }}\wedge {\rm{D}}(s)[{A}^{{\hat{\alpha }}_{1}{\hat{\alpha }}_{2}}(s)\wedge {\rm{D}}(s){Y}^{{\hat{\alpha }}_{3}}\wedge \cdots \wedge {\rm{D}}(s){Y}^{{\hat{\alpha }}_{d}}]\end{array}\end{eqnarray*}$
$\begin{eqnarray*}\begin{array}{l}\quad -(d-2)\delta {A}_{\,\,\hat{\gamma }}^{\hat{\alpha }}{Y}^{\hat{\gamma }}\wedge {A}^{\hat{\beta }{\hat{\alpha }}_{3}}(s)\wedge {F}^{{\hat{\alpha }}_{1}{\hat{\alpha }}_{2}}(s)\\ \quad \wedge {\rm{D}}(s){Y}^{{\hat{\alpha }}_{4}}\wedge \cdots \wedge {\rm{D}}(s){Y}^{{\hat{\alpha }}_{d}}\\ \quad +(d-2)\delta {Y}^{\hat{\alpha }}{\rm{D}}(s)[{A}^{\hat{\beta }{\hat{\alpha }}_{1}}\wedge {F}^{{\hat{\alpha }}_{2}{\hat{\alpha }}_{3}}(s)\\ \quad \left.\wedge {\rm{D}}(s){Y}^{{\hat{\alpha }}_{4}}\wedge \cdots \wedge {\rm{D}}(s){Y}^{{\hat{\alpha }}_{d}}]\right\}\\ \quad +{\displaystyle \int }_{{\rm{\Sigma }}}{\displaystyle \int }_{0}^{1}{\rm{d}}s\,{\epsilon }_{\hat{\alpha }\hat{\beta }{\hat{\alpha }}_{1}\cdots {\hat{\alpha }}_{d}}\left\{\delta {A}^{\hat{\alpha }\hat{\beta }}\right.\\ \quad \wedge {A}^{{\hat{\alpha }}_{1}{\hat{\alpha }}_{2}}(s)\wedge {\rm{D}}(s){Y}^{{\hat{\alpha }}_{3}}\wedge \cdots \wedge {\rm{D}}(s){Y}^{{\hat{\alpha }}_{d}}\\ \quad -(d-2)\delta {Y}^{\hat{\alpha }}{A}^{\hat{\beta }{\hat{\alpha }}_{1}}\wedge {F}^{{\hat{\alpha }}_{2}{\hat{\alpha }}_{3}}(s)\wedge {\rm{D}}(s){Y}^{{\hat{\alpha }}_{4}}\end{array}\,\end{eqnarray*}$
$\begin{eqnarray*}\begin{array}{l}\quad \left.\wedge \cdots \wedge {\rm{D}}(s){Y}^{{\hat{\alpha }}_{d}}\right\}\\ ={\displaystyle \int }_{M}{\displaystyle \int }_{0}^{1}{\rm{d}}s\,{\epsilon }_{\hat{\alpha }\hat{\beta }{\hat{\alpha }}_{1}\cdots {\hat{\alpha }}_{d}}\left\{\delta {A}^{\hat{\alpha }\hat{\beta }}\right.\\ \quad \wedge {F}^{{\hat{\alpha }}_{1}{\hat{\alpha }}_{2}}(s)\wedge {\rm{D}}(s){Y}^{{\hat{\alpha }}_{3}}\wedge \cdots \wedge {\rm{D}}(s){Y}^{{\hat{\alpha }}_{d}}\\ \quad +\delta {A}^{\hat{\alpha }\hat{\beta }}\wedge s{\partial }_{s}{F}^{{\hat{\alpha }}_{1}{\hat{\alpha }}_{2}}(s)\wedge {\rm{D}}(s){Y}^{{\hat{\alpha }}_{3}}\\ \wedge \cdots \wedge {\rm{D}}(s){Y}^{{\hat{\alpha }}_{d}}\\ \quad -(d-2)\delta {A}^{\hat{\alpha }\hat{\beta }}\wedge {A}^{{\hat{\alpha }}_{1}{\hat{\alpha }}_{2}}(s)\wedge {[F(s)Y]}^{{\hat{\alpha }}_{3}}\\ \quad \wedge {\rm{D}}(s){Y}^{{\hat{\alpha }}_{4}}\wedge \cdots \wedge {\rm{D}}(s){Y}^{{\hat{\alpha }}_{d}}\end{array}\,\end{eqnarray*}$
$\begin{eqnarray*}\begin{array}{l}\quad -(d-2)\delta {A}_{\,\,\hat{\gamma }}^{\hat{\alpha }}{Y}^{\hat{\gamma }}\wedge {A}^{\hat{\beta }{\hat{\alpha }}_{3}}(s)\\ \quad \wedge {F}^{{\hat{\alpha }}_{1}{\hat{\alpha }}_{2}}(s)\wedge {\rm{D}}(s){Y}^{{\hat{\alpha }}_{4}}\wedge \cdots \wedge {\rm{D}}(s){Y}^{{\hat{\alpha }}_{d}}\\ \,+(d-2)\delta {Y}^{\hat{\alpha }}{\partial }_{s}{F}^{\hat{\beta }{\hat{\alpha }}_{1}}(s)\wedge {F}^{{\hat{\alpha }}_{2}{\hat{\alpha }}_{3}}(s)\wedge {\rm{D}}(s){Y}^{{\hat{\alpha }}_{4}}\\ \quad \wedge \cdots \wedge {\rm{D}}(s){Y}^{{\hat{\alpha }}_{d}}\\ \quad -(d-2)(d-3)\delta {Y}^{\hat{\alpha }}{A}^{\hat{\beta }{\hat{\alpha }}_{1}}\wedge {F}^{{\hat{\alpha }}_{2}{\hat{\alpha }}_{3}}(s)\wedge {[F(s)Y]}^{{\hat{\alpha }}_{4}}\\ \quad \left.\wedge {\rm{D}}(s){Y}^{{\hat{\alpha }}_{5}}\wedge \cdots \wedge {\rm{D}}(s){Y}^{{\hat{\alpha }}_{d}}\right\}\\ \quad +{\displaystyle \int }_{{\rm{\Sigma }}}{\displaystyle \int }_{0}^{1}{\rm{d}}s\,{\epsilon }_{\hat{\alpha }\hat{\beta }{\hat{\alpha }}_{1}\cdots {\hat{\alpha }}_{d}}\left\{\delta {A}^{\hat{\alpha }\hat{\beta }}\right.\end{array}\end{eqnarray*}$
$\begin{eqnarray*}\begin{array}{l}\quad \wedge {A}^{{\hat{\alpha }}_{1}{\hat{\alpha }}_{2}}(s)\wedge {\rm{D}}(s){Y}^{{\hat{\alpha }}_{3}}\wedge \cdots \wedge {\rm{D}}(s){Y}^{{\hat{\alpha }}_{d}}\\ \quad -\frac{2(d-2)}{{{\ell }}^{2}}\delta {Y}^{\hat{\alpha }}{Y}^{\hat{\beta }}{F}^{{\hat{\alpha }}_{1}{\hat{\alpha }}_{2}}(s)\wedge {(AY)}^{{\hat{\alpha }}_{3}}\\ \quad \wedge {\rm{D}}(s){Y}^{{\hat{\alpha }}_{4}}\wedge \cdots \wedge {\rm{D}}(s){Y}^{{\hat{\alpha }}_{d}}\\ \quad -\frac{2(d-2)}{{{\ell }}^{2}}\delta {Y}^{\hat{\alpha }}{Y}^{\hat{\beta }}{A}^{{\hat{\alpha }}_{1}{\hat{\alpha }}_{2}}\wedge {[F(s)Y]}^{{\hat{\alpha }}_{3}}\\ \quad \left.\wedge {\rm{D}}(s){Y}^{{\hat{\alpha }}_{4}}\wedge \cdots \wedge {\rm{D}}(s){Y}^{{\hat{\alpha }}_{d}}\right\}\\ \quad ={\displaystyle \int }_{M}{\displaystyle \int }_{0}^{1}{\rm{d}}s\,{\epsilon }_{\hat{\alpha }\hat{\beta }{\hat{\alpha }}_{1}\cdots {\hat{\alpha }}_{d}}\left\{\delta {A}^{\hat{\alpha }\hat{\beta }}\right.\end{array}\,\end{eqnarray*}$
$\begin{eqnarray*}\begin{array}{l}\quad \wedge {F}^{{\hat{\alpha }}_{1}{\hat{\alpha }}_{2}}(s)\wedge {\rm{D}}(s){Y}^{{\hat{\alpha }}_{3}}\wedge \cdots \wedge {\rm{D}}(s){Y}^{{\hat{\alpha }}_{d}}\\ \quad +\delta {A}^{\hat{\alpha }\hat{\beta }}\wedge s{\partial }_{s}{F}^{{\hat{\alpha }}_{1}{\hat{\alpha }}_{2}}(s)\\ \quad \wedge {\rm{D}}(s){Y}^{{\hat{\alpha }}_{3}}\wedge \cdots \wedge {\rm{D}}(s){Y}^{{\hat{\alpha }}_{d}}\\ \quad +(d-2)\delta {A}^{\hat{\alpha }\hat{\beta }}\wedge {F}^{{\hat{\alpha }}_{1}{\hat{\alpha }}_{2}}(s)\wedge s{\partial }_{s}[{\rm{D}}(s){Y}^{{\hat{\alpha }}_{3}}]\\ \quad \wedge {\rm{D}}(s){Y}^{{\hat{\alpha }}_{4}}\wedge \cdots \wedge {\rm{D}}(s){Y}^{{\hat{\alpha }}_{d}}\\ \quad +(d-2)\delta {Y}^{\hat{\alpha }}{\partial }_{s}{F}^{\hat{\beta }{\hat{\alpha }}_{1}}(s)\wedge {F}^{{\hat{\alpha }}_{2}{\hat{\alpha }}_{3}}(s)\\ \quad \wedge {\rm{D}}(s){Y}^{{\hat{\alpha }}_{4}}\wedge \cdots \wedge {\rm{D}}(s){Y}^{{\hat{\alpha }}_{d}}\end{array}\,\end{eqnarray*}$
$\begin{eqnarray*}\begin{array}{l}\quad +\frac{(d-2)(d-3)}{2}\delta {Y}^{\hat{\alpha }}{F}^{\hat{\beta }{\hat{\alpha }}_{1}}(s)\wedge {F}^{{\hat{\alpha }}_{2}{\hat{\alpha }}_{3}}(s)\\ \quad \left.\wedge {\partial }_{s}[{\rm{D}}(s){Y}^{{\hat{\alpha }}_{4}}]\wedge {\rm{D}}(s){Y}^{{\hat{\alpha }}_{5}}\wedge \cdots \wedge {\rm{D}}(s){Y}^{{\hat{\alpha }}_{d}}\right\}\\ \quad +{\displaystyle \int }_{{\rm{\Sigma }}}{\displaystyle \int }_{0}^{1}{\rm{d}}s\,{\epsilon }_{\hat{\alpha }\hat{\beta }{\hat{\alpha }}_{1}\cdots {\hat{\alpha }}_{d}}\left\{\delta {A}^{\hat{\alpha }\hat{\beta }}\right.\\ \quad \wedge {A}^{{\hat{\alpha }}_{1}{\hat{\alpha }}_{2}}(s)\wedge {\rm{D}}(s){Y}^{{\hat{\alpha }}_{3}}\wedge \cdots \wedge {\rm{D}}(s){Y}^{{\hat{\alpha }}_{d}}\\ \quad -\frac{2(d-2)}{{{\ell }}^{2}}\delta {Y}^{\hat{\alpha }}{Y}^{\hat{\beta }}{F}^{{\hat{\alpha }}_{1}{\hat{\alpha }}_{2}}(s)\wedge {\partial }_{s}\\ \quad \times \,{\rm{D}}(s){Y}^{{\hat{\alpha }}_{3}}\wedge \cdots \wedge {\rm{D}}(s){Y}^{{\hat{\alpha }}_{d}}\\ \quad -\frac{2(d-2)}{{{\ell }}^{2}}\delta {Y}^{\hat{\alpha }}{Y}^{\hat{\beta }}{A}^{{\hat{\alpha }}_{1}{\hat{\alpha }}_{2}}\wedge {\rm{D}}(s){\rm{D}}(s)\\ \quad \left.\times \,{Y}^{{\hat{\alpha }}_{3}}\wedge {\rm{D}}(s){Y}^{{\hat{\alpha }}_{4}}\wedge \cdots \wedge {\rm{D}}(s){Y}^{{\hat{\alpha }}_{d}}\right\}\end{array}\,\end{eqnarray*}$
$\begin{eqnarray*}\begin{array}{l}={\displaystyle \int }_{M}{\displaystyle \int }_{0}^{1}{\rm{d}}s\,{\epsilon }_{\hat{\alpha }\hat{\beta }{\hat{\alpha }}_{1}\cdots {\hat{\alpha }}_{d}}\left\{{\partial }_{s}[s\delta {A}^{\hat{\alpha }\hat{\beta }}\right.\\ \quad \wedge {F}^{{\hat{\alpha }}_{1}{\hat{\alpha }}_{2}}(s)\wedge {\rm{D}}(s){Y}^{{\hat{\alpha }}_{3}}\wedge \cdots \wedge {\rm{D}}(s){Y}^{{\hat{\alpha }}_{d}}]\\ \quad +\frac{d-2}{2}\delta {Y}^{\hat{\alpha }}{\partial }_{s}[{F}^{\hat{\beta }{\hat{\alpha }}_{1}}(s)\wedge {F}^{{\hat{\alpha }}_{2}{\hat{\alpha }}_{3}}(s)\\ \quad \left.\wedge {\rm{D}}(s){Y}^{{\hat{\alpha }}_{4}}\wedge \cdots \wedge {\rm{D}}(s){Y}^{{\hat{\alpha }}_{d}}]\right\}\\ \quad +{\displaystyle \int }_{{\rm{\Sigma }}}{\displaystyle \int }_{0}^{1}{\rm{d}}s\,{\epsilon }_{\hat{\alpha }\hat{\beta }{\hat{\alpha }}_{1}\cdots {\hat{\alpha }}_{d}}\left\{\delta {A}^{\hat{\alpha }\hat{\beta }}\right.\\ \quad \wedge {A}^{{\hat{\alpha }}_{1}{\hat{\alpha }}_{2}}(s)\wedge {\rm{D}}(s){Y}^{{\hat{\alpha }}_{3}}\wedge \cdots \wedge {\rm{D}}(s){Y}^{{\hat{\alpha }}_{d}}\\ \quad -\frac{2}{{{\ell }}^{2}}\delta {Y}^{\hat{\alpha }}{Y}^{\hat{\beta }}{\partial }_{s}[{F}^{{\hat{\alpha }}_{1}{\hat{\alpha }}_{2}}(s)\end{array}\,\end{eqnarray*}$
$\begin{eqnarray*}\begin{array}{l}\quad \wedge {\rm{D}}(s){Y}^{{\hat{\alpha }}_{3}}\wedge \cdots \wedge {\rm{D}}(s){Y}^{{\hat{\alpha }}_{d}}]\\ \quad +\frac{2}{{{\ell }}^{2}}\delta {Y}^{\hat{\alpha }}{Y}^{\hat{\beta }}{[{\partial }_{s}F(s)-{\rm{D}}(s)A]}^{{\hat{\alpha }}_{1}{\hat{\alpha }}_{2}}\\ \quad \wedge {\rm{D}}(s){Y}^{{\hat{\alpha }}_{3}}\wedge \cdots \wedge {\rm{D}}(s){Y}^{{\hat{\alpha }}_{d}}\\ +\frac{2}{{{\ell }}^{2}}\delta {Y}^{\hat{\alpha }}{Y}^{\hat{\beta }}{\rm{D}}(s)[{A}^{{\hat{\alpha }}_{1}{\hat{\alpha }}_{2}}\\ \quad \left.\wedge {\rm{D}}(s){Y}^{{\hat{\alpha }}_{3}}\wedge \cdots \wedge {\rm{D}}(s){Y}^{{\hat{\alpha }}_{d}}]\right\}\\ ={\displaystyle \int }_{M}{\epsilon }_{\hat{\alpha }\hat{\beta }{\hat{\alpha }}_{1}\cdots {\hat{\alpha }}_{d}}\left\{\delta {A}^{\hat{\alpha }\hat{\beta }}\right.\\ \quad \wedge {F}^{{\hat{\alpha }}_{1}{\hat{\alpha }}_{2}}\wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{3}}\wedge \cdots \wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{d}}+\frac{2}{{{\ell }}^{2}}\end{array}\,\end{eqnarray*}$
$\begin{eqnarray*}\begin{array}{l}\quad \left.\times \,\delta {Y}^{\hat{\alpha }}{Y}^{\hat{\beta }}{\rm{D}}[{F}^{{\hat{\alpha }}_{1}{\hat{\alpha }}_{2}}\wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{3}}\wedge \cdots \wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{d}}]\right\}\\ \quad +{\displaystyle \int }_{{\rm{\Sigma }}}{\epsilon }_{\hat{\alpha }\hat{\beta }{\hat{\alpha }}_{1}\cdots {\hat{\alpha }}_{d}}\left\{{\displaystyle \int }_{0}^{1}{\rm{d}}s\right.\\ \quad \times \,\delta {A}^{\hat{\alpha }\hat{\beta }}\wedge {A}^{{\hat{\alpha }}_{1}{\hat{\alpha }}_{2}}(s)\wedge {\rm{D}}(s){Y}^{{\hat{\alpha }}_{3}}\wedge \cdots \wedge {\rm{D}}(s){Y}^{{\hat{\alpha }}_{d}}\end{array}\end{eqnarray*}$
$\begin{eqnarray}\begin{array}{l}\quad -\frac{2}{{{\ell }}^{2}}\delta {Y}^{\hat{\alpha }}{Y}^{\hat{\beta }}\left[{F}^{{\hat{\alpha }}_{1}{\hat{\alpha }}_{2}}\wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{3}}\wedge \cdots \wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{d}}\right.\\ \quad -{\displaystyle \int }_{0}^{1}{\rm{d}}s\,{\rm{D}}(s)[{A}^{{\hat{\alpha }}_{1}{\hat{\alpha }}_{2}}\wedge {\rm{D}}(s)\\ \quad \times \,\left.\left.{Y}^{{\hat{\alpha }}_{3}}\wedge \cdots \wedge {\rm{D}}(s){Y}^{{\hat{\alpha }}_{d}}]\right]\right\},\end{array}\end{eqnarray}$
where we have used that $\begin{eqnarray}\begin{array}{l}{\partial }_{s}{\rm{D}}(s){Y}^{\hat{\alpha }}={(AY)}^{\hat{\alpha }},\\ {\rm{D}}(s){A}^{{\hat{\alpha }}_{1}{\hat{\alpha }}_{2}}(s)={[F(s)+A(s)\wedge A(s)]}^{{\hat{\alpha }}_{1}{\hat{\alpha }}_{2}}\\ \quad =s{\partial }_{s}{F}^{{\hat{\alpha }}_{1}{\hat{\alpha }}_{2}}(s),\\ {\rm{D}}(s){A}^{{\hat{\alpha }}_{1}{\hat{\alpha }}_{2}}={s}^{-1}{[F(s)+A(s)\wedge A(s)]}^{{\hat{\alpha }}_{1}{\hat{\alpha }}_{2}}\\ \qquad ={\partial }_{s}{F}^{{\hat{\alpha }}_{1}{\hat{\alpha }}_{2}}(s),\end{array}\end{eqnarray}$
as well as the resummations $\begin{eqnarray}\begin{array}{l}{\epsilon }_{\hat{\alpha }\hat{\beta }{\hat{\alpha }}_{1}\cdots {\hat{\alpha }}_{d}}\delta {A}^{\hat{\alpha }\hat{\beta }}\wedge {A}^{{\hat{\alpha }}_{1}{\hat{\alpha }}_{2}}(s)\wedge {F}^{{\hat{\alpha }}_{3}\hat{\gamma }}(s)\\ \quad \times \,{Y}_{\hat{\gamma }}\wedge {\rm{D}}(s){Y}^{{\hat{\alpha }}_{4}}\wedge \cdots \wedge {\rm{D}}(s){Y}^{{\hat{\alpha }}_{d}}\\ =-{\epsilon }_{\hat{\alpha }\hat{\beta }{\hat{\alpha }}_{1}\cdots {\hat{\alpha }}_{d}}\delta {A}^{\hat{\alpha }\hat{\gamma }}{Y}_{\hat{\gamma }}\wedge {A}^{\hat{\beta }{\hat{\alpha }}_{1}}(s)\\ \quad \wedge {F}^{{\hat{\alpha }}_{2}{\hat{\alpha }}_{3}}(s)\wedge {\rm{D}}(s){Y}^{{\hat{\alpha }}_{4}}\wedge \cdots \wedge {\rm{D}}(s){Y}^{{\hat{\alpha }}_{d}}\\ \quad -{\epsilon }_{\hat{\alpha }\hat{\beta }{\hat{\alpha }}_{1}\cdots {\hat{\alpha }}_{d}}\delta {A}^{\hat{\alpha }\hat{\beta }}\wedge {F}^{{\hat{\alpha }}_{1}{\hat{\alpha }}_{2}}(s)\\ \quad \wedge s{\partial }_{s}[{\rm{D}}(s){Y}^{{\hat{\alpha }}_{3}}]\wedge {\rm{D}}(s){Y}^{{\hat{\alpha }}_{4}}\wedge \cdots \wedge {\rm{D}}(s){Y}^{{\hat{\alpha }}_{d}},\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{l}{\epsilon }_{\hat{\alpha }\hat{\beta }{\hat{\alpha }}_{1}\cdots {\hat{\alpha }}_{d}}\delta {Y}^{\hat{\alpha }}{A}^{\hat{\beta }{\hat{\alpha }}_{1}}\wedge {F}^{{\hat{\alpha }}_{2}{\hat{\alpha }}_{3}}(s)\\ \quad \wedge {F}^{{\hat{\alpha }}_{4}\hat{\gamma }}(s){Y}_{\gamma }\wedge {\rm{D}}(s){Y}^{{\hat{\alpha }}_{5}}\wedge \cdots \wedge {\rm{D}}(s){Y}^{{\hat{\alpha }}_{d}}\\ =-\frac{1}{2}{\epsilon }_{\hat{\alpha }\hat{\beta }{\hat{\alpha }}_{1}\cdots {\hat{\alpha }}_{d}}\delta {Y}^{\hat{\alpha }}{F}^{\hat{\beta }{\hat{\alpha }}_{1}}(s)\\ \quad \wedge {F}^{{\hat{\alpha }}_{2}{\hat{\alpha }}_{3}}(s)\wedge {\partial }_{s}[{\rm{D}}(s){Y}^{{\hat{\alpha }}_{4}}]\\ \quad \wedge {\rm{D}}(s){Y}^{{\hat{\alpha }}_{5}}\wedge \cdots \wedge {\rm{D}}(s){Y}^{{\hat{\alpha }}_{d}},\end{array}\end{eqnarray}$
$\begin{eqnarray}\,\begin{array}{l}-{{\ell }}^{2}{\epsilon }_{\hat{\alpha }\hat{\beta }{\hat{\alpha }}_{1}\cdots {\hat{\alpha }}_{d}}\delta {Y}^{\hat{\alpha }}{F}^{\hat{\beta }{\hat{\alpha }}_{1}}\wedge {F}^{{\hat{\alpha }}_{2}{\hat{\alpha }}_{3}}\\ \wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{4}}\wedge \cdots \wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{d}}\\ =-\frac{4}{d-2}{\epsilon }_{\hat{\alpha }\hat{\beta }{\hat{\alpha }}_{1}\cdots {\hat{\alpha }}_{d}}\delta {Y}^{\hat{\alpha }}{Y}^{\hat{\beta }}{\rm{D}}[{F}^{{\hat{\alpha }}_{1}{\hat{\alpha }}_{2}}\\ \wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{3}}\wedge \cdots \wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{d}}],\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{l}-{{\ell }}^{2}\delta {Y}^{\hat{\alpha }}{A}^{\hat{\beta }{\hat{\alpha }}_{1}}\wedge {F}^{{\hat{\alpha }}_{2}{\hat{\alpha }}_{3}}(s)\\ \quad \wedge {\rm{D}}(s){Y}^{{\hat{\alpha }}_{4}}\wedge \cdots \wedge {\rm{D}}(s){Y}^{{\hat{\alpha }}_{d}}\\ =\delta {Y}^{\hat{\alpha }}{A}^{\hat{\beta }{\hat{\alpha }}_{1}}\wedge {F}^{{\hat{\alpha }}_{2}{\hat{\alpha }}_{3}}(s)\wedge {\rm{D}}(s){Y}^{{\hat{\alpha }}_{4}}\\ \quad \wedge \cdots \wedge {\rm{D}}(s){Y}^{{\hat{\alpha }}_{d}}{Y}^{\hat{\gamma }}{Y}_{\hat{\gamma }}\\ =-\frac{2}{d-2}\delta {Y}^{\hat{\alpha }}{Y}^{\hat{\beta }}{\partial }_{s}[{F}^{{\hat{\alpha }}_{1}{\hat{\alpha }}_{2}}(s)\\ \quad \wedge {\rm{D}}(s){Y}^{{\hat{\alpha }}_{3}}\wedge \cdots \wedge {\rm{D}}(s){Y}^{{\hat{\alpha }}_{d}}]\\ \quad +\frac{2}{d-2}\delta {Y}^{\hat{\alpha }}{Y}^{\hat{\beta }}{\rm{D}}(s)[{A}^{{\hat{\alpha }}_{1}{\hat{\alpha }}_{2}}\\ \quad \wedge {\rm{D}}(s){Y}^{{\hat{\alpha }}_{3}}\wedge \cdots \wedge {\rm{D}}(s){Y}^{{\hat{\alpha }}_{d}}].\end{array}\end{eqnarray}$
On the other hand, the variation of ${S}_{(d+1)}^{{\rm{top}}}$ only gives rise to the boundary term $\begin{eqnarray}\begin{array}{l}\frac{{\kappa }^{2}{{\ell }}^{3}(d+1)!}{d}\delta {S}_{(d+1)}^{{\rm{top}}}\\ =\delta {\displaystyle \int }_{M}{\epsilon }_{{\hat{\alpha }}_{1}\cdots {\hat{\alpha }}_{d+2}}{\rm{d}}{Y}^{{\hat{\alpha }}_{1}}\wedge \cdots \wedge {\rm{d}}{Y}^{{\hat{\alpha }}_{d+1}}{Y}^{{\hat{\alpha }}_{d+2}}\\ ={\displaystyle \int }_{M}{\epsilon }_{{\hat{\alpha }}_{1}\cdots {\hat{\alpha }}_{d+2}}[(d+1){\rm{d}}\delta {Y}^{{\hat{\alpha }}_{1}}\wedge {\rm{d}}{Y}^{{\hat{\alpha }}_{2}}\\ \quad \wedge \cdots \wedge {\rm{d}}{Y}^{{\hat{\alpha }}_{d+1}}{Y}^{{\hat{\alpha }}_{d+2}}+{\rm{d}}{Y}^{{\hat{\alpha }}_{1}}\wedge {\rm{d}}{Y}^{{\hat{\alpha }}_{2}}\\ \quad \wedge \cdots \wedge {\rm{d}}{Y}^{{\hat{\alpha }}_{d+1}}\delta {Y}^{{\hat{\alpha }}_{d+2}}]\\ ={(-1)}^{d}(d+1){\displaystyle \int }_{{\rm{\Sigma }}}{\epsilon }_{{\hat{\alpha }}_{1}\cdots {\hat{\alpha }}_{d+2}}\delta {Y}^{{\hat{\alpha }}_{1}}{Y}^{{\hat{\alpha }}_{2}}\\ \quad \times \,{\rm{d}}{Y}^{{\hat{\alpha }}_{3}}\wedge \cdots \wedge {\rm{d}}{Y}^{{\hat{\alpha }}_{d+2}}\\ -{(-1)}^{d}(d+2){\displaystyle \int }_{M}{\epsilon }_{{\hat{\alpha }}_{1}\cdots {\hat{\alpha }}_{d+2}}\delta {Y}^{{\hat{\alpha }}_{1}}{\rm{d}}{Y}^{{\hat{\alpha }}_{2}}\\ \quad \wedge \cdots \wedge {\rm{d}}{Y}^{{\hat{\alpha }}_{d+1}}\wedge {\rm{d}}{Y}^{{\hat{\alpha }}_{d+2}}\\ ={(-1)}^{d}(d+1){\displaystyle \int }_{{\rm{\Sigma }}}{\epsilon }_{{\hat{\alpha }}_{1}\cdots {\hat{\alpha }}_{d+2}}\delta {Y}^{{\hat{\alpha }}_{1}}{Y}^{{\hat{\alpha }}_{2}}\\ \times \,{\rm{d}}{Y}^{{\hat{\alpha }}_{3}}\wedge \cdots \wedge {\rm{d}}{Y}^{{\hat{\alpha }}_{d+2}},\end{array}\end{eqnarray}$
where we have used the resummation $\begin{eqnarray}\begin{array}{l}-{{\ell }}^{2}{\epsilon }_{{\hat{\alpha }}_{1}\cdots {\hat{\alpha }}_{d+2}}\delta {Y}^{{\hat{\alpha }}_{1}}{\rm{d}}{Y}^{{\hat{\alpha }}_{2}}\wedge \cdots \wedge {\rm{d}}{Y}^{{\hat{\alpha }}_{d+2}}\\ ={\epsilon }_{{\hat{\alpha }}_{1}\cdots {\hat{\alpha }}_{d+2}}\delta {Y}^{{\hat{\alpha }}_{1}}{\rm{d}}{Y}^{{\hat{\alpha }}_{2}}\\ \quad \wedge \cdots \wedge {\rm{d}}{Y}^{{\hat{\alpha }}_{d+2}}{Y}^{\hat{\gamma }}{Y}_{\hat{\gamma }}=0.\end{array}\end{eqnarray}$
In total, we have
$\begin{eqnarray}\begin{array}{l}\frac{4{\kappa }^{2}(d-2)!}{{(-1)}^{d+1}{\ell }}\delta {S}_{(d+1)}^{{\rm{con}}}\\ =2{\kappa }^{2}{\ell }(d-1)!\frac{2{(-1)}^{d+1}}{(d-1){{\ell }}^{2}}(\delta {S}_{(d+1)}^{{\rm{CS}}}-\delta {S}_{(d+1)}^{{\rm{top}}})\\ ={\displaystyle \int }_{M}{\epsilon }_{\hat{\alpha }\hat{\beta }{\hat{\alpha }}_{1}\cdots {\hat{\alpha }}_{d}}\left\{\delta {A}^{\hat{\alpha }\hat{\beta }}\right.\\ \quad \wedge {F}^{{\hat{\alpha }}_{1}{\hat{\alpha }}_{2}}\wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{3}}\wedge \cdots \wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{d}}\\ \quad +\frac{2}{{{\ell }}^{2}}\delta {Y}^{\hat{\alpha }}{Y}^{\hat{\beta }}{\rm{D}}[{F}^{{\hat{\alpha }}_{1}{\hat{\alpha }}_{2}}\\ \quad \left.\wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{3}}\wedge \cdots \wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{d}}]\right\}\\ \quad +{\displaystyle \int }_{{\rm{\Sigma }}}{\epsilon }_{\hat{\alpha }\hat{\beta }{\hat{\alpha }}_{1}\cdots {\hat{\alpha }}_{d}}\left\{{\displaystyle \int }_{0}^{1}{\rm{d}}s\,\delta {A}^{\hat{\alpha }\hat{\beta }}\right.\\ \quad \wedge {A}^{{\hat{\alpha }}_{1}{\hat{\alpha }}_{2}}(s)\wedge {\rm{D}}(s){Y}^{{\hat{\alpha }}_{3}}\wedge \cdots \wedge {\rm{D}}(s){Y}^{{\hat{\alpha }}_{d}}\\ \quad +\frac{2}{{{\ell }}^{2}}\delta {Y}^{\hat{\alpha }}{Y}^{\hat{\beta }}\left[-{F}^{{\hat{\alpha }}_{1}{\hat{\alpha }}_{2}}\wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{3}}\right.\\ \quad \wedge \cdots \wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{d}}+\frac{2}{(d-1){{\ell }}^{2}}{\rm{d}}{Y}^{{\hat{\alpha }}_{1}}\wedge \cdots \wedge {\rm{d}}{Y}^{{\hat{\alpha }}_{d}}\\ \quad +{\displaystyle \int }_{0}^{1}{\rm{d}}s\,{\rm{D}}(s)[{A}^{{\hat{\alpha }}_{1}{\hat{\alpha }}_{2}}\wedge {\rm{D}}(s){Y}^{{\hat{\alpha }}_{3}}\\ \quad \left.\left.\wedge \cdots \wedge {\rm{D}}(s){Y}^{{\hat{\alpha }}_{d}}]\right]\right\}.\end{array}\end{eqnarray}$
Especially, we have verified $\begin{eqnarray}\begin{array}{l}{{ \mathcal J }}_{(d+1)}^{{\rm{cov}}}={{ \mathcal J }}_{(d+1)}^{{\rm{CS}}}={{ \mathcal J }}_{(d+1)}^{{\rm{con}}}\end{array}\end{eqnarray}$
which confirms that ${S}_{(d+1)}^{{\rm{cov}}},{S}_{(d+1)}^{{\rm{CS}}}$ and ${S}_{(d+1)}^{{\rm{con}}}$ gives rise to the same bulk EOM.C.4. The boundary effective action
Starting from either consistent or covariant current, we can construct the regularized effective action. Begin with the covariant current, we get
$\begin{eqnarray}\begin{array}{l}{W}_{(d)}={\displaystyle \int }_{{\rm{\Sigma }}}{{ \mathcal L }}_{(d)}^{{\rm{eff}}}={\displaystyle \int }_{{\rm{\Sigma }}}{\displaystyle \int }_{0}^{1}{\rm{d}}s\,{A}^{\hat{\alpha }\hat{\beta }}\\ \quad \wedge {({{ \mathcal J }}_{(d)}^{{\rm{cov}}})}_{\hat{\alpha }\hat{\beta }}(A(s),Y)\\ =-\frac{1}{2{\kappa }^{2}{\ell }(d-1)!}{\displaystyle \int }_{{\rm{\Sigma }}}{\displaystyle \int }_{0}^{1}{\rm{d}}s\,{\epsilon }_{\hat{\alpha }\hat{\beta }{\hat{\alpha }}_{1}\cdots {\hat{\alpha }}_{d}}\\ \quad \times {A}^{\hat{\alpha }\hat{\beta }}\wedge {\rm{D}}(s){Y}^{{\hat{\alpha }}_{1}}\wedge \cdots \wedge {\rm{D}}(s){Y}^{{\hat{\alpha }}_{d-1}}{Y}^{{\hat{\alpha }}_{d}}.\end{array}\end{eqnarray}$
C.4.1. ${\rm{d}}{{ \mathcal L }}_{(d)}^{{\rm{eff}}}={{ \mathcal L }}_{(d+1)}^{{\rm{con}}}-{{ \mathcal L }}_{(d+1)}^{{\rm{cov}}}$
By using the resummations
$\begin{eqnarray}\begin{array}{l}{\epsilon }_{\hat{\alpha }\hat{\beta }{\hat{\alpha }}_{1}\cdots {\hat{\alpha }}_{d}}{A}^{\hat{\alpha }\hat{\beta }}\wedge {F}^{{\hat{\alpha }}_{1}\hat{\gamma }}(s)\\ \quad \times \,{Y}_{\hat{\gamma }}\wedge {\rm{D}}(s){Y}^{{\hat{\alpha }}_{2}}\wedge \cdots \wedge {\rm{D}}(s){Y}^{{\hat{\alpha }}_{d-1}}{Y}^{{\hat{\alpha }}_{d}}\\ =-{\epsilon }_{\hat{\alpha }\hat{\beta }{\hat{\alpha }}_{1}\cdots {\hat{\alpha }}_{d}}{F}^{\hat{\alpha }\hat{\beta }}(s)\wedge {\partial }_{s}({\rm{D}}(s)\\ \quad \times \,{Y}^{{\hat{\alpha }}_{1}})\wedge {\rm{D}}(s){Y}^{{\hat{\alpha }}_{2}}\wedge \cdots \wedge {\rm{D}}(s){Y}^{{\hat{\alpha }}_{d-1}}{Y}^{{\hat{\alpha }}_{d}}\\ \quad -\frac{{{\ell }}^{2}{(-1)}^{d}}{2}{\epsilon }_{\hat{\alpha }\hat{\beta }{\hat{\alpha }}_{1}\cdots {\hat{\alpha }}_{d}}{A}^{\hat{\alpha }\hat{\beta }}\\ \quad \wedge {F}^{{\hat{\alpha }}_{1}{\hat{\alpha }}_{2}}(s)\wedge {\rm{D}}(s){Y}^{{\hat{\alpha }}_{3}}\wedge \cdots \wedge {\rm{D}}(s){Y}^{{\hat{\alpha }}_{d}},\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{l}-{{\ell }}^{2}{\epsilon }_{\hat{\alpha }\hat{\beta }{\hat{\alpha }}_{1}\cdots {\hat{\alpha }}_{d}}{A}^{\hat{\alpha }\hat{\beta }}\wedge {\rm{D}}(s){Y}^{{\hat{\alpha }}_{1}}\wedge \cdots \wedge {\rm{D}}(s){Y}^{{\hat{\alpha }}_{d}}\\ ={\epsilon }_{\hat{\alpha }\hat{\beta }{\hat{\alpha }}_{1}\cdots {\hat{\alpha }}_{d}}{A}^{\hat{\alpha }\hat{\beta }}\wedge {\rm{D}}(s)\\ \quad \times \,{Y}^{{\hat{\alpha }}_{1}}\wedge \cdots \wedge {\rm{D}}(s){Y}^{{\hat{\alpha }}_{d}}{Y}^{\hat{\gamma }}{Y}_{\hat{\gamma }}\\ ={(-1)}^{d}2{\epsilon }_{\hat{\alpha }\hat{\beta }{\hat{\alpha }}_{1}\cdots {\hat{\alpha }}_{d}}{\partial }_{s}[{\rm{D}}(s){Y}^{\hat{\alpha }}]\wedge {\rm{D}}(s){Y}^{\hat{\beta }}\\ \quad \wedge {\rm{D}}(s){Y}^{{\hat{\alpha }}_{1}}\wedge \cdots \wedge {\rm{D}}(s){Y}^{{\hat{\alpha }}_{d-1}}{Y}^{{\hat{\alpha }}_{d}}.\end{array}\end{eqnarray}$
We find that the exterior derivative of ${{ \mathcal L }}_{(d)}^{{\rm{eff}}}$ is $\begin{eqnarray*}\begin{array}{l}-2{\kappa }^{2}{\ell }(d-1)!{\rm{d}}{{ \mathcal L }}_{(d)}^{{\rm{eff}}}\\ ={\rm{d}}\left[{\displaystyle \int }_{0}^{1}{\rm{d}}s\,{\epsilon }_{\hat{\alpha }\hat{\beta }{\hat{\alpha }}_{1}\cdots {\hat{\alpha }}_{d}}{A}^{\hat{\alpha }\hat{\beta }}\wedge {\rm{D}}(s)\right.\\ \quad \left.\times \,{Y}^{{\hat{\alpha }}_{1}}\wedge \cdots \wedge {\rm{D}}(s){Y}^{{\hat{\alpha }}_{d-1}}{Y}^{{\hat{\alpha }}_{d}}\right]\\ ={\displaystyle \int }_{0}^{1}{\rm{d}}s\,{\epsilon }_{\hat{\alpha }\hat{\beta }{\hat{\alpha }}_{1}\cdots {\hat{\alpha }}_{d}}{\rm{D}}(s)[{A}^{\hat{\alpha }\hat{\beta }}\\ \quad \wedge {\rm{D}}(s){Y}^{{\hat{\alpha }}_{1}}\wedge \cdots \wedge {\rm{D}}(s){Y}^{{\hat{\alpha }}_{d-1}}{Y}^{{\hat{\alpha }}_{d}}]\\ ={\displaystyle \int }_{0}^{1}{\rm{d}}s\,{\epsilon }_{\hat{\alpha }\hat{\beta }{\hat{\alpha }}_{1}\cdots {\hat{\alpha }}_{d}}\left\{{\partial }_{s}{F}^{\hat{\alpha }\hat{\beta }}(s)\right.\end{array}\,\end{eqnarray*}$
$\begin{eqnarray*}\begin{array}{l}\quad \wedge {\rm{D}}(s){Y}^{{\hat{\alpha }}_{1}}\wedge \cdots \wedge {\rm{D}}(s){Y}^{{\hat{\alpha }}_{d-1}}{Y}^{{\hat{\alpha }}_{d}}\\ \quad -(d-1){A}^{\hat{\alpha }\hat{\beta }}\wedge {[F(s)Y]}^{{\hat{\alpha }}_{1}}\wedge {\rm{D}}(s){Y}^{{\hat{\alpha }}_{2}}\\ \quad \wedge \cdots \wedge {\rm{D}}(s){Y}^{{\hat{\alpha }}_{d-1}}{Y}^{{\hat{\alpha }}_{d}}\\ \quad +{(-1)}^{d}{A}^{\hat{\alpha }\hat{\beta }}\wedge {\rm{D}}(s){Y}^{{\hat{\alpha }}_{1}}\wedge \cdots \wedge {\rm{D}}(s)\\ \left.\times \,{Y}^{{\hat{\alpha }}_{d-1}}\wedge {\rm{D}}(s){Y}^{{\hat{\alpha }}_{d}}]\right\}\\ ={\displaystyle \int }_{0}^{1}{\rm{d}}s\,{\epsilon }_{\hat{\alpha }\hat{\beta }{\hat{\alpha }}_{1}\cdots {\hat{\alpha }}_{d}}\left\{{\partial }_{s}{F}^{\hat{\alpha }\hat{\beta }}(s)\right.\end{array}\,\end{eqnarray*}$
$\begin{eqnarray*}\begin{array}{l}\quad \wedge {\rm{D}}(s){Y}^{{\hat{\alpha }}_{1}}\wedge \cdots \wedge {\rm{D}}(s){Y}^{{\hat{\alpha }}_{d-1}}{Y}^{{\hat{\alpha }}_{d}}\\ \quad +(d-1){F}^{\hat{\alpha }\hat{\beta }}(s)\wedge {\partial }_{s}({\rm{D}}(s){Y}^{{\hat{\alpha }}_{1}})\wedge {\rm{D}}(s){Y}^{{\hat{\alpha }}_{2}}\\ \quad \wedge \cdots \wedge {\rm{D}}(s){Y}^{{\hat{\alpha }}_{d-1}}{Y}^{{\hat{\alpha }}_{d}}\\ \quad +\frac{{{\ell }}^{2}{(-1)}^{d}(d-1)}{2}{A}^{\hat{\alpha }\hat{\beta }}\wedge {F}^{{\hat{\alpha }}_{1}{\hat{\alpha }}_{2}}(s)\\ \quad \wedge {\rm{D}}(s){Y}^{{\hat{\alpha }}_{3}}\wedge \cdots \wedge {\rm{D}}(s){Y}^{{\hat{\alpha }}_{d}}\\ \quad +\frac{2}{{{\ell }}^{2}}{\partial }_{s}[{\rm{D}}(s){Y}^{\hat{\alpha }}]\wedge {\rm{D}}(s){Y}^{\hat{\beta }}\wedge {\rm{D}}(s)\\ \quad \left.\times {Y}^{{\hat{\alpha }}_{1}}\wedge \cdots \wedge {\rm{D}}(s){Y}^{{\hat{\alpha }}_{d-1}}{Y}^{{\hat{\alpha }}_{d}}]\right\}\\ ={\displaystyle \int }_{0}^{1}{\rm{d}}s\,{\epsilon }_{\hat{\alpha }\hat{\beta }{\hat{\alpha }}_{1}\cdots {\hat{\alpha }}_{d}}\left\{{\partial }_{s}[{F}^{\hat{\alpha }\hat{\beta }}(s)\right.\end{array}\,\end{eqnarray*}$
$\begin{eqnarray*}\begin{array}{l}\quad \wedge {\rm{D}}(s){Y}^{{\hat{\alpha }}_{1}}\wedge \cdots \wedge {\rm{D}}(s){Y}^{{\hat{\alpha }}_{d-1}}{Y}^{{\hat{\alpha }}_{d}}]\\ \quad +\frac{{{\ell }}^{2}{(-1)}^{d}(d-1)}{2}{A}^{\hat{\alpha }\hat{\beta }}\wedge {F}^{{\hat{\alpha }}_{1}{\hat{\alpha }}_{2}}(s)\\ \quad \wedge {\rm{D}}(s){Y}^{{\hat{\alpha }}_{3}}\wedge \cdots \wedge {\rm{D}}(s){Y}^{{\hat{\alpha }}_{d}}\\ \quad -\frac{2}{{{\ell }}^{2}(d+1)}{\partial }_{s}[{\rm{D}}(s){Y}^{\hat{\alpha }}\wedge {\rm{D}}(s){Y}^{\hat{\beta }}\wedge {\rm{D}}(s)\\ \quad \left.\times {Y}^{{\hat{\alpha }}_{1}}\wedge \cdots \wedge {\rm{D}}(s){Y}^{{\hat{\alpha }}_{d-1}}{Y}^{{\hat{\alpha }}_{d}}]\right\}\\ ={\epsilon }_{\hat{\alpha }\hat{\beta }{\hat{\alpha }}_{1}\cdots {\hat{\alpha }}_{d}}\left\{\left[{F}^{\hat{\alpha }\hat{\beta }}\right.\right.\\ \quad \left.-\frac{2}{(d+1){{\ell }}^{2}}{\rm{D}}{Y}^{\hat{\alpha }}\wedge {\rm{D}}{Y}^{\hat{\beta }}\right]\end{array}\,\end{eqnarray*}$
$\begin{eqnarray}\begin{array}{l}\quad \wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{1}}\wedge \cdots \wedge {\rm{D}}{Y}^{{\hat{\alpha }}_{d-1}}{Y}^{{\hat{\alpha }}_{d}}\\ \quad +\frac{2}{(d+1){{\ell }}^{2}}{\rm{d}}{Y}^{\hat{\alpha }}\wedge {\rm{d}}{Y}^{\hat{\beta }}\wedge {\rm{d}}{Y}^{{\hat{\alpha }}_{1}}\\ \quad \wedge \cdots \wedge {\rm{d}}{Y}^{{\hat{\alpha }}_{d-1}}{Y}^{{\hat{\alpha }}_{d}}\\ \quad -\frac{{{\ell }}^{2}{(-1)}^{d+1}(d-1)}{2}{\displaystyle \int }_{0}^{1}{\rm{d}}s\,{A}^{\hat{\alpha }\hat{\beta }}\wedge {F}^{{\hat{\alpha }}_{1}{\hat{\alpha }}_{2}}(s)\\ \quad \left.\wedge {\rm{D}}(s){Y}^{{\hat{\alpha }}_{3}}\wedge \cdots \wedge {\rm{D}}(s){Y}^{{\hat{\alpha }}_{d}}\right\}\\ =2{\kappa }^{2}{\ell }\,(d-1)!({{ \mathcal L }}_{(d+1)}^{{\rm{cov}}}-{{ \mathcal L }}_{(d+1)}^{{\rm{con}}}).\end{array}\end{eqnarray}$
It implies that $\begin{eqnarray}{S}_{(d+1)}^{{\rm{cov}}}={S}_{(d+1)}^{{\rm{con}}}-{W}_{(d)}.\end{eqnarray}$
C.4.2. SO(2, d) gauge transformation of ${{ \mathcal L }}_{(d)}^{{\rm{eff}}}$
By using the resummation C.60 ) of ${{ \mathcal L }}_{(d+1)}^{{\rm{con}}}$. That is
$\begin{eqnarray}\begin{array}{l}-{{\ell }}^{2}{\epsilon }_{\hat{\alpha }\hat{\beta }{\hat{\alpha }}_{1}\cdots {\hat{\alpha }}_{d}}{\hat{A}}^{\hat{\alpha }\hat{\beta }}\wedge {A}^{{\hat{\alpha }}_{1}{\hat{\alpha }}_{2}}\\ \quad \wedge {\rm{D}}(s,\hat{s}){Y}^{{\hat{\alpha }}_{3}}\wedge \cdots \wedge {\rm{D}}(s,\hat{s}){Y}^{{\hat{\alpha }}_{d}}\\ ={\epsilon }_{\hat{\alpha }\hat{\beta }{\hat{\alpha }}_{1}\cdots {\hat{\alpha }}_{d}}{\hat{A}}^{\hat{\alpha }\hat{\beta }}\wedge {A}^{{\hat{\alpha }}_{1}{\hat{\alpha }}_{2}}\\ \quad \wedge {\rm{D}}(s,\hat{s}){Y}^{{\hat{\alpha }}_{3}}\wedge \cdots \wedge {\rm{D}}(s,\hat{s}){Y}^{{\hat{\alpha }}_{d}}{Y}^{\hat{\gamma }}{Y}_{\hat{\gamma }}\\ =-2{(-1)}^{d}{\epsilon }_{\hat{\alpha }\hat{\beta }{\hat{\alpha }}_{1}\cdots {\hat{\alpha }}_{d}}{A}^{\hat{\alpha }\hat{\beta }}\wedge {(\hat{A}Y)}^{{\hat{\alpha }}_{1}}\\ \quad \wedge {\rm{D}}(s,\hat{s}){Y}^{{\hat{\alpha }}_{2}}\wedge \cdots \wedge {\rm{D}}(s,\hat{s}){Y}^{{\hat{\alpha }}_{d-1}}{Y}^{{\hat{\alpha }}_{d}}\\ \quad +2{(-1)}^{d}{\epsilon }_{\hat{\alpha }\hat{\beta }{\hat{\alpha }}_{1}\cdots {\hat{\alpha }}_{d}}{\hat{A}}^{\hat{\alpha }\hat{\beta }}\wedge {(AY)}^{{\hat{\alpha }}_{1}}\\ \quad \wedge {\rm{D}}(s,\hat{s}){Y}^{{\hat{\alpha }}_{2}}\wedge \cdots \wedge {\rm{D}}(s,\hat{s}){Y}^{{\hat{\alpha }}_{d-1}}{Y}^{{\hat{\alpha }}_{d}},\end{array}\end{eqnarray}$
we find the finite gauge transformation of ${{ \mathcal L }}_{(d)}^{{\rm{eff}}}$ is $\begin{eqnarray*}\begin{array}{l}-2{\kappa }^{2}{\ell }\,(d-1)!{{ \mathcal L }}_{(d)}^{{\rm{eff}}}[\tilde{A},\tilde{Y}]\\ ={\displaystyle \int }_{0}^{1}{\rm{d}}s\,{\epsilon }_{\hat{\alpha }\hat{\beta }{\hat{\alpha }}_{1}\cdots {\hat{\alpha }}_{d}}{\tilde{A}}^{\hat{\alpha }\hat{\beta }}\\ \quad \wedge \tilde{{\rm{D}}(s)}{\tilde{Y}}^{{\hat{\alpha }}_{1}}\wedge \cdots \wedge \tilde{{\rm{D}}(s)}{\tilde{Y}}^{{\hat{\alpha }}_{d-1}}{Y}^{{\hat{\alpha }}_{d}}\\ ={\displaystyle \int }_{0}^{1}{\rm{d}}s\,{\epsilon }_{\hat{\alpha }\hat{\beta }{\hat{\alpha }}_{1}\cdots {\hat{\alpha }}_{d}}{(A-\hat{A})}^{\hat{\alpha }\hat{\beta }}\\ \quad \wedge {[{\rm{D}}(s)Y+(1-s)\hat{A}Y]}^{{\hat{\alpha }}_{1}}\\ \quad \wedge \cdots \wedge {[{\rm{D}}(s)Y+(1-s)\hat{A}Y]}^{{\hat{\alpha }}_{d-1}}{Y}^{{\hat{\alpha }}_{d}}\\ ={\displaystyle \int }_{0}^{1}{\rm{d}}s\,{\epsilon }_{\hat{\alpha }\hat{\beta }{\hat{\alpha }}_{1}\cdots {\hat{\alpha }}_{d}}{A}^{\hat{\alpha }\hat{\beta }}\wedge {\rm{D}}(s,1-s){Y}^{{\hat{\alpha }}_{1}}\wedge \cdots \end{array}\,\end{eqnarray*}$
$\begin{eqnarray*}\begin{array}{l}\quad \wedge {\rm{D}}(s,1-s){Y}^{{\hat{\alpha }}_{d-1}}{Y}^{{\hat{\alpha }}_{d}}\\ \quad -{\displaystyle \int }_{0}^{1}{\rm{d}}\hat{s}\,{\epsilon }_{\hat{\alpha }\hat{\beta }{\hat{\alpha }}_{1}\cdots {\hat{\alpha }}_{d}}{\hat{A}}^{\hat{\alpha }\hat{\beta }}\\ \quad \wedge {\rm{D}}(1-\hat{s},\hat{s}){Y}^{{\hat{\alpha }}_{1}}\wedge \cdots \wedge {\rm{D}}(1-\hat{s},\hat{s}){Y}^{{\hat{\alpha }}_{d-1}}{Y}^{{\hat{\alpha }}_{d}}\\ ={\displaystyle \int }_{0}^{1}{\rm{d}}s\,{\epsilon }_{\hat{\alpha }\hat{\beta }{\hat{\alpha }}_{1}\cdots {\hat{\alpha }}_{d}}{A}^{\hat{\alpha }\hat{\beta }}\wedge {\rm{D}}(s,0){Y}^{{\hat{\alpha }}_{1}}\\ \quad \wedge \cdots \wedge {\rm{D}}(s,0){Y}^{{\hat{\alpha }}_{d-1}}{Y}^{{\hat{\alpha }}_{d}}\\ \quad +{\displaystyle \int }_{0}^{1}{\rm{d}}s{\displaystyle \int }_{0}^{1-s}{\rm{d}}\hat{s}\,{\epsilon }_{\hat{\alpha }\hat{\beta }{\hat{\alpha }}_{1}\cdots {\hat{\alpha }}_{d}}{\partial }_{\hat{s}}[{A}^{\hat{\alpha }\hat{\beta }}\end{array}\end{eqnarray*}$
$\begin{eqnarray*}\begin{array}{l}\quad \wedge {\rm{D}}(s,\hat{s}){Y}^{{\hat{\alpha }}_{1}}\wedge \cdots \wedge {\rm{D}}(s,\hat{s}){Y}^{{\hat{\alpha }}_{d-1}}{Y}^{{\hat{\alpha }}_{d}}]\\ \quad -{\displaystyle \int }_{0}^{1}{\rm{d}}\hat{s}\,{\epsilon }_{\hat{\alpha }\hat{\beta }{\hat{\alpha }}_{1}\cdots {\hat{\alpha }}_{d}}{\hat{A}}^{\hat{\alpha }\hat{\beta }}\wedge {\rm{D}}(0,\hat{s}){Y}^{{\hat{\alpha }}_{1}}\\ \quad \wedge \cdots \wedge {\rm{D}}(0,\hat{s}){Y}^{{\hat{\alpha }}_{d-1}}{Y}^{{\hat{\alpha }}_{d}}\\ \quad -{\displaystyle \int }_{0}^{1}{\rm{d}}\hat{s}{\displaystyle \int }_{0}^{1-\hat{s}}{\rm{d}}s\,{\epsilon }_{\hat{\alpha }\hat{\beta }{\hat{\alpha }}_{1}\cdots {\hat{\alpha }}_{d}}{\partial }_{s}[{\hat{A}}^{\hat{\alpha }\hat{\beta }}\\ \quad \wedge {\rm{D}}(s,\hat{s}){Y}^{{\hat{\alpha }}_{1}}\wedge \cdots \wedge {\rm{D}}(s,\hat{s}){Y}^{{\hat{\alpha }}_{d-1}}{Y}^{{\hat{\alpha }}_{d}}]\\ \quad ={\displaystyle \int }_{0}^{1}{\rm{d}}s\,{\epsilon }_{\hat{\alpha }\hat{\beta }{\hat{\alpha }}_{1}\cdots {\hat{\alpha }}_{d}}{A}^{\hat{\alpha }\hat{\beta }}\wedge {\rm{D}}(s,1-s){Y}^{{\hat{\alpha }}_{1}}\end{array}\,\end{eqnarray*}$
$\begin{eqnarray*}\begin{array}{l}\quad \wedge \cdots \wedge {\rm{D}}(s,1-s){Y}^{{\hat{\alpha }}_{d-1}}{Y}^{{\hat{\alpha }}_{d}}\\ \quad -{\displaystyle \int }_{0}^{1}{\rm{d}}\hat{s}\,{\epsilon }_{\hat{\alpha }\hat{\beta }{\hat{\alpha }}_{1}\cdots {\hat{\alpha }}_{d}}{\hat{A}}^{\hat{\alpha }\hat{\beta }}\wedge {\rm{D}}(1-\hat{s},\hat{s}){Y}^{{\hat{\alpha }}_{1}}\\ \quad \wedge \cdots \wedge {\rm{D}}(1-\hat{s},\hat{s}){Y}^{{\hat{\alpha }}_{d-1}}{Y}^{{\hat{\alpha }}_{d}}\\ ={\displaystyle \int }_{0}^{1}{\rm{d}}s\,{\epsilon }_{\hat{\alpha }\hat{\beta }{\hat{\alpha }}_{1}\cdots {\hat{\alpha }}_{d}}{A}^{\hat{\alpha }\hat{\beta }}\wedge {\rm{D}}(s){Y}^{{\hat{\alpha }}_{1}}\\ \quad \wedge \cdots \wedge {\rm{D}}(s){Y}^{{\hat{\alpha }}_{d-1}}{Y}^{{\hat{\alpha }}_{d}}\\ \quad -{\displaystyle \int }_{0}^{1}{\rm{d}}\hat{s}\,{\epsilon }_{\hat{\alpha }\hat{\beta }{\hat{\alpha }}_{1}\cdots {\hat{\alpha }}_{d}}{\hat{A}}^{\hat{\alpha }\hat{\beta }}\wedge \hat{{\rm{D}}(}\hat{s}){Y}^{{\hat{\alpha }}_{1}}\end{array}\,\end{eqnarray*}$
$\begin{eqnarray*}\begin{array}{l}\quad \wedge \cdots \wedge \hat{{\rm{D}}(}\hat{s}){Y}^{{\hat{\alpha }}_{d-1}}{Y}^{{\hat{\alpha }}_{d}}\\ \quad +{\displaystyle \int }_{0}^{1}{\rm{d}}s{\displaystyle \int }_{0}^{1-s}{\rm{d}}\hat{s}\,{\epsilon }_{\hat{\alpha }\hat{\beta }{\hat{\alpha }}_{1}\cdots {\hat{\alpha }}_{d}}\left\{{\partial }_{\hat{s}}[{A}^{\hat{\alpha }\hat{\beta }}\right.\\ \quad \wedge {\rm{D}}(s,\hat{s}){Y}^{{\hat{\alpha }}_{1}}\wedge \cdots \wedge {\rm{D}}(s,\hat{s}){Y}^{{\hat{\alpha }}_{d-1}}{Y}^{{\hat{\alpha }}_{d}}]\\ \quad -{\partial }_{s}[{\hat{A}}^{\hat{\alpha }\hat{\beta }}\wedge {\rm{D}}(s,\hat{s}){Y}^{{\hat{\alpha }}_{1}}\\ \quad \left.\wedge \cdots \wedge {\rm{D}}(s,\hat{s}){Y}^{{\hat{\alpha }}_{d-1}}{Y}^{{\hat{\alpha }}_{d}}]\right\}\\ ={\displaystyle \int }_{0}^{1}{\rm{d}}s\,{\epsilon }_{\hat{\alpha }\hat{\beta }{\hat{\alpha }}_{1}\cdots {\hat{\alpha }}_{d}}{A}^{\hat{\alpha }\hat{\beta }}\wedge {\rm{D}}(s){Y}^{{\hat{\alpha }}_{1}}\end{array}\,\end{eqnarray*}$
$\begin{eqnarray*}\begin{array}{l}\quad \wedge \cdots \wedge {\rm{D}}(s){Y}^{{\hat{\alpha }}_{d-1}}{Y}^{{\hat{\alpha }}_{d}}\\ \quad -{\displaystyle \int }_{0}^{1}{\rm{d}}\hat{s}\,{\epsilon }_{\hat{\alpha }\hat{\beta }{\hat{\alpha }}_{1}\cdots {\hat{\alpha }}_{d}}{\hat{A}}^{\hat{\alpha }\hat{\beta }}\wedge \hat{{\rm{D}}(}\hat{s}){Y}^{{\hat{\alpha }}_{1}}\\ \quad \wedge \cdots \wedge \hat{{\rm{D}}(}\hat{s}){Y}^{{\hat{\alpha }}_{d-1}}{Y}^{{\hat{\alpha }}_{d}}\\ \quad +{\displaystyle \int }_{0}^{1}{\rm{d}}s{\displaystyle \int }_{0}^{1-s}{\rm{d}}\hat{s}\,{\epsilon }_{\hat{\alpha }\hat{\beta }{\hat{\alpha }}_{1}\cdots {\hat{\alpha }}_{d}}\left\{(d-1){A}^{\hat{\alpha }\hat{\beta }}\right.\\ \quad \wedge {(\hat{A}Y)}^{{\hat{\alpha }}_{1}}\wedge {\rm{D}}(s,\hat{s}){Y}^{{\hat{\alpha }}_{2}}\wedge \cdots \wedge {\rm{D}}(s,\hat{s}){Y}^{{\hat{\alpha }}_{d-1}}{Y}^{{\hat{\alpha }}_{d}}\end{array}\,\end{eqnarray*}$
$\begin{eqnarray}\begin{array}{l}\quad -(d-1){\hat{A}}^{\hat{\alpha }\hat{\beta }}\wedge {(AY)}^{{\hat{\alpha }}_{1}}\wedge {\rm{D}}(s,\hat{s}){Y}^{{\hat{\alpha }}_{2}}\\ \quad \left.\wedge \cdots \wedge {\rm{D}}(s,\hat{s}){Y}^{{\hat{\alpha }}_{d-1}}{Y}^{{\hat{\alpha }}_{d}}\right\}\\ =-2{\kappa }^{2}{\ell }\,(d-1)!{{ \mathcal L }}_{(d)}^{{\rm{eff}}}[A,Y]\\ \quad -{\displaystyle \int }_{0}^{1}{\rm{d}}\hat{s}\,{\epsilon }_{\hat{\alpha }\hat{\beta }{\hat{\alpha }}_{1}\cdots {\hat{\alpha }}_{d}}{\hat{A}}^{\hat{\alpha }\hat{\beta }}\\ \quad \wedge \hat{{\rm{D}}(}\hat{s}){Y}^{{\hat{\alpha }}_{1}}\wedge \cdots \wedge \hat{{\rm{D}}(}\hat{s}){Y}^{{\hat{\alpha }}_{d-1}}{Y}^{{\hat{\alpha }}_{d}}\\ \quad +\frac{{{\ell }}^{2}(d-1)}{2{(-1)}^{d}}{\displaystyle \int }_{0}^{1}{\rm{d}}s{\displaystyle \int }_{0}^{1-s}{\rm{d}}\hat{s}\,{\epsilon }_{\hat{\alpha }\hat{\beta }{\hat{\alpha }}_{1}\cdots {\hat{\alpha }}_{d}}\\ \quad \times \,{\hat{A}}^{\hat{\alpha }\hat{\beta }}\wedge {A}^{{\hat{\alpha }}_{1}{\hat{\alpha }}_{2}}\wedge {\rm{D}}(s,\hat{s}){Y}^{{\hat{\alpha }}_{3}}\wedge \cdots \wedge {\rm{D}}(s,\hat{s}){Y}^{{\hat{\alpha }}_{d}}.\end{array}\end{eqnarray}$
It correctly reproduces the boundary term appeared in the gauge transformation ($\begin{eqnarray}{{\rm{\Delta }}}_{U}{{ \mathcal L }}_{(d+1)}^{{\rm{con}}}={\rm{d}}({{\rm{\Delta }}}_{U}{{ \mathcal L }}_{(d)}^{{\rm{eff}}}).\end{eqnarray}$
This result is consistent with the facts that ${\rm{d}}{{ \mathcal L }}_{(d)}^{{\rm{eff}}}={{ \mathcal L }}_{(d+1)}^{{\rm{con}}}-{{ \mathcal L }}_{(d+1)}^{{\rm{cov}}}$ and ${{ \mathcal L }}_{(d+1)}^{{\rm{cov}}}$ is manifestly gauge invariant.At the infinitesimal level, we have
$\begin{eqnarray}\begin{array}{l}2{\kappa }^{2}{\ell }\,(d-1)!{\delta }_{u}{{ \mathcal L }}_{(d)}^{{\rm{eff}}}\\ ={\displaystyle \int }_{0}^{1}{\rm{d}}s\,{\epsilon }_{\hat{\alpha }\hat{\beta }{\hat{\alpha }}_{1}\cdots {\hat{\alpha }}_{d}}\left[{\rm{d}}{u}^{\hat{\alpha }\hat{\beta }}\right.\\ \quad \wedge {\rm{d}}{Y}^{{\hat{\alpha }}_{1}}\wedge \cdots \wedge {\rm{d}}{Y}^{{\hat{\alpha }}_{d-1}}{Y}^{{\hat{\alpha }}_{d}}\\ \quad -\frac{{{\ell }}^{2}(d-1)}{2{(-1)}^{d}}{\displaystyle \int }_{0}^{1-s}{\rm{d}}\hat{s}\,{\rm{d}}{u}^{\hat{\alpha }\hat{\beta }}\wedge {A}^{{\hat{\alpha }}_{1}{\hat{\alpha }}_{2}}\\ \quad \left.\wedge {\rm{D}}(s){Y}^{{\hat{\alpha }}_{3}}\wedge \cdots \wedge {\rm{D}}(s){Y}^{{\hat{\alpha }}_{d}}\right]\\ ={\rm{d}}\left\{{\epsilon }_{\hat{\alpha }\hat{\beta }{\hat{\alpha }}_{1}\cdots {\hat{\alpha }}_{d}}{u}^{\hat{\alpha }\hat{\beta }}\left[{\rm{d}}{Y}^{{\hat{\alpha }}_{1}}\right.\right.\\ \quad \wedge \cdots \wedge {\rm{d}}{Y}^{{\hat{\alpha }}_{d-1}}{Y}^{{\hat{\alpha }}_{d}}\\ \quad -{\displaystyle \int }_{0}^{1}{\rm{d}}s\frac{{{\ell }}^{2}(d-1)(1-s)}{2{(-1)}^{d}}{A}^{{\hat{\alpha }}_{1}{\hat{\alpha }}_{2}}\\ \quad \left.\left.\wedge {\rm{D}}(s){Y}^{{\hat{\alpha }}_{3}}\wedge \cdots \wedge {\rm{D}}(s){Y}^{{\hat{\alpha }}_{d}}\right]\right\}\\ \quad +{(-1)}^{d}{\epsilon }_{\hat{\alpha }\hat{\beta }{\hat{\alpha }}_{1}\cdots {\hat{\alpha }}_{d}}{u}^{\hat{\alpha }\hat{\beta }}\\ \quad \times \,\left[{\rm{d}}{Y}^{{\hat{\alpha }}_{1}}\wedge \cdots \wedge {\rm{d}}{Y}^{{\hat{\alpha }}_{d}}\right.\\ \quad +{\displaystyle \int }_{0}^{1}{\rm{d}}s\frac{{{\ell }}^{2}(d-1)(1-s)}{2}{\rm{d}}\left({A}^{{\hat{\alpha }}_{1}{\hat{\alpha }}_{2}}\right.\\ \quad \left.\left.\wedge {\rm{D}}(s){Y}^{{\hat{\alpha }}_{3}}\wedge \cdots \wedge {\rm{D}}(s){Y}^{{\hat{\alpha }}_{d}}\right)\right],\end{array}\end{eqnarray}$
thus the boundary consistent anomaly is $\begin{eqnarray}\begin{array}{l}2{\kappa }^{2}{\ell }\,(d-1)!{({{ \mathcal A }}_{(d)}^{{\rm{con}}})}_{\hat{\alpha }\hat{\beta }}\\ ={(-1)}^{d}{\epsilon }_{\hat{\alpha }\hat{\beta }{\hat{\alpha }}_{1}\cdots {\hat{\alpha }}_{d}}\left[{\rm{d}}{Y}^{{\hat{\alpha }}_{1}}\wedge \cdots \wedge {\rm{d}}{Y}^{{\hat{\alpha }}_{d}}\right.\\ \quad +{\displaystyle \int }_{0}^{1}{\rm{d}}s\frac{{{\ell }}^{2}(d-1)(1-s)}{2}{\rm{d}}\left({A}^{{\hat{\alpha }}_{1}{\hat{\alpha }}_{2}}\right.\\ \quad \left.\left.\wedge {\rm{D}}(s){Y}^{{\hat{\alpha }}_{3}}\wedge \cdots \wedge {\rm{D}}(s){Y}^{{\hat{\alpha }}_{d}}\right)\right].\end{array}\end{eqnarray}$
It satisfies that $\begin{eqnarray}\begin{array}{rcl}{\rm{d}}[{u}^{\hat{\alpha }\hat{\beta }}{({{ \mathcal A }}_{(d)}^{{\rm{con}}})}_{\hat{\alpha }\hat{\beta }}] & = & {\delta }_{u}{I}_{(d+1)}^{{\rm{con}}}\\ & = & {\delta }_{u}{I}_{(d+1)}^{{\rm{CS}}}-{\delta }_{u}{{ \mathcal L }}_{(d+1)}^{{\rm{top}}}.\end{array}\end{eqnarray}$