1. Introduction
The idea of unifying the Standard Model (SM) gauge interactions into a Grand Unified Theory (GUT) has an old history [1]. In order for such a GUT to be both tenable and internally consistent, it necessitates a symmetry group characterized by a simple Lie group, one that encompasses the SM’s group SU(3)C × SU(2)L × U(1)Y as a subgroup, and crucially, permits complex representations to enable the emergence of a chiral theory. The fundamental examples within the unitary and orthogonal group families are the well-recognized SU(5) and SO(10) groups. Later, with the advent of heterotic string theory, the gauge group E6 ⊂ E8 has become more important. Interestingly, the trinification group SU(3) × SU(3) × SU(3) is also one of the subgroups of E6 which has recently gained renewed interest from E6 symmetry breaking. In the trinification models, the quarks, leptons and Higgs fields all arise from the bifundamental representations of the gauge group SU(3)C × SU(3)L × SU(3)R.
Apart from choosing the GUT gauge group, embedding the SM into string theory also involves a certain choice of Calabi–Yau compactification. The SM fermions belong to the chiral representations of the gauge group SU(3)C × SU(2)L × U(1)Y such that all the gauge anomalies are canceled. In Type II string theory, simply placing the parallel D-branes in flat space does not yield chiral fermions. Instead, to realize the chiral fermions we need to consider either the D-branes on orbifold singularities [2] or the intersecting D-branes on generalized orbifolds called orientifolds [3, 4]. In orientifolds, both the discrete internal symmetries of the world-sheet theory and the products of internal symmetries with world-sheet parity reversal become gauged. In Type IIA string theory, setting the intersecting D6-branes over orientifolds provides a geometric framework to generate the gauge structures, chiral spectra, and various couplings. In particular, the four-dimensional gauge couplings depend on the volume of the cycles wrapped by the D6-branes, and the gravitational coupling is determined by their total internal volume. Similarly, the cubic Yukawa couplings depend exponentially on the triangular areas of open worldsheet intersections which naturally explain the observed fermion mass hierarchies. The general flavor structure and selection rules for intersecting D6-brane models have been investigated in [5, 6].
In [7] one of us (TL) had constructed the first consistent three-family supersymmetric trinification model with gauge group SU(3)C × SU(3)L × SU(3)R from intersecting D6-branes over Type IIA ${{\mathbb{T}}}^{6}/({{\mathbb{Z}}}_{2}\times {{\mathbb{Z}}}_{2})$ orientifold. It was pointed out that unlike the Pati–Salam gauge group, the RR tadpole cancellation and K-theory conditions for the trinification gauge group are much more difficult to satisfy. More recently, in the context of supersymmetric Pati–Salam models, the complete landscape with three families [8–12] and four families [13] has already been enumerated. In fact, the SM does not provide a theoretical prediction regarding the number of families. Allowing for a fourth family naturally leads to the flavor democracy hypothesis as discussed in [14]. The particles from the massive fourth generation would alter the cross sections and the decay channels of the Higgs particles and when the Yukawa couplings of the fourth generation particles are large enough, these particles are natural candidates for electroweak symmetry breaking [15].
In this paper we present results of the four-family supersymmetric trinification models from the intersecting D6-branes on a Type IIA ${{\mathbb{T}}}^{6}/({{\mathbb{Z}}}_{2}\times {{\mathbb{Z}}}_{2})$ orientifold. The plan of the paper is as follows. In section 2 , we review the model building rules for ${ \mathcal N }=1$ supersymmetric trinification models with gauge group SU(3)C × SU(3)L × SU(3)R on a ${{\mathbb{T}}}^{6}/({{\mathbb{Z}}}_{2}\times {{\mathbb{Z}}}_{2})$ orientifold. In section 2.2 we discuss the gauge symmetry breaking via the brane-splitting mechanism to yield the SM gauge group. In section 3 we discuss the salient phenomenological features and comment on the decoupling of the various exotic particles from the low energy spectra through strong dynamics in a particular model. Finally, we conclude in section 4 .
2. Trinification model building from ${{\mathbb{T}}}^{6}/({{\mathbb{Z}}}_{2}\times {{\mathbb{Z}}}_{2})$ orientifold
In the orientifold ${{\mathbb{T}}}^{6}/({{\mathbb{Z}}}_{2}\times {{\mathbb{Z}}}_{2})$, ${{\mathbb{T}}}^{6}$ is treated as a product of three two-tori with the orbifold group $({{\mathbb{Z}}}_{2}\times {{\mathbb{Z}}}_{2})$, having the generators θ and ω that are associated with the twist vectors (1/2, −1/2, 0) and (0, 1/2, −1/2) respectively such that their action on zi is given by
$\begin{eqnarray}\begin{array}{rcl}\theta :\quad ({z}_{1},{z}_{2},{z}_{3}) & \to & (-{z}_{1},-{z}_{2},{z}_{3}),\\ \omega :\quad ({z}_{1},{z}_{2},{z}_{3}) & \to & ({z}_{1},-{z}_{2},-{z}_{3}).\end{array}\end{eqnarray}$
Orientifold projection is the gauged ${\rm{\Omega }}{ \mathcal R }$ symmetry, where Ω denotes the world-sheet parity that interchanges the left- and right-movers of a closed string and acts on the ends of an open string as $\begin{eqnarray}\begin{array}{rcl}\mathrm{Closed}:{\rm{\Omega }}:({\sigma }_{1},{\sigma }_{2}) & \mapsto & (2\pi -{\sigma }_{1},{\sigma }_{2}),\\ \,\mathrm{Open}:{\rm{\Omega }}:(\tau ,\sigma ) & \mapsto & (\tau ,\pi -\sigma ),\end{array}\end{eqnarray}$
and ${ \mathcal R }$ represents the complex conjugation which acts on the complex coordinates zi. This leads to the generations of four different kinds of orientifold 6-planes (O6-planes) related to ${\rm{\Omega }}{ \mathcal R }$, ${\rm{\Omega }}{ \mathcal R }\theta $, ${\rm{\Omega }}{ \mathcal R }\omega $, and ${\rm{\Omega }}{ \mathcal R }\theta \omega $ respectively. These orientifold projections are only consistent with either the rectangular or the tilted complex structures of the factorized two-tori. The wrapping numbers for the rectangular and tilted two-tori are denoted as ${n}_{a}^{i}[{a}_{i}]+{m}_{a}^{i}[{b}_{i}]$ and ${n}_{a}^{i}[a{{\prime} }_{i}]+{m}_{a}^{i}[{b}_{i}]$ respectively, where $[{a}_{i}^{\prime} ]=[{a}_{i}]+\tfrac{1}{2}[{b}_{i}]$. Then for the rectangular and tilted two-torus, generic 1-cycle satisfies ${l}_{a}^{i}\equiv {m}_{a}^{i}$, and ${l}_{a}^{i}\equiv 2{\tilde{m}}_{a}^{i}\,=2{m}_{a}^{i}+{n}_{a}^{i}$ respectively. The homology cycles for a stack a of Na D6-branes along the cycle $({n}_{a}^{i},{l}_{a}^{i})$ and their ${\rm{\Omega }}{ \mathcal R }$ images $a^{\prime} $ are defined as $\begin{eqnarray}\begin{array}{rcl}[{{\rm{\Pi }}}_{a}] & = & \displaystyle \prod _{i=1}^{3}\left({n}_{a}^{i}[{a}_{i}]+{2}^{-{\beta }_{i}}{l}_{a}^{i}[{b}_{i}]\right),\\ \left[{{\rm{\Pi }}}_{a^{\prime} }\right] & = & \displaystyle \prod _{i=1}^{3}\left({n}_{a}^{i}[{a}_{i}]-{2}^{-{\beta }_{i}}{l}_{a}^{i}[{b}_{i}]\right),\end{array}\end{eqnarray}$
where βi = 0 or βi = 1 for the rectangular or tilted ith two-tori respectively.The homology 3-cycles wrapped by the four O6-planes are defined as
$\begin{eqnarray}\begin{array}{rcl}{\rm{\Omega }}{ \mathcal R }:\quad [{{\rm{\Pi }}}_{{\rm{\Omega }}{ \mathcal R }}] & = & {2}^{3}[{a}_{1}]\times [{a}_{2}]\times [{a}_{3}],\\ {\rm{\Omega }}{ \mathcal R }\omega :\quad [{{\rm{\Pi }}}_{{\rm{\Omega }}{ \mathcal R }\omega }] & = & -{2}^{3-{\beta }_{2}-{\beta }_{3}}[{a}_{1}]\times [{b}_{2}]\times [{b}_{3}],\\ {\rm{\Omega }}{ \mathcal R }\theta \omega :\quad [{{\rm{\Pi }}}_{{\rm{\Omega }}{ \mathcal R }\theta \omega }] & = & -{2}^{3-{\beta }_{1}-{\beta }_{3}}[{b}_{1}]\times [{a}_{2}]\times [{b}_{3}],\\ {\rm{\Omega }}{ \mathcal R }\theta :[{{\rm{\Pi }}}_{{\rm{\Omega }}{ \mathcal R }\theta }] & = & -{2}^{3-{\beta }_{1}-{\beta }_{2}}[{b}_{1}]\times [{b}_{2}]\times [{a}_{3}].\end{array}\end{eqnarray}$
The intersection numbers can be calculated in terms of wrapping numbers as, $\begin{eqnarray}\begin{array}{rcl}{I}_{{ab}} & = & [{{\rm{\Pi }}}_{a}][{{\rm{\Pi }}}_{b}]={2}^{-k}\displaystyle \prod _{i=1}^{3}({n}_{a}^{i}{l}_{b}^{i}-{n}_{b}^{i}{l}_{a}^{i}),\\ {I}_{{ab}^{\prime} } & = & [{{\rm{\Pi }}}_{a}]\left[{{\rm{\Pi }}}_{b^{\prime} }\right]=-{2}^{-k}\displaystyle \prod _{i=1}^{3}({n}_{a}^{i}{l}_{b}^{i}+{n}_{b}^{i}{l}_{a}^{i}),\\ {I}_{{aa}^{\prime} } & = & [{{\rm{\Pi }}}_{a}]\left[{{\rm{\Pi }}}_{a^{\prime} }\right]=-{2}^{3-k}\displaystyle \prod _{i=1}^{3}({n}_{a}^{i}{l}_{a}^{i}),\\ {I}_{{aO}6} & = & [{{\rm{\Pi }}}_{a}][{{\rm{\Pi }}}_{{\rm{O}}6}]={2}^{3-k}(-{l}_{a}^{1}{l}_{a}^{2}{l}_{a}^{3}+{l}_{a}^{1}{n}_{a}^{2}{n}_{a}^{3}\\ & & +{n}_{a}^{1}{l}_{a}^{2}{n}_{a}^{3}+{n}_{a}^{1}{n}_{a}^{2}{l}_{a}^{3}),\end{array}\end{eqnarray}$
where $k={\sum }_{i=1}^{3}{\beta }_{i}$ and $[{{\rm{\Pi }}}_{{\rm{O}}6}]=[{{\rm{\Pi }}}_{{\rm{\Omega }}{ \mathcal R }}]+[{{\rm{\Pi }}}_{{\rm{\Omega }}{ \mathcal R }\omega }]+[{{\rm{\Pi }}}_{{\rm{\Omega }}{ \mathcal R }\theta \omega }]\,+[{{\rm{\Pi }}}_{{\rm{\Omega }}{ \mathcal R }\theta }]$.2.1. Constraints from tadpole cancellation and supersymmetry
As D6-branes and O6-orientifold planes contain the Ramond–Ramond charges, therefore they must satisfy the Gauss’s law in compact space implying the sum of D-brane and cross-cap RR-charges must vanish [16]
$\begin{eqnarray}\displaystyle \sum _{a}{N}_{a}[{{\rm{\Pi }}}_{a}]+\displaystyle \sum _{a}{N}_{a}\left[{{\rm{\Pi }}}_{a^{\prime} }\right]-4[{{\rm{\Pi }}}_{{\rm{O}}6}]=0,\end{eqnarray}$
where the last contribution comes from the O6-planes, which have −4 RR charges in D6-brane charge units. The constraint from RR tadpole is sufficient to cancel the $\mathrm{SU}{\left({N}_{a}\right)}^{3}$ cubic non-Abelian anomaly, while Green–Schwarz mechanism, mediated by untwisted RR fields [17], plays a crucial rule in canceling the U(1) mixed gauge and gravitational or ${[\mathrm{SU}({N}_{a})]}^{2}{\rm{U}}(1)$ gauge anomaly.For the sake of convenience, let us define the following products of wrapping numbers
$\begin{eqnarray}\begin{array}{rclrcl}{A}_{a} & \equiv & -{n}_{a}^{1}{n}_{a}^{2}{n}_{a}^{3},\quad {B}_{a}\equiv {n}_{a}^{1}{l}_{a}^{2}{l}_{a}^{3}, & {C}_{a} & \equiv & {l}_{a}^{1}{n}_{a}^{2}{l}_{a}^{3},\quad {D}_{a}\equiv {l}_{a}^{1}{l}_{a}^{2}{n}_{a}^{3},\\ {\tilde{A}}_{a} & \equiv & -{l}_{a}^{1}{l}_{a}^{2}{l}_{a}^{3},\quad {\tilde{B}}_{a}\equiv {l}_{a}^{1}{n}_{a}^{2}{n}_{a}^{3}, & {\tilde{C}}_{a} & \equiv & {n}_{a}^{1}{l}_{a}^{2}{n}_{a}^{3},\quad {\tilde{D}}_{a}\equiv {n}_{a}^{1}{n}_{a}^{2}{l}_{a}^{3}.\end{array}\end{eqnarray}$
Cancellation of RR tadpoles requires the orientifold plane numbers also called ‘filler branes’ that satisfy, trivially, the four-dimensional ${ \mathcal N }=1$ supersymmetry conditions. Thus, the no-tadpole condition is given as $\begin{eqnarray}\begin{array}{rcl}-{2}^{k}{N}^{(1)}+\displaystyle \sum _{a}{N}_{a}{A}_{a} & = & -{2}^{k}{N}^{(2)}+\displaystyle \sum _{a}{N}_{a}{B}_{a}\\ =-{2}^{k}{N}^{(3)}+\displaystyle \sum _{a}{N}_{a}{C}_{a} & = & -{2}^{k}{N}^{(4)}+\displaystyle \sum _{a}{N}_{a}{D}_{a}=-16,\end{array}\end{eqnarray}$
where 2N(i) represents the number of filler branes wrapping along the ith O6-plane. The filler branes belong to the hidden sector USp group and carry the same wrapping numbers as one of the O6-planes as shown in table 1. USp group is hence referred with respect to the non-zero A, B, C or D-type.Table 1. The wrapping numbers for four O6-planes. |
| Orientifold action | O6-plane | (n1, l1) × (n2, l2) × (n3, l3) |
|---|---|---|
| ${\rm{\Omega }}{ \mathcal R }$ | 1 | $({2}^{{\beta }_{1}},0)\times ({2}^{{\beta }_{2}},0)\times ({2}^{{\beta }_{3}},0)$ |
| ${\rm{\Omega }}{ \mathcal R }\omega $ | 2 | $({2}^{{\beta }_{1}},0)\times (0,-{2}^{{\beta }_{2}})\times (0,{2}^{{\beta }_{3}})$ |
| ${\rm{\Omega }}{ \mathcal R }\theta \omega $ | 3 | $(0,-{2}^{{\beta }_{1}})\times ({2}^{{\beta }_{2}},0)\times (0,{2}^{{\beta }_{3}})$ |
| ${\rm{\Omega }}{ \mathcal R }\theta $ | 4 | $(0,-{2}^{{\beta }_{1}})\times (0,{2}^{{\beta }_{2}})\times ({2}^{{\beta }_{3}},0)$ |
Preserving ${ \mathcal N }=1$ supersymmetry in four dimensions shows that the rotation angle of any D6-brane with respect to the orientifold plane must be an element of SU(3), i.e.
$\begin{eqnarray}{\theta }_{1}^{a}+{\theta }_{2}^{a}+{\theta }_{3}^{a}=0\mathrm{mod}2\pi ,\end{eqnarray}$
with ${\theta }_{j}^{a}=\arctan ({2}^{-{\beta }_{j}}{\chi }_{j}{l}_{j}^{a}/{n}_{j}^{a})$. θi is the angle between the D6-brane and orientifold-plane in the ith two-tori and ${\chi }_{i}={R}_{i}^{2}/{R}_{i}^{1}$ are the complex structure moduli for the ith two-tori. ${ \mathcal N }=1$ supersymmetry conditions are given as $\begin{eqnarray}\begin{array}{c}{x}_{A}{\tilde{A}}_{a}+{x}_{B}{\tilde{B}}_{a}+{x}_{C}{\tilde{C}}_{a}+{x}_{D}{\tilde{D}}_{a}=0,\\ \quad \frac{{A}_{a}}{{x}_{A}}+\frac{{B}_{a}}{{x}_{B}}+\frac{{C}_{a}}{{x}_{C}}+\frac{{D}_{a}}{{x}_{D}}\lt 0,\end{array}\end{eqnarray}$
where ${x}_{A}=\lambda ,{x}_{B}={2}^{{\beta }_{2}+{\beta }_{3}}\cdot \lambda /{\chi }_{2}{\chi }_{3}$, ${x}_{C}={2}^{{\beta }_{1}+{\beta }_{3}}\cdot \lambda /{\chi }_{1}{\chi }_{3}$, ${x}_{D}={2}^{{\beta }_{1}+{\beta }_{2}}\cdot \lambda /{\chi }_{1}{\chi }_{2}$.Orientifolds also contain the discrete D-brane RR charges represented by the ${{\mathbb{Z}}}_{2}$ K-theory groups, which are invisible by the ordinary homology [18–21], should also be taken into account [22] during model building. The K-theory conditions read [7]
$\begin{eqnarray}\begin{array}{l}\displaystyle \sum _{a}{N}_{a}{A}_{a}=\displaystyle \sum _{a}{N}_{a}{B}_{a}=\displaystyle \sum _{a}{N}_{a}{C}_{a}\\ =\displaystyle \sum _{a}{N}_{a}{D}_{a}=0\mathrm{mod}4.\end{array}\end{eqnarray}$
The multiplicities of the particles for intersecting D6-branes models are shown in table 2. By following the convention of [23], the N number of D6-brane stacks corresponds to U(N/2) and USp(N) respectively. A positive intersection number in our convention refers to the left-chiral supermultiplet, and vice versa.
Table 2. General spectrum for intersecting D6-branes at generic angles, where ${ \mathcal M }$ is the multiplicity, and ${a}_{\square \square }$ and denote respectively the symmetric and antisymmetric representations of ${\rm{U}}\left({N}_{a}/2\right)$. Positive intersection numbers in our convention refer to the left-handed chiral supermultiplets. |
| Sector | Representation |
|---|---|
| aa | U(Na/2) vector multiplet |
| 3 adjoint chiral multiplets | |
| ab + ba | ${ \mathcal M }\left(\tfrac{{N}_{a}}{2},\tfrac{\overline{{N}_{b}}}{2}\right)$ = ${I}_{{ab}}({\,\square }_{a}\,,{\overline{\square }}_{b}\,)$ |
| ab’ + b’a | ${ \mathcal M }\left(\tfrac{{N}_{a}}{2},\tfrac{{N}_{b}}{2}\right)$ = ${I}_{{ab}^{\prime} }({\,\square }_{a}\,,{\square }_{b}\,)$ |
| aa’ + a’a | ${ \mathcal M }({a}_{\square \square })$ = $\tfrac{1}{2}({I}_{{aa}^{\prime} }-\tfrac{1}{2}{I}_{{aO}6})$ |
| ${ \mathcal M }$() = $\tfrac{1}{2}({I}_{{aa}^{\prime} }+\tfrac{1}{2}{I}_{{aO}6})$ |
Within trinification models, quarks, leptons, and the Higgs boson all originate from the bifundamental representations of the gauge group SU(3)C × SU(3)L × SU(3)R. Accordingly, we require three stacks of six D6-branes to yield three SU(3) gauge group factors. If the number of generations is ng, and nh be the number of higgs, then we have
$\begin{eqnarray}\begin{array}{rcl}{n}_{g} & = & {I}_{{ab}}+{I}_{{ab}^{\prime} }=-({I}_{{ac}}+{I}_{{ac}^{\prime} }),\\ {n}_{h} & = & \left|{I}_{{bc}}+{I}_{{bc}^{\prime} }\right|-{n}_{g}.\end{array}\end{eqnarray}$
2.2. Gauge symmetry breaking
The U(3)C × U(3)L × U(3)R gauge symmetry is broken down to the SU(3)C × SU(3)L × SU(3)R gauge symmetry via the Green–Schwartz mechanism [24–26]. The resulting SU(3)C × SU(3)L × SU(3)R gauge symmetry can be broken down to the $\mathrm{SU}{(3)}_{C}\times \mathrm{SU}{(2)}_{L}\times {\rm{U}}{(1)}_{{Y}_{L}}\times {\rm{U}}{(1)}_{{I}_{3R}}\times {\rm{U}}{(1)}_{{Y}_{R}}$ gauge symmetry by the splittings of the U(3)L and U(3)R stacks of the D6-branes.
The SM electromagnetic charge generator QY is defined as
$\begin{eqnarray}{Q}_{Y}\equiv {I}_{3L}-\displaystyle \frac{{Y}_{L}}{2}+{I}_{3R}-\displaystyle \frac{{Y}_{R}}{2},\end{eqnarray}$
where the generators for ${\rm{U}}{(1)}_{{I}_{3L}}$ and ${\rm{U}}{(1)}_{{I}_{3R}}$, and ${\rm{U}}{(1)}_{{Y}_{L}}$ and ${\rm{U}}{(1)}_{{Y}_{R}}$ in SU(3)L and SU(3)R gauge symmetries are $\begin{eqnarray}{{\boldsymbol{T}}}_{{\rm{U}}{(1)}_{{I}_{3L,R}}}=\left(\begin{array}{ccc}\displaystyle \frac{1}{2} & 0 & 0\\ 0 & -\displaystyle \frac{1}{2} & 0\\ 0 & 0 & 0\end{array}\right),\end{eqnarray}$
$\begin{eqnarray}{{\boldsymbol{T}}}_{{\rm{U}}{(1)}_{{Y}_{L,R}}}=\left(\begin{array}{ccc}\displaystyle \frac{1}{3} & 0 & 0\\ 0 & \displaystyle \frac{1}{3} & 0\\ 0 & 0 & -\displaystyle \frac{2}{3}\end{array}\right).\end{eqnarray}$
And the explicit particle components in the bifundamental quarks ${{\boldsymbol{Q}}}_{L,R}^{i}$, leptons Li, and higgs Hi representations are
$\begin{eqnarray}({\bf{3}},\bar{{\bf{3}}},{\bf{1}}):\ \ {{\boldsymbol{Q}}}_{L}^{i}=\left(\begin{array}{ccc}d & u & h\\ d & u & h\\ d & u & h\end{array}\right)\,,\,\,\end{eqnarray}$
$\begin{eqnarray}(\bar{{\bf{3}}},{\bf{1}},{\bf{3}}):\ \ {{\boldsymbol{Q}}}_{R}^{i}=\left(\begin{array}{ccc}{d}^{c} & {d}^{c} & {d}^{c}\\ {u}^{c} & {u}^{c} & {u}^{c}\\ {h}^{c} & {h}^{c} & {h}^{c}\end{array}\right)\,,\,\,\end{eqnarray}$
$\begin{eqnarray}({\bf{1}},{\bf{3}},\bar{{\bf{3}}}):\ \ {{\boldsymbol{L}}}^{i}\ \mathrm{or}\ {{\boldsymbol{H}}}^{k}=\left(\begin{array}{ccc}N & {E}^{c} & \nu \\ E & {N}^{c} & e\\ {\nu }^{c} & {e}^{c} & S\end{array}\right)\,.\,\,\end{eqnarray}$
Giving VEVs to the singlet Higgs fields νc and S, we can break the ${\rm{U}}{(1)}_{{Y}_{L}}\times {\rm{U}}{(1)}_{{I}_{3R}}\times {\rm{U}}{(1)}_{{Y}_{R}}$ gauge symmetry down to the U(1)Y hypercharge interaction
$\begin{eqnarray}\langle {\nu }^{c}\rangle \ne 0,\qquad \langle S\rangle \ne 0.\end{eqnarray}$
The electric charges for h and hc are respectively $-\tfrac{1}{3}$ and $\tfrac{1}{3}$, for E and Ec are respectively −1 and 1; and for N, Nc, and S are zero. The complete gauge symmetry breaking chains are $\begin{eqnarray}\begin{array}{l}\mathrm{SU}{(3)}_{C}\times \mathrm{SU}{(3)}_{L}\times \mathrm{SU}{(3)}_{R}\\ \mathop{\longrightarrow }\limits^{{\rm{Splitting}}}\quad \mathrm{SU}{(3)}_{C}\times \mathrm{SU}{(2)}_{L}\times {\rm{U}}{(1)}_{{Y}_{L}}\\ \times {\rm{U}}{(1)}_{{I}_{3R}}\times {\rm{U}}{(1)}_{{Y}_{R}}\\ \mathop{\longrightarrow }\limits^{{\rm{VEVs}}}\quad \mathrm{SU}{(3)}_{C}\times \mathrm{SU}{(2)}_{L}\times {\rm{U}}{(1)}_{Y}.\end{array}\end{eqnarray}$
The process of dynamical supersymmetry breaking has been studied in [27] for D6-brane models from Type IIA orientifolds. The complex structure moduli U can be obtained from the supersymmetry conditions as [10]10 ). Actually, one can determine χ1, χ2, χ3 up to an overall coefficient, namely an action of dilation on these parameters. So one has to stabilize this dilation to determine all the moduli parameters. The holomorphic gauge kinetic function for any D6-brane stack x wrapping a calibrated 3-cycle is given as [29], 13 ), the holomorphic gauge kinetic function for the hypercharge is also taken as a linear combination of the kinetic gauge functions from all of the stacks as [25, 32],
$\begin{eqnarray}{U}^{i}=\displaystyle \frac{4{\rm{i}}{\chi }^{i}+2{\beta }_{i}^{2}{\chi }_{i}^{2}}{4+{\beta }_{i}^{2}{\chi }_{i}^{2}},\qquad \because {\chi }^{i}\equiv \displaystyle \frac{{R}_{2}^{i}}{{R}_{1}^{i}}.\end{eqnarray}$
These upper case moduli in string theory basis can be transformed in to lower case s, t, u moduli in field theory basis as [28] $\begin{eqnarray}\begin{array}{rcl}\mathrm{Re}(s) & = & \displaystyle \frac{{{\rm{e}}}^{-{\phi }_{4}}}{2\pi }\displaystyle \frac{\sqrt{\mathrm{Im}({U}^{1})\,\mathrm{Im}({U}^{2})\,\mathrm{Im}({U}^{3})}}{\left|{U}^{1}{U}^{2}{U}^{3}\right|},\\ \mathrm{Re}({u}^{1}) & = & \displaystyle \frac{{{\rm{e}}}^{-{\phi }_{4}}}{2\pi }\sqrt{\displaystyle \frac{\mathrm{Im}({U}^{1})}{\mathrm{Im}({U}^{2})\,\mathrm{Im}({U}^{3})}}\left|\displaystyle \frac{{U}^{2}{U}^{3}}{{U}^{1}}\right|,\\ \mathrm{Re}({u}^{2}) & = & \displaystyle \frac{{{\rm{e}}}^{-{\phi }_{4}}}{2\pi }\sqrt{\displaystyle \frac{\mathrm{Im}({U}^{2})}{\mathrm{Im}({U}^{1})\,\mathrm{Im}({U}^{3})}}\left|\displaystyle \frac{{U}^{1}{U}^{3}}{{U}^{2}}\right|,\\ \mathrm{Re}({u}^{3}) & = & \displaystyle \frac{{{\rm{e}}}^{-{\phi }_{4}}}{2\pi }\sqrt{\displaystyle \frac{\mathrm{Im}({U}^{3})}{\mathrm{Im}({U}^{1})\,\mathrm{Im}({U}^{2})}}\left|\displaystyle \frac{{U}^{1}{U}^{2}}{{U}^{3}}\right|.\end{array}\end{eqnarray}$
where φ4 is the four dimensional dilaton which is related to the supergravity moduli as $\begin{eqnarray}2\pi {{\rm{e}}}^{{\phi }_{4}}={\left(\mathrm{Re}(s)\mathrm{Re}({u}_{1})\mathrm{Re}({u}_{2})\mathrm{Re}({u}_{3})\right)}^{-1/4}.\end{eqnarray}$
Note that the three moduli parameter χ1, χ2, χ3 are not independent, as they can be expressed in terms of xA, xB, xC, xD and the latter parameters are related by the supersymmetric condition equation ( $\begin{eqnarray}{f}_{x}=\displaystyle \frac{1}{2\pi {{\ell }}_{s}^{3}}\left[{{\rm{e}}}^{-\phi }{\int }_{{{\rm{\Pi }}}_{x}}{\rm{Re}}({{\rm{e}}}^{-{\rm{i}}{\theta }_{x}}{{\rm{\Omega }}}_{3})-{\rm{i}}{\int }_{{{\rm{\Pi }}}_{x}}{C}_{3}\right],\end{eqnarray}$
where the integral involving 3-form Ω3 gives $\begin{eqnarray}{\int }_{{{\rm{\Pi }}}_{x}}{{\rm{\Omega }}}_{3}=\displaystyle \frac{1}{4}\displaystyle \prod _{i=1}^{3}({n}_{x}^{i}{R}_{1}^{i}+{2}^{-{\beta }_{i}}{\rm{i}}\,{l}_{x}^{i}{R}_{2}^{i}).\end{eqnarray}$
It can then be shown that $\begin{eqnarray}\begin{array}{l}{f}_{x}=\displaystyle \frac{1}{4{\kappa }_{x}}\left({n}_{x}^{1}\,{n}_{x}^{2}\,{n}_{x}^{3}\,s-\displaystyle \frac{{n}_{x}^{1}\,{l}_{x}^{2}\,{l}_{x}^{3}\,{u}^{1}}{{2}^{({\beta }_{2}+{\beta }_{3})}}\right.\\ \left.\quad -\displaystyle \frac{{l}_{x}^{1}\,{n}_{x}^{2}\,{l}_{x}^{3}\,{u}^{2}}{{2}^{({\beta }_{1}+{\beta }_{3})}}-\displaystyle \frac{{l}_{x}^{1}\,{l}_{x}^{2}\,{n}_{x}^{3}\,{u}^{3}}{{2}^{({\beta }_{1}+{\beta }_{2})}}\right),\end{array}\end{eqnarray}$
where the factor κx is the Kac–Moody level of the corresponding gauge Kac–Moody algebra associated to the D6-brane stacks such that κx = 1 for U(Nx) and κx = 2 for USp(2Nx) or SO(2Nx) [30, 31]. The gauge coupling constant related to any stack x of D6-branes is $\begin{eqnarray}{g}_{x}^{-2}=\left|\mathrm{Re}({f}_{x})\right|.\end{eqnarray}$
Since, the SM hypercharge U(1)Y is a linear combination of several U(1)s as defined in equation ( $\begin{eqnarray}{f}_{Y}={f}_{b}+{f}_{c},\end{eqnarray}$
where we have taken into account that our U(1)s are not canonically normalized. And the coupling constant for the hypercharge gY can be determined as $\begin{eqnarray}{g}_{Y}^{-2}=\left|\mathrm{Re}({f}_{Y})\right|.\end{eqnarray}$
Hence, the tree-level gauge couplings have the relation $\begin{eqnarray}{g}_{a}^{2}=\alpha {g}_{b}^{2}=\beta \displaystyle \frac{5}{3}{g}_{Y}^{2}=\gamma (\pi {{\rm{e}}}^{-{\phi }_{4}}),\end{eqnarray}$
where α, β and γ denote the strengths of the strong coupling, the weak coupling, and the hypercharge coupling respectively. Moreover, the Kähler potential takes the following form $\begin{eqnarray}K=-\mathrm{ln}(S+\overline{S})-\displaystyle \sum _{i=1}^{3}\mathrm{ln}({U}^{i}+{\overline{U}}^{i}).\end{eqnarray}$
And the one-loop beta functions for each hidden sector gauge group $\begin{eqnarray}\begin{array}{l}{\beta }_{i}^{g}=-3\left(\displaystyle \frac{{N}^{(i)}}{2}+1\right)+2\left|{I}_{{ai}}\right|\\ \,+\,\left|{I}_{{bi}}\right|+\left|{I}_{{ci}}\right|+3\left(\displaystyle \frac{{N}^{(i)}}{2}-1\right)\\ =-6+2\left|{I}_{{ai}}\right|+\left|{I}_{{bi}}\right|+\left|{I}_{{ci}}\right|\,\end{array}\end{eqnarray}$
are required to be negative [23].Since we have large RR charges from three SU(3) groups, the construction becomes very difficult. Furthermore, unlike the Pati–Salam gauge symmetry where we avoid the nonvanishing torsion charges by taking an even number of D-branes, here in trinification gauge group, due to the appearance of three odd gauge factors of SU(3) the K-theory conditions turn out to be highly restrictive. Consequently, we were not able to construct any three-family trinification model for the ${{\mathbb{T}}}^{6}/({{\mathbb{Z}}}_{2}\times {{\mathbb{Z}}}_{2})$ compactification which requires atleast one two-torus to be tilted. We next turn our attention to the construction of four-family models which only require rectangular two-tori where RR tadpoles can be satisfied with lesser difficulty.
3. Particle spectra of four-family supersymmetric trinification models
Employing novel random and supervised scanning algorithm [33, 34] we have systematically scanned for the ${ \mathcal N }=1$ supersymmetric trinification models with the gauge group SU(3)C × SU(3)L × SU(3)R. We have obtained some new models where the largest observed wrapping number is 2. After imposing various T-duality equivalence conditions [23], there only remain three inequivalent models as listed in the appendix .
Table 3. The spectrum of chiral and vector-like superfields, and their quantum numbers under the gauge symmetry SU(3)C × SU(3)L × SU(3)R × U(1) × USp(8) for the model 1. |
| Model 1 | Quantum number | QC | QL | QR | Qem | B-L | Field |
|---|---|---|---|---|---|---|---|
| ab | $1\times (3,\overline{3},1,1,1)$ | 1 | −1 | 0 | $-\tfrac{1}{3},\tfrac{2}{3},-1,0$ | $\tfrac{1}{3},-1$ | QL, LL |
| ab′ | 3 × (3, 3, 1, 1, 1) | 1 | 1 | 0 | $-\tfrac{1}{3},\tfrac{2}{3},-1,0$ | $\tfrac{1}{3},-1$ | QL, LL |
| ac | $1\times (\overline{3},1,3,1,1)$ | −1 | 0 | 1 | $\tfrac{1}{3},-\tfrac{2}{3},1,0$ | $-\tfrac{1}{3},1$ | QR, LR |
| ac′ | $3\times (\overline{3},1,\overline{3},1,1)$ | −1 | 0 | −1 | $\tfrac{1}{3},-\tfrac{2}{3},1,0$ | $-\tfrac{1}{3},1$ | QR, LR |
| bc | $8\times (1,\overline{3},3,1,1)$ | 0 | −1 | 1 | 1, 0, 0, −1 | 0 | H |
| bd | $3\times (1,\overline{3},1,1,1)$ | 0 | −1 | 0 | $\mp \tfrac{1}{2}$ | 0 | |
| bd′ | $3\times (1,\overline{3},1,\overline{1},1)$ | 0 | −1 | 0 | $\mp \tfrac{1}{2}$ | 0 | |
| cd | $3\times (1,1,3,\overline{1},1)$ | 0 | 0 | 1 | $\pm \tfrac{1}{2}$ | 0 | |
| cd′ | 3 × (1, 1, 3, 1, 1) | 0 | 0 | 1 | $\pm \tfrac{1}{2}$ | 0 | |
| b3 | 1 × (1, 3, 1, 1, 8) | 0 | 1 | 0 | $\pm \tfrac{1}{2}$ | 0 | |
| c3 | $1\times (1,1,\overline{3},1,8)$ | 0 | 1 | 0 | $\mp \tfrac{1}{2}$ | 0 | |
| ${b}_{\square \square }$ | 2 × (1, 6, 1, 1, 1) | 0 | 2 | 0 | 0, ±1 | 0 | |
| 2 × (1, 3, 1, 1, 1) | 0 | −2 | 0 | 0 | 0 | ||
| ${c}_{\overline{\square \square }}$ | $2\times (1,1,\overline{6},1,1)$ | 0 | 0 | −2 | 0, ±1 | 0 | |
| 2 × (1, 1, 3, 1, 1) | 0 | 0 | 2 | 0 | 0 |
Table 4. The spectrum of chiral and vector-like superfields, and their quantum numbers under the gauge symmetry $\mathrm{SU}{(3)}_{C}\times \mathrm{SU}{(3)}_{L}\times \mathrm{SU}{(3)}_{R}\times {\rm{U}}(1)\times \mathrm{USp}{\left(2\right)}^{3}$ for the model 2. |
| Model 2 | Quantum number | QC | QL | QR | Qem | B-L | Field |
|---|---|---|---|---|---|---|---|
| ab | $1\times (3,\overline{3},1,1,1,1,1)$ | 1 | −1 | 0 | $-\displaystyle \frac{1}{3}$, $\displaystyle \frac{2}{3}$, −1, 0 | $\displaystyle \frac{1}{3}$, −1 | QL, LL |
| ab′ | 3 × (3, 3, 1, 1, 1, 1, 1) | 1 | 1 | 0 | -$\displaystyle \frac{1}{3},\displaystyle \frac{2}{3}$, −1, 0 | $\displaystyle \frac{1}{3}$, −1 | QL, LL |
| ac | $1\times (\overline{3},1,3,1,1,1,1)$ | −1 | 0 | 1 | $\displaystyle \frac{1}{3}$, $-\displaystyle \frac{2}{3}$, 1, 0 | $-\displaystyle \frac{1}{3}$, 1 | QR, LR |
| ac′ | $3\times (\overline{3},1,\overline{3},1,1,1,1)$ | −1 | 0 | −1 | $\displaystyle \frac{1}{3},-\displaystyle \frac{2}{3},1,0$ | $-\displaystyle \frac{1}{3},1$ | QR, LR |
| bc | $4\times (1,\overline{3},3,1,1,1,1)$ | 0 | 1 | 1 | 1, 0, 0, −1 | 0 | H |
| bd | $3\times (1,3,1,\overline{1},1,1,1)$ | 0 | 1 | 0 | $\pm \displaystyle \frac{1}{2}$ | 0 | |
| bd′ | 3 × (1, 3, 1, 1, 1, 1, 1) | 0 | 1 | 0 | $\pm \displaystyle \frac{1}{2}$ | 0 | |
| cd | $18\times (1,1,3,\overline{1},1,1,1)$ | 0 | 0 | 1 | $\pm \displaystyle \frac{1}{2}$ | 0 | |
| cd′ | $10\times (1,1,\overline{3},\overline{1},1,1,1)$ | 0 | 0 | −1 | $\mp \displaystyle \frac{1}{2}$ | 0 | |
| ad | $5\times (3,1,1,\overline{1},1,1,1)$ | 1 | 0 | 0 | $\displaystyle \frac{1}{6},-\displaystyle \frac{1}{2}$ | $\tfrac{1}{3},-1$ | |
| ad′ | 9 × (3, 1, 1, 1, 1, 1, 1) | 1 | 0 | 0 | $\tfrac{1}{6}$, $-\tfrac{1}{2}$ | $\tfrac{1}{3},-1$ | |
| a1 | $1\times (\overline{3},1,1,1,2,1,1)$ | −1 | 0 | 0 | $-\tfrac{1}{6},\tfrac{1}{2}$ | $-\tfrac{1}{3},1$ | |
| b1 | $1\times (1,\overline{3},1,1,2,1,1)$ | 0 | −1 | 0 | $\mp \tfrac{1}{2}$ | 0 | |
| d1 | $8\times (1,1,1,\overline{1},2,1,1)$ | 0 | 0 | 0 | 0 | 0 | |
| a3 | 2 × (3, 1, 1, 1, 1, 2, 1) | 1 | 0 | 0 | $\tfrac{1}{6},-\tfrac{1}{2}$ | $\tfrac{1}{3},-1$ | |
| c3 | 2 ×(1, 1, 3, 1, 1, 2, 1) | 0 | 0 | 1 | $\pm \tfrac{1}{2}$ | 0 | |
| b4 | 1×(1, 3, 1, 1, 1, 1, 2) | 0 | 1 | 0 | $\pm \tfrac{1}{2}$ | 0 | |
| c4 | $1\times (1,1,\overline{3},1,1,1,2)$ | 0 | 0 | −1 | $\mp \tfrac{1}{2}$ | 0 | |
| d4 | $2\times (1,1,1,\overline{1},1,1,2)$ | 0 | 0 | 0 | 0 | 0 | |
| ${a}_{\overline{\square \square }}$ | $2\times (\overline{6},1,1,1,1,1,1)$ | −2 | 0 | 0 | $-\tfrac{1}{3},\tfrac{1}{3},1$ | $-\tfrac{2}{3},2$ | |
| 2 × (3, 1, 1, 1, 1, 1, 1) | 2 | 0 | 0 | $\tfrac{1}{3},-1$ | $\tfrac{2}{3},-2$ | ||
| ${c}_{\overline{\square \square }}$ | $2\times (1,1,\overline{6},1,1,1,1)$ | 0 | 0 | −2 | 0, ±1 | 0 | |
| 2 × (1, 1, 3, 1, 1, 1, 1) | 0 | 0 | 2 | 0 | 0 |
Table 5. The spectrum of chiral and vector-like superfields, and their quantum numbers under the gauge symmetry $\mathrm{SU}{(3)}_{C}\times \mathrm{SU}{(3)}_{L}\times \mathrm{SU}{(3)}_{R}\times {\rm{U}}(1)\times \mathrm{USp}{\left(2\right)}^{3}$ for the model 2-dual. |
| Model 2-dual | Quantum number | QC | QL | QR | Qem | B-L | Field |
|---|---|---|---|---|---|---|---|
| ab | $1\times (\overline{3},3,1,1,1,1,1)$ | −1 | 1 | 0 | $\displaystyle \frac{1}{3},-\displaystyle \frac{2}{3},1,0$ | $-\displaystyle \frac{1}{3}$, 1 | QL, LL |
| ab′ | $3\times (\overline{3},\overline{3},1,1,1,1,1)$ | −1 | −1 | 0 | $\displaystyle \frac{1}{3},-\displaystyle \frac{2}{3}$, 1, 0 | $-\displaystyle \frac{1}{3}$, 1 | QL, LL |
| ac | $1\times (3,1,\overline{3},1,1,1,1)$ | −1 | 0 | 1 | $-\displaystyle \frac{1}{3},\displaystyle \frac{2}{3},-1$, 0 | $\displaystyle \frac{1}{3}$, −1 | QR, LR |
| ac′ | 3 × (3, 1, 3, 1, 1, 1, 1) | 1 | 0 | 1 | $-\displaystyle \frac{1}{3}$, $\displaystyle \frac{2}{3},-1$, 0 | $\displaystyle \frac{1}{3},-1$ | QR, LR |
| bc | $4\times (1,3,\overline{3},1,1,1,1)$ | 0 | 1 | −1 | 1, 0, 0, −1 | 0 | H |
| cd | $3\times (1,1,3,\overline{1},1,1,1)$ | 0 | 0 | 1 | $\pm \displaystyle \frac{1}{2}$ | 0 | |
| cd′ | 3 × (1, 1, 3, 1, 1, 1, 1) | 0 | 0 | 1 | $\pm \displaystyle \frac{1}{2}$ | 0 | |
| bd | $18\times (1,3,1,\overline{1},1,1,1)$ | 0 | 1 | 0 | $\pm \displaystyle \frac{1}{2}$ | 0 | |
| bd′ | $10\times (1,\overline{3},1,\overline{1},1,1,1)$ | 0 | −1 | 0 | $\mp \displaystyle \frac{1}{2}$ | 0 | |
| ad | $5\times (3,1,1,\overline{1},1,1,1)$ | 1 | 0 | 0 | $\displaystyle \frac{1}{6},-\displaystyle \frac{1}{2}$ | $\tfrac{1}{3},-1$ | |
| ad′ | 9 × (3, 1, 1, 1, 1, 1, 1) | 1 | 0 | 0 | $\tfrac{1}{6}$, $-\tfrac{1}{2}$ | $\tfrac{1}{3},-1$ | |
| a1 | $1\times (\overline{3},1,1,1,2,1,1)$ | −1 | 0 | 0 | $-\tfrac{1}{6},\tfrac{1}{2}$ | $-\tfrac{1}{3},1$ | |
| c1 | $1\times (1,1,\overline{3},1,2,1,1)$ | 0 | 0 | −1 | $\mp \tfrac{1}{2}$ | 0 | |
| d1 | $8\times (1,1,1,\overline{1},2,1,1)$ | 0 | 0 | 0 | 0 | 0 | |
| a3 | 2 × (3, 1, 1, 1, 1, 2, 1) | 1 | 0 | 0 | $\tfrac{1}{6},-\tfrac{1}{2}$ | $\tfrac{1}{3},-1$ | |
| b3 | 2 × (1, 3, 1, 1, 1, 2, 1) | 0 | 1 | 0 | $\pm \tfrac{1}{2}$ | 0 | |
| c4 | 1 × (1, 1, 3, 1, 1, 1, 2) | 0 | 0 | 1 | $\pm \tfrac{1}{2}$ | 0 | |
| b4 | $1\times (1,\overline{3},1,1,1,1,2)$ | 0 | −1 | 0 | $\mp \tfrac{1}{2}$ | 0 | |
| d4 | $2\times (1,1,1,\overline{1},1,1,2)$ | 0 | 0 | 0 | 0 | 0 | |
| ${a}_{\overline{\square \square }}$ | $2\times (\overline{6},1,1,1,1,1,1)$ | −2 | 0 | 0 | $-\tfrac{1}{3}$, $\tfrac{1}{3},1$ | $-\tfrac{2}{3},2$ | |
| 2 × (3, 1, 1, 1, 1, 1, 1) | 2 | 0 | 0 | $\tfrac{1}{3},-1$ | $\tfrac{2}{3},-2$ | ||
| ${b}_{\overline{\square \square }}$ | $2\times (1,\overline{6},1,1,1,1,1)$ | 0 | −2 | 0 | 0, ±1 | 0 | |
| 2 × (1, 3, 1, 1, 1, 1, 1) | 0 | 2 | 0 | 0 | 0 |
Table 6. The spectrum of chiral and vector-like superfields, and their quantum numbers under the gauge symmetry SU(3)C × SU(3)L × SU(3)R × USp(2) × USp(6) for the model 3. |
| Model 3 | Quantum number | QC | QL | QR | Qem | B-L | Field |
|---|---|---|---|---|---|---|---|
| ab | $3\times (\overline{3},3,1,1,1,1)$ | −1 | 1 | 0 | $-\displaystyle \frac{1}{3}$, $\displaystyle \frac{2}{3},-1$, 0 | $\displaystyle \frac{1}{3},-1$ | QL, LL |
| ab′ | $1\times (\overline{3},\overline{3},1,1,1,1)$ | −1 | −1 | 0 | $-\displaystyle \frac{1}{3}$, $\displaystyle \frac{2}{3},-1$, 0 | $\displaystyle \frac{1}{3},-1$ | QL, LL |
| ac | $3\times (3,1,\overline{3},1,1,1)$ | 1 | 0 | −1 | $\displaystyle \frac{1}{3}$, $-\displaystyle \frac{2}{3}$, 1, 0 | $-\displaystyle \frac{1}{3},1$ | QR, LR |
| ac′ | 1 × (3, 1, 3, 1, 1, 1) | 1 | 0 | 1 | $\displaystyle \frac{1}{3},-\displaystyle \frac{2}{3}$, 1, 0 | $-\tfrac{1}{3}$, 1 | QR, LR |
| bc | $4\times (1,\overline{3},3,1,1,1)$ | 0 | −1 | 1 | 1, 0, 0, −1 | 0 | H |
| ad | $3\times (\overline{3},1,1,2,1,1)$ | −1 | 0 | 0 | $\tfrac{1}{6},-\tfrac{1}{2}$ | $\tfrac{1}{3},-1$ | |
| ad′ | $3\times (\overline{3},1,1,\overline{2},1,1)$ | −1 | 0 | 0 | $\tfrac{1}{6},-\tfrac{1}{2}$ | $\tfrac{1}{3},-1$ | |
| cd | $6\times (1,1,3,\overline{1},1,1)$ | 0 | 0 | 1 | $\pm \tfrac{1}{2}$ | 0 | |
| a2 | $2\times (\overline{3},1,1,1,2,1)$ | −1 | 0 | 0 | $\tfrac{1}{6},-\tfrac{1}{2}$ | $\tfrac{1}{3},-1$ | |
| b3 | $1\times (1,\overline{3},1,1,1,6)$ | 0 | −1 | 0 | $\pm \tfrac{1}{2}$ | 0 | |
| c2 | $2\times (1,1,2,1,\overline{2},1)$ | 0 | 0 | 1 | $\pm \tfrac{1}{2}$ | 0 | |
| c3 | $1\times (1,1,\overline{2},1,1,6)$ | 0 | 0 | −1 | $\mp \tfrac{1}{2}$ | 0 | |
| d2 | $1\times (1,1,1,\overline{2},2,1)$ | 0 | 0 | 0 | 0 | 0 | |
| d3 | $2\times (1,1,1,\overline{2},1,6)$ | 0 | 0 | 0 | 0 | 0 | |
| ${a}_{\square \square }$ | 2 × (6, 1, 1, 1, 1, 1, 1) | 2 | 0 | 0 | $-\tfrac{1}{3}$, $\tfrac{1}{3},1$ | $-\tfrac{2}{3},2$ | |
| 2 × (3, 1, 1, 1, 1, 1, 1) | −2 | 0 | 0 | $\tfrac{1}{3},-1$ | $\tfrac{2}{3},-2$ | ||
| ${c}_{\overline{\square \square }}$ | $2\times (1,1,\overline{6},1,1,1)$ | 0 | 0 | −2 | 0, ±1 | 0 | |
| 2 × (1, 1, 3, 1, 1, 1) | 0 | 0 | 2 | 0 | 0 | ||
| $16\times (1,1,1,\overline{1},1,1)$ | 0 | 0 | 0 | 0 | 0 |
Table 7. The spectrum of chiral and vector-like superfields, and their quantum numbers under the gauge symmetry SU(3)C × SU(3)L × SU(3)R × USp(2) × USp(6) for the model 3-dual. |
| Model 3-dual | Quantum number | QC | QL | QR | Qem | B-L | Field |
|---|---|---|---|---|---|---|---|
| ab | $3\times (3,\overline{3},1,1,1,1)$ | 1 | −1 | 0 | $\displaystyle \frac{1}{3}$, $-\displaystyle \frac{2}{3}$, 1, 0 | $-\displaystyle \frac{1}{3},1$ | QL, LL |
| ab′ | 1 × (3, 3, 1, 1, 1, 1) | 1 | 1 | 0 | $-\displaystyle \frac{1}{3}$, $\displaystyle \frac{2}{3},-1$, 0 | $\displaystyle \frac{1}{3},-1$ | QL, LL |
| ac | $3\times (\overline{3},1,3,1,1,1)$ | −1 | 0 | 1 | $-\displaystyle \frac{1}{3}$, $\displaystyle \frac{2}{3},-1$, 0 | $\displaystyle \frac{1}{3},-1$ | QR, LR |
| ac′ | $1\times (\overline{3},1,\overline{3},1,1,1)$ | 1 | 0 | 1 | $\displaystyle \frac{1}{3},-\displaystyle \frac{2}{3}$, 1, 0 | $-\tfrac{1}{3}$, 1 | QR, LR |
| bc | $4\times (1,3,\overline{3},1,1,1)$ | 0 | 1 | −1 | 1, 0, 0, −1 | 0 | H |
| ad | $3\times (\overline{3},1,1,2,1,1)$ | −1 | 0 | 0 | $\tfrac{1}{6},-\tfrac{1}{2}$ | $\tfrac{1}{3},-1$ | |
| ad′ | $3\times (\overline{3},1,1,\overline{2},1,1)$ | −1 | 0 | 0 | $\tfrac{1}{6},-\tfrac{1}{2}$ | $\tfrac{1}{3},-1$ | |
| bd | $6\times (1,3,1,\overline{1},1,1)$ | 0 | 0 | 1 | $\pm \tfrac{1}{2}$ | 0 | |
| a2 | $2\times (\overline{3},1,1,1,2,1)$ | −1 | 0 | 0 | $\tfrac{1}{6},-\tfrac{1}{2}$ | $\tfrac{1}{3},-1$ | |
| c3 | $1\times (1,1,\overline{3},1,1,6)$ | 0 | −1 | 0 | $\pm \tfrac{1}{2}$ | 0 | |
| b2 | $2\times (1,2,1,1,\overline{2},1)$ | 0 | 0 | 1 | $\pm \tfrac{1}{2}$ | 0 | |
| b3 | $1\times (1,\overline{2},1,1,1,6)$ | 0 | 0 | −1 | $\mp \tfrac{1}{2}$ | 0 | |
| d2 | $1\times (1,1,1,\overline{2},2,1)$ | 0 | 0 | 0 | 0 | 0 | |
| d3 | $2\times (1,1,1,\overline{2},1,6)$ | 0 | 0 | 0 | 0 | 0 | |
| a${a}_{\square \square }$ | 2 × (6, 1, 1, 1, 1, 1, 1) | 2 | 0 | 0 | $-\tfrac{1}{3}$, $\tfrac{1}{3},1$ | $-\tfrac{2}{3},2$ | |
| 2 × (3, 1, 1, 1, 1, 1, 1) | −2 | 0 | 0 | $\tfrac{1}{3},-1$ | $\tfrac{2}{3},-2$ | ||
| ${b}_{\overline{\square \square }}$ | $2\times (1,\overline{6},1,1,1,1)$ | 0 | 0 | −2 | 0, ±1 | 0 | |
| 2 × (1, 3, 1, 1, 1, 1) | 0 | 0 | 2 | 0 | 0 | ||
| $16\times (1,1,1,\overline{1},1,1)$ | 0 | 0 | 0 | 0 | 0 |
All models feature several exotic states that need to be decoupled at some higher energy scale for the models to be realistic.
4. Conclusion
We have revisited the construction of ${ \mathcal N }=1$ supersymmetric trinification grand unified models arising from intersecting D6-branes at angles on a ${{\mathbb{T}}}^{6}/({{\mathbb{Z}}}_{2}\times {{\mathbb{Z}}}_{2})$ orientifold from IIA string theory. We have discussed the gauge symmetry breaking from the trinification symmetry down to the standard model gauge group. Notably, due to large RR charges from product of three SU(3) gauge factors and the associated non-trivial K-theory constraints we did not find any three-family trinification models in our search. However, we have found three inequivalent four-family models where the highest wrapping number observed is 2. We have also listed the detailed perturbative particle spectra and the tree-level gauge coupling relations at the string-scale. All models feature exotic states from the hidden sector.