| ⋆ | Hodge star, given by equation (2.2) |
| ${\rm{d}}$ | Exterior derivative |
| $\hat{\delta }$ | Codifferential: ${\left(-1\right)}^{{np}+n+1}\star {\rm{d}}\star $ |
| Δ | Laplace–de Rham: $\hat{\delta }{\rm{d}}+{\rm{d}}\hat{\delta }$ |
| $\square $ | d’Alembertian: ${g}^{\mu \nu }{{\rm{\nabla }}}_{\mu }{{\rm{\nabla }}}_{\nu }$ |
| ${\mathbb{P}}$ | $\hat{\delta }{\rm{d}}$ |
| O1 | $\star {\rm{d}}\,\star \,{\rm{d}}$ |
| O2 | ${\rm{d}}\,\star \,{\rm{d}}\star $ |
| ${{\mathbb{O}}}_{{jk}}$ | ${\alpha }_{1j}{O}_{1}^{j}+{\alpha }_{2k}{O}_{2}^{k}$ |
| $\hat{O}$ | ${\alpha }_{1j}{O}_{1}^{j}+{\alpha }_{2l}{O}_{2}^{l}$ |
| $\tilde{O}$ | ${\beta }_{1s}{O}_{1}^{s}+{\beta }_{2t}{O}_{2}^{t}$ |