# Determination of Torsion constraints in ${\cal N} = 1$, D = 10 Superspace

For the on-shell theory, containing the graviton $$e_m^{\ \ \ a}$$, gravitino $$\psi_m^{\ \ \ \alpha}$$, dilaton $$\phi$$, dilatino $$\lambda$$ and 3-form $$H_{n m p}$$, one has to demand that the SUSY algebra is closed to the right value always, meaning

$$\{ D_\alpha, D_\beta\} = 2(\gamma^a)_{\alpha \beta} D_a$$.

The relation in superspace

$$[ D_A, D_B \}V_C = T_{A B}^{\ \ \ \ \ D} D_D V_C + R_{A B C}^{\ \ \ \ \ \ \ \ \ D} V_D$$

fixes then some of the torsion constraints, meaning that $$T_{\alpha \beta}^{\ \ \ \ c} = 2(\gamma^c)_{\alpha \beta}$$ or $$T_{\alpha \beta}^{\ \ \ \ \gamma} = 0$$. This also happens to some of the curvature components.

In research papers, the undetermined components of the torsion are then determined to solve consistently the Bianchi identity

$$D_{[A} T_{B C \}}^{\ \ \ \ D} - R_{[A B C\}}^{\ \ \ \ \ \ D} + T_{[A B}^{\ \ \ \ E} T_{|E| C\}}^{\ \ \ \ D} = 0$$

along with other BI for $$H_{a b c}$$, for example.

But this Bianchi identity is derived from the cartan structure equation $$T^A = D E^A$$, where $$D$$ is the covariant derivative with torsionful spin connection and $$E^A$$ is the orthogonal 1-form $$E^A = dZ^M E_M^{\ \ \ A}$$. If one chooses a parametrization for the $$\theta = 0$$ of the vielbein $$E_M^{\ \ \ A}$$(which fixes higher $$\theta$$ orders), which usually is

$$E_m^{\ \ \ a} \rvert = e_m^{\ \ \ a}, \ \ \ \ E_m^{\ \ \ \alpha} \rvert = \psi_m^{\ \ \ \ \alpha}, \ \ \ \ E_\mu^{\ \ \ a} \rvert = 0, \ \ \ \ E_\mu^{\ \ \ \alpha} \rvert = \delta_\mu^\alpha$$,

wouldn't all the torsion components be determined immediately by the Cartan structure equation? I don't understand why papers always solve Bianchi identities for the components of the torsion but never use $$T^A = D E^A$$, which leads me to believe torsion components can't be derived from that equation, but I don't get why.

Can someone also recommend some good bibliography on $$\mathcal{N}=1$$, $$D = 10$$ superspace? Research papers don't explore some of the subtleties I don't understand.