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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.

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