$\mathrm{D=10}$ Super-Yang-Mills Theory and Poincar\'e Duality in Supermanifolds

We consider super Yang-Mills theory on supermanifolds $\mathcal{M}^{(D|m)}$ using integral forms. The latter are used to define a geometric theory of integration and are essential for a consistent action principle. The construction relies on Picture Changing Operators $\mathbb{Y}^{(0|m)}$, analogous to those introduced in String Theory, that admit the geometric interpretation of Poincar\'e duals of closed submanifolds of superspace $\mathcal{S}^{(D|0)} \subset \mathcal{M}^{(D|m)}$ having maximal bosonic dimension $D$. We discuss the case of Super-Yang-Mills theory in $D=10$ with $\mathcal{N}=1$ supersymmetry and we show how to retrieve its pure-spinor formulation from the rheonomic lagrangian $\mathcal{L}_{rheo}$ of D'Auria, Fr\'e and Da Silva, choosing a suitable $\mathbb{Y}^{(0|m)}_{ps}$. From the same lagrangian $\mathcal{L}_{rheo}$, with another choice $\mathbb{Y}^{(0|m)}_{comp}$ of the PCO, one retrieves the component form of the SYM action. Equivalence of the formulations is ensured when the corresponding PCO.s are cohomologous, which is true, in this case, of $\mathbb{Y}^{(0|m)}_{ps}$ and $\mathbb{Y}^{(0|m)}_{comp}$.