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

Abstract

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 Poincaré 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, Fré and Da Silva, choosing a suitable ${\mathbb{Y}}_{ps}^{(0|m)}$. From the same lagrangian ${\mathcal{L}}_{rheo}$, with another choice ${\mathbb{Y}}_{comp}^{(0|m)}$ 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 ${\mathbb{Y}}_{ps}^{(0|m)}$ and ${\mathbb{Y}}_{comp}^{(0|m)}$.

22 pages, Latex

References in corpus (2)

• Supergravity Actions with Integral Forms
• Constrained Supermanifolds for AdS M-Theory Backgrounds