Yang–Mills Theory and a Superquadric

Summary

We construct a real-analytic CR supermanifold $\mathcal{R}$, holomorphically embedded into a superquadric $\mathcal{Q}\subset {\mathbf{ P}}^{3\vert 3} \times {\mathbf{ P}}^{{_\ast}3\vert 3}$. A CR distribution $\mathcal{F}$ on $\mathcal{R}$ enables us to define a tangential CR complex $\left ({\Omega }_{\mathcal{F}}^{\bullet },\bar{\partial }\right )$.We define a $\bar{\partial }$-closed trace functional $\int \nolimits \nolimits : {\Omega }_{\mathcal{F}}^{\bullet }\rightarrow \mathbb{C}$ and conjecture that a Chern-Simons theory associated with a triple $\left ({\Omega }_{\mathcal{F}}^{\bullet }\otimes {\mathrm{Mat}}_{n},\bar{\partial },\int \nolimits \nolimits \ \mathrm{tr}\right )$ is equivalent to N = 3, D = 4 Yang–Mills theory with a gauge group U(n). We give some evidences to this conjecture.

2000 Mathematics Subject Classifications: 53C80, 53C28, 81R25, 32C11, 58C50