Super Chern–Simons theory and flat super connections on a torus

Abstract

We study the moduli space of a super Chern-Simons theory on a manifold with the topology ${\bf R}\times \S$, where $\S$ is a compact surface. The moduli space is that of flat super connections modulo gauge transformations on $\S$, and we study in detail the case when $\S$ is atorus and the supergroup is $OSp(m|2n)$. The bosonic moduli space is determined by the flat connections for the maximal bosonic subgroup $O(m)\times Sp(2n)$, while the fermionic moduli appear only for special parts of the bosonic moduli space, which are determined by a vanishing determinant of a matrix associated to the bosonic part of the holonomy. If the CS supergroup is the exponential of a super Lie algebra, the fermionic moduli appear only for the bosonic holonomies whose generators have zero determinant in the fermion-fermion block of the super-adjoint representation. A natural symplectic structure on the moduli space is induced by the super Chern-Simons theory and it is determined by the Poisson bracket algebra of the holonomies. We show that the symplectic structure of homogenous connections is useful for understanding the properties of the moduli space and the holonomy algebra, and we illustrate this on the example of the $OSp(1|2)$ supergroup.