# Pictures from Super Chern-Simons Theory

## TL;DR

This paper develops a comprehensive framework for super-Chern-Simons theory on supermanifolds, introducing new mathematical tools like Picture Changing Operators and pseudo-forms, and demonstrates their equivalence to traditional formulations.

## Contribution

It introduces a novel approach to super-Chern-Simons theory using pseudo-forms and PCOs, establishing equivalence between different picture formulations and exploring interaction structures with $A_$ algebra.

## Key findings

- Equivalence of different picture formulations of super-Chern-Simons theory.
- Introduction of pseudo-forms with infinite components.
- Construction of a non-associative 2-product with $A_$ structure.

## Abstract

We study super-Chern-Simons theory on a generic supermanifold. After a self-contained review of integration on supermanifolds, the complexes of forms (superforms, pseudo-forms and integral forms) and the extended Cartan calculus are discussed. We then introduce Picture Changing Operators. We provide several examples of computation of PCO's acting on different type of forms. We illustrate also the action of the $\eta$ operator, crucial ingredient to define the interactions of super Chern-Simons theory. Then, we discuss the action for super Chern-Simons theory on any supermanifold, first in the factorized form (3-form $\times$ PCO) and then, we consider the most general expression. The latter is written in term of psuedo-forms containing an infinite number of components. We show that the free equations of motion reduce to the usual Chern-Simons equations yielding the proof of the equivalence between the formulations at different pictures of the same theory. Finally, we discuss the interaction terms. They require a suitable definition in order to take into account the picture number. That implies the construction of a 2-product which is not associative that inherits an $A_\infty$ algebra structure. That shares several similarities with a recent construction of a super string field theory action by Erler, Konopka and Sachs.