# Homotopy Algebras of Differential (Super)forms in Three and Four Dimensions

## TL;DR

This paper explores the structure of $A_{
abla}$-algebras of differential forms in relation to gauge theories, constructing homotopic complexes in 3D and 4D that connect to superfield formulations and Chern-Simons theory.

## Contribution

It introduces new $A_{
abla}$-algebra structures for differential forms in three and four dimensions, linking them to gauge theories and superfield formulations.

## Key findings

- Constructed homotopic complexes related to gauge theories.
- Linked $A_{
abla}$-structures to superfield formulations of Chern-Simons theory.
- Demonstrated explicit transfer of $A_{
abla}$-structures in reformulated gauge theories.

## Abstract

We consider various $A_{\infty}$-algebras of differential (super)forms, which are related to gauge theories and demonstrate explicitly how certain reformulations of gauge theories lead to the transfer of the corresponding $A_{\infty}$-structures. In $N=2$ 3D space we construct the homotopic counterpart of the de Rham complex, which is related to the superfield formulation of the $N=2$ Chern-Simons theory.