# $L_\infty$-Bootstrap Approach to Non-Commutative Gauge Theories

## TL;DR

This paper introduces an $L_$-algebra framework to construct consistent non-commutative gauge theories from commutative ones, providing explicit examples and discussing the resulting non-Lagrangian equations of motion.

## Contribution

It develops a systematic $L_$-algebra method to derive non-commutative gauge theories, including explicit constructions for $$-like and octonionic deformations, advancing the mathematical foundation of NC gauge theories.

## Key findings

- Explicit $L_$-algebra formulations for NC gauge theories.
- Derived non-Lagrangian equations of motion for NC Chern-Simons theory.
- Constructed non-commutative deformations of Abelian gauge transformations.

## Abstract

A consistent description of gauge theories on coordinate dependent non-commutative (NC) space-time is a long-standing problem with a number of solutions, none of which is free from criticism. In this work, we discuss the approach proposed in arXiv:1803.00732, based on the conjecture that any consistent gauge theory can be described in terms of the $L_\infty$-structure. Starting with a well-defined commutative gauge theory, we represent it, together with the non-commutative deformation, as a part of a bigger $L_\infty$-algebra by setting some initial brackets $\ell_1$, $\ell_2$, etc. Then, solving the $L_\infty$-relations we determine the missing brackets $\ell_n$ and close the $L_\infty$-algebra defining the NC gauge theory which reproduces in the commutative limit the original one. We provide the recurrence relations for the construction of the pure gauge algebra $L^{\rm gauge}_\infty$, using which we find an explicit form of the NC $\mathfrak{su}(2)$-like and non-associative octonionic-like deformations of the Abelian gauge transformations. The construction of the $L^{\rm full}_\infty$-algebra describing the dynamics is discussed using the example of the NC Chern-Simons theory. The obtained equations of motion are non-Lagrangian, which indicates the difference between our approach and the previous ones.