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

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.