Four-dimensional noncommutative deformations of $U(1)$ gauge theory and $L_{\infty}$ bootstrap

TL;DR

This paper develops four-dimensional noncommutative deformations of $U(1)$ gauge theory, incorporating coordinate-dependent structures and an $L_{ abla}$-algebra framework, advancing the understanding of noncommutative gauge models with exact all-order expressions.

Contribution

It introduces a novel class of four-dimensional noncommutative $U(1)$ gauge theories with explicit all-order deformation expressions and links to $L_{ abla}$-algebra and symplectic embedding formalisms.

Findings

Deformed gauge transformations and field strengths are explicitly constructed.

The models have flat commutative limits and include all orders in deformation.

Connection established between the formalism and $L_{ abla}$-bootstrap.

Abstract

We construct a family of four-dimensional noncommutative deformations of $U(1)$ gauge theory following a general scheme, recently proposed in JHEP 08 (2020) 041 for a class of coordinate-dependent noncommutative algebras. This class includes the $\mathfrak{su}(2)$, the $\mathfrak{su}(1,1)$ and the angular (or $\lambda$-Minkowski) noncommutative structures. We find that the presence of a fourth, commutative coordinate $x^0$ leads to substantial novelties in the expression for the deformed field strength with respect to the corresponding three-dimensional case. The constructed field theoretical models are Poisson gauge theories, which correspond to the semi-classical limit of fully noncommutative gauge theories. Our expressions for the deformed gauge transformations, the deformed field strength and the deformed classical action exhibit flat commutative limits and they are exact in the sense that all orders in the deformation parameter are present. We review the connection of the formalism with the $L_{\infty}$ bootstrap and with symplectic embeddings, and derive the $L_{\infty}$-algebra, which underlies our model.