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

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 $L_{\infty }^{\mathrm{gauge}}$, 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 $L_{\infty }^{\mathrm{full}}$-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.

12 pages, Contribution to Proceedings of LMS/EPSRC Durham Symposium Higher Structures in M-Theory, August 2018, v2: typos corrected, references added, v3: references fixed

References in corpus (8)

• Symmetry, Gravity and Noncommutativity
• T-duality and closed string non-commutative (doubled) geometry
• Gauge theories on the kappa-Minkowski spacetime
• Position-dependent noncommutativity in quantum mechanics
• Twist to close
• Star products made (somewhat) easier
• Non-associativity in non-geometric string and M-theory backgrounds, the algebra of octonions, and missing momentum modes
• $G_2$-structures and quantization of non-geometric M-theory backgrounds

Cited by in corpus (1)

• $k$-Minkowski-deformation of $U(1)$ gauge theory