$L_{\infty }$-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.

arXiv comment12 pages, Contribution to Proceedings of LMS/EPSRC Durham Symposium Higher Structures in M-Theory, August 2018, v2: typos corrected, references added, v3: references fixed

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• Non-commutative deformation of Chern-Simons theory
• $k$-Minkowski-deformation of $U(1)$ gauge theory
• Poisson gauge theory
• Braided Symmetries in Noncommutative Field Theory
• A novel approach to non-commutative gauge theory
• Nonassociative black ellipsoids distorted by R-fluxes and four dimensional thin locally anisotropic accretion disks
• On the L${}_{\infty }$ structure of Poisson gauge theory
• Nonassociative geometry of nonholonomic phase spaces with star R-flux string deformations and (non) symmetric metrics
• Light Propagation in $\kappa$-Minkowski Space-Time: Gauge Ambiguities and Invariance
• Non-commutative gauge symmetry from strong homotopy algebras
• Four-dimensional noncommutative deformations of $U(1)$ gauge theory and $L_{\infty }$ bootstrap