Abstract
We apply the modern Batalin–Vilkovisky quantization techniques of Costello and Gwilliam to noncommutative field theories in the finite-dimensional case of fuzzy spaces. We further develop a generalization of this framework to theories that are equivariant under a triangular Hopf algebra symmetry, which in particular leads to quantizations of finite-dimensional analogues of the field theories proposed recently through the notion of ‘braided L_infty -algebras’. The techniques are illustrated by computing perturbative correlation functions for scalar and Chern–Simons theories on the fuzzy 2-sphere, as well as for braided scalar field theories on the fuzzy 2-torus.
Highlights
Noncommutative quantum field theories are well-known to exhibit many novel features not present in conventional quantum field theory, see e.g. [30] for a review
In the following we provide an elementary and self-contained review of the BV quantization techniques developed by Costello and Gwilliam [10,16]
In this paper we have initiated the study of noncommutative quantum field theories using modern tools from BV quantization [10,16]
Summary
Noncommutative quantum field theories are well-known to exhibit many novel features not present in conventional quantum field theory, see e.g. [30] for a review. We treat only fuzzy field theories, which are by definition finite-dimensional systems, i.e. matrix models, and so can be quantized in a completely rigorous way while avoiding the analytic issues involved when dealing with continuum field theories. These examples will serve to nicely illustrate our formalism while avoiding much technical clutter. We study in detail the example of 4-theory where we reproduce the known 2-point function at 1-loop order obtained through more traditional techniques [9] This nicely illustrates how BV quantization captures the known distinction between planar and nonplanar loop corrections in the standard noncommutative field theories, see e.g. Unlike the fuzzy sphere, we are not aware of any construction of a differential calculus on the fuzzy torus that could be used to define versions of the standard gauge theories, in contrast to its continuum version, i.e. the noncommutative torus; for a rigorous discussion of this point, see [23]
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