Batalin-Vilkovisky quantization of fuzzy field theories

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 analogs 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.

arXiv commentv4: Final version accepted for publication in Letters in Mathematical Physics

References in corpus (21)

• Quantum Field Theory on Noncommutative Spaces
• Noncommutative Perturbative Dynamics
• Wilson Loops in Noncommutative Yang Mills
• Brane Dynamics in Background Fluxes and Non-commutative Geometry
• Untwisting Noncommutative R^d and the Equivalence of Quantum Field Theories
• Scaling Limits of the Fuzzy Sphere at one Loop
• Statistics and UV-IR Mixing with Twisted Poincare Invariance
• $L_{\infty }$-Algebras of Classical Field Theories and the Batalin-Vilkovisky Formalism
• Field Theory on the q-deformed Fuzzy Sphere I
• Quantum and Braided Lie Algebras
• Remarks on Chern-Simons invariants
• Braided Quantum Field Theory
• Field Theory on the q-deformed Fuzzy Sphere II: Quantization
• From Large N Matrices to the Noncommutative Torus
• Gauge Theories on Deformed Spaces
• Finite spectral triples for the fuzzy torus
• Noncommutative geometry and the BV formalism: application to a matrix model
• The BV formalism: theory and application to a matrix model
• Lectures on shifted Poisson geometry
• Mapping spaces and automorphism groups of toric noncommutative spaces
• The BRST cohomology and a generalized Lie algebra cohomology: analysis of a matrix model

Cited by in corpus (3)

• Braided $L_{\infty }$-Algebras, Braided Field Theory and Noncommutative Gravity
• Braided Symmetries in Noncommutative Field Theory
• On multimatrix models motivated by random noncommutative geometry II: A Yang-Mills-Higgs matrix model