Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

The Batalin-Vilkovisky formalism in noncommutative effective field theory

Alastair Hamilton

TL;DR

The paper develops a noncommutative BV framework by marrying Kontsevich's NC geometry with Costello's BV EFT approach, enabling quantization and RG analysis of NC gauge theories. It establishes a scale-dependent quantum master equation, demonstrates RG flow compatibility with the QME, and interprets algebraic structures on field cohomology as classes in ribbon-graph complexes related to moduli spaces of Riemann surfaces. A noncommutative analogue of Chern–Simons theory is constructed and quantized in flat backgrounds, with large $N$ phenomena controlled by a NC–commutative correspondence and a vanishing criterion from LQT. The work also elucidates how NC EFTs generate NC and commutative gauge theories across scales via OTFT/Frobenius algebra connections, offering a robust framework for NC large-$N$ gauge dynamics and their geometric/topological implications.

Abstract

We address the treatment of gauge theories within the framework that is formed from combining the machinery of noncommutative symplectic geometry, as introduced by Kontsevich, with Costello's approach to effective gauge field theories within the Batalin-Vilkovisky formalism; discussing the problem of quantization in this context, and identifying the relevant cohomology theory controlling this process. We explain how the resulting noncommutative effective gauge field theories produce classes in a compactification of the moduli space of Riemann surfaces, when we pass to the large length scale limit. Within this setting, the large $N$ correspondence of 't Hooft -- describing a connection between open string theories and gauge theories -- appears as a relation between the noncommutative and commutative geometries. We use this correspondence to investigate and ultimately quantize a noncommutative analogue of Chern-Simons theory.

The Batalin-Vilkovisky formalism in noncommutative effective field theory

TL;DR

The paper develops a noncommutative BV framework by marrying Kontsevich's NC geometry with Costello's BV EFT approach, enabling quantization and RG analysis of NC gauge theories. It establishes a scale-dependent quantum master equation, demonstrates RG flow compatibility with the QME, and interprets algebraic structures on field cohomology as classes in ribbon-graph complexes related to moduli spaces of Riemann surfaces. A noncommutative analogue of Chern–Simons theory is constructed and quantized in flat backgrounds, with large phenomena controlled by a NC–commutative correspondence and a vanishing criterion from LQT. The work also elucidates how NC EFTs generate NC and commutative gauge theories across scales via OTFT/Frobenius algebra connections, offering a robust framework for NC large- gauge dynamics and their geometric/topological implications.

Abstract

We address the treatment of gauge theories within the framework that is formed from combining the machinery of noncommutative symplectic geometry, as introduced by Kontsevich, with Costello's approach to effective gauge field theories within the Batalin-Vilkovisky formalism; discussing the problem of quantization in this context, and identifying the relevant cohomology theory controlling this process. We explain how the resulting noncommutative effective gauge field theories produce classes in a compactification of the moduli space of Riemann surfaces, when we pass to the large length scale limit. Within this setting, the large correspondence of 't Hooft -- describing a connection between open string theories and gauge theories -- appears as a relation between the noncommutative and commutative geometries. We use this correspondence to investigate and ultimately quantize a noncommutative analogue of Chern-Simons theory.

Paper Structure

This paper contains 88 sections, 50 theorems, 341 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Background
  3. Noncommutative effective field theory
  4. Batalin-Vilkovisky formalism
  5. The large $N$ string/gauge theory correspondence
  6. Algebraic structures on the cohomology of the fields---links to classes in the moduli space of Riemann surfaces
  7. Noncommutative Chern-Simons theory
  8. Acknowledgements
  9. Layout of the paper
  10. Notation and conventions
  11. Free theories
  12. Free BV-theories and gauge-fixing conditions
  13. Basic definitions
  14. Families of gauge-fixing conditions
  15. Heat kernels and convolution operators
  16. ...and 73 more sections

Key Result

Lemma 2.10

Let $\mathcal{E}$ be a free BV-theory and $\mathcal{A}$ be as above:

Figures (3)

  • Figure 1: A correspondence defining an equivalence of categories.
  • Figure 2: Contracting loops in a ribbon graph defines the cobracket on $\mathscr{H}\left(\mathcal{E},\mathcal{A}\right)$.
  • Figure 3: Contracting edges in a ribbon graph defines the bracket on $\mathscr{H}\left(\mathcal{E},\mathcal{A}\right)$.

Theorems & Definitions (180)

  • Definition 2.1
  • Definition 2.2
  • Remark 2.3
  • Example 2.4
  • Definition 2.5
  • Example 2.6
  • Example 2.7
  • Remark 2.8
  • Definition 2.9
  • Lemma 2.10
  • ...and 170 more