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Introduction to Classical Gauge Field Theory and to Batalin-Vilkovisky Quantization

Glenn Barnich, Fabrizio Del Monte

TL;DR

These lectures develop a cohesive, algebraic formulation of classical gauge field theory and BV quantization using jet-bundle methods. They introduce locality via the horizontal complex, Noether identities via the Koszul–Tate resolution, and gauge symmetries through prolongations and variational symmetries. The BV formalism with master equation and BRST differential provides a gauge-invariant framework for renormalization, gauge fixing, and anomaly analysis, including Slavnov–Taylor/Zinn-Justin equations. The text emphasizes the cohomological organization of conservation laws, surface charges, and algebraic structure of gauge symmetries, and outlines perturbative implications for renormalization and deformations of gauge theories.

Abstract

Lectures held at the 22nd "Saalburg" Summer School (2016)

Introduction to Classical Gauge Field Theory and to Batalin-Vilkovisky Quantization

TL;DR

These lectures develop a cohesive, algebraic formulation of classical gauge field theory and BV quantization using jet-bundle methods. They introduce locality via the horizontal complex, Noether identities via the Koszul–Tate resolution, and gauge symmetries through prolongations and variational symmetries. The BV formalism with master equation and BRST differential provides a gauge-invariant framework for renormalization, gauge fixing, and anomaly analysis, including Slavnov–Taylor/Zinn-Justin equations. The text emphasizes the cohomological organization of conservation laws, surface charges, and algebraic structure of gauge symmetries, and outlines perturbative implications for renormalization and deformations of gauge theories.

Abstract

Lectures held at the 22nd "Saalburg" Summer School (2016)

Paper Structure

This paper contains 39 sections, 10 theorems, 250 equations.

Table of Contents

  1. Locality and the horizontal complex
  2. Jet-bundles
  3. Total and Euler-Lagrange derivatives
  4. Exercise 1
  5. Local functionals
  6. Exercise 2
  7. The Horizontal Complex
  8. Exercise 3
  9. Dynamics and the Koszul-Tate complex
  10. Stationary surface
  11. Exercise 4
  12. Noether identities
  13. Exercise 5
  14. Irreducible gauge theories
  15. The Koszul-Tate complex
  16. ...and 24 more sections

Key Result

Lemma 1

The Euler-Lagrange derivative of a local function is zero if and only if it is a total divergence, i.e., for some local functions $j^\mu$.

Theorems & Definitions (38)

  • Remark 1
  • Remark 2
  • Remark 3
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Theorem 1
  • proof
  • Remark 4
  • ...and 28 more