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On the Evaluation of the Ray-Singer Torsion Path Integral

Matthias Blau, Mbambu Kakona, George Thompson

Abstract

There are very few explicit evaluations of path integrals for topological gauge theories in more than 3 dimensions. Here we provide such a calculation for the path integral representation of the Ray-Singer Torsion of a flat connection on a vector bundle on base manifolds that are themselves $S^{1}$ bundles of any dimension. The calculation relies on a suitable algebraic choice of gauge which leads to a convenient factorisation of the path integral into horizontal and vertical parts.

On the Evaluation of the Ray-Singer Torsion Path Integral

Abstract

There are very few explicit evaluations of path integrals for topological gauge theories in more than 3 dimensions. Here we provide such a calculation for the path integral representation of the Ray-Singer Torsion of a flat connection on a vector bundle on base manifolds that are themselves bundles of any dimension. The calculation relies on a suitable algebraic choice of gauge which leads to a convenient factorisation of the path integral into horizontal and vertical parts.
Paper Structure (10 sections, 138 equations)

This paper contains 10 sections, 138 equations.

Table of Contents

  1. Introduction
  2. Connections on Bundles over S1 Bundles
  3. Factorisation of the Partition Function
  4. Z[A] and the Zero Mode Measure
  5. A Formal Expression for Z[phi]
  6. Flat Connections on a Pull Back Bundle E=pi*V
  7. Evaluation of the Path Integral Representation of Z[phi]
  8. Some Generalisations and Observations
  9. Some Background on the Geometry of S1 Bundles
  10. Gauge Transformations, Gauge Fixing and BV Triangles

Theorems & Definitions (7)

  • Example 6.1
  • Example 6.2
  • Example 6.3
  • Example 7.1
  • Example A.1
  • Example B.1
  • Example B.2