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Volume-Dependent Field Theories

Richard Wedeen

Abstract

We develop the axiom system proposed by Kontsevich and Segal to define volume-dependent field theories (VFTs), a class of non-topological quantum field theories whose dependence on the background metric factors through the associated density. We construct a well-defined Lorentzian limit of a Wick-rotated VFT defined on smooth, possibly degenerate Lorentzian bordisms with incoming and outgoing boundary both nonempty. When the VFT is reflection positive, this extends the main theorem given by Kontsevich and Segal.

Volume-Dependent Field Theories

Abstract

We develop the axiom system proposed by Kontsevich and Segal to define volume-dependent field theories (VFTs), a class of non-topological quantum field theories whose dependence on the background metric factors through the associated density. We construct a well-defined Lorentzian limit of a Wick-rotated VFT defined on smooth, possibly degenerate Lorentzian bordisms with incoming and outgoing boundary both nonempty. When the VFT is reflection positive, this extends the main theorem given by Kontsevich and Segal.
Paper Structure (28 sections, 85 theorems, 359 equations)

This paper contains 28 sections, 85 theorems, 359 equations.

Table of Contents

  1. Introduction
  2. Geometric Axioms for QFT
  3. The Bordism Category of Germs
  4. Preliminaries on Germs
  5. F-Bordisms
  6. The Category of Nuclear Pairs
  7. Field Theory
  8. Sheaves of Complex Manifolds
  9. Nuclearity
  10. Holomorphicity
  11. Coherence
  12. The Action of Diffeomorphisms
  13. Definition of Field Theory
  14. Reflection Positivity
  15. Volume-dependent Field Theories
  16. ...and 13 more sections

Key Result

Theorem 1.1

Let $X$ be a compact, connected $n$-manifold and let $\omega_0, \omega_1$ be real and positive densities of the same total volume. Then there exists a diffeomorphism $\varphi \in \mathop{\mathrm{Diff}}\nolimits(X)$ such that

Theorems & Definitions (241)

  • Theorem 1.1: Moser
  • Theorem 1.2: KS Theorem 5.2
  • Lemma 2.1
  • proof
  • Proposition 2.2
  • proof
  • Corollary 2.3
  • proof
  • Remark 2.4
  • Remark 2.5
  • ...and 231 more