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The Fermionic integral on loop space and the Pfaffian line bundle

Florian Hanisch, Matthias Ludewig

Abstract

As the loop space of a Riemannian manifold is infinite-dimensional, it is a non-trivial problem to make sense of the "top degree component" of a differential form on it. In this paper, we show that a formula from finite dimensions generalizes to assign a sensible "top degree component" to certain composite forms, obtained by wedging with the exponential (in the exterior algebra) of the canonical 2-form on the loop space. The result is a section on the Pfaffian line bundle on the loop space. We then identify this with a section of the line bundle obtained by transgression of the spin lifting gerbe. These results are a crucial ingredient for defining the fermionic part of the supersymmetric path integral on the loop space.

The Fermionic integral on loop space and the Pfaffian line bundle

Abstract

As the loop space of a Riemannian manifold is infinite-dimensional, it is a non-trivial problem to make sense of the "top degree component" of a differential form on it. In this paper, we show that a formula from finite dimensions generalizes to assign a sensible "top degree component" to certain composite forms, obtained by wedging with the exponential (in the exterior algebra) of the canonical 2-form on the loop space. The result is a section on the Pfaffian line bundle on the loop space. We then identify this with a section of the line bundle obtained by transgression of the spin lifting gerbe. These results are a crucial ingredient for defining the fermionic part of the supersymmetric path integral on the loop space.

Paper Structure

This paper contains 22 sections, 11 theorems, 155 equations.

Table of Contents

  1. Introduction
  2. Acknowledgements.
  3. Preliminaries I: Loop Space Differential Forms
  4. The loop space.
  5. Differential forms on infinite-dimensional manifolds.
  6. The bundle $\boldsymbol{L_{\mathrm{alt}}^\ell(T\text{\normalfont\sffamily L} X, \mathbb{R})}$.
  7. Preliminaries II: Spin Geometry
  8. Spin structures.
  9. The real spinor bundle.
  10. Comparison to the complex spinor bundle.
  11. Atiyah's formula
  12. The parallel transport in the spinor bundle.
  13. The determinant of the covariant derivative.
  14. The Pfaffian and the Spin Line Bundle
  15. The Pfaffian line bundle.
  16. ...and 7 more sections

Key Result

Theorem 1.1

Let $X$ be a spin manifold. Then under the canonical trivialization of the Pfaffian line bundle provided by the spin structure, we have the formula for the top degree component, initially defined using TopDegreeFiniteDimensionsIntro respectively its generalization to non-invertible $A$. Here $[\gamma\|_\bullet^\bullet]^\Sigma$ denotes parallel transport in the spinor bundle along the loop $\gamma

Theorems & Definitions (26)

  • Theorem 1.1
  • Lemma 1.2
  • Proof 1
  • Lemma 2.1
  • Proof 2
  • Corollary 2.2
  • Proposition 2.3
  • Remark 2.4
  • Remark 3.1
  • Definition 3.2: The isomorphism
  • ...and 16 more