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Regularizing Calabi-Yau topological conformal field theories using cutoff heat kernels

Yashasvi Aulak

Abstract

In this paper we construct a family of topological conformal field theories (TCFTs) associated to a Calabi-Yau space by modifying the heat kernel and sections of the Calabi-Yau space. This is done by restricting to certain eigenspaces of the Laplacian. We then present two a-priori distinct ways to regularize the Calabi-Yau TCFT by using these modified heat kernels, and then show that they are equivalent. Finally, we relate the regularized TCFTs for different cutoffs.

Regularizing Calabi-Yau topological conformal field theories using cutoff heat kernels

Abstract

In this paper we construct a family of topological conformal field theories (TCFTs) associated to a Calabi-Yau space by modifying the heat kernel and sections of the Calabi-Yau space. This is done by restricting to certain eigenspaces of the Laplacian. We then present two a-priori distinct ways to regularize the Calabi-Yau TCFT by using these modified heat kernels, and then show that they are equivalent. Finally, we relate the regularized TCFTs for different cutoffs.

Paper Structure

This paper contains 14 sections, 27 theorems, 61 equations.

Table of Contents

  1. Introduction
  2. Background
  3. Calabi-Yau Spaces
  4. The heat kernel
  5. Moduli space of metrized ribbon graphs
  6. labeling graphs in moduli space with Calabi-Yau space valued forms
  7. Topological conformal field theory
  8. Single eigenvalue TCFTs
  9. The cutoff heat kernel
  10. Single eigenvalue TCFT
  11. Constructing TCFTs using single-eigenvalue TCFTs
  12. Adding two TCFTs
  13. Cutoff TCFT
  14. Category Theory Background

Key Result

Proposition 2.2

Let $Q^\dagger$ be the Hermitian adjoint to $Q$ with respect to the metric. Define the Laplacian/Hamiltonian by $H=[Q,Q^\dagger].$ Then,

Theorems & Definitions (76)

  • Definition 2.1
  • Proposition 2.2
  • Example 2.3
  • Definition 2.4
  • Proposition 2.5
  • Definition 2.6
  • Definition 2.7
  • Definition 2.8
  • Definition 2.9
  • Definition 2.10
  • ...and 66 more