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Chern-Gauss-Bonnet theorem via BV localization

Vyacheslav Lysov

Abstract

We present a new proof for the Chern-Gauss-Bonnet theorem. We represent the Euler class integral as the partition function for zero-dimensional field theory with on-shell supersymmetry. We rewrite the supersymmetric partition function as a BV integral and deform the Lagrangian submanifold. The new Lagrangian submanifold localizes the BV integral to the critical points of the Morse function.

Chern-Gauss-Bonnet theorem via BV localization

Abstract

We present a new proof for the Chern-Gauss-Bonnet theorem. We represent the Euler class integral as the partition function for zero-dimensional field theory with on-shell supersymmetry. We rewrite the supersymmetric partition function as a BV integral and deform the Lagrangian submanifold. The new Lagrangian submanifold localizes the BV integral to the critical points of the Morse function.
Paper Structure (9 sections, 8 theorems, 56 equations)

This paper contains 9 sections, 8 theorems, 56 equations.

Table of Contents

  1. Introduction
  2. Chern-Gauss-Bonnet theorem
  3. Morse theory
  4. Superspace integral representation for Euler class
  5. Supersymmetric partition function
  6. BV formalism
  7. BV description of the curvature theory
  8. BV localization
  9. Proof of the Chern-Gauss-Bonnet theorem

Key Result

Theorem 2.1

(Gauss-Bonnet) Let $\Sigma$ be two-dimensional compact, closed manifold with Riemann metric $g$, then the integral of the Ricci scalar $R$ is related to the Euler characteristics $\chi(\Sigma)$ of the surface $\Sigma$ via

Theorems & Definitions (18)

  • Theorem 2.1
  • Theorem 2.2
  • Theorem 3.1
  • Lemma 4.1
  • proof
  • Remark 4.2
  • Lemma 5.1
  • proof
  • Definition 6.1
  • Remark 6.2
  • ...and 8 more