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Supersymmetry and localization

Albert Schwarz, Oleg Zaboronsky

TL;DR

The paper develops a general localization framework for integrals on compact (super)manifolds with an odd symmetry $Q$, proving that under $div_{dV} Q=0$ and $Q^2 \in {\cal K}(M)$ the integral of a $Q$-invariant function localizes to the zero set $R_Q$ of the number part $m(Q)$. It extends the classical Duistermaat–Heckman formula to supergeometry by interpreting the DH integral as a $Q$-localized superintegral, and further shows that multiple anticommuting $Q_i$ localize to the intersection of their zero loci. The work derives exact stationary-phase expressions for localized integrals, expressing results as finite sums over $K_Q$ with superdeterminants, and identifies conditions under which stationary-phase approximations are exact. These results have potential implications for BV formalism, dimensional reduction, and $OSp(n|m)$-invariant computations in field theory.

Abstract

We study conditions under which an odd symmetry of the integrand leads to localization of the corresponding integral over a (super)manifold. We also show that in many cases these conditions guarantee exactness of the stationary phase approximation of such integrals.

Supersymmetry and localization

TL;DR

The paper develops a general localization framework for integrals on compact (super)manifolds with an odd symmetry , proving that under and the integral of a -invariant function localizes to the zero set of the number part . It extends the classical Duistermaat–Heckman formula to supergeometry by interpreting the DH integral as a -localized superintegral, and further shows that multiple anticommuting localize to the intersection of their zero loci. The work derives exact stationary-phase expressions for localized integrals, expressing results as finite sums over with superdeterminants, and identifies conditions under which stationary-phase approximations are exact. These results have potential implications for BV formalism, dimensional reduction, and -invariant computations in field theory.

Abstract

We study conditions under which an odd symmetry of the integrand leads to localization of the corresponding integral over a (super)manifold. We also show that in many cases these conditions guarantee exactness of the stationary phase approximation of such integrals.

Paper Structure

This paper contains 5 sections, 7 theorems, 42 equations.

Table of Contents

  1. Introduction
  2. The Duistermaat-Heckman formula in the language of supergeometry.
  3. Localization of integrals over supermanifolds.
  4. Exactness of stationary phase approximation.
  5. Appendix.

Key Result

Theorem 1

Let $M$ be a compact supermanifold with a volume form $dV$. Let $Q$ be an odd vector field on $M$ which satisfies the following conditions: Then for any neighborhood $U( R_{Q})$ of $R_{Q}$ in $M$ there exists an even $Q$-invariant function $g_{0}$ which is equal to 1 on $\hbox{some neighborhood} O(R_{Q}) \subset U(R_{Q}) \hbox{of} R_{Q}$ and vanishes outside of $U(R_{Q})$ . For every $Q$-invarian

Theorems & Definitions (7)

  • Theorem 1
  • Lemma 1
  • Theorem 2
  • Theorem 3
  • Lemma 2
  • Lemma 3
  • Theorem 4