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Local index theory for geometric first-order differential operators

Alberto Richtsfeld

TL;DR

This work proves a unified local index theorem for chiral geometric operators, extending Gilkey’s invariant-theory approach to non-Dirac first-order operators. By framing geometric operators as twists of universal elliptic symbols on G_n-structures and employing a heat-kernel expansion with natural, curvature-polynomial coefficients, the authors identify the short-time limit of the heat-supertrace with a Chern–Weil density. They introduce higher Dirac and higher signature operators as concrete, non-Dirac examples and establish their index formulas, including the Rarita–Schwinger operator, via a cohesive framework that recovers classical results (e.g., the A-roof genus, L-genus) and yields explicit topological invariants. The methodology blends invariant theory, representation theory, and heat-kernel techniques to provide a concise, general proof suitable for chiral geometric operators and their twists. This broadens the scope of local index theory and offers a practical path for computing indices of a wide class of geometric first-order operators in even dimensions.

Abstract

We introduce the concept of chiral geometric operators and use Gilkey's invariance theory to prove the local index theorem for these operators. In other words, we demonstrate that the supertrace of the heat kernel of a given geometric operator converges as time approaches zero and that this limit is the Chern-Weil form of the Atiyah-Singer integrand. In addition to classical Dirac-type operators that appear in geometry, chiral geometric operators include all higher Dirac operators. This includes in particular the Rarita-Schwinger operator. We also construct a new class of such operators on four-manifolds called higher signature operators.

Local index theory for geometric first-order differential operators

TL;DR

This work proves a unified local index theorem for chiral geometric operators, extending Gilkey’s invariant-theory approach to non-Dirac first-order operators. By framing geometric operators as twists of universal elliptic symbols on G_n-structures and employing a heat-kernel expansion with natural, curvature-polynomial coefficients, the authors identify the short-time limit of the heat-supertrace with a Chern–Weil density. They introduce higher Dirac and higher signature operators as concrete, non-Dirac examples and establish their index formulas, including the Rarita–Schwinger operator, via a cohesive framework that recovers classical results (e.g., the A-roof genus, L-genus) and yields explicit topological invariants. The methodology blends invariant theory, representation theory, and heat-kernel techniques to provide a concise, general proof suitable for chiral geometric operators and their twists. This broadens the scope of local index theory and offers a practical path for computing indices of a wide class of geometric first-order operators in even dimensions.

Abstract

We introduce the concept of chiral geometric operators and use Gilkey's invariance theory to prove the local index theorem for these operators. In other words, we demonstrate that the supertrace of the heat kernel of a given geometric operator converges as time approaches zero and that this limit is the Chern-Weil form of the Atiyah-Singer integrand. In addition to classical Dirac-type operators that appear in geometry, chiral geometric operators include all higher Dirac operators. This includes in particular the Rarita-Schwinger operator. We also construct a new class of such operators on four-manifolds called higher signature operators.
Paper Structure (11 sections, 22 theorems, 188 equations)

This paper contains 11 sections, 22 theorems, 188 equations.

Table of Contents

  1. Introduction
  2. Invariant theory
  3. Geometric operators
  4. Geometric operators and their properties
  5. Generalized gradients and examples of geometric operators
  6. Chiral geometric operators
  7. Examples of chiral geometric operators
  8. Higher Dirac operators
  9. Higher signature operators
  10. The heat kernel expansion
  11. The local index theorem

Key Result

Theorem 1

A joint geometric invariant $\omega$ of mixed weight $(k,l)$ vanishes identically if $k>0$ or $l\neq 0$ holds. If $k=l=0$ then $\omega$ is an element of $\mathrm{Pont}\otimes \mathrm{Chern}$:

Theorems & Definitions (51)

  • Definition 2.1: atiyahHeatEquationIndex1973atiyahErrataPaperHeat1975freedAtiyahSingerIndex2021
  • Remark 2.2
  • Remark 2.3
  • Definition 2.4
  • Theorem : Gilkey, atiyahHeatEquationIndex1973gilkeyCurvatureEigenvaluesLaplacian1973gilkeyInvarianceTheoryHeat1994
  • Definition 3.1: atiyahIndexEllipticOperators1968
  • Remark 3.2
  • Definition 3.3
  • Definition 3.4
  • Lemma 3.5
  • ...and 41 more