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Equivariant Index Theorem on $\mathbb{R}^n$ in the Context of Continuous Fields of $C^*$-algebras

Baiying Ren, Hang Wang, Zijing Wang

TL;DR

The paper develops a framework to compute equivariant indices on ${\mathbb R^n}$ using a continuous field of ${C^*}$-algebras, bridging analytic index theory with algebraic index theory in the presence of a compact group action ${G\le SO(n)}$. By constructing a $G$-invariant graph projection and a family of equivariant cyclic cocycles ${\omega_g}$ and ${\epsilon_g}$, the authors derive fixed-point type formulas for the equivariant index ${\mathrm{ind}_{(g)}(P_a)}$, including a semiclassical limit that recovers the fixed-point expression. The main results give explicit formulas for both general fixed-point sets and isolated fixed points, and a detailed Bott–Dirac operator example on ${\mathbb R}^{2n}$ showing the equivariant index equals 1 for all ${g\in SO(2n)}$, reinforcing connections to the Baum–Connes gamma-element. The work combines Shubin pseudodifferential operator calculus, continuous fields of ${C^*}$-algebras, and cyclic cohomology to produce computable, representation-theoretic indices with potential implications for isometry groups and KK-theory.

Abstract

We prove an equivariant index theorem on the Euclidean space using a continuous field of $C^*$-algebras. This generalizes the work of Elliott, Natsume and Nest, which is a special case of the algebraic index theorem by Nest-Tsygan. Using our formula, the equivariant index of the Bott-Dirac operator on $\mathbb{R}^{2n}$ can be explicitly calculated.

Equivariant Index Theorem on $\mathbb{R}^n$ in the Context of Continuous Fields of $C^*$-algebras

TL;DR

The paper develops a framework to compute equivariant indices on using a continuous field of -algebras, bridging analytic index theory with algebraic index theory in the presence of a compact group action . By constructing a -invariant graph projection and a family of equivariant cyclic cocycles and , the authors derive fixed-point type formulas for the equivariant index , including a semiclassical limit that recovers the fixed-point expression. The main results give explicit formulas for both general fixed-point sets and isolated fixed points, and a detailed Bott–Dirac operator example on showing the equivariant index equals 1 for all , reinforcing connections to the Baum–Connes gamma-element. The work combines Shubin pseudodifferential operator calculus, continuous fields of -algebras, and cyclic cohomology to produce computable, representation-theoretic indices with potential implications for isometry groups and KK-theory.

Abstract

We prove an equivariant index theorem on the Euclidean space using a continuous field of -algebras. This generalizes the work of Elliott, Natsume and Nest, which is a special case of the algebraic index theorem by Nest-Tsygan. Using our formula, the equivariant index of the Bott-Dirac operator on can be explicitly calculated.
Paper Structure (9 sections, 13 theorems, 147 equations)

This paper contains 9 sections, 13 theorems, 147 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Proof of the Equivariant Index Theorem
  4. The Equivariant Analytic Index
  5. Continuous Field of $C^*$-algebras
  6. Equivariant Continuous Vector Field
  7. A Field of Cyclic Cocycles on Any Sufficiently Smooth Field of Projections
  8. Example of Bott-Dirac operator
  9. Appendix

Key Result

Theorem 1.1

Suppose that $P_a$ is an elliptic pseudodifferential operator associtated to a symbol $a$ of positive order (cf. Definition def5) on $\mathbb{R}^n$, then where $T^*\mathbb{R}^n$ is oriented by $dx_1\wedge d{\xi}_1 \wedge \dots \wedge dx_n\wedge d{\xi}_n$, $\hat{e_a}=e_a-\left( \right)$ and the convergence of the integral on the right hand side follows from the fact that $a$ has positive order.

Theorems & Definitions (30)

  • Theorem 1.1: ENN2; Section 3
  • Theorem 1.2
  • Theorem 1.3
  • Remark 1
  • Definition 2.1: Sh
  • Definition 2.2
  • Remark 2
  • Remark 3
  • Proposition 3.1
  • proof
  • ...and 20 more