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An L^2-Index Theorem for Dirac Operators on S^1 * R^3

Tom M. W. Nye, Michael A. Singer

TL;DR

<3-5 sentence high-level summary>Problem: determine the $L^2$-index of a Dirac operator coupled to a unitary connection on a framed bundle over $S^1\times\overline{B}^3$ with boundary data ensuring Fredholmness. Approach: reduce to Callias-type indices via Fourier analysis in the $S^1$ direction and prove the general case by an excision argument; relate the index to the adiabatic limit of the boundary $\eta$-invariant. Main result: an explicit index formula $\operatorname{ind}(D_{\mathbb A}^+) = -c_2({\mathbb E},f)[X] - \sum_k c_1(E^+_{(k)})[S^2_{\infty}]$, equivalently $\int_X \mathrm{ch}({\mathbb E}) - \tfrac{1}{2}\overline{\eta}_{\mathrm{lim}}$, plus a boundary-η invariant interpretation. Application: counts zero modes in caloron backgrounds and reveals periodic rank jumps in the Nahm transform.

Abstract

An expression is found for the $L^2$-index of a Dirac operator coupled to a connection on a $U_n$ vector bundle over $S^1\times{\mathbb R}^3$. Boundary conditions for the connection are given which ensure the coupled Dirac operator is Fredholm. Callias' index theorem is used to calculate the index when the connection is independent of the coordinate on $S^1$. An excision theorem due to Gromov, Lawson, and Anghel reduces the index theorem to this special case. The index formula can be expressed using the adiabatic limit of the $η$-invariant of a Dirac operator canonically associated to the boundary. An application of the theorem is to count the zero modes of the Dirac operator in the background of a caloron (periodic instanton).

An L^2-Index Theorem for Dirac Operators on S^1 * R^3

TL;DR

<3-5 sentence high-level summary>Problem: determine the -index of a Dirac operator coupled to a unitary connection on a framed bundle over with boundary data ensuring Fredholmness. Approach: reduce to Callias-type indices via Fourier analysis in the direction and prove the general case by an excision argument; relate the index to the adiabatic limit of the boundary -invariant. Main result: an explicit index formula , equivalently , plus a boundary-η invariant interpretation. Application: counts zero modes in caloron backgrounds and reveals periodic rank jumps in the Nahm transform.

Abstract

An expression is found for the -index of a Dirac operator coupled to a connection on a vector bundle over . Boundary conditions for the connection are given which ensure the coupled Dirac operator is Fredholm. Callias' index theorem is used to calculate the index when the connection is independent of the coordinate on . An excision theorem due to Gromov, Lawson, and Anghel reduces the index theorem to this special case. The index formula can be expressed using the adiabatic limit of the -invariant of a Dirac operator canonically associated to the boundary. An application of the theorem is to count the zero modes of the Dirac operator in the background of a caloron (periodic instanton).

Paper Structure

This paper contains 17 sections, 5 theorems, 72 equations.

Table of Contents

  1. Introduction
  2. Statement of results
  3. A sketch of the proof
  4. Application: counting the zero modes of the Dirac operator in the background of a caloron.
  5. Acknowledgements
  6. On the topology of calorons
  7. Topological classification of framed bundles
  8. Chern-Weil theory for framed bundles
  9. Boundary conditions versus asymptotics
  10. Proof of Theorem 1.1
  11. First proof
  12. Second proof
  13. Remarks about the $L^2$-condition
  14. Proof of the index theorem
  15. Proof when $c_2({\mathbb E},f)[X] =0$
  16. ...and 2 more sections

Key Result

Theorem 1

Let ${\mathbb A}$ be a caloron configuration framed by an admissible pair $(A_\infty,\Phi_{\infty})$. Then the operator in sobop is Fredholm if and only if $1 - \exp(2\pi\Phi_{\infty}/\mu_{0})$ is invertible.

Theorems & Definitions (9)

  • Definition 1
  • Definition 2
  • Theorem 1
  • Theorem 2
  • Corollary 1
  • Proposition 1
  • proof
  • Proposition 2
  • proof