Zero Modes and the Atiyah-Singer Index in Noncommutative Instantons

TL;DR

This paper investigates zero modes in noncommutative instantons, explicitly constructing them via ADHM, and relates the fermionic zero modes to the Atiyah-Singer index, highlighting effects of noncommutativity on conformal zero modes.

Contribution

It provides explicit constructions of bosonic and fermionic zero modes in noncommutative instantons using ADHM, and connects these to the Atiyah-Singer index, revealing effects of noncommutativity.

Findings

Explicit zero mode constructions in noncommutative instantons

Fermionic zero modes match the Atiyah-Singer index

Noncommutativity affects superconformal zero modes

Abstract

We study the bosonic and fermionic zero modes in noncommutative instanton backgrounds based on the ADHM construction. In k instanton background in U(N) gauge theory, we show how to explicitly construct 4Nk (2Nk) bosonic (fermionic) zero modes in the adjoint representation and 2k (k) bosonic (fermionic) zero modes in the fundamental representation from the ADHM construction. The number of fermionic zero modes is also shown to be exactly equal to the Atiyah-Singer index of the Dirac operator in the noncommutative instanton background. We point out that (super)conformal zero modes in non-BPS instantons are affected by the noncommutativity. The role of Lorentz symmetry breaking by the noncommutativity is also briefly discussed to figure out the structure of U(1) instantons.