Instantons on Noncommutative R^4 and Projection Operators

TL;DR

This paper explores the noncommutative ADHM construction of instantons on R^4, revealing how zero-modes project out states in Fock space and extending the analysis from U(1) to U(N) gauge groups.

Contribution

It provides a detailed study of zero-mode projections in noncommutative instantons and introduces a construction of zero-modes for U(N) gauge groups, clarifying the projection mechanism.

Findings

Zero-modes project out states in Fock space in noncommutative ADHM construction.

Projection mechanisms are similar for U(1) and U(N) gauge groups.

Physical interpretation in IIB matrix model is discussed.

Abstract

I carefully study noncommutative version of ADHM construction of instantons, which was proposed by Nekrasov and Schwarz. Noncommutative ${\bf R}^4$ is described as algebra of operators acting in Fock space. In ADHM construction of instantons, one looks for zero-modes of Dirac-like operator. The feature peculiar to noncommutative case is that these zero-modes project out some states in Fock space. The mechanism of these projections is clarified when the gauge group is U(1). I also construct some zero-modes when the gauge group is U(N) and demonstrate that the projections also occur, and the mechanism is similar to the U(1) case. A physical interpretation of the projections in IIB matrix model is briefly discussed.