Comments on the U(2) noncommutative instanton

Abstract

We discuss the 't Hoof ansatz for instanton solutions in noncommutative U (2) Yang-Mills theory. We show that the extension of the ansatz leading to singular solutions in the commutative case, yields to non self-dual (or self-antidual) configurations in noncommutative space-time. A proposal leading to selfdual solutions with Q = 1 topological charge (the equivalent of the regular BPST ansatz) can be engineered, but in that case the gauge field and the curvature are not Hermitian (although the resulting Lagrangian is real).

Noncommutative Instantons via Dressing and Splitting Approaches

“…Our calculations demonstrate that noncommutativity with a self-dual θ µν causes no difficulties in constructing solutions. Potential singularities like (r 2 −2θ) −1 , as occurring for the noncommutative 't Hooft instanton [16,22], are regulated in our case by the instanton size. Moreover, in the framework of the dressing and splitting approaches described in this paper we were able to solve the reality problem of the gauge field which was encountered by the authors of [16] in generalizing the BPST ansatz [45].…”

“…With these ingredients we parametrize S † as follows, 16) and expect conditions on v 1 , v 2 , and K. Due to the nilpotency property (3.12a) of S and S † we find [v 1 , v 2 ] = 0 and hence, T † 1 T 2 = T † 2 T 1 = 0. The condition (3.13a) then tells us that v 1 and v 2 are anti-holomorphic functions depending onz 1 andz 2 only.…”

“…The proper noncommutative 't Hooft multi-instanton field strength was written down explicitly but its associated gauge potential could be given only implicitly. In order to get around these difficulties, Correa et al [16] suggested to use the Belavin-Polyakov-Schwarz-Tyupkin (BPST) [45] ansatz for constructing the noncommutative U (2) one-instanton. In contrast to [22] they did obtain an explicit expression for the self-dual gauge potential but the reality of the gauge potential and field strength was lost.…”

Remarks on the noncommutative Atiyah-Drinfeld-Hitchin-Manin construction

“…In this appendix we apply our method to the construction of noncommutative 't Hooft instantons [17,18]. We will only discuss the ASD 't Hooft instantons on ASD R 4 NC (i.e., ζ = θ 1 + θ 2 = 0 while θ 1,2 = 0).…”

Zero modes and the Atiyah-Singer index in noncommutative instantons

“…Thus the noncommutativity of space doesn't make any serious obstacle for the ADHM construction of noncommutative instantons and indeed it turns out that it is a really powerful tool even for noncommutative instantons. Recently much progress has been made in this direction [16,17,68,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35]. This paper is aiming to explain how to construct bosonic and fermionic zero modes in noncommutative instanton backgrounds based on the ADHM construction and how to relate them to the Atiyah-Singer index.…”