q-Deformed su(2) instantons by q-quaternions

TL;DR

This paper constructs q-deformed instanton solutions in a noncommutative space using q-quaternions, revealing a noncommutative moduli space and gauge transformations dependent on noncommutative parameters.

Contribution

It introduces a method to construct explicit q-deformed instantons on R_q^4 using q-quaternions, expanding the understanding of noncommutative gauge theories.

Findings

Constructed explicit (anti)instanton solutions on R_q^4.

Identified noncommutative parameters as moduli of instantons.

Suggested the moduli space and gauge transformations are inherently noncommutative.

Abstract

Interpreting the coordinates of the quantum Euclidean space R_q^4 [the SO_q(4)-covariant noncommutative space] as the entries of a ``q-quaternion matrix'' we construct (anti)instanton solutions of a would-be q-deformed su(2) Yang-Mills theory on R_q^4. Since the (anti)selfduality equations are covariant under the quantum group of deformed rotations, translations and scale change, by applying the latter we can respectively generate ``gauge equivalent'' or ``inequivalent'' solutions from the one centered at the origin and with unit size. We also construct multi-instanton solutions. As these solutions depend on noncommuting parameters playing the roles of `sizes' and `coordinates of the centers' of the instantons, this indicates that the moduli space of a complete theory should be a noncommutative manifold. Similarly, as the (global) gauge transformations relating ``gauge equivalent'' solutions depend on the generators of two copies of SU_q(2), this suggests that gauge transformations should be allowed to depend on additional noncommutative parameters.