q-Quaternions and q-deformed su(2) instantons

TL;DR

This paper constructs explicit instanton solutions in a q-deformed noncommutative space using q-quaternions, revealing a noncommutative moduli space and extending classical gauge theory concepts into quantum group frameworks.

Contribution

It introduces a method to generate (anti)instanton solutions in q-deformed su(2) Yang-Mills theory on quantum Euclidean space using q-quaternions, and explores their moduli space structure.

Findings

Constructed explicit (anti)instanton solutions in q-deformed space.

Demonstrated the generation of new solutions via quantum group symmetries.

Indicated the moduli space is a noncommutative manifold.

Abstract

We construct (anti)instanton solutions of a would-be q-deformed su(2) Yang-Mills theory on the quantum Euclidean space R_q^4 [the SO_q(4)-covariant noncommutative space] by reinterpreting the function algebra on the latter as a q-quaternion bialgebra. Since the (anti)selfduality equations are covariant under the quantum group of deformed rotations, translations and scale change, by applying the latter we can generate new solutions from the one centered at the origin and with unit size. We also construct multi-instanton solutions. As they 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 will be a noncommutative manifold. Similarly, gauge transformations should be allowed to depend on additional noncommutative parameters.