Solution of Self-Duality Equation in Quantum-Group Gauge Theory and Quantum Harmonics

TL;DR

This paper develops a quantum-group gauge theory on quantum Euclidean space, formulates a self-duality equation, constructs a deformed instanton solution, and explores a harmonic interpretation using quantum harmonics.

Contribution

It introduces the quantum-group self-duality equation and constructs a deformed instanton solution within quantum gauge theory on quantum Euclidean space.

Findings

Constructed a quantum-group self-duality equation (QGSDE).

Obtained a deformed analog of the BPST-instanton solution.

Formulated quantum harmonic gauge equations using quantum harmonics.

Abstract

We discuss the gauge theory for quantum group $SU_q (2)\times U(1) $ on the quantum Euclidean space. This theory contains three physical gauge fields and one$\; U(1)-$gauge field with a zero field strength. We construct the quantum-group self-duality equation (QGSDE) in terms of differential forms and with the help of the field-strength decomposition. A deformed analog of the BPST-instanton solution is obtained. We consider a harmonic (twistor) interpretation of QGSDE in terms of $SU_q (2)/U(1) $ quantum harmonics. The quantum harmonic gauge equations are formulated in the framework of a left-covariant 3D differential calculus on the quantum group $SU_q (2)$.