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

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 $SU_q (2)$.