Differential and Twistor Geometry of the Quantum Hopf Fibration

Abstract

We study a quantum version of the SU(2) Hopf fibration $${S^7 \to S^4}$$ and its associated twistor geometry. Our quantum sphere $${S^7_q}$$ arises as the unit sphere inside a q-deformed quaternion space $${\mathbb{H}^2_q}$$ . The resulting four-sphere $${S^4_q}$$ is a quantum analogue of the quaternionic projective space $${\mathbb{HP}^1}$$ . The quantum fibration is endowed with compatible non-universal differential calculi. By investigating the quantum symmetries of the fibration, we obtain the geometry of the corresponding twistor space $${\mathbb{CP}^3_q}$$ and use it to study a system of anti-self-duality equations on $${S^4_q}$$ , for which we find an ‘instanton’ solution coming from the natural projection defining the tautological bundle over $${S^4_q}$$ .