Differential and Twistor Geometry of the Quantum Hopf Fibration

TL;DR

This paper develops a quantum analog of the classical SU(2) Hopf fibration and its twistor geometry, introducing quantum spheres and analyzing their differential calculus and instanton solutions.

Contribution

It constructs a quantum version of the Hopf fibration and twistor space, providing new insights into quantum geometry and gauge theory on noncommutative spaces.

Findings

Quantum spheres $S^7_q$ and $S^4_q$ modeled as deformations of classical spaces

Development of compatible differential calculi on quantum fibrations

Identification of instanton solutions in the quantum setting

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 $S^4_q$.