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 $S_q^7$ arises as the unit sphere inside a q-deformed quaternion space ${\mathbb{H}}_q^2$. The resulting four-sphere $S_q^4$ 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 ${\mathbb{CP}}_q^3$ and use it to study a system of anti-self-duality equations on $S_q^4$, for which we find an `instanton' solution coming from the natural projection defining the tautological bundle over $S_q^4$.

v2: 38 pages; completely rewritten. The crucial difference with respect to the first version is that in the present one the quantum four-sphere, the base space of the fibration, is NOT a quantum homogeneous space. This has important consequences and led to very drastic changes to the paper. To appear in CMP

References in corpus (4)

• Anti-selfdual Connections on the Quantum Projective Plane: Monopoles
• Quantisation of twistor theory by cocycle twist
• The 3D Spin Geometry of the Quantum Two-Sphere
• q-Quaternions and q-deformed su(2) instantons

Cited by in corpus (2)

• An algebraic framework for noncommutative bundles with homogeneous fibres
• Warped Products and Yang-Mills equations on non commutative spaces