Communications in Mathematical Physics

, Volume 315, Issue 2, pp 489–530 | Cite as

Differential and Twistor Geometry of the Quantum Hopf Fibration

  • Simon BrainEmail author
  • Giovanni Landi


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 \({S^4_q}\) .


Hopf Algebra Quantum Group Twistor Space Differential Calculus Hodge Structure 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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Authors and Affiliations

  1. 1.Unité de Recherche en MathématiquesUniversité du Luxembourg (Campus Kirchberg)LuxembourgGrand Duchy of Luxembourg
  2. 2.Dipartimento di MatematicaUniversità di TriesteTriesteItaly
  3. 3.INFNTriesteItaly

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