Seminario

Seminario Geometría

The projective equivalence of the Hitchin and the Knizhnik–Zamolodchikov connections

Ponente:  Tim Henke (Universidade de Porto)
Fecha:  miércoles 14 de mayo de 2025 - 11:30
Lugar:  Aula Naranja, ICMAT

Resumen:

The Pauly Isomorphism identifies the geometric quantisation of the moduli space of parabolic bundles over a pointed Riemann surface with the space of conformal blocks associated to the same surface. This isomorphism is the mathematical formalisation of the physical CS/WZNW duality. Both spaces depend crucially on the complex structure of the surface, but the physical duality is understood to be purely topological. This implies that the Pauly Isomorphism should be projectively flat with respect to variations of the complex structure. These are given by the Hitchin connection for the moduli space and the Knizhnik–Zamolodchikov connection on the sheaf of conformal blocks. For higher genera the equivalence follows from the non-parabolic case. We treat the genus 0 case that is crucial for developing topological invariants from the moduli theory.

I will explain the construction of the different spaces, define the bundles on both sides of the isomorphism and sketch the ideas that go into the proof.