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SUMMARY:Hodge structures on conformal blocks
DTSTART:20251218T130000Z
DTEND:20251218T150000Z
DTSTAMP:20260611T052400Z
UID:indico-event-15169@indico.math.cnrs.fr
DESCRIPTION:Speakers: Pierre Godfard (UNC Chapel Hill)\n\nModular functors
  are collections of vector bundles with flat connections on (twisted) modu
 li spaces of curves\, individually known as conformal blocks\, that satisf
 y strong compatibility conditions with respect to natural maps between the
 se moduli spaces. Such structures arise naturally in the representation th
 eory of affine Lie algebras and quantum groups\, where the conformal block
 s are known to be semisimple.\nRecently\, Hodge structures on the genus-0 
 conformal blocks associated to affine Lie algebras have been studied by Be
 lkale\, Fakhruddin\, and Mukhopadhyay through a motivic construction. In p
 articular\, they computed genus-0 Hodge numbers for $sl_n$.\nI will discus
 s an axiomatic proof of the existence and uniqueness of such Hodge structu
 res and of the semisimplicity of conformal blocks\, for any modular functo
 r (i.e. any modular fusion category). If the flat bundles of conformal blo
 cks were rigid and semisimple\, a result of Simpson in non-Abelian Hodge t
 heory would imply that they support Hodge structures. However\, they are n
 ot rigid in general. I will explain how a different form of rigidity for m
 odular fusion categories—Ocneanu rigidity—can be used\, together with 
 non-Abelian Hodge theory\, to tackle these questions. Finally\, I will dis
 cuss an application to the computation of Hodge numbers for $sl_2$ modular
  functors of odd level in higher genus and how these numbers are part of (
 new) cohomological field theories (CohFTs).\n\nhttps://indico.math.cnrs.fr
 /event/15169/
URL:https://indico.math.cnrs.fr/event/15169/
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