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SUMMARY:Quantum Geometry of Moduli Spaces of Local Systems and Representat
 ion Theory
DTSTART:20190708T123000Z
DTEND:20190708T143000Z
DTSTAMP:20260419T053400Z
UID:indico-event-4376@indico.math.cnrs.fr
CONTACT:cecile@ihes.fr
DESCRIPTION:Speakers: Alexander Goncharov (Yale University & IHES)\n\nLect
 ures 1-3 are mostly based on our recent work with Linhui Shen.\n\nGiven a 
 surface S with punctures and special points on the boundary considered mod
 ulo isotopy\, and a split semi-simple adjoint group G\, we define and quan
 tize moduli spaces Loc(G\,S) G-local systems on S\, generalising character
  varieties.\n\nTo achieve this\, we introduce a new moduli space P(G\, S) 
 closely related to Loc(G\,S). We prove that it has a cluster Poisson varie
 ty structure\, equivariant under the action of a discrete group\, containi
 ng the mapping class group of S. This generalises results of V. Fock and t
 he author\, and I. Le.\n\nFor any cluster Poisson variety X\, we consider 
 the quantum Langlands modular double of the algebra of regular functions o
 n X. If the Planck constant h is either real or unitary\, we equip it with
  a structure of a *-algebra\, and construct its principal series of repres
 entations.\n\nCombining this\, we get principal series representations of 
 the quantum Langlands modular double of the algebras of regular functions 
 on moduli spaces P(G\, S) and Loc(G\,S).\n\nWe discuss applications to rep
 resentations theory\, geometry\, and mathematical physics.\n\nIn particula
 r\, when S has no boundary\, we get a local system of infinite dimensional
  vector spaces over the punctured determinant line bundle on the moduli sp
 ace M(g\,n). Assigning to a complex structure on S the coinvariants of osc
 illatory representations of W-algebras sitting at the punctures of S\, we 
 get another local system on the same spa. We conjecture there exists a nat
 ural non-degenerate pairing between these local systems\, providing confor
 mal blocks for Liouville / Toda theories.\n\nIn Lecture 4 we discuss spect
 ral description of non-commutative local systems on S\, providing a non-co
 mmutative cluster structure of the latter. It is based on our joint work w
 ith Maxim Kontsevich.\n\nhttps://indico.math.cnrs.fr/event/4376/
LOCATION:Amphithéâtre Léon Motchane (IHES)
URL:https://indico.math.cnrs.fr/event/4376/
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