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SUMMARY:Quantum Geometry of Moduli Spaces of Local Systems and Representat
ion Theory
DTSTART;VALUE=DATE-TIME:20190701T123000Z
DTEND;VALUE=DATE-TIME:20190701T143000Z
DTSTAMP;VALUE=DATE-TIME:20190716T232622Z
UID:indico-event-4377@indico.math.cnrs.fr
DESCRIPTION:Lectures 1-3 are mostly based on our recent work with Linhui S
hen.\n\nGiven a surface S with punctures and special points on the boundar
y considered modulo isotopy\, and a split semi-simple adjoint group G\, we
define and quantize moduli spaces Loc(G\,S) G-local systems on S\, genera
lising 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 clust
er Poisson variety structure\, equivariant under the action of a discrete
group\, containing the mapping class group of S. This generalises results
of V. Fock and the author\, and I. Le.\n\nFor any cluster Poisson variety
X\, we consider the quantum Langlands modular double of the algebra of reg
ular functions on 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 representations.\n\nCombining this\, we get principal series rep
resentations of the quantum Langlands modular double of the algebras of re
gular functions on moduli spaces P(G\, S) and Loc(G\,S).\n\nWe discuss app
lications to representations theory\, geometry\, and mathematical physics.
\n\nIn particular\, when S has no boundary\, we get a local system of infi
nite dimensional vector spaces over the punctured determinant line bundle
on the moduli space M(g\,n). Assigning to a complex structure on S the coi
nvariants of oscillatory representations of W-algebras sitting at the punc
tures of S\, we get another local system on the same spa. We conjecture th
ere exists a natural non-degenerate pairing between these local systems\,
providing conformal blocks for Liouville / Toda theories.\n\nIn Lecture 4
we discuss spectral description of non-commutative local systems on S\, pr
oviding a non-commutative cluster structure of the latter. It is based on
our joint work with Maxim Kontsevich.\n\nhttps://indico.math.cnrs.fr/event
/4377/
LOCATION:IHES Amphithéâtre Léon Motchane
URL:https://indico.math.cnrs.fr/event/4377/
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