Quantum geometry of moduli spaces of local systems and representation theory

Abstract

Let G be a split semi-simple adjoint group over Q, and S an oriented surface with punctures and special boundary points. We introduce a moduli space P G,S parametrizing G-local system on S with some boundary data, and prove that it carries a cluster Poisson structure, equivariant under the action of the cluster modular group Γ G,S . We prove that the group Γ G,S contains the mapping class group of S, the group of outer automorphisms of G, the product of Weyl groups over punctures, and the product of braid groups over boundary components.Let G be the universal cover of G. Then there is a moduli space A G,S , parametrizing unipotent G-local systems on S with an extra boundary data. We prove that it carries a Γ G,S -equivariant cluster K 2 -structure, and the pair (A G,S , P G,S ) is a cluster ensemble. These results generalize the works of V. Fock & the first author, and of I. Le.We quantize cluster Poisson varieties X for any Planck constant such that > 0 or | | = 1. Starting from a non-commutative algebra O q (X ) deforming the algebra of functions on X , we define a * -algebra structure on its Langlands modular double:The involution * preserves the factors if > 0, and swipes them if | | = 1. We construct a principal series of representations of the * -algebra A (X ) in a dence subspace S X of a Hilbert space, equivariant under a unitary projective representation of the cluster modular group Γ X , assumingThis extends works of V. Fock & the first author when > 0.Combining this, we get a Γ G,S -equivariant quantization of the space P G,S . It is given by the *algebra A (P G,S ) and its principal series representations in a Hilbert space H G,S , equivariant under the Γ G,S −action. We construct realizations of the principal series * -representations. In particular, when S is punctured disc with two special points we get principal series * -representations of the Langlands modular double of the quantum group U q (g).We conjecture that for a genus g surface S with with n > 0 punctures, there is a canonical nondegenerate continuous pairing between the infinite dimensional vector bundle on the punctured determinant line bundle on M g,n given by coinvariants of oscillatory representations of the W -algebra of g ∨ , and the bundle of topological spaces provided by Γ G,S −equivariant quantization of the space P G,

Skein and cluster algebras of unpunctured surfaces for $\mathfrak{sl}_3$

“…Although we mainly deal with (the quantum aspects of) the former in the text, we borrow some notations from the latter to indicate connections to relevant geometries. For a comparison of their quantizations given by [BZ05] and [FG08], see [GS19,Section 13.3].…”

“…Our goal is to find higherrank analogues of the Muller's result for semisimple Lie algebras g other than sl 2 . Indeed, for the simply-connected algebraic group G with Lie algebra g, the moduli space A G,Σ has also a canonical K 2 -structure, which is encoded in a seed pattern s(g, Σ) ( [FG06a] for sl n ; [Le19] for classical Lie algebras; [GS19] for general). On the other hand, the higher-rank analogues of the skein theory has been studied by Kuperberg [Kup96] for rank two Lie algebras, Murakami-Ohtsuki-Yamada [MOY98], Sikora [Sik05] and Morrison [Mor07] for sl n .…”

Quantum decorated character stacks

“…Now, let ∆ be the ideal triangulation underlying ∆ ‚ . Denote by X q ∆ the quantum chart of the cluster Poisson variety defined by ∆, see [GS19], and set X s to be the quantum cluster X -variable labelled by the edge s. We have: and then applying the three-term relation in F q pConf f r 4 q…”

“…The first one, X G,S , is the moduli space of framed G-local systems on a decorated surface S, defined for an arbitrary split reductive group G. The second one, A G,S , is the moduli space of decorated twisted G-local systems on S, defined for any simply-connected reductive group G. Among other results, it was shown in [FG06] that X P GLn,S and A SLn,S are respectively cluster Poisson and cluster K 2 -varieties 1 , and moreover, form a cluster ensemble. In [GS19], [Le19], [Ip18], these results were extended to arbitrary Dynkin types. In [GS19] the moduli space X G,S was promoted to a new one, P G,S , parameterizing framed G-local systems with pinnings.…”

“…In [GS19], [Le19], [Ip18], these results were extended to arbitrary Dynkin types. In [GS19] the moduli space X G,S was promoted to a new one, P G,S , parameterizing framed G-local systems with pinnings. This moduli space also has a cluster Poisson structure, but in contrast with X G,S , allows for frozen variables at the boundary of S and thus admits gluing maps that are not available for the spaces X G,S .…”