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Quantum geometry of moduli spaces of local systems and representation theory

Alexander Goncharov, Linhui Shen

Abstract

Let G be a split semi-simple adjoint group, and S a colored decorated surface, given by an oriented surface with punctures, special boundary points, and a specified collection of boundary intervals. 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 M(G,S), containing the mapping class group of S, the group of outer automorphisms of G, and the product of Weyl / braid groups over punctures / boundary components. We prove that the dual moduli space A(G,S) carries a M(G,S)-equivariant cluster 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 h s.t. h>0 or |h|=1. First, we define a *-algebra structure on the Langlands modular double A(h; X) of the algebra of functions on X. We construct a principal series of representations of the *-algebra A(h; X), equivariant under a unitary projective representation of the cluster modular group M(X). This extends works of V. Fock and the first author when h>0. Combining this, we get a M(G,S)-equivariant quantization of the moduli space P(G,S), given by the *-algebra A(h; P(G,S)) and its principal series representations. We construct realizations of the principal series *-representations. In particular, when S is punctured disc with two special points, we get a principal series *-representations of the Langlands modular double of the quantum group Uq(g). We conjecture that there is a nondegenerate pairing between the local system of coinvariants of oscillatory representations of the W-algebra and the one provided by the projective representation of the mapping class group of S.

Quantum geometry of moduli spaces of local systems and representation theory

Abstract

Let G be a split semi-simple adjoint group, and S a colored decorated surface, given by an oriented surface with punctures, special boundary points, and a specified collection of boundary intervals. 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 M(G,S), containing the mapping class group of S, the group of outer automorphisms of G, and the product of Weyl / braid groups over punctures / boundary components. We prove that the dual moduli space A(G,S) carries a M(G,S)-equivariant cluster 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 h s.t. h>0 or |h|=1. First, we define a *-algebra structure on the Langlands modular double A(h; X) of the algebra of functions on X. We construct a principal series of representations of the *-algebra A(h; X), equivariant under a unitary projective representation of the cluster modular group M(X). This extends works of V. Fock and the first author when h>0. Combining this, we get a M(G,S)-equivariant quantization of the moduli space P(G,S), given by the *-algebra A(h; P(G,S)) and its principal series representations. We construct realizations of the principal series *-representations. In particular, when S is punctured disc with two special points, we get a principal series *-representations of the Langlands modular double of the quantum group Uq(g). We conjecture that there is a nondegenerate pairing between the local system of coinvariants of oscillatory representations of the W-algebra and the one provided by the projective representation of the mapping class group of S.

Paper Structure

This paper contains 341 sections, 209 theorems, 1267 equations, 98 figures.

Table of Contents

  1. Summary
  2. Main results.
  3. Application I: Donaldson-Thomas transformations for the spaces ${\mathscr P}_{{\rm G}, {\mathbb{S}}}$.
  4. Intermezzo: Tropical points in Higher Teichmuller theory.
  5. Application II: Duality Conjectures and canonical linear basis in ${\mathcal{O}}({\mathscr P}_{{\rm G}, {\mathbb{S}}})$.
  6. Application III: Cluster convexity, length functions, and Earthquake flow in Higher Teichmuller theory
  7. The main tool: cluster quantization.
  8. Application III: quantization of moduli spaces ${\mathscr P}_{{\rm G}, {\mathbb{S}}}$.
  9. Application IV: quantization of generalized character varieties.
  10. A hypothetical link to the quantum geometric Langlands.
  11. Application V: Poisson Lie groups via moduli spaces.
  12. Application VI: Quantum groups and their principal series of representations.
  13. Applications VII: quantization of ${\rm Loc}_{{\rm G}, {\mathbb{S}}}$ and representation theory $\&$ Mathematical Physics.
  14. Application VIII: Finite dimensional ${\mathscr U}_q(\mathfrak{g})-$modules and WZW theory from ${\mathcal{O}}_q({\mathcal{P}}_{{\rm G}, {\mathbb{S}}})-$modules.
  15. The irregular case.
  16. ...and 326 more sections

Key Result

Theorem 1.2

For any semi-simple Lie algebra $\mathfrak{g}$, Theorem UEAB provides a monoidal functor In the simply-laced case, it is an equivalence thanks to S22. Conjecturally, this is always the case.

Figures (98)

  • Figure 1: Three colored decorated surfaces, related to the Poisson Lie groups ${\rm G}^*$, ${\rm G}$ and ${\mathbb D}({\rm G})$.
  • Figure 2: A colored decorated surface ${\mathbb{S}}$ with 2 punctures (blue), 5 special points (red), 4 colored boundary intervals (red dashed), and 1 non-colored boundary interval (black). The moduli space ${\mathscr P}_{{\rm G}, {\mathbb{S}}}$ parametrizes ${\rm G}$-local systems ${\mathcal{L}}$ on ${\mathbb{S}}$ with invariant flags ${\rm B}$ near all punctures and special points, and pinnings over colored boundary segments.
  • Figure 3: Gluing a quadrangle out of two triangles with two vertical colored sides.
  • Figure 4: Gluing framed local systems with pinnings on two triangles.
  • Figure 5: The moduli space ${\mathscr A}_{{\rm G}, {\mathbb{S}}}$ for a colored decorated surface ${\mathbb{S}}$, without non-colored boundary intervals, parametrizes ${\rm G}$-local systems ${\mathcal{L}}$ on ${\mathbb{S}}$ with an invariant flag ${\rm A}$ near each special point or puncture.
  • ...and 93 more figures

Theorems & Definitions (456)

  • Definition 1.1
  • Theorem 1.2
  • Conjecture 1.3
  • Conjecture 1.4
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Theorem 2.6
  • ...and 446 more