[Paper Review] Quantum geometry of moduli spaces of local systems and representation theory

A. B. Goncharov, Linhui Shen|arXiv (Cornell University)|Apr 23, 2019

Advanced Algebra and Geometry19 references36 citations

TL;DR

The paper constructs cluster Poisson and K2-structures on moduli spaces of G-local systems on decorated surfaces, proves their equivariant quantization, and derives representation-theoretic and duality applications.

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.

Motivation & Objective

• Motivate the study by linking moduli of G-local systems to cluster structures and higher Teichmüller theory.
• Define and study PG,S and AG′,S as cluster Poisson and cluster K2-structures respectively.
• Develop a ΓG,S-equivariant quantization of PG,S and relate it to Langlands duality.
• Demonstrate applications to Donaldson–Thomas transformations, duality conjectures, and canonical bases in function algebras.

Proposed method

• Introduce and equip the moduli space PG,S with a canonical ΓG,S-equivariant cluster Poisson structure.
• Prove that the dual moduli space AG′,S carries a ΓG,S-equivariant cluster K2-structure forming a cluster ensemble with PG,S.
• Quantize PG,S to obtain a ΓG,S-equivariant ∗-representation of the Langlands modular double Aℏ(PG,S) and its principal series representation.
• Show DT-transformations are cluster transformations and compute them explicitly using the cluster structure.
• Apply the framework to recover principal series representations of quantum groups and construct canonical linear bases in O(PG,S).

Experimental results

Research questions

• RQ1 What cluster structures exist on PG,S and its dual AG′,S for a split semi-simple adjoint G?
• RQ2 How does the ΓG,S symmetry act on these cluster structures and their quantizations?
• RQ3 Can one quantize PG,S to obtain ΓG,S-equivariant ∗-representations corresponding to Langlands dual data?
• RQ4 What are the implications of these structures for DT-transformations, duality conjectures, and canonical bases in O(PG,S)?
• RQ5 How do these constructions connect to higher Teichmüller theory, conformal blocks, and quantum group representations?

Key findings

• PG,S carries a canonical ΓG,S-equivariant cluster Poisson structure.
• AG′,S carries a ΓG,S-equivariant cluster K2-structure, forming a cluster ensemble with PG,S.
• Aℏ(PG,S) provides a ΓG,S-equivariant quantization via a principal series ∗-representation.
• The Donaldson–Thomas transformation of PG,S is a cluster Poisson transformation and is computed explicitly.
• A canonical linear basis in O(PG,S) is constructed, parametrized by tropical points of AG∨,S, and the duality conjectures are established in this framework.
• Applications include principal series representations of quantum groups and connections to Langlands duality and modular functors.