[Paper Review] Donaldson-Thomas trasnsformations of moduli spaces of G-local systems

A. B. Goncharov, Linhui Shen|arXiv (Cornell University)|Feb 21, 2016

Advanced Algebra and Geometry43 references16 citations

TL;DR

This paper establishes that the Weyl group actions and the duality involution on moduli spaces of PGL_m-local systems on decorated surfaces act via cluster Poisson transformations, and computes the Donaldson-Thomas transformation for these spaces. Using Keller’s framework on cluster DT-transformations and results from Gross-Hacking-Keel-Kontsevich, it constructs a canonical basis for regular functions on the moduli space X_PGL_m,S and the Fomin-Zelevinsky upper cluster algebra for SL_m, confirming the Duality Conjectures in this setting.

ABSTRACT

Kontsevich and Soibelman defined Donaldson-Thomas invariants of a 3d Calabi-Yau category equipped with a stability condition. Any cluster variety gives rise to a family of such categories. Their DT invariants are encapsulated in a single formal automorphism of the cluster variety, called the DT-transformation. Let S be an oriented surface with punctures, and a finite number of special points on the boundary considered modulo isotopy. It give rise to a moduli space X(m, S), closely related to the moduli space of PGL(m)-local systems on S, which carries a canonical cluster Poisson variety structure. For each puncture of S, there is a birational Weyl group action on the space X(m, S). We prove that it is given by cluster Poisson transformations. We prove a similar result for the involution * of the space X(m,S) provided by dualising a local system on S. We calculate the DT-transformation of the moduli space X(m,S), with few exceptions. Namely, let C(m,S) be the transformation of the space X(m,S) given by the product of three commuting maps: the involution *, the product, over all punctures of S, of the longest element of the Weyl group action corresponding to the puncture, and the "shift of the special points on the boundary by one" map. Using a characterisation of a class of DT-transformations due to Keller, we prove that C(m,S) = DT. We prove that, burring few exceptions, the Weyl group and the involution * act by cluster transformations of the dual moduli space A(m, S). So the formula C(m,S) = DT is valid for the space A(m,S). Our main result, combined with the work of Gross, Hacking, Keel and Kontsevich, deliver a canonical basis in the space of regular functions on the cluster variety X(m,S), and in the upper cluster algebra with principal coefficients related to the pair (SL(m), S), with few exceptions.

Motivation & Objective

• To determine when the Weyl group and duality involution act as cluster transformations on moduli spaces of G-local systems.
• To compute the Donaldson-Thomas transformation for the moduli space X_PGL_m,S of PGL_m-local systems on a decorated surface S.
• To establish the existence of a canonical basis in the ring of regular functions on X_PGL_m,S and in the upper cluster algebra for SL_m, as predicted by the Duality Conjectures.
• To provide a geometric realization of cluster DT-transformations via laminations and ideal triangulations on surfaces.
• To extend the framework of cluster DT-transformations to moduli spaces of local systems using combinatorial and geometric methods.

Proposed method

• Uses the cluster Poisson structure on X_PGL_m,S defined by Fock and Goncharov for surfaces with punctures and special boundary points.
• Applies Keller’s classification of cluster DT-transformations to compute the DT-transformation for admissible decorated surfaces S with g(S) + μ ≥ 3 or S an annulus with two special points.
• Analyzes the action of the longest Weyl group element w_0 on ideal triangulations and laminations via sequences of flips and framing transports.
• Utilizes tropicalization of Weyl group actions and the mapping class group action to verify cluster nature of transformations.
• Employs the duality involution ∗ on local systems and shows it acts as a cluster transformation via the Schützenberger involution on the dual A-space.
• Combines results with GHKK’s construction of canonical bases to prove existence of canonical bases in X_PGL_m,S and the upper cluster algebra for SL_m.

Experimental results

Research questions

• RQ1 Does the Weyl group action on the moduli space X_PGL_m,S of PGL_m-local systems on a decorated surface S act as a cluster transformation?
• RQ2 Is the duality involution ∗ on X_PGL_m,S induced by dualizing local systems a cluster transformation?
• RQ3 What is the explicit form of the Donaldson-Thomas transformation for the cluster Poisson variety X_PGL_m,S when S is admissible?
• RQ4 How do the Weyl group and duality actions lift to the dual A-space A_SL_m,S, and are they cluster transformations there?
• RQ5 Can the canonical basis conjectured by Fomin-Zelevinsky and Fock-Goncharov be constructed geometrically via cluster DT-transformations and laminations?

Key findings

• The Weyl group action on X_PGL_m,S is given by cluster Poisson transformations for any admissible decorated surface S.
• The duality involution ∗ on X_PGL_m,S acts as a cluster transformation, and its action is realized via the Schützenberger involution on the dual A-space.
• The Donaldson-Thomas transformation of X_PGL_m,S is explicitly computed using Keller’s framework on cluster DT-transformations.
• For admissible S, the canonical basis of regular functions on X_PGL_m,S is constructed via the GHKK method, confirming the Duality Conjectures.
• The upper cluster algebra with principal coefficients for the pair (SL_m, S) admits a canonical basis, as predicted by the Duality Conjectures.
• The cluster DT-transformation for X_PGL_m,S is realized geometrically through laminations and ideal triangulations, with explicit coordinate changes verified via flips and framing transports.