Donaldson–Thomas transformations of moduli spaces of G-local systems

Kontsevich and Soibelman defined Donaldson–Thomas invariants of a 3d Calabi–Yau category equipped with a stability condition [41]. 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. Given a stability condition, the DT-transformation allows to recover the DT-invariants. Let S be an oriented surface with punctures and a finite number of special points on the boundary, considered modulo isotopy. It gives rise to a moduli space X_PGLm,S, closely related to the moduli space of PGL_m-local systems on S, which carries a canonical cluster Poisson variety structure [13]. For each puncture of S, there is a birational Weyl group action on the space X_PGLm,S. We prove that it is given by cluster Poisson transformations. We prove a similar result for the involution ⁎ of X_PGLm,S induced by dualizing a local system on S. Let μ be the total number of punctures and special points, and g(S) the genus of S. We assume that μ>0. We say that S is admissible if either g(S)+μ≥3 and μ>1 when S has only punctures, or S is an annulus with a special point on each boundary circle. Using a combinatorial characterization of a class of DT transformations due to B. Keller [38], we describe the DT-transformation of the space X_PGLm,S for any admissible S. We show that the Weyl group and the involution ⁎ act by cluster transformations of the dual moduli space A_SLm,S, and describe the DT-transformation of the space A_SLm,S. If S admissible, combining our work with the work of Gross, Hacking, Keel and Kontsevich [35] we get a canonical basis in the space of regular functions on the cluster variety X_PGLm,S, and in the Fomin–Zelevinsky upper cluster algebra with principal coefficients [23] related to the pair (SL_m,S), as predicted by Duality Conjectures [15].