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. 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 XPGLm,S, closely related to the moduli space of PGLm-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 XPGLm,S. We prove that it is given by cluster Poisson transformations. We prove a similar result for the involution ⁎ of XPGLm,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 , we describe the DT-transformation of the space XPGLm,S for any admissible S. We show that the Weyl group and the involution ⁎ act by cluster transformations of the dual moduli space ASLm,S, and describe the DT-transformation of the space ASLm,S. If S admissible, combining our work with the work of Gross, Hacking, Keel and Kontsevich  we get a canonical basis in the space of regular functions on the cluster variety XPGLm,S, and in the Fomin–Zelevinsky upper cluster algebra with principal coefficients  related to the pair (SLm,S), as predicted by Duality Conjectures .
- Canonical bases
- Cluster varieties
- Motivic Donaldson–Thomas invariants
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