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Donaldson-Thomas transformations of moduli spaces of G-local systems. (English) Zbl 1434.13022
Summary: M. Kontsevich and Y. Soibelman defined Donaldson-Thomas invariants of a 3d Calabi-Yau category equipped with a stability condition [“Stability structures, motivic Donaldson-Thomas invariants and cluster transformations”, Preprint, arXiv:0811.2435]. 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 \(\mathbb{S}\) 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 \(\mathcal{X}_{\mathrm{PGL}_{\mathrm{m}}, \mathbb{S}}\), closely related to the moduli space of \(\mathrm{PGL}_{\mathrm{m}}\)-local systems on \(\mathbb{S}\), which carries a canonical cluster Poisson variety structure [V. Fock and A. Goncharov, Publ. Math., Inst. Hautes Étud. Sci. 103, 1–211 (2006; Zbl 1099.14025)]. For each puncture of \(\mathbb{S}\), there is a birational Weyl group action on the space \(\mathcal{X}_{\mathrm{PGL}_{\mathrm{m}}, \mathbb{S}}\). We prove that it is given by cluster Poisson transformations. We prove a similar result for the involution of \(\mathcal{X}_{\mathrm{PGL}_{\mathrm{m}}, \mathbb{S}}\) induced by dualizing a local system on \(\mathbb{S}\).
Let \(\mu\) be the total number of punctures and special points, and \(g(\mathbb{S})\) the genus of \(\mathbb{S}\). We assume that \(\mu > 0\). We say that \(\mathbb{S}\) is admissible if either \(g(\mathbb{S}) + \mu \geq 3\) and \(\mu > 1\) when \(\mathbb{S}\) has only punctures, or \(\mathbb{S}\) 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 [“Quiver mutation and combinatorial DT-invariants”, Preprint, arXiv:1709.03143], we describe the DT-transformation of the space \(\mathcal{X}_{\mathrm{PGL}_{\mathrm{m}}, \mathbb{S}}\) for any admissible \(\mathbb{S}\).
We show that the Weyl group and the involution act by cluster transformations of the dual moduli space \(\mathcal{A}_{\mathrm{SL}_{\mathrm{m}}, \mathbb{S}}\), and describe the DT-transformation of the space \(\mathcal{A}_{\mathrm{SL}_m, \mathbb{S}}\).
If \(\mathbb{S}\) admissible, combining our work with the work of M. Gross et al. [J. Am. Math. Soc. 31, No. 2, 497–608 (2018; Zbl 1446.13015)] we get a canonical basis in the space of regular functions on the cluster variety \(\mathcal{X}_{\mathrm{PGL}_{\mathrm{m}}, \mathbb{S}}\), and in the Fomin-Zelevinsky upper cluster algebra with principal coefficients [S. Fomin and A. Zelevinsky, Compos. Math. 143, No. 1, 112–164 (2007; Zbl 1127.16023)] related to the pair \((\mathrm{SL}_{\mathrm{m}},\mathbb{S})\), as predicted by Duality Conjectures [V. V. Fock and A. B. Goncharov, Ann. Sci. Éc. Norm. Supér. (4) 42, No. 6, 865–930 (2009; Zbl 1180.53081)].

MSC:
13F60 Cluster algebras
14F08 Derived categories of sheaves, dg categories, and related constructions in algebraic geometry
14N35 Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects)
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