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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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References:
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