TY - JOUR

T1 - Donaldson–Thomas transformations of moduli spaces of G-local systems

AU - Goncharov, Alexander

AU - Shen, Linhui

N1 - Funding Information:
This work was supported by the NSF grants DMS-1301776 and DMS-1564385. A.G. is grateful to IHES for the hospitality and support during the Summer of 2015.
Publisher Copyright:
© 2017 Elsevier Inc.

PY - 2018/3/17

Y1 - 2018/3/17

N2 - 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 XPGLm,S, closely related to the moduli space of PGLm-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 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 [38], 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 [35] 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 [23] related to the pair (SLm,S), as predicted by Duality Conjectures [15].

AB - 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 XPGLm,S, closely related to the moduli space of PGLm-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 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 [38], 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 [35] 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 [23] related to the pair (SLm,S), as predicted by Duality Conjectures [15].

KW - Canonical bases

KW - Cluster varieties

KW - Motivic Donaldson–Thomas invariants

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U2 - 10.1016/j.aim.2017.06.017

DO - 10.1016/j.aim.2017.06.017

M3 - Article

AN - SCOPUS:85021700448

SN - 0001-8708

VL - 327

SP - 225

EP - 348

JO - Advances in Mathematics

JF - Advances in Mathematics

ER -