Donaldson-Thomas transformations for moduli spaces of G-local systems on surfaces
Hot Topics: Cluster algebras and wall-crossing March 28, 2016 - April 01, 2016
Location: SLMath: Eisenbud Auditorium
moduli spaces
local systems
arithmetic mirror symmetry
frame bundle
projective bundles
higher Teichmuller theory
cluster varieties
13P10 - Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases)
14E22 - Ramification problems in algebraic geometry [See also 11S15]
11J54 - Small fractional parts of polynomials and generalizations
14486
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.
Let S be an oriented surface with punctures, and a finite number of special points on the boundary considered modulo isotopy. It give rise to a moduli space X(m, S), closely related to the moduli space of PGL(m)-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 X(m, S).
We calculate the DT-transformation of the moduli space X(m,S), with few exceptions.
We discuss the relationship between DT-transformations and duality conjectures in general.
In particular, our main result, combined with the work of Gross, Hacking, Keel and Kontsevich,
gives rise to a canonical basis in the space of regular functions on the cluster variety X(m,S), and in the upper cluster algebra with principal coefficients related to the pair (SL(m), S).
Shen Notes
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