DonaldsonThomas transformations for moduli spaces of Glocal systems on surfaces
Hot Topics: Cluster algebras and wallcrossing 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]
16E50  von Neumann regular rings and generalizations (associative algebraic aspects)
16E40  (Co)homology of rings and associative algebras (e.g., Hochschild, cyclic, dihedral, etc.)
14485
Kontsevich and Soibelman defined DonaldsonThomas invariants of a 3d CalabiYau 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 DTtransformation.
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 DTtransformation of the moduli space X(m,S), with few exceptions.
We discuss the relationship between DTtransformations 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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