Donaldson-Thomas transformations for moduli spaces of G-local systems on surfaces

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)

Please report video problems to itsupport@slmath.org.

See more of our Streaming videos on our main VMath Videos page.