This is a joint work with Linhui Shen (Northwestern University) 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, the DT-transformation.
Let S be an oriented surface with punctures, and a finite number of special boundary points 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. This space carries a canonical cluster Poisson variety structure.
We calculate the DT-transformation of the moduli space X(m,S), with few exceptions.
This, 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), conjectured by V. Fock and the speaker.
