Donaldson-Thomas Transformation of Double Bruhat Cells in Semisimple Lie Groups

Double Bruhat cells $G^{u,v}$ were studied by Fomin and Zelevinsky. They provide important examples of cluster algebras and cluster Poisson varieties. Cluster varieties produce examples of 3d Calabi-Yau categories with stability conditions, and their Donaldson-Thomas invariants, defined by Kontsevich and Soibelman, are encoded by a formal automorphism on the cluster variety known as the Donaldson-Thomas transformation. Goncharov and Shen conjectured in that for any semisimple Lie group $G$, the Donaldson-Thomas transformation of the cluster Poisson variety $H\backslash G^{u,v}/H$ is a slight modification of Fomin and Zelevinsky's twist map. In this paper we prove this conjecture, using crucially Fock and Goncharov's cluster ensembles and the amalgamation construction. Our result, combined with the work of Gross, Hacking, Keel, and Kontsevich, proves the duality conjecture of Fock and Goncharov in the case of $H\backslash G^{u,v}/H$.