Abstract
Let m and n be two integers such that 1<m<n−1. The configuration space Confn×(Pm−1), which is the moduli space of n points in the projective space Pm−1 satisfying certain linearly independent condition, is closely related to the Grassmannian Grm,n, and is birationally equivalent to the cluster Poisson variety XAm−1□An−m−1.A Donaldson-Thomas transformation is a special formal automorphism on a cluster Poisson variety and encodes the Donaldson-Thomas invariants of the moduli space of stability conditions on the associated 3d Calabi-Yau category. The existence of a cluster Donaldson-Thomas transformation is part of Gross-Hacking-Keel-Kontsevich's sufficient condition for the Fock-Goncharov cluster duality on the cluster ensemble. In this paper we construct the cluster Donaldson-Thomas transformation on the cluster Poisson variety XAm−1□An−m−1 and realize it as an isomorphism on the configuration space Confn×(Pm−1), proving a conjecture of Goncharov and Shen. Via our construction we also prove the Fock-Goncharov cluster duality conjecture on the associated cluster ensemble and obtain a periodicity result on the Donaldson-Thomas transformation associated to the Am−1□An−m−1 quiver.
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