Source: Dissertation Abstracts International, Volume: 79-12(E), Section: B.

Adviser: Alexander Goncharov.

A Donaldson-Thomas transformation is a special formal automorphism on a cluster Poisson variety which encodes the Donaldson-Thomas invariants of the moduli space of stability conditions on the associated 3d Calabi-Yau category. Existence of a cluster Donaldson-Thomas transformation is part of a sufficient condition that implies the Fock-Goncharov duality on the cluster ensemble, which gives rise to canonical bases for the algebras of regular functions on the two cluster varieties. Such canonical bases often have important applications in algebraic geometry and representation theory.

The main original contribution of this thesis is the construction the cluster Donaldson-Thomas transformations on two families of cluster Poisson varieties: one is associated to Grassmannians, and the other one is associated to double Bruhat cells in semisimple Lie groups.

Let m and n be two integers such that 1 < m < n --1. The configuration space Confxn ( P m--1) is defined to be the moduli space of n points in the projective space P m--1 satisfying certain general position relation. It is known that the configuration space Confxn ( P m-1) is closely related to the Grassmannian Grm,n, and is birationally equivalent to the cluster Poisson variety (X m,n)uf associated to the quiver Am --1 &timesb; An--m, --1. In the first half of this thesis we construct the cluster Donaldson-Thomas transformation on the cluster Poisson variety associated to Am--1 &timesb; An--m --1 and realize it as a biregular isomorphism on the configuration space Confxn ( P m--1). Two corollaries came out of our result: one is the proof of Fock-Goncharov duality conjecture on the associated cluster ensemble (Am,n, Xm,n), and the other is a new proof of the already-proven Zamolodchikov's periodicity conjecture in the case of the Am --1 &timesb; An--m --1 quiver.

Let G be a semisimple Lie group, let B +/- be a pair of opposite Borel subgroups, and let H = B+&cap;B_ be the corresponding maximal torus. It is known that for a pair of Weyl group elements ( u, v), the double quotient H \Gu,v / H = H \B +uB+ &cap; B_vB _/ H is birationally equivalent to a cluster Poisson variety Xu,vuf. In the second half of this thesis we construct the cluster Donaldson-Thomas transformation on this cluster Poisson variety and show that it can be realized geometrically as a modified version of Forain and Zelevinsky's twist map. A direct corollary of this result is the proof of Fock-Goncharov duality conjecture on the associated cluster ensemble (Au,v Xu,v ).