Quiver mutation and combinatorial DT-invariants

TL;DR

This paper introduces a combinatorial approach to construct Donaldson-Thomas invariants for quivers using mutation and quantum dilogarithms, linking algebraic geometry, quantum cluster algebras, and combinatorics.

Contribution

It develops a method to compute combinatorial DT-invariants for large classes of quivers, connecting them to quantum dilogarithms and existing geometric invariants.

Findings

Constructed combinatorial DT-invariants for various quivers.

Showed these invariants match algebraic DT-invariants in many cases.

Linked invariants to quantum cluster algebras and associahedra.

Abstract

A quiver is an oriented graph. Quiver mutation is an elementary operation on quivers. It appeared in physics in Seiberg duality in the nineties and in mathematics in the definition of cluster algebras by Fomin-Zelevinsky in 2002. We show, for large classes of quivers Q, using quiver mutation and quantum dilogarithms, one can construct the combinatorial DT-invariant, a formal power series intrinsically associated with Q. When defined, it coincides with the "total" Donaldson-Thomas invariant of Q (with a generic potential) provided by algebraic geometry (work of Joyce, Kontsevich-Soibelman, Szendroi and many others). We illustrate combinatorial DT-invariants on many examples and point out their links to quantum cluster algebras and to (infinite) generalized associahedra.