Quiver mutation and combinatorial DT-invariants

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.

12 pages, slightly corrected version of a submission to FPSAC 2013

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Cited by in corpus (15)

• Cyclic Sieving and Cluster Duality of Grassmannian
• A survey on maximal green sequences
• Cluster structures and subfans in scattering diagrams
• Littlewood-Richardson coefficients via mirror symmetry for cluster varieties
• Donaldson-Thomas Transformation of Double Bruhat Cells in Semisimple Lie Groups
• Positive Braid Links with Infinitely Many Fillings
• Donaldson-Thomas Transformation of Double Bruhat Cells in General Linear Groups
• The No Gap Conjecture for tame hereditary algebras
• On the relationship between green-to-red sequences, local-acyclicity, and upper cluster algebras
• Quantum dilogarithm identities for n-cycle quivers
• Quantum dilogarithms and partition q-series
• Interpolating factorizations for acyclic Donaldson--Thomas invariants
• A family of structure isomorphisms for the Cohomological Hall Algebra of an acyclic quiver
• Sign-coherence of c-vectors and maximal green sequences for acyclic sign-skew-symmetric matrices
• A recursion on maximal chains in the Tamari lattices