Stability conditions and quantum dilogarithm identities for Dynkin quivers

Abstract

We study fundamental group of the exchange graphs for the bounded derived category D(Q) of a Dynkin quiver Q and the finite-dimensional derived category D(Γ_N Q) of the Calabi-Yau-N Ginzburg algebra associated to Q. In the case of D(Q), we prove that its space of stability conditions (in the sense of Bridgeland) is simply connected; as applications, we show that its Donanldson-Thomas invariant can be calculated via a quantum dilogarithm function on exchange graphs. In the case of D(Γ_N Q), we show that faithfulness of the Seidel-Thomas braid group action (which is known for Q of type A or N = 2) implies the simply connectedness of its space of stability conditions.

Journal (almost) equivalent version

References in corpus (13)

• Calabi-Yau algebras
• BPS Quivers and Spectra of Complete N=2 Quantum Field Theories
• Exchange graphs and Ext quivers
• Spaces of stability conditions
• Cohomological Hall algebra, exponential Hodge structures and motivic Donaldson-Thomas invariants
• Deformed Calabi-Yau Completions
• Stability conditions and the $A_2$ quiver
• The modular curve as the space of stability conditions of a CY3 algebra
• Wall-crossing of the motivic Donaldson-Thomas invariants
• C-sortable words as green mutation sequences
• Autoequivalences of derived categories on the minimal resolutions of $A_n$-singularities on surfaces
• Classifying τ-tilting modules over preprojective algebras of Dynkin type
• Tilting and Refined Donaldson-Thomas Invariants

Cited by in corpus (29)

• Exchange graphs and Ext quivers
• Decorated marked surfaces: spherical twists versus braid twists
• Stability conditions and the $A_2$ quiver
• Derived automorphism groups of K3 surfaces of Picard rank 1
• Topological aspects of completely integrable foliations
• Quiver mutation and combinatorial DT-invariants
• Discreteness of silting objects and t-structures in triangulated categories
• Contractible stability spaces and faithful braid group actions
• C-sortable words as green mutation sequences
• Cambrian combinatorics on quiver representations (type A)
• Linearity of stability conditions
• q-Stability conditions on Calabi-Yau-X categories
• $q$-Stability conditions via $q$-quadratic differentials for Calabi-Yau-$\mathbb{X}$ categories
• Cluster exchange groupoids and framed quadratic differentials
• Global dimension function on stability conditions and Gepner equations
• On Maximal Green Sequences
• Stability conditions on ${\text{CY}}_N$ categories associated to $A_n$-quivers and period maps
• Stability conditions for affine type A
• Maximal green sequences for cluster-tilted algebras of finite representation type
• T-stabilities for a weighted projective line
• Topological structure of spaces of stability conditions and topological Fukaya type categories
• Discrete derived categories II: The silting pairs CW complex and the stability manifold
• Building maximal green sequences via component preserving mutations
• The braid group for a quiver with superpotential
• Horizontal and vertical mutation fans
• Systolic inequalities, Ginzburg dg algebras and Milnor fibers
• Frobenius morphisms and stability conditions
• Stability conditions on Brauer tree algebras
• Quintic periods and stability conditions via homological mirror symmetry