Topological strings, strips and quivers

Abstract

We find a direct relation between quiver representation theory and open topological string theory on a class of toric Calabi-Yau manifolds without compact four-cycles, also referred to as strip geometries. We show that various quantities that characterize open topological string theory on these manifolds, such as partition functions, Gromov-Witten invariants, or open BPS invariants, can be expressed in terms of characteristics of the moduli space of representations of the corresponding quiver. This has various deep consequences; in particular, expressing open BPS invariants in terms of motivic Donaldson-Thomas invariants, immediately proves integrality of the former ones. Taking advantage of the relation to quivers we also derive explicit expressions for classical open BPS invariants for an arbitrary strip geometry, which lead to a large set of number theoretic integrality statements. Furthermore, for a specific framing, open topological string partition functions for strip geometries take form of generalized $q$-hypergeometric functions, which leads to a novel representation of these functions in terms of quantum dilogarithms and integral invariants. We also study quantum curves and A-polynomials associated to quivers, various limits thereof, and their specializations relevant for strip geometries. The relation between toric manifolds and quivers can be regarded as a generalization of the knots-quivers correspondence to more general Calabi-Yau geometries.

arXiv comment47 pages, 9 figures

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• Branches, quivers, and ideals for knot complements
• Tangle addition and the knots-quivers correspondence
• Branes, quivers and wave-functions
• Mirror Symmetry and Mixed Chern-Simons Levels for Abelian 3d $\mathcal{N}=2$ theories
• Nahm sums, quiver A-polynomials and topological recursion
• On explicit formulae of LMOV invariants
• Quiver diagonalization and open BPS states
• Revisiting the Melvin-Morton-Rozansky Expansion, or There and Back Again
• Refined open topological strings revisited
• 3d $\mathcal{N}=2$ Brane Webs and Quiver Matrices
• Open/Closed BPS Correspondence and Integrality
• Combinatorics of Nahm sums, quiver resultants and the K-theoretic condition
• Quantum fields, strings, and physical mathematics
• Permutohedra for knots and quivers
• GW/DT invariants and 5D BPS indices for strips from topological recursion
• Enumerative geometry of surfaces and topological strings
• Quantum Curve for strip geometries, Topological Recursion and open GW/DT invariants