Topological strings, strips and quivers

Miłosz Panfil,Piotr Sułkowski

doi:10.1007/jhep01(2019)124

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.

Highlights

Among various techniques to compute topological string amplitudes, a very powerful one relies on links with Chern-Simons theory
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, referred to as strip geometries
We show that various quantities that characterize open topological string theory on these manifolds, such as partition functions, GromovWitten invariants, or open BPS invariants, can be expressed in terms of characteristics of the moduli space of representations of the corresponding quiver

Summary

Topological string amplitudes and BPS invariants

A-model topological string amplitudes depend on Kahler parameters Q = {Qk} of a given target Calabi-Yau manifold M , and open moduli x = {xi} that characterize branes They are defined in terms of the genus expansion in the topological string coupling , and various terms in such expansion encode closed or open Gromov-Witten invariants. Spacetime interpretation of BPS counting implies that in presence of branes open topological string amplitudes have product decomposition. Where kP = {kiP } and mkP are respectively weights of the representation P and their multiplicities, the open partition function (2.3) can be written in the product form ψopen(Q, x) =. A single trace TrP X in (2.3) represents one stack of branes; for multiple stacks the open amplitude would in general take form ψopen(Q, x) =.

Topological vertex and strip geometries

Brane amplitudes and generalized q-hypergeometric functions

Quantum mirror curves and generalized hypergeometric equations

Motivic and numerical Donaldson-Thomas invariants for quivers

Partial classical limit

Quantum curves and A-polynomials for quivers

Topological strings and quivers

Brane amplitudes as quiver generating functions

Quantum curves and A-polynomials

BPS invariants and their structure

Examples

A Quiver generating functions and motivic Donaldson-Thomas invariants