Open string amplitudes of closed topological vertex

TL;DR

This paper derives fermionic and $q$-difference equation representations for open string amplitudes in the closed topological vertex, linking them to quantum mirror curves in topological string theory.

Contribution

It introduces a fermionic operator formalism for open string amplitudes in the closed topological vertex and derives associated $q$-difference equations as quantum mirror curves.

Findings

Fermionic expression for open string amplitudes obtained

$q$-difference equations derived for generating functions

Connection established between amplitudes and quantum mirror curves

Abstract

The closed topological vertex is the simplest ``off-strip'' case of non-compact toric Calabi-Yau threefolds with acyclic web diagrams. By the diagrammatic method of topological vertex, open string amplitudes of topological string theory therein can be obtained by gluing a single topological vertex to an ``on-strip'' subdiagram of the tree-like web diagram. If non-trivial partitions are assigned to just two parallel external lines of the web diagram, the amplitudes can be calculated with the aid of techniques borrowed from the melting crystal models. These amplitudes are thereby expressed as matrix elements, modified by simple prefactors, of an operator product on the Fock space of 2D charged free fermions. This fermionic expression can be used to derive $q$-difference equations for generating functions of special subsets of the amplitudes. These $q$-difference equations may be interpreted as the defining equation of a quantum mirror curve.