Open string amplitudes of closed topological vertex

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.

q-Difference Kac-Schwarz Operators in Topological String Theory

“…Comparing the expansion of Φ 0 (x) with the expansion of the generating function Ψ 1 (x) presented in our previous work [61], one can confirm that Φ 0 (x) agrees with Ψ 1 (x) except for the constant multiplier…”

“…A unified expression of the quantum mirror curves for general strip geometry is presented in our previous work [60]. Moreover, we extended these results to the simplest example of "off-strip geometry" called closed topological vertex [61].…”

“…Nevertheless, computations are mostly parallel to the case of strip geometry. Actually, the previous work [61] can be reproduced in a more systematic way in the language of the operators G, A, B.…”

Quantum mirror curve of periodic chain geometry

“…Actual computation is based on the vertex operator formalism which is well established in the context of the melting crystal [36,37]. In section 3, we will provide the prescription to obtain the mirror curves and its toric diagrams corresponding to the web diagrams by the method argued in [34,38]. In section 4, we conclude with some remarks and discussions.…”

“…Now we shall derive the quantum operator H(x, y) from the partition function of the topological string with the single brane by the method in [34,38]. The coupling constant g s plays a role of the Planck constant, so that we can obtain the classical mirror curve by taking the limit g s → 0.…”