Total partition function with fermionic number fluxes of local toric Calabi--Yau threefold and KP integrability

TL;DR

This paper proves that the total open string partition function for local toric Calabi--Yau threefolds, incorporating fermionic fluxes, is a tau-function of the multi-component KP hierarchy, confirming a prior prediction.

Contribution

It constructs the total partition function using a fermionic Fock space operator and proves its KP integrability, advancing understanding of string theory and integrable systems.

Findings

Total partition function is a trace of an operator on fermionic Fock space.

Partition function is a tau-function of multi-component KP hierarchy.

Confirmed the prediction relating open string partition functions to KP integrability.

Abstract

Aganagic, Dijkgraaf, Klemm, Mari\~{n}o and Vafa \cite{adkmv} predicted that the open string partition function on a smooth toric Calabi--Yau threefold should be a tau-function of multi-component KP hierarchy after considering the contributions from nonzero fermion number fluxes through loops in the toric diagram. In this paper, we prove their prediction in the case of local toric Calabi--Yau threefolds. More precisely, we construct the total partition function of local toric Calabi--Yau threefolds using an operator on the fermionic Fock space which we developed in an earlier work \cite{wyz} to represent the topological vertex, and show that the total partition function is the trace of an operator on the fermionic Fock space. As an application, we prove the KP integrability of the total partition function.