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

Abstract: Aganagic, Dijkgraaf, Klemm, Mariño 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.