Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

Zhiyuan Wang, Chenglang Yang, Jian Zhou

TL;DR

The paper proves that the total open string partition function for one-loop local toric Calabi–Yau threefolds, incorporating all fermion-number-flux sectors, is a tau-function of the KP hierarchy. It achieves this by constructing Z^{total} as a trace on the fermionic Fock space, deriving a determinantal formula, and providing explicit affine coordinates; it also derives a quantum spectral curve in a principal specialization.Two detailed examples, the (-2,...,-2) model and local P^2, demonstrate the method: constant terms, affine coordinates, and a q-difference (quantum) spectral curve are computed, with Plücker relations checked in the latter. The results generalize ADKMV-type integrability to looped toric diagrams, highlighting new flux-induced shifts and their impact on the KP structure. The approach yields a robust fermionic framework for relating open Gromov–Witten invariants to integrable hierarchies in toric Calabi–Yau geometry, with potential connections to mirror symmetry and nonperturbative enumerative predictions.

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.

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

TL;DR

The paper proves that the total open string partition function for one-loop local toric Calabi–Yau threefolds, incorporating all fermion-number-flux sectors, is a tau-function of the KP hierarchy. It achieves this by constructing Z^{total} as a trace on the fermionic Fock space, deriving a determinantal formula, and providing explicit affine coordinates; it also derives a quantum spectral curve in a principal specialization.Two detailed examples, the (-2,...,-2) model and local P^2, demonstrate the method: constant terms, affine coordinates, and a q-difference (quantum) spectral curve are computed, with Plücker relations checked in the latter. The results generalize ADKMV-type integrability to looped toric diagrams, highlighting new flux-induced shifts and their impact on the KP structure. The approach yields a robust fermionic framework for relating open Gromov–Witten invariants to integrable hierarchies in toric Calabi–Yau geometry, with potential connections to mirror symmetry and nonperturbative enumerative predictions.

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.

Paper Structure

This paper contains 25 sections, 18 theorems, 217 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Schur functions and skew Schur functions
  4. Free fermions and the fermionic Fock space
  5. Tau-functions of the KP hierarchy and affine coordinates
  6. Some operators on the fermionic Fock space
  7. Open Gromov--Witten Theory of Local Toric Calabi--Yau Threefolds
  8. The topological vertex formalism
  9. Bogoliubov transform representation of the topological vertex
  10. Gluing of the topological vertices
  11. A new fermionic representation of the topological vertex
  12. Open string amplitudes of local toric Calabi--Yau geometry
  13. Partition function from fermionic point of view
  14. Total Partition Function for Local Toric Calabi--Yau Threefolds
  15. Construction of the contribution of fermion number flux $N$
  16. ...and 10 more sections

Key Result

Theorem 1.1

Denote by $Z(\mathbf{t};Q_i,\gamma_i)$ the partition function of open string amplitudes of $X$ with one outer brane, where $X$ is a formal Calabi--Yau manifold whose toric diagram consists of a polygon together with an external leg attached to each vertex, and $\gamma_i$, $Q_i$ are the framing facto a tau-function of the KP hierarchy, and

Figures (1)

  • Figure 1: Toric diagrams for local ${\mathbb P}^2$ and local ${\mathbb P}^1\times {\mathbb P}^1$

Theorems & Definitions (38)

  • Theorem 1.1
  • Corollary 1.2
  • Theorem 1.3
  • Proposition 1.4
  • Theorem 3.1: wyz
  • Definition 4.1
  • Theorem 4.2
  • Remark 4.3
  • Remark 4.4
  • Lemma 4.5
  • ...and 28 more