Crystal Model for the Closed Topological Vertex Geometry

TL;DR

This paper shows that the topological string partition function for the closed topological vertex $Z^{\mathcal{C}}$ can be reproduced by a finite cube Calabi-Yau crystal model, extending CY crystals beyond the standard $\mathbb{C}^3$ case. By deriving $Z^{\mathcal{C}}$ from the topological vertex and demonstrating an off-strip computation, it connects a cube-counting generating function $Z^{cube}$ to $Z^{\mathcal{C}}$ via $Z^{cube}=M(q)\,Z^{\mathcal{C}}$ with identifications $Q_1=q^M$, $Q_2=q^L$, $Q_3=q^N$. The work also analyzes a flop transition to a related geometry $\mathcal{C}^{flop}$, verifying the invariance of the full partition function and clarifying how classical contributions transform. Overall, the results suggest a path to general off-strip crystal descriptions of toric Calabi-Yau geometries and highlight connections to GV and DT invariants in this combinatorial framework.

Abstract

The topological string partition function for the neighbourhood of three spheres meeting at one point in a Calabi-Yau threefold, the so-called 'closed topological vertex', is shown to be reproduced by a simple Calabi-Yau crystal model which counts plane partitions inside a cube of finite size. The model is derived from the topological vertex formalism. This derivation can be understood as 'moving off the strip' in the terminology of hep-th/0410174, and offers a possibility to simplify topological vertex techniques to a broader class of Calabi-Yau geometries. To support this claim a flop transition of the closed topological vertex is considered and the partition function of the resulting geometry is computed in agreement with general expectations.

Figures (7)

• Figure 1: The closed topological vertex $\mathcal{C}$.
• Figure 2: Toric diagram (left) and its dual (right) for the double-$\mathbb{P}^1$.
• Figure 3: The cube crystal model of finite size $M \times L \times N$.
• Figure 4: The closed topological vertex as a strip with an additional $\mathbb{P}^1$ (of the size determined by $Q_3$) attached.
• Figure 5: The closed topological vertex $\mathcal{C}$ with its dual diagram (left) and the geometry after the flop $\mathcal{C}^{flop}$ (right).