Flop invariance of the topological vertex

TL;DR

The paper addresses how Gromov–Witten invariants of toric Calabi–Yau threefolds transform under flops by expressing partition functions via the topological vertex and proving a flop transformation formula. The authors reduce the comparison to a key combinatorial identity on skew Schur functions, derive explicit local contributions, and show that the primed partition functions for birationally related TCY models agree under a natural change of variables. They connect the results to Nekrasov’s partition function in a geometric engineering limit and apply the framework to blowups of toric surfaces, yielding precise relations between GW invariants of canonical bundles before and after blowups. Overall, the work provides a rigorous, combinatorially grounded account of flop invariance for TCY partition functions and links to gauge-theory partition functions, advancing understanding of birational invariance in noncompact settings.

Abstract

We prove transformation formulae for generating functions of Gromov-Witten invariants on general toric Calabi-Yau threefolds under flops. Our proof is based on a combinatorial identity on the topological vertex and analysis of fans of toric Calabi-Yau threefolds.

Figures (6)

• Figure 1: Fan (section at $z=1$) and toric graph
• Figure 2: $\vec{\lambda}_v$
• Figure 3: Fans (sections at $z=1$): $\Sigma$ (left), $\bar{\Sigma}$ (middle) and $\Sigma^+$ (right). The generators $\vec{\omega}_1,\ldots,\vec{\omega}_4$ of $\rho_1,\dots,\rho_4$ satisfy the relation $\vec{\omega}_1+\vec{\omega}_3=\vec{\omega}_2+\vec{\omega}_4$.
• Figure 4: Toric graphs $\Gamma_X$ (left) and $\Gamma_{X^+}$ (right).
• Figure 5: TCY threefold which contains two disjoint ${\mathbb P}^1 \times {\mathbb P}^1$ connected by a $(-1,-1)$-curve (left) and its flop (right).

Theorems & Definitions (28)

• Definition 2.1
• Remark 2.2
• Proposition 2.3
• proof
• Corollary 2.4
• proof
• Lemma 2.5
• proof
• Lemma 2.6
• proof