Refined Gromov-Witten invariants
Abstract
We study the enumerative geometry of stable maps to Calabi-Yau 5-folds $Z$ with a group action preserving the Calabi-Yau form. In the central case $Z=X \times \mathbb{C}^2$, where $X$ is a Calabi-Yau 3-fold with a group action scaling the holomorphic volume form non-trivially, we conjecture that the disconnected equivariant Gromov-Witten generating series of $Z$ returns the Nekrasov-Okounkov equivariant K-theoretic PT partition function of $X$ and, under suitable rigidity conditions, its refined BPS index. We show that in the unrefined limit the conjecture reproduces known statements about the higher genus Gromov-Witten theory of $X$; we prove it for $X$ the resolved conifold; and we establish a refined cycle-level local/relative correspondence for local del Pezzo surfaces, implying the Nekrasov-Shatashvili limit of the conjecture when $X$ is the local projective plane. We further establish B-model physics predictions of Huang-Klemm for refined higher genus mirror symmetry for local $\mathbb{P}^2$. In particular, we prove that our refined Gromov-Witten generating series obey extended holomorphic anomaly equations, are quasi-modular functions of $\Gamma_1(3)$, have leading asymptotics at the conifold point given by the logarithm of the Barnes double-Gamma function, and satisfy a version of the higher genus Crepant Resolution Correspondence with the refined orbifold Gromov-Witten theory of $[\mathbb{C}^3/\mu_3]$. This refines results, and partially proves conjectures, of Lho-Pandharipande, Coates-Iritani, and Bousseau-Fan-Guo-Wu.
- Publication:
-
arXiv e-prints
- Pub Date:
- September 2024
- DOI:
- 10.48550/arXiv.2410.00118
- arXiv:
- arXiv:2410.00118
- Bibcode:
- 2024arXiv241000118B
- Keywords:
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- Mathematics - Algebraic Geometry;
- High Energy Physics - Theory;
- Mathematical Physics;
- 14N35;
- 53D45;
- 14J33;
- 14H81;
- 14J81
- E-Print:
- 76 pages