The Enumerative Geometry and Arithmetic of Banana Nano-Manifolds

TL;DR

This paper explores the geometry and arithmetic of special Calabi-Yau threefolds called banana nano-manifolds, constructing examples, computing enumerative invariants, and revealing connections to Siegel modular forms and Frobenius traces.

Contribution

It constructs four new rigid banana nano-manifolds with minimal Hodge numbers and computes their Donaldson-Thomas invariants and Gromov-Witten potentials, linking them to modular forms and Frobenius traces.

Findings

The Gromov-Witten potential is a genus 2 Siegel modular form.

The Frobenius trace corresponds to a unique weight 4 cusp form.

Four rigid banana nano-manifolds with specific Hodge numbers are constructed.

Abstract

A banana manifold is a Calabi-Yau threefold fibered by Abelian surfaces whose singular fibers contain banana configurations: three rational curves meeting each other in two points. A nano-manifold is a Calabi-Yau threefold $X$ with very small Hodge numbers: $h^{1,1}(X)+h^{2,1}(X)\leq 6$. We construct four rigid banana nano-manifolds $\tilde{X}_N$, $N\in \{5,6,8,9 \}$, each with Hodge numbers given by $(h^{1,1},h^{2,1})=(4,0)$. We compute the Donaldson-Thomas partition function for banana curve classes and show that the associated genus $g$ Gromov-Witten potential is a genus 2 meromorphic Siegel modular form of weight $2g-2$ for a certain discrete subgroup $P^{*}_{N} \subset Sp_{4}(\mathbb{R})$. We also compute the weight 4 modular form whose $p$th Fourier coefficient is given by the trace of the action of Frobenius on $H^{3}_{et }(\tilde{X}_N ,{\mathbb{Q}}_{l})$ for almost all prime $p$. We observe that it is the unique weight 4 cusp form on $\Gamma_{0}(N)$.