The Enumerative Geometry and Arithmetic of Banana Nano-Manifolds
Jim Bryan, Stephen Pietromonaco

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 with very small Hodge numbers: . We construct four rigid banana nano-manifolds , , each with Hodge numbers given by . We compute the Donaldson-Thomas partition function for banana curve classes and show that the associated genus Gromov-Witten potential is a genus 2 meromorphic Siegel modular form of weight for a certain discrete subgroup . We also compute the weight 4 modular form whose th Fourier coefficient is given by the trace of the action of Frobenius on for almost all prime…
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Taxonomy
TopicsAdvanced Combinatorial Mathematics · Advanced Mathematical Theories and Applications · Advanced Algebra and Logic
