An Introduction to Modern Enumerative Geometry with Applications to the Banana Manifold

Abstract

The banana manifold $X_{\text{ban}}$ is a smooth projective Calabi-Yau threefold fibered over ${\mathbb{P}}^1$ by abelian surfaces. Each singular fiber contains a "banana configuration of curves" which generates the rank-three lattice $\Gamma$ of curve classes supported in the fibers of $X_{\text{ban}}\to {\mathbb{P}}^1$. The Donaldson-Thomas partition function of $X_{\text{ban}}$ in fiber classes was computed by J. Bryan (arXiv:1902.08695) to be the infinite product \[Z_{\text{DT}}(X_{\text{ban}})_Γ= \prod_{d_{1},d_{2},d_{3}\geq0}\prod_{k\in\mathbb{Z}}\big(1-Q_{1}^{d_{1}}Q_{2}^{d_{2}}Q_{3}^{d_{3}}t^{k}\big)^{-12c(||\underline{\bf{d}}||,k)}\] where $||\underline{\mathbf{d}}||=2d_1{d}_2+2d_1{d}_3+2d_2{d}_3-d_1^2-d_2^2-d_3^2$, and $c(||\underline{\mathbf{d}}||,k)$ are coefficients of the equivariant elliptic genus of ${\mathbb{C}}^2$. We observe that under a change of variables, $Z_{\text{DT}}(X_{\text{ban}}{)}_{\Gamma }$ behaves formally like a Borcherds lift of the equivariant elliptic genus. The main result of this thesis is that the associated Gromov-Witten potentials $F_g$ in genus $g\ge 2$ are meromorphic genus two Siegel modular forms of weight $2g-2$. They arise as Maass lifts of weak Jacobi forms of weight $2g-2$ and index 1 arising in an expansion of the elliptic genus in the equivariant parameter. We show the equivariant elliptic genus of ${\mathbb{C}}^2$ encodes the Gopakumar-Vafa invariants of $X_{\text{ban}}$. Therefore, one can regard $X_{\text{ban}}$ as an example where the generating functions of Gromov-Witten and Donaldson-Thomas invariants in fiber classes are produced by standard lifts of a modular object encoding the Gopakumar-Vafa invariants. We note that because this is a Masters thesis, the first six chapters offer an extended introduction to the relevant background material, while the original results are presented in the final chapter.

MSc Thesis, University of British Columbia, 2018. Comments welcome!