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

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 $$Z_{\text{DT}}(X_{\text{ban}})_{\Gamma}= \prod_{d_{1},d_{2},d_{3}\geq0}\prod_{k\in\mathbb{Z}}\big(1-Q_{1}^{d_{1}}Q_{2}^{d_{2}}Q_{3}^{d_{3}}t^{k}\big)^{-12c(||\underline{\bf{d}}||,k)}$$ where $||\underline{\bf{d}}||=2d_{1}d_{2}+2d_{1}d_{3}+2d_{2}d_{3}-d_{1}^{2}-d_{2}^{2}-d_{3}^{2}$, and $c(||\underline{\bf{d}}||,k)$ 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 $g\geq2$ 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.