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 $\mathrm{\Gamma }$ 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 $$Z_{\text{DT}}(X_{\text{ban}}{)}_{\mathrm{\Gamma }}=\prod _{d_1,d_2,d_3\ge 0}\prod _{k\in \mathbb{Z}}(1-Q_1^{d_1}{Q}_2^{d_2}{Q}_3^{d_3}{t}^k{)}^{-12c(\mathrm{\mid }\mathrm{\mid }\underset{‾}{\mathbf{d}}\mathrm{\mid }\mathrm{\mid },k)}$$ where $\mathrm{\mid }\mathrm{\mid }\underset{‾}{\mathbf{d}}\mathrm{\mid }\mathrm{\mid }=2d_1{d}_2+2d_1{d}_3+2d_2{d}_3-d_1^2-d_2^2-d_3^2$, and $c(\mathrm{\mid }\mathrm{\mid }\underset{‾}{\mathbf{d}}\mathrm{\mid }\mathrm{\mid },k)$ are coefficients of the equivariant elliptic genus of ${\mathbb{C}}^2$. We observe that under a change of variables, $Z_{\text{DT}}(X_{\text{ban}}{)}_{\mathrm{\Gamma }}$ 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.