The Donaldson-Thomas partition function of the banana manifold

A banana manifold is a compact Calabi-Yau threefold, fibered by Abelian surfaces, whose singular fibers have a singular locus given by a “banana configuration of curves”. A basic example is given by $X_{ban}$, the blowup along the diagonal of the fibered product of a generic rational elliptic surface $S\to {\mathbb{P}}^1$ with itself. In this paper we give a closed formula for the Donaldson-Thomas partition function of the banana manifold $X_{ban}$ restricted to the 3-dimensional lattice $\mathrm{\Gamma }$ of curve classes supported in the fibers of $X_{ban}\to $\mathbb{P}$^{1}$. It is given by $$Z_{\mathrm{\Gamma }}(X_{ban})=\prod _{d_1,d_2,d_3\ge 0}\prod _k{(1-p^k{Q}_1^{d_1}{Q}_2^{d_2}{Q}_3^{d_3})}^{-12c(\mathrm{\mid }\mathrm{\mid }\mathbf{d}\mathrm{\mid }\mathrm{\mid },k)}$$ where $||$\mathbf{d}$ || = 2d_{1}d_{2}+ 2d_{2}d_{3}+ 2d_{3}d_{1}-d_{1}^{2}-d_{2}^{2}-d_{3}^{2}$,andthecoefficients$c(a,k)$ have a generating function given by an explicit ratio of theta functions. This formula has interesting properties and is closely realated to the equivariant elliptic genera of $\mathrm{Hilb}({\mathbb{C}}^2)$. In an appendix with S. Pietromonaco, it is shown that the corresponding genus $g$ Gromov-Witten potential $F_g$ is a genus 2 Siegel modular form of weight $2g-2$ for $g\geq 2$;namelyitistheSkoruppa-MaassliftofamultipleofanEisensteinseries:$\frac{6|B_{2g}|}{g(2g-2)!} E_{2g}(\tau )$.