The Donaldson-Thomas partition function of the banana manifold

Abstract

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 $\hat{I}“$ of curve classes supported in the fibers of $X_{ban}\to {\mathbb{P}}^1$. It is given by \[ Z_Γ(X_{ban}) = \prod_{d_{1},d_{2},d_{3}\geq 0} \prod_{k} \left(1-p^{k}Q_{1}^{d_{1}}Q_{2}^{d_{2}}Q_{3}^{d_{3}}\right)^{-12c(||\mathbf{d} ||,k)} \] where $||\mathbf{d}||=2d_1{d}_2+2d_2{d}_3+2d_3{d}_1-d_1^2-d_2^2-d_3^2$, and the coefficients $c(a,k)$ 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 $g\ge 2$; namely it is the Skoruppa-Maass lift of a multiple of an Eisenstein series: $\frac{6|B_{2g}|}{g(2g-2)!}{E}_{2g}(\ddot{I}„)$.

With an Appendix by Jim Bryan and Stephen Pietromonaco