Pandharipande-Thomas theory of elliptic threefolds, quasi-Jacobi forms and holomorphic anomaly equations

Abstract: Let $\pi : X \to B$ be an elliptically fibered threefold satisfying $c_3(T_X \otimes \omega_X)=0$. We conjecture that the $\pi$-relative generating series of Pandharipande-Thomas invariants of $X$ are quasi-Jacobi forms and satisfy two holomorphic anomaly equations. For elliptic Calabi-Yau threefolds our conjectures specialize to the Huang-Katz-Klemm conjecture. The proposed formulas constitute the first case of holomorphic anomaly equations in Pandharipande-Thomas theory. We prove our conjectures for the equivariant Pandharipande-Thomas theory of $\mathbb{C}^2 \times E$ when specialized to the anti-diagonal action. For $K3 \times \mathbb{C}$ we state reduced versions of our conjectures. As a corollary we find an explicit conjectural formula for the stationary theory generalizing the Katz-Klemm-Vafa formula for K3 surfaces. Further evidence is available for $\mathbb{P}^2 \times E$ based on earlier work of the second author. To deal with elliptic threefolds with $c_3(T_X \otimes \omega_X) \neq 0$ we show that the moduli space of $\pi$-stable pairs is represented by a proper algebraic space. We conjecture that the associated $\pi$-stable pair invariants form quasi-Jacobi forms.