Communications in Number Theory and Physics

Volume 10 (2016)

Number 4

On direct integration for mirror curves of genus two and an almost meromorphic Siegel modular form

Pages: 587 – 701



Albrecht Klemm (Bethe Center for Theoretical Physics and Hausdorff Center for Mathematics, Universität Bonn, Germany)

Maximilian Poretschkin (Rittenhouse Laboratory, University of Pennsylvania, Philadelphia, Penn., U.S.A.)

Thorsten Schimannek (Bethe Center for Theoretical Physics, Universität Bonn, Germany)

Martin Westerholt-Raum (Max Planck Institute for Mathematics, Bonn, Germany)


This work considers aspects of almost holomorphic and meromorphic Siegel modular forms from the perspective of physics and mathematics. The first part is concerned with (refined) topological string theory and the direct integration of the holomorphic anomaly equations. Here, a central object to compute higher genus amplitudes, which serve as the generating functions of various enumerative invariants, is provided by the so-called propagator.We derive a universal expression for the propagator for geometries that have mirror curves of genus two which is given by the derivative of the logarithm of Igusa’s cusp form $\chi_{10}$. In addition, we illustrate our findings by solving the refined topological string on the resolutions of the three toric orbifolds $\mathbb{C}^3 / \mathbb{Z}_3$, $\mathbb{C}^3 / \mathbb{Z}_5$ and $\mathbb{C}^3 / \mathbb{Z}_6$.

In the second part, we give explicit expressions for lowering and raising operators on Siegel modular forms, and define almost holomorphic Siegel modular forms based on them. Extending the theory of Fourier-Jacobi expansions to almost holomorphic Siegel modular forms, we can show that there is no analogue of the almost holomorphic elliptic Eisenstein series $\widetilde{E}_2$. In the case of genus $2$, we provide an almost meromorphic substitute for it. This, in particular, leads us to a generalization of Ramanujan’s differential equation for $\widetilde{E}_2$.

The two parts are intertwined by the observation that the meromorphic analogue of $\widetilde{E}_2$ coincides with the physical propagator. In addition, the generalized Ramanujan identities match precisely the physical consistency conditions that need to be imposed on the propagator.

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