  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 of weight 10. In addition, we illustrate our findings by
solving the refined topological string on the resolutions of the three toric
orbifolds of order three, five and six.
  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 and building up on recent work by
Pitale, Saha, and Schmidt, we can show that there is no analogue of the almost
holomorphic elliptic second Eisenstein series. 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 the second
Eisenstein series.
  The two parts are intertwined by the observation that the meromorphic
analogue of the almost holomorphic second Eisenstein series 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.
