Title: Curve Counting for Abelian Surface Fibrations and Modular Forms
Abstract:
A long-standing prediction of string theory and mirror symmetry is that
certain formal generating series of curve-counting invariants are in fact
expansions of quasi-modular objects. In this talk I will discuss on-going
work with Aaron Pixton aiming to understand this modularity for Calabi-Yau
threefolds fibered by Abelian surfaces (of Picard rank 2 or 3). I will
focus on two very explicit examples: the banana manifold and Schoen
nano-manifold. In both cases, we are interested in the Gromov-Witten (GW)
potentials F_{g,k} where we assemble into a generating series the GW
invariants of genus g for curve classes of degree k over the base. For the
banana manifold, the GW potentials are formal series in 19 variables, which
we conjecture to be Siegel-Jacobi forms for the E_{8} lattice, as
introduced by Ziegler in the late 80s. As evidence, we prove an elliptic
transformation law. For the Schoen nano-manifold, the GW potentials are
formal series in only 2 variables, which we conjecture to be symmetric
tensor products of quasi-modular forms with level. In degree k=1, this is
consistent with work of Bryan-Oberdieck via degeneration to a CHL model.