GWT · Gromov-Witten Theory: Mirror Symmetry, Modular Forms, and Integrable Systems

The Gromov-Witten invariants of a space X record the number of curves in X of a given genus and degree which meet a given collection of cycles in X. They have important applications in algebraic geometry, symplectic geometry, and theoretical physics. The program proposed here will allow us to compute Gromov-Witten invariants, and particularly higher-genus Gromov-Witten invariants, for a very broad class of spaces. Recent progress, partly due to the Principal Investigator, has led to a greatly-improved mathematical understanding of the string-theoretic duality known as Mirror Symmetry. This allows us to compute genus-zero Gromov-Witten invariants (those where the curves involved are spheres) for a wide range of target spaces. But at the moment there are very few effective tools for computing higher-genus Gromov-Witten invariants (those where the curves involved are tori, or n-holed tori for n>1). We will solve this problem by extending mathematical Mirror Symmetry to cover this case. In doing so we will draw on and make rigorous recent insights from topological string theory. These insights have revealed close and surprising connections between Gromov-Witten theory, modular forms, and the theory of integrable systems.