Counting curves with symmetry

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Enumerative algebraic geometry, a.k.a. "curve counting", has been a major part of algebraic geometry for the last 30 years. It refers to subtle invariants of spaces obtained by counting the number of ways that curves (or sometimes more general objects) can sit inside a space. The subject has its origin in ideas coming from theoretical physics (string theory) where the curves can be thought of as the world-sheets of strings. There is an infinite number of curve counting invariants for any given space, and collection of numbers is encoded into a single function called the partition function of the space. Computing the partition function of a space (especially for a certain kind of six dimensional space called a compact Calabi-Yau threefold), is a major goal of both algebraic geometry and physics. The goal of this project is to enrich our understanding of curve counting invariants for spaces which have extra symmetries (including certain "hidden" symmetries called "derived" symmetries). Broadly speaking one can count curves which are invariant under the symmetry. This will lead to new curve counting invariants, which will in turn give us greater understanding of the original invariants. Preliminary investigations suggest that for certain spaces, the partition functions associated to these new curve counting theories will be given by modular forms --- extraordinary functions that arise in number theory and have been studied for well over a hundred years. This provides a powerful link between three very different subjects: geometry, physics, and number theory. One of the holy grails in this subject is to obtain a complete and explicit formula for the partition function of any compact Calabi-Yau threefold. One aim of this project is to do exactly that: find an explicit formula for the partition function, in terms of modular forms, of a very special Calabi-Yau threefold, namely the Schoen manifold. The idea is that this space has an enormous number of symmetries, including many of the "hidden" symmetries alluded to above. This, along with new geometric ideas introduced, should allow us to give a complete computation of this partition function and express it in terms of modular forms. No one has every been able to compute the partition function of a compact Calabi-Yau threefold (the geometry most relevant in string theory). Having such a function would give us great insight into the corresponding physical theory and thus would have a significant impact in both geometry and physics.