Stringy Mathematics

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In the past few years, there has been significant interplay between string theory and mathematics. For example, advances in supersymmetric field theory have shed light on four manifold invariants, while advances in algebraic geometry have clarified the origins of mirror symmetry. The aim of this project is to further strengthen this interplay by research focused on three areas of interest to both mathematicians and physicists. The broader impact of this project centers on improved interdisciplinary ties fostered through lectures at schools and workshops, and through direct collaboration. The first research goal is to develop a recently discovered analogue of mirror symmetry for the heterotic string. Conventional mirror symmetry applies to theories with (2; 2) super-symmetry. However, these compactifications constitute a special subclass of more general heterotic string compactifications with only (0; 2) supersymmetry. Many of the interesting structures of (2; 2) theories, like quantum cohomology (or chiral) rings, generalize to this richer setting. Using the dual description, the chiral ring can be determined exactly in many examples, leading to predictions about heterotic string instanton corrections. The study of (0; 2) duality is a topic in its infancy, and there are many directions to explore: for example, S-duality maps heterotic world-sheet instantons into D-instantons of type I open string theory. This suggests a relation between open and closed string instantons, which is likely to be fascinating both to physicists and to geometers. The second area of focus involves compactifications with flux. In the presense of flux, a string target space need not be Ricci-flat. Compact examples of this kind involving just NS-NS fluxes have been found in recent years. These are compactifications with torsion. It is clear that there should be dual descriptions for vacua of this kind (in the sense of mirror symmetry), but there is little currently known about these duals. Since generic string compactifications involve fluxes, constructing dual descriptions for these cases is likely to both enhance our understanding of the string moduli space, and lead to novel questions in mathematics. The third direction revolves around a relation between matrix integrals and modular forms. The twisted partition function for type IIB D-instantons in ten dimensions is computed by evaluating a complicated matrix integral. Yet these matrix integrals are encoded in a particular modular form, which appears in the effective action for the type IIB string. This modular form is completely determined by supersymmetry. The connection between the U-duality group, SL(2;Z), and the matrix integrals is intriguing and puzzling: why is it true? Does it generalize to lower dimensions where the U-duality group is larger? Does it extend to other solitons like monopoles? There are tantalizing hints that the answer to the last two questions is affirmative, but much remains to be understood.