The Gromov-Witten invariants of curves, surfaces, and 3-folds

Tulane University, New Orleans LA

Investigators

Abstract

Abstract Award: DMS-0072492 Principal Investigator: Jim A. Bryan This project is primarily concerned with Gromov-Witten invariants. The investigator will study the Gromov-Witten invariants of a 3-fold X by determining the local contributions of a suitably rigid curve C in X. In particular, the investigator will study the integrality properties of such contributions and their relationship to the number of certain BPS states in M-theory as defined via the formula of Gopakumar and Vafa. For nodal curves C, the investigator will continue his work with Katz and Leung; for smooth higher genus curves C, the investigator will continue work begun with R. Pandharipande. In collaborations with Leung, the investigator will also seek to define a new invariant of symplectic 4-manifolds which would specialize to the modified Gromov-Witten invariants defined by Behrend and Fantechi for algebraic surfaces with positive geometric genus (which in turn generalized the modified invariants defined by the investigator and Leung for K3 and Abelian surfaces). Such an invariant would be better suited to study the enumerative geometry of irrational surfaces than the ordinary Gromov-Witten invariants. In the physics of string theory, particles are replaced with one-dimensional objects (``strings'') and so the path that a particle traces out over time becomes a surface in space-time (a ``world-sheet''). The equations of string theory then tell us that the surface should be a holomorphic surface (a Riemann surface) mapped into space-time in a holomorphic manner. This has led to the purely mathematical notion of Gromov-Witten invariants. Gromov-Witten invariants study holomorphic mappings of Riemann surfaces into higher dimensional geometric objects (for example, projective manifolds). They have become important invariants in geometry, topology, and algebraic geometry as well as being central in string theory. Bryan's project specifically addresses the problem of how the algebraic geometry of a projective manifold is encoded in the Gromov-Witten invariants and how they are tied to string theory. Algebraic geometry is a classical subject in pure mathematics that has recently found application in such diverse subjects as robotics and coding theory; string theory is the leading candidate for a "theory of everything", i.e. a single physical theory that describes all known physical phenomenon.