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

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AbstractAward: DMS-0072492Principal Investigator: Jim A. BryanThis project is primarily concerned with Gromov-Witteninvariants. The investigator will study the Gromov-Witteninvariants of a 3-fold X by determining the local contributionsof a suitably rigid curve C in X. In particular, the investigatorwill study the integrality properties of such contributions andtheir relationship to the number of certain BPS states inM-theory as defined via the formula of Gopakumar and Vafa. Fornodal curves C, the investigator will continue his work with Katzand Leung; for smooth higher genus curves C, the investigatorwill continue work begun with R. Pandharipande. In collaborationswith Leung, the investigator will also seek to define a newinvariant of symplectic 4-manifolds which would specialize to themodified Gromov-Witten invariants defined by Behrend and Fantechifor algebraic surfaces with positive geometric genus (which inturn generalized the modified invariants defined by theinvestigator and Leung for K3 and Abelian surfaces). Such aninvariant would be better suited to study the enumerativegeometry of irrational surfaces than the ordinary Gromov-Witteninvariants.In the physics of string theory, particles are replaced withone-dimensional objects (``strings'') and so the path that aparticle traces out over time becomes a surface in space-time (a``world-sheet''). The equations of string theory then tell usthat the surface should be a holomorphic surface (a Riemannsurface) mapped into space-time in a holomorphic manner. This hasled to the purely mathematical notion of Gromov-Witteninvariants. Gromov-Witten invariants study holomorphic mappingsof Riemann surfaces into higher dimensional geometric objects(for example, projective manifolds). They have become importantinvariants in geometry, topology, and algebraic geometry as wellas being central in string theory. Bryan's project specificallyaddresses the problem of how the algebraic geometry of aprojective manifold is encoded in the Gromov-Witten invariantsand how they are tied to string theory. Algebraic geometry is aclassical subject in pure mathematics that has recently foundapplication in such diverse subjects as robotics and codingtheory; string theory is the leading candidate for a "theory ofeverything", i.e. a single physical theory that describes allknown physical phenomenon.