Three problems on Gromov-Witten invariants of algebraic varieties

University Of Minnesota-Twin Cities, Minneapolis MN

Investigators

Abstract

DMS-0303614 Ionut Ciocan-Fontanine The PI proposes to pursue a program based on the theory of derived moduli spaces towards calculating higher genus Gromov-Witten invariants of certain Calabi-Yau threefolds, such as the quintic in projective 4-space. While computations of the genus zero invariants have been quite successful, leading to proofs of physicists' Mirror Symmetry - based predictions in that case, there has been no progress so far in higher genus. The main obstacle has been a lack of concrete understanding of the virtual fundamental classes of the moduli spaces of higher genus stable maps. In recent work, the PI and Kapranov have developed an approach to virtual classes via differential-graded (dg) manifolds. Dg-manifolds appear as derived versions of algebro-geometric moduli spaces. The PI and Kapranov constructed such a structure on the moduli spaces of stable maps. The greater flexibility of the dg-point of view restores some of the features that facilitated the genus zero computations. The ultimate goal is to prove the higher genus ``mirror theorem'', as predicted by the physicists Bershadsky, Cecotti, Ooguri, and Vafa. A crucial part of the program involves a novel construction of a virtual class in situations outside the reach of earlier approaches. This is of great independent interest in algebraic geometry, as it allows to extend the theory of virtual classes to all moduli spaces, as opposed to just the (rather small) subset to which it currently applies. Ciocan-Fontanine also proposes to establish a relationship between the genus zero Gromov-Witten theory of a GIT quotient X//G by a reductive algebraic group G, and that of the associated abelian quotient X//T, where T is a maximal torus in G. Precise conjectures are made on what the relationship is, inspired by results obtained recently by the PI and his collaborators in the case when X is a complex vector space and G is the general linear group. They are consistent with expectations on how the physical theories associated to X//G and X//T are related and will be of interest to physicists, as well as to algebraic geometers. The third problem proposed by the PI is to give a proof of a combinatorial formula for the three-point, genus zero Gromov-Witten invariants of complex Grassmannians. This is the only outstanding unsolved problem remaining in ``Quantum Schubert Calculus.'' This is research in the field of algebraic geometry, which is one of the oldest branches of modern mathematics. In recent years, the methods and ideas of algebraic geometry, especially the study of moduli spaces, have been employed in string theory, a very active part of theoretical physics. Developments in string theory have sparked a fruitful interaction between the two communities of researchers and have led to the discovery and study of many unexpected new phenomena. The theory of Gromov-Witten invariants and Mirror Symmetry are particularly striking examples.