Studies in Gromov-Witten Theory

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The first part of the project concerns a relationship, discovered by Ciocan-Fontanine and his collaborators, between the genus zero Gromov-Witten theories of nonsingular projective quotients of a projective manifold X by a complex reductive Lie group G, and by a maximal torus T in G. This relationship can be viewed as a kind of highly nontrivial functoriality in Gromov-Witten theory. Such instances of functorial behavior are quite rare and therefore very interesting. Ciocan-Fontanine proposes to significantly broaden the study of this abelian/nonabelian correspondence in several different directions. Specifically, the correspondence will be extended to higher genus Gromov-Witten invariants, and a version of the correspondence in terms of bounded derived categories of coherent sheaves will be explored. Further, some new and nontrivial examples will be studied. A second project deals with several applications of the derived moduli spaces introduced several years ago by Ciocan-Fontanine and Kapranov to the study of Gromov-Witten and Donaldson-Thomas invariants of some classes of projective varieties. For example, a construction of derived versions of the moduli of sheaves appearing in DT-theory is proposed. In the crucial rank one case, it is expected that a variant of the construction will lead to a very concrete expression of the virtual fundamental classes in terms of known complexes of bundles on simple varieties (products of Grassmannians). Using this expression, Ciocan-Fontanine will investigate the DT-theory of hypersurfaces in projective 4-space, about which little is known at the present time.This is research in the field of algebraic geometry, a highly developed branch 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 theories of Gromov-Witten and Donaldson-Thomas invariants (and the exciting conjectural relations between them) are particularly striking examples.