Three problems on Gromov-Witten invariants of algebraic varieties

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DMS-0303614Ionut Ciocan-Fontanine The PI proposes to pursue a program based on thetheory of derived moduli spaces towards calculating higher genusGromov-Witten invariants of certain Calabi-Yau threefolds, such as thequintic in projective 4-space. While computations of the genus zeroinvariants have been quite successful, leading to proofs of physicists'Mirror Symmetry - based predictions in that case, there has been noprogress so far in higher genus. The main obstacle has been a lack ofconcrete understanding of the virtual fundamental classes of the modulispaces of higher genus stable maps. In recent work, the PI and Kapranovhave developed an approach to virtual classes via differential-graded (dg)manifolds. Dg-manifolds appear as derived versions of algebro-geometricmoduli spaces. The PI and Kapranov constructed such a structure on themoduli spaces of stable maps. The greater flexibility of the dg-pointof view restores some of the features that facilitated the genus zerocomputations. The ultimate goal is to prove the higher genus ``mirrortheorem'', as predicted by the physicists Bershadsky, Cecotti, Ooguri,and Vafa. A crucial part of the program involves a novel construction ofa virtual class in situations outside the reach of earlier approaches.This is of great independent interest in algebraic geometry, as it allowsto extend the theory of virtual classes to all moduli spaces, as opposedto just the (rather small) subset to which it currently applies.Ciocan-Fontanine also proposes to establish a relationship between thegenus zero Gromov-Witten theory of a GIT quotient X//G by a reductivealgebraic 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 therelationship is, inspired by results obtained recently by the PI and hiscollaborators in the case when X is a complex vector space and G is thegeneral linear group. They are consistent with expectations on how thephysical theories associated to X//G and X//T are related and will be ofinterest to physicists, as well as to algebraic geometers.The third problem proposed by the PI is to give a proof of a combinatorialformula for the three-point, genus zero Gromov-Witten invariants of complexGrassmannians. 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 theoldest branches of modern mathematics. In recent years, the methods andideas of algebraic geometry, especially the study of moduli spaces, havebeen employed in string theory, a very active part of theoretical physics.Developments in string theory have sparked a fruitful interaction betweenthe two communities of researchers and have led to the discovery and studyof many unexpected new phenomena. The theory of Gromov-Witten invariantsand Mirror Symmetry are particularly striking examples.