Moduli spaces, homotopy Frobenius algebras and mirror symmetry

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AbstractAward: DMS-0706945Principal Investigator: Kevin J. CostelloThe moduli spaces of Riemann surfaces play a central role in manyareas of mathematics and theoretical physics, including geometricand algebraic topology, symplectic topology, string theory, andnumber theory. This project explores another manifestation ofthe moduli spaces of Riemann surfaces, which is maybe not sowidely known. This is their appearance in homological algebra,and more precisely in the study of cyclic A-infinity algebras (akind of "Frobenius algebra up to homotopy"). The philosophyunderlying much of this project is that everything one can sayabout the homotopy types of the various moduli spaces can beexpressed in terms of the homotopy theory of cyclic A-infinityalgebras. One precise manifestation of this philosophy (whichwill be proved in this project) is that the moduli space ofsurfaces arises as certain "homology operations" for cyclicA-infinity algebras. This theoretical result will be used toinvestigate some concrete higher-genus aspects of the mirrorsymmetry conjecture, which plays a prominent role in current workon algebraic and symplectic geometry. In particular, thisproject will attempt to compute the conjectural mirror partner ofthe higher-genus Gromov-Witten invariants in some examples.The space of all possible two-dimensional shapes -- known as themoduli space of surfaces -- has long been a fundamental object ofstudy in many areas of mathematics, from geometry to numbertheory. This space also plays an important role in stringtheory, the putative "theory of everything". This project isconcerned with setting up a correspondence betweentwo-dimensional geometry (the moduli space of surfaces) and akind of abstract algebra. This correspondence will be used totest certain mathematical conjectures coming from string theory.String theorists have predicted that two different simplifiedmodels of string theory-- known as the A model and the B model --are mathematically equivalent. This prediction has stimulated agreat deal of mathematical work in the last 15 years. The PIwill tackle some computations in the B model which have beenheretofore out of reach. The results of these computations willthen be compared with known computations in the A model,hopefully leading to further verification of the predictions ofstring theory.