Integrable systems and Gromov-Witten theory of non-orientable surfaces

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AbstractAward: DMS-0406077Principal Investigator: Motohico MulaseThe proposed project is aimed at generalizing one of the recentdevelopments in the area of Gromov-Witten theory of compactRiemann surfaces to the case of non-orientable surfaces. In thearea of complex and symplectic geometry, integrable systems, andmathematical physics, amazing new developments have been achievedin recent years. At the time of their discovery in 1985,Donaldson invariants of differentiable four manifolds, Jonespolynomials of knots and links, and Gromov's idea of pseudo-holomorphic curves were considered as independent entities.Since the time of Atiyah's provocative lecture in 1988, theirhidden inter-relations have been emerged. Through the work ofmany mathematicians and physicists, a clearer picture of theconnection of these theories is revealed. Most recently, moredirect relations have been discovered, through calculation of thegenerating function of these invariants and their associationwith integrable systems and representation theory. Among them isthe work of Okounkov and Pandharipande, who determined theGromov-Witten invariants of an arbitrary compact Riemann surfaceas the target space. Their fundamental results include a proof ofthe conjectured Virasoro constraints, identification of thegenerating function of the invariants of the Riemann sphere as asolution to the two-dimensional Toda lattice equations throughFermionic Fock representation, and a new proof of theWitten-Kontsevich theory of intersection of cohomology classes onthe moduli space of Riemann surfaces. We propose to establishGromov-Witten theory of real algebraic curves without boundary,and obtain the counterpart of the above theorems for the case ofnon-orientable surfaces.The topological and geometric structures of various kinds ofspaces have attracted intensive research in mathematics andmathematical physics for many decades. Poincare's idea onhomology and homotopy theories have proven to be usefulthroughout the 20th century. The applications of these theoriesare seen in physics, chemistry, and understanding the dynamicalproperties of DNA. Only toward the end of the previous centurymathematics has encountered a true generalization of the ideas ofPoincare, applicable to the particularly interesting spaces(called symplectic manifolds) that appear naturally in any kindsof mechanics and dynamics. The new invariants, known as theGromov-Witten invariants of symplectic spaces, have shownextremely mysterious connections to almost all areas ofmathematics. Mathematicians feel quite happily that theyunderstand the homology and homotopy theories very well. Comparedto that level, our current understanding of the Gromov-Wittentheory is at the best very limited, and the whole subject isstill filled with mysteries. We believe that to have a deeperunderstanding of the theory, we should generalize it further andinclude the consideration of new cases never done before. Theproposed project is aimed at discovering a generalization of thetheory in a particular context, using non-orientable surfaces.