Principal Investigator: Motohico Mulase
The proposed project is aimed at generalizing one of the recent developments in the area of Gromov-Witten theory of compact Riemann surfaces to the case of non-orientable surfaces. In the area of complex and symplectic geometry, integrable systems, and mathematical physics, amazing new developments have been achieved in recent years. At the time of their discovery in 1985, Donaldson invariants of differentiable four manifolds, Jones polynomials 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, their hidden inter-relations have been emerged. Through the work of many mathematicians and physicists, a clearer picture of the connection of these theories is revealed. Most recently, more direct relations have been discovered, through calculation of the generating function of these invariants and their association with integrable systems and representation theory. Among them is the work of Okounkov and Pandharipande, who determined the Gromov-Witten invariants of an arbitrary compact Riemann surface as the target space. Their fundamental results include a proof of the conjectured Virasoro constraints, identification of the generating function of the invariants of the Riemann sphere as a solution to the two-dimensional Toda lattice equations through Fermionic Fock representation, and a new proof of the Witten-Kontsevich theory of intersection of cohomology classes on the moduli space of Riemann surfaces. We propose to establish Gromov-Witten theory of real algebraic curves without boundary, and obtain the counterpart of the above theorems for the case of non-orientable surfaces.
The topological and geometric structures of various kinds of spaces have attracted intensive research in mathematics and mathematical physics for many decades. Poincare's idea on homology and homotopy theories have proven to be useful throughout the 20th century. The applications of these theories are seen in physics, chemistry, and understanding the dynamical properties of DNA. Only toward the end of the previous century mathematics has encountered a true generalization of the ideas of Poincare, applicable to the particularly interesting spaces (called symplectic manifolds) that appear naturally in any kinds of mechanics and dynamics. The new invariants, known as the Gromov-Witten invariants of symplectic spaces, have shown extremely mysterious connections to almost all areas of mathematics. Mathematicians feel quite happily that they understand the homology and homotopy theories very well. Compared to that level, our current understanding of the Gromov-Witten theory is at the best very limited, and the whole subject is still filled with mysteries. We believe that to have a deeper understanding of the theory, we should generalize it further and include the consideration of new cases never done before. The proposed project is aimed at discovering a generalization of the theory in a particular context, using non-orientable surfaces.
