String topology

Abstract: We prove that the pair-of-pants product on the Floer homology of the cotangent bundle of a compact manifold M corresponds to the Chas–Sullivan loop product on the singular homology of the loop space of M. We also prove related results concerning the Floer homological interpretation of the Pontrjagin product and of the Serre fibration. The techniques include a Fredholm theory for Cauchy–Riemann operators with jumping Lagrangian boundary conditions of conormal type, and a new cobordism argument replacing the standard gluing technique.

Abstract: Let M be a closed, connected manifold, and LM its loop space. In this paper we describe closed string topology operations in h_*(LM), where h_* is a generalized homology theory that supports an orientation of M. We will show that these operations give h_*(LM) the structure of a unital, commutative Frobenius algebra without a counit. Equivalently they describe a positive boundary, two dimensional topological quantum field theory associated to h_*(LM). This implies that there are operations corresponding to any surface with p incoming and q outgoing boundary components, so long as q >0. The absence of a counit follows from the nonexistence of an operation associated to the disk, D^2, viewed as a cobordism from the circle to the empty set. We will study homological obstructions to constructing such an operation, and show that in order for such an operation to exist, one must take h_*(LM) to be an appropriate homological pro-object associated to the loop space. Motivated by this, we introduce a prospectrum associated to LM when M has an almost complex structure. Given such a manifold its loop space has a canonical polarization of its tangent bundle, which is the fundamental feature needed to define this prospectrum. We refer to this as the "polarized Atiyah - dual" of LM . An appropriate homology theory applied to this prospectrum would be a candidate for a theory that supports string topology operations associated to any surface, including closed surfaces.

Abstract: Part I. Contributions: 1. A variant of K-theory Michael Atiyah and Michael Hopkins 2. Two-vector bundles and forms of elliptic cohomology Nils Baas, Bjorn Dundas and John Rognes 3. Geometric realisation of the Segal-Sugawara construction David Ben-Zvi and Edward Frenkel 4. Differential isomorphism and equivalence of algebraic varieties Yuri Berest and George Wilson 5. A polarized view of string topology Ralph Cohen and Veronique Godin 6. Random matrices and Calabi-Yau geometry Robbert Dijkgraaf 7. A survey of the topological properties of symplectomorphism groups Dusa McDuff 8. K-theory from a physical perspective Gregory Moore 9. Heisenberg groups and algebraic topology Jack Morava 10. What is an elliptic object? Stephan Stolz and Peter Teichner 11. Open and closed string field theory interpreted in classical algebraic topology Dennis Sullivan 12. K-theory of the moduli of principal bundles on a surface and deformations of the Verlinde algebra Constantin Teleman 13. Cohomology of the stable mapping class group Michael S. Weiss 14. Conformal field theory in four and six dimensions Edward Witten Part II. The Definition of Conformal Field Theory by Graeme Segal: 15. Definition of a conformal field theory Graeme Segal.