Core Research Area SM: String topology and mirror symmetry
String topology and mirror symmetry are both directly inspired by developments in string theory, but are not typically mentioned together in one sentence. The topics in this core research area bring these subjects together in studies involving holomorphic curves and the algebraic structures arising from them.
String topology, introduced by Chas and Sullivan [CS1], [CS2] is an interpretation of open-closed string field theory in the language of algebraic topology (see also [OL] for background). The most satisfactory treatments to date identify the (co)homology of the free loop space with some version of Hochschild (co)homology, where the set of possible operations is now essentially understood [Wa]. By far the most important of these operations for symplectic applications are the loop product and the loop and string brackets, including their homotopy versions. The basic link between string topology and symplectic topology comes via the study of holomorphic curves with boundary on a Lagrangian submanifold. Noticing that the compactification of such moduli spaces can be described in terms of string topology of this submanifold, Fukaya [Fu2] has outlined a proof that any prime, orientable closed Lagrangian 3-manifold in C3 must be diffeomorphic to a product S1 x Σ for some closed surface Σ. To extend his approach to the study of other Lagrangian embedding problems, it seems central to understand the string topology for other classes of closed manifolds. A specific thesis topic in this direction could aim to treat connected sums, particularly of K(π,1)'s, with the goal of proving vanishing results for the operations which could serve as a starting point for a variant of Fukaya's arguments.
The work on mirror symmetry considered here concerns the toric degeneration point of view developed in the joint program of one of the applicants with Mark Gross [GS1], [GS2], [GS3]. At the center of this program is a class of maximal degenerations of certain algebraic varieties whose central fibre is a union of toric varieties. An example is the Dwork family X0X1X2X3+t(X04+X14+X24+X34)=0 of quartic surfaces in P3 with central fibre t=0 the union of four coordinate planes. Our program provides canonical such families out of purely discrete data [GS3], much as toric varieties are encoded by fans, but our families involve an inductively constructed wall structure and hence are highly non-linear. Our wall structure is a higher dimensional and tropical analogue of the gradient flow trees in two dimensions of [KS]. The discrete data consists of a topological manifold B along with a decomposition into integral polyhedra P and the structure of a fan at each vertex of P. Mirror symmetry then works as a perfect duality on the discrete data. The program can be partially extended to also include more singular degenerations that occur naturally from the classification theoretic point of view [GHK]. Note that the construction of the family is entirely complex-geometric without any explicit reference to mirror symmetry. The wall structure is nevertheless conjectured to encode certain Floer-theoretic invariants of the mirror side, as verified in some easy cases in [GPS], [GHK] and [CLT], [Lau]. This makes contact with the Floer-theoretic mirror construction [Fu1], [Ab2].
During the first funding period there have been two major new techniques introduced into the program. (1) The existence of a canonical basis of sections of the ample line bundle that our construction comes with [GS5]. A comprehensive treatment of these generalized theta functions is currently being finalized. There is a conjectured relation to symplectic cohomology of the mirror side [GHK],[Pa],[Fa]. (2) A theory of log Gromov-Witten invariants [GS4], [AC]. This theory is necessary for computing Gromov-Witten invariants from the central fibre. An ongoing project jointly with Dan Abramovich, Qile Chen and Mark Gross concerns a gluing formula for log Gromov-Witten invariants generalizing [Li], which treated the case with only two irreducible components meeting along a smooth divisor. In this context a generalization of log Gromov-Witten invariants occur naturally that admit negative virtual touching orders relative a divisor. These are called punctured invariants. The (straightforward) generalization of log Gromov-Witten theory to this case is currently being in the process of rewriting. In a degeneration situation an unobstructed punctured curve is expected to deform to a map from a Riemann surface with boundary circles mapping to a vanishing cycle, see [GS6] for some discussion.
Possible topics to be considered here include algebraic structures in symplectic cohomology via logarithmic Gromov-Witten theory, the symplectic interpretation of walls in our construction, the geometric quantization interpretation of our theta functions and the investigation of generalized local mirror symmetry suggested by Aganagic and Vafa in the context of knot contact homology [AV][AENV].
[CS2] M. Chas and D. Sullivan:
Closed String Operators in Topology Leading to Lie
Bialgebras and higher String Algebra,
in: O.A. Laudal, R. Piene (eds.), The Legacy of Niels Henrik
Abel, Springer, 2004, 771--784.
[Fu1] K. Fukaya:
Multivalued Morse theory, asymptotic analysis and mirror symmetry,
in ``Graphs and patterns in mathematics and theoretical physics,''
205--278, Proc. Sympos. Pure Math. 73,
Amer. Math. Soc. 2005.
[Fu2] K. Fukaya:
Application of Floer homology of Langrangian submanifolds to
in: Paul Biran et al. (eds.), Morse Theoretic Methods in
Nonlinear Analysis and in Symplectic Topology, Springer,
[KS] M. Kontsevich, Y. Soibelman,
Affine structures and non-Archimedean analytic spaces.,
in: The unity of mathematics (P. Etingof, V. Retakh,
I.M. Singer, eds.), 321--385, Progr. Math. 244,