Frobenius manifolds are complex manifolds with a rich additional structure, a multiplication and a flat metric on the holomorphic tangent bundle, with several strong compatibility conditions. They turn up in very different areas, and always in a highly transcendental way, in singularity theory, in quantum cohomology and in hierarchies of integrable systems. They give rise to surprising and deep connections between these areas.
The first appearance of Frobenius manifolds was made in singularity theory, as moduli spaces of functions and possibly additional data, namely as parameter spaces of universal unfoldings of functions with isolated singularities, or as Hurwitz spaces. Some of them are related by one version of mirror symmetry to the quantum cohomology of certain Fano manifolds, for others one has a relation to certain integrable systems.
Whereas there have been organized many activities around quantum cohomology and integrable systems and relations to physics, the workshop will be the first one devoted to the relations between Frobenius manifolds and singularity theory. He shall bring together the specialists, review the state of the art and help exchanging ideas. There are a number of recent developments which make this an especially good time for the workshop.
The following recent developments will be especially at the focus of the workshop.
On the quantum cohomology side one has higher genus Gromov-Witten invariants with gravitational descendents. Their structures have been formalized some years ago. This again involves a quantization of data of an irregular connection. Nowadays one has also here an extremely rich picture from which Frobenius manifolds form only one corner. The question on the role of these higher genus data for the Frobenius manifolds from singularities is wide open.
- Harmonic Frobenius manifolds. They come equipped with the structure of a harmonic bundle, which they inherit from an additional real structure in an underlying meromorphic connection. This is natural for Frobenius manifolds from tame functions, and especially for those relevant for mirror symmetry. This additional structure is related to tt^* geometry and to polarizable twistor D-modules.
- Logarithmic Frobenius manifolds. They formalize those arising in quantum cohomology and mirror symmetry. They allow a good formulation of (one version of) mirror symmetry, and they make contact to variations of Hodge structures and to tame or wild harmonic bundles.
- The search for more mirror partners. On the singularity side this amounts to the construction of logarithmic Frobenius manifolds from functions on divisors and their unfoldings.
- Orbifold quantum cohomology. This provides on the quantum cohomology side candidates for mirror partners. It allows much more freedom in studying interesting examples of mirror symmetry and Frobenius manifolds.
- Z-lattice structure on the A-side and B-side. On the B-side it comes from topology, on the A-side it comes from mirror symmetry and the B- side, in any case it is interweaved with the data of associated meromorphic connections and it has arithmetic aspects.
- The associated meromorphic connections, regular and irregular. There are the classical regular singular data as Picard-Fuchs equations, Gauss-Manin connections, and their monodromy groups. And there are the less studied irregular connections with their Stokes data. Although it is common knowledge that they are related, the precise relations are not well established, and neither the properties of those connections and their Stokes data which really turn up in singularity theory and in mirror symmetry.
- Homological mirror symmetry for singularities. This relates matrix factorization with a derived Fukaya-Seidel category. The latter one is in turn related to distinguished bases of vanishing cycles and thus to the Stokes data of the irregular connections from singularities.
- Hurwitz spaces. They form a distinguished class of Frobenius manifolds where one has a very explicit control on the data, including the associated meromorphic connections and the Stokes data. They thus form a prime class of examples.
- Relations to integrable systems. The direct link between singularities and integrable systems is altogether rather young. It starts with the irregular connections associated to singularities, but quantizes them and leads to yet to be explored extremely rich structures.
- Higher genus data of cohomological field theories.C. Hertling