# Achievements of Research Area C

## Mirror symmetry, topological and integrable quantum systems.

In many topological field (TFT) and string theories (TST) the formalism of quantization can be made mathematically rigorous. The quantum correlators encode geometric invariants and are restricted by quantum symmetries, which are sometimes sufficient to determine them. Dualities between physical theories lead to far reaching mathematical conjectures concerning the geometric invariants.
In particular Huybrechts’ research formulates and confirms rigorously predictions following from mirror duality for the structure of the derived category of Calabi-Yau manifolds. The paper [E:HMS09] with Macrì (HCM-Postdoc, Bonn Junior Fellow) and Stellari proves the mirror analogue of a result of Donaldson. This required a completely new set of techniques involving derived categories of generic fibers of formal deformations and deformation theory of bounded complexes, the foundations of which have been developed in [C:HMS11]. In joint work with Thomas, this was pursued to prove the deformation invariance of a new curve count invariant recently introduced by Pandharipande and Thomas, cf. [C:HT10].
The work by Klemm et al proved [C:KMPS10] the well known Yau-Zaslow  conjecture from the heterotic-String/Type II duality, which suggests modular properties for counting functions of degenerate holomorphic curves on K3. The duality between TST and large N gauge theory leads to the conjecture that 3d Chern-Simons gauge theories calculate open and closed Gromov- Witten invariants as employed by Klemm in the topological vertex formalism. Mirror symmetry maps the above duality to a duality between TST and matrix models, as was formulated precisely and checked for non-compact Calabi-Yau manifolds in [C:BKMP09, C:BKMP10].
Zagier et al. found that quantum knot invariants, e.g. the Kashaev invariants have surprising modular forms properties [C:Zag10]. Catharina Stroppel established some categorical  connections between the realization of Khovanov homology in Lagrangian Floer homology and in representation theory of Lie algebras and the geometry of Springer fibers respectively [C:WS]. With Korff she proved an isomorphism, between quantum cohomology of Grassmannians specialized at q = 1 and the $\hat{\mathfrak{sl}}(n)$-WZNW fusion ring using representations of quantum groups and the theory of non-commutative symmetric functions [C:KS10].

## Operator algebras and quantum theory.

Lück has been working on the calculation of the K-theory of groups, e.g. crystallographic groups [C:DL].
The paper by Lesch, Moscovici and Pflaum ‘Connes-Chern character for manifolds with boundary and eta cochains’ is the result of a joint effort over several years and settles one of the goals formulated in the original application: namely, to understand the relative pairings between cyclic (co)homology classes associated to elliptic operators. A predecessor is [C:LMP09].
The category of finite dimensional representations of the Lie superalgebra gl(mjn) was studied by Catharina Stroppel who obtained [C:BS] a quite unexpected, very explicit, description of this category including character formulas and an interesting grading on this category arising from mixed Hodge structures on the category of perverse sheaves on  Grassmannians.
Albeverio has worked on representation theory and operator algebras, quantum field theory, quantum mechanics and inverse problems, number theory, and Feynman path integrals. Functional analytic methods have led to a breakthrough in the study of quantum anharmonic crystals [C:AKKR09], related methods have been providing a rigorous setting for implementing  Witten’s approach to index theory [C:ADK08]. A breakthrough in the study of quantum wave guides shrinking to graphs, preserving coupling at the vertex has been achieved in [C:ACF07].

## Field theory.

Stolz and Teichner have developed functorial field theories (such theories were introduced by Atiyah, Kontsevich, Segal) in the super symmetric Euclidean case, where spacetime has super dimension d|1 with d = 0,1,2. In [C:HKST11] they show that the space of all 0|1-dimensional Euclidean field theories is the classifying space for de Rham cohomology and in [C:HST10] that 1|1-dimensional Euclidean field theories gives 8-periodic K-theory. This establishes a new relation between algebraic topology and quantum field theory and leads in particular to a precise mathematical understanding of deformations of functorial field theories.
In modern field theories, triangulated categories are an important tool. Schwede investigated the relationship between triangulated categories and their model. [C:MSS07], joint with Muro and Strickland, exhibited the first examples of triangulated categories without models. In another direction [C:Sch10], the notion of “n-order” was introduced. This invariant allowed to prove that the stable homotopy category, localized at an odd prime, is not algebraic.

## Organization of scientific events.

Besides prolific writing, the members of the Research Area C have been very active in the organization of scientific events including the Trimester Program ‘Geometry and Physics’ (held at the HIM in summer 2008). Moreover, no less than nine workshops (2 part of the trimester program) related to Research Area C were organized. Several very prominent mathematicians (e.g. Connes, Hopkins, Witten) visited Bonn for activities related to Research Area C. The activities also gave young researchers excellent opportunities: e.g. Himpel, a postdoc at Mathematical Insitute, acted as the local (main) organizer of the HCM workshop ‘Chern-Simons Gauge Theory: 20 years after’. The workshop brought together 72 researchers, top specialists in the field, from all over the world.