# Goals of Research Area C

## Mirror symmetry, topological and integrable quantum systems.

Extending the mirror symmetry to open topological strings on Calabi-Yau manifolds with D-branes lead to the formulation of homological mirror symmetry. This gives rise to more general geometric invariants and relates the stability theory for objects in different categories and opens new perspectives for this theory in algebraic geometry on which Huybrechts and collaborators will continue their  fundamental research, which is central for many developments discussed below.
E.g. recent developments culminated in the Kontsevich-Soibelmann wall crossing formula for BPS states and Zagier et al. found a relation between wall crossing and mock modularity in the counting functions of BPS states for N =4 black holes. Klemm, Alim (HCM-Postdoc) et al. found a holomophic anomaly equation related to mock modularity in N = 2 theories. Understanding the geometric origin of the latter and the number-theoretical meaning of the special families of mock modular forms is an important research goal.
The work of Klemm et al. provides new relations of geometric invariants to matrix models, which were independently introduced by Zagier as a tool to answer combinatorial questions in graph theory related to the moduli spaces of Riemann surfaces. We plan to investigate asymptotic expansions and non-perturbative effects in TFT using matrix model techniques. Zagier plans further to study Nahm’s conjectures relating the modularity properties of q-series to algebraic theory, and the possible links between these questions and the loop equation in the matrix model theory.
In the context of the duality between Chern-Simons theory and TST one needs to understand and/or prove the experimentally obtained arithmeticity and modularity properties of quantum invariants (WRT–, Kashaev invariants,...) of 3-manifolds. Related topics are the quantum dilogarithm, state integrals, the quantized A-polynomial, and Faddeev’s quantum modular double. The expertise of Catharina Stroppel and Zagier on knot homologies will be complemented by the new cooperation partner Gukov, who works also on the relation of TFT to geometric  representation theory, i.e. ramification, geometric Langlands, D-modules, braid group actions on categories etc. Further studies of ‘Surface operators’ are instrumental to establish the connection between gauge theory and geometric representation theory in the ramified geometric Langlands program and in the homology theory for knot invariants.

## Field theory.

Stolz and Teichner conjecture that the space of 2|1-dimensional Euclidean field theories classifies the cohomology theory of topological modular forms. This theory is related to the moduli space of derived elliptic curves and is 242-periodic. All homotopy groups have been computed by Hopkins and Miller which is one of the deepest results in modern algebraic topology. The conjecture would relate this computation to deformations of 2|1-dimensional Euclidean field theories.
Schwede has started systematic investigations into rigidity questions for triangulated categories of relevance to algebraic topology. Here a triangulated category is rigid if it has a unique model, up to Quillen equivalence of model categories. Schwede’s rigidity theorem for the stable homotopy category was the first such result, complemented later by Roitzheim’s rigidity theorem. On the other hand, Franke had constructed exotic algebraic models for the n-th chromatic localization of the stable homotopy category of a “large” prime p. The proofs of the known rigidity results give some evidence that for “small” primes, the n-th chromatic localization should be rigid. We plan to investigate this conjecture, starting with special cases such as n = 2 and p = 3, where recent computational advances about the local homotopy groups of spheres by Henn, Goerss, Mahowald and Rezk provide a new handle to attack the question.

## Operator algebras.

The theory of operator algebras (initiated originally e.g. by von Neumann, Gelfand) is at the heart of quantum theory. Modern noncommutative geometry has revealed a deep connection with topology, differential geometry and partial differential equations. K–theory is the standard cohomology theory for operator algebras. KK–theory, a bifunctor on the category of C*–algebras, is a far reaching generalization of K-homology, operator K-theory and index theory. The ongoing projects deal with computational aspects of K–theory as well as fundamental questions related to K– and KK–theory.
The Farrell-Jones Conjecture aims at the algebraic K-theory and L-theory of group rings. Its companion in non-commutative geometry is the celebrated Baum-Connes Conjecture which deals with the topological K-theory of the group C*-algebra. These two conjectures have rather parallel formulations in terms of assembly maps and classifying spaces for families of subgroups.
We plan to continue working on the computation of the topological K-theory of certain groups. The Baum-Connes Conjecture identifies the topological K-theory $K_*(C^*_r(G))$ of the group C*-algebra with the equivariant topological K-theory $K_*^G(\underline{E}G)$ of the classifying space for proper actions. Some machinery, e.g., equivariant Chern characters and equivariant spectral sequences, have already been developed to carry out computations of $K_*^G(\underline{E}G)$ and successfully been applied to interesting groups. There are still many important cases open, where new techniques are required and shall be designed.
The unique feature of KK–theory is the celebrated Kasparov product. One is still lacking an explicit description in the case of unbounded operators including important examples arising in geometry and from partial differential operators, which are a priori unbounded. Kaad (HCMPostdoc) and Lesch work on finding explicit unbounded representatives for the product in KK– theory, with the goal of a unified approach to several known index theoretic constructions, for instance the (Atiyah-)Robbin-Salamon spectral flow formula. It should then be possible to extend this formula to the scattering context in the sense of Gesztesy et al. and relate this to the trace defect formula in the Melrose b–calculus.