Mathematical models are indispensable tools in all theoretical fields of science. Although rare, there are instances in which the model admits exact solutions. Such systems are called integrable. Integrable systems have been studied since 19th century and before, but during the last 1/4 of the 20th century it has developed into a mathematical field on its own.
Dependning on the type of systems there is a distinction between classical and quantum. Classical integrable systems deal with non-linear partial differential/difference equations, including soliton equations, discrete/ultradiscrete analogs and Painlevé equations. Quantum integrable systems arise in statistical mechanics and field theory. They are best studied in 2 spacetime dimensions, either in the continuum or on the lattice. 2d conformal field theory is a good example. Integrable structures arise also in higher dimensions and in string theory, in various different contexts. On the mathematical side, there are close links with other branches: representation theory, low dimensional topology, operator algebras and combinatorics, to name a few. The methods for studying them vary from algebraic, geometric to analytic ones. Often some algebraic symmetry (typically hidden and infinite dimensional) give clues to understanding and solving the systems.
Group Members
Kentaro Hori
I am interested in (2,2) and (0,2) superconformal field theories and conformally invariant boundary conditions.
Ooguri is studying conformal field theories in diverse dimensions that are relevant to dynamics of strings and branes in superstring theory. He is also applying conformal field theory techniques to study the landscape of string vacua.
Domenico Orlando
Spin chains (XXZ model and related two-dimensional lattices) in connection to dimer models and topological strings.
Susanne Reffert
I am interested in (quantum) dimer models, (quantum) crystal melting and spin chains.
My research area is representation theory of quantum groups and infinite dimensional Lie algebras, and related topics.
Tadashi Takayanagi
I am interested in the solvable conformal field theories in their applications to the descriptions of tachyon condensation in string theory. I am also interested in solvable matrix models and their application to non-perturbative formulations of quantum gravity.