The mathematics of string theory and quantum field theory
Scientists in the Department of Theoretical Physics and at the Mathematical Science Institute are collaborating in understanding the mathematics behind String Theory and Quantum Field Theory. This involves applying fairly recent mathematical disciplines, such as non-commutative geometry, but also involves developing and studying new mathematics suggested by, and needed for, a further development of the theory.
Examples of research in this area include:
investigation of finite dimensional Lie algebras and their representation theory, such as the 2-dimensional conformal algebra (Virasoro algebra) and generalisations thereof (W-algebras)
representation theory of quantum groups, quantum affine algebras and Yangians, their use in the integrable structure of QFT (in particular in 2D Conformal Field Theories) and applications in condensed matter systems (fractional statistics, exclusion statistics, non-abelian statistics, fractional quantum Hall effect)
use of homological algebra in analysing free field realisations and BRST cohomology (semi-infinite cohomology)
use of algebraic topology (cohomology, K-theory) in classifying extended objects in String Theory (D-branes) and analysing states of matter (topological insulators, superconductors)
analysis of dualities in String Theory and Quantum Field Theory, such as T-duality, S-duality, Mirror Symmetry, and Langlands duality, and their mathematical implications (such as isomorphisms of (twisted) K-theory)
study of generalisations of geometry (generalised geometry) and the underlying algebraic structures (Lie algebroids, Courant algebroids).