Quantum field theories describe and unify the fundamental forces of nature, with the possible exception of gravity. These theories are relativistic, hence their symmetries include translations and Lorentz boosts. A quantum field theory may have more symmetries, however. The conformal field theories (CFTs), in particular, admit a symmetry under rescaling (and more). Such theories accurately describe the thermodynamic limit of local observables in statistical lattice models such as percolation and the Ising model. Moreover, they provide the foundation upon which the quantisation of string theory is built.
In two dimensions, the algebra of infinitesimal conformal transformations is infinite-dimensional. A 2D CFT is therefore very strongly constrained by its symmetries and, in favourable cases, these symmetries can be used to exactly solve the theory. Aside from applications to physics, the development of 2D CFT has also led to an enormous amount of mathematical progress in a wide variety of seemingly disparate fields.
Research being undertaken at the ANU aims to better understand the mathematical structures underlying CFT. The properties of so-called logarithmic CFTs are of particular interest. In terms of representation theory, the quantum state space is built from representations of the algebra of infinitesimal symmetries that are indecomposable rather than irreducible. Logarithmic theories arise naturally when considering non-local observables, such as crossing probabilities and fractal dimensions, in many statistical lattice models (percolation, Ising). They also appear in string theories when spacetime is enlarged to admit fermionic directions, the famous AdS/CFT correspondence, and the study of black holes in three spacetime dimensions. In mathematics, a recent exciting development is that logarithmic CFT should be closely related to the theory of Schramm-Loewner Evolutions.