The Construction and Application of Integrable Models within Four-Dimensional Quantum Field Theories

In certain quantum field theories, which are said to be integrable one may find exact solutions, traditionally quantum field theories have perturbative solutions. A classic example of such a theory is the Ising models of spins on a lattice. In two dimensions the Ising models has a critical point at which it is described by a massless fermionic theory, which can be shown to be conformally invariant. One may insert defects into such theories in order to find descriptions in the presence of a boundary, it often being useful to identify their critical phenomena and universality classes. I am currently studying a mechanism to generate such theories using a 4 dimensional gauge theory, where the holonomies in this theory can be mapped to a 2 dimensional plane, the point at which these holonomies cross give a quantity called an R matrix which may be calculated from the gauge theory. The hope is to modify this gauge theory such that one can generate a description of a class of models which have trivial R matrices and are currently not described by the theory.