Birman-Wenzl-Murakami Algebra and Logarithmic Superconformal Minimal Models

Two-dimensional exactly solvable loop models, built on the Temperley-Lieb algebra, have previously been introduced to study statistical systems with non-local degrees of freedom such as polymers and percolation. In the continuum scaling limit, these models describe logarithmic minimal Conformal Field Theories (CFTs). In this thesis, we introduce and study new two-dimensional exactly solvable superconformal loop models generalising the logarithmic minimal models. These models include superconformal polymers and superconformal percolation as the first members. The superconformal loop models are constructed using the generators of a one-parameter specialization of the Birman-WenzlMurakami (BMW) algebra arising from fusion of the Temperley-Lieb algebra. The BMW algebra extends the Temperley-Lieb algebra by allowing overand under-crossings of loop segments. Since the BMW is non-commutative and unwieldy, a computer system is implemented in Mathematica to carry out general algebraic calculations in the full two-parameter BMW algebra. In particular, the program produces matrix representations by acting on suitable vector spaces of link states. The face weights of the superconformal loop models are built algebraically by moving to compound 2 × 2 face weights constructed by a fusion process. In the continuum scaling limit, the new models describe logarithmic superconformal minimal Conformal Field Theories (CFTs). Finite-size corrections are studied to obtain the central charges, conformal dimensions and finitised conformal characters associated with the corresponding superconformal logarithmic minimal models. The analytic and numerical findings are in agreement with general theory giving these logarithmic CFTs as a “logarithmic limit” of the rational superconformal minimal models.