Two Remarks on Recent Developments in Solvable Models

Abstract

In the exact study of two-dimensional statistical mechanics and quantum field theory, two approaches are presently available: solvable lattice models (SLMs) and conformal field theory (CFT). The ideas and methods in these approaches are quite independent of each other. On the one hand, the SLM deals with generically non-critical models on the lattice, and is built upon solutions to a set of algebraic equations called the star-triangle relation. The CFT on the other hand deals with critical and continuous systems. The main tool is the symmetry under infinite-dimensional Lie algebras, notably the Virasoro algebra. Notwithstanding their conceptual difference, these theories exhibit unexpected similarities. On the basis of the recent works on SLM, this chapter discusses two similar structures that occur in different contexts: one is the representation of the braid group, and the other is the modular covariance. The chapter also discusses modular covariant characters.