Conformal Invariance, Unitarity and Two Dimensional Critical Exponents

Abstract

We show that conformai invariance and unitarity severely limit the possible values of critical exponents in two dimensional systems by finding the discrete series of unitarisable representations of the Virasoro algebra. The realization of conformai symmetry in a given system is parametrized by a real number c, the coefficient of the trace anomaly. For c<1 the only values allowed by unitarity are c=1–6/m(m+1), m=2,3,4 ⋯ . For each of these values of c unitarity determines a finite set of rational numbers that must contain all possible critical exponents. These finite sets account for the known critical exponents of the following two dimensional models: Ising(m=3), tricritical Ising(m=4), 3-state Potts(m=5), and tricritical 3-state Potts(m=6).