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Two Topics in Two-Dimensional Quantum Field Theory

G. Rivlis

Published Jan 1, 1991• 0 citations
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Two topics in two-dimensional quantum field theory are presented. The first is a classification of 2- and 3-field rational conformal field theories. Using the fact that the fusion algebra of a RCFT is specified in terms of integers that are related to modular transformation properties, we classify 2- and 3-field chiral RCFT's. We show that the only possibilities for the non-trivial fusion rule in the 2-field case are φ x φ = 1 or φ x φ = 1 + φ. Similar results are obtained for the 3-field case. A partial classification of possible conformal dimensions and central charges for these theories is also obtained. The second topic is in two-dimensional quantum gravity. Explicit computation of the non-perturbative correlation functions of the (1, q) models of KdV-gravity is presented. This computation includes contributions from high genus as well as correlation functions of descendant fields. A ghost number conservation law for these models is derived from purely algebraic considerations. A hint of further selection rules is found.
Two dimensional toy models display, in a gentler setting, manysalient aspects of Quantum Field Theory. Here I discuss a concrete two dimensional case, the Thirring model, which illustrates several important concepts of this theory: the anomalous dimension of the fields; the exact solvability; the anomalies of the Ward-Takahashi identities. Besides, I give a glimpse of the decisive role that this model plays in the study of an apparently unrelated topic: correlation critical exponents of two dimensional lattice systems of Statistical Mechanics.