Supplementary Notes on VOA and 2D SCFT

Abstract

Sections 2--3 of this note were prepared primarily for the author's private use, as a means of organizing the materials in (generally non-holomorphic) conformal field theory (CFT) on 2-dimensions in the language of the vertex operator algebra, and also of collecting subtle details. The choice of topics is intended to supplement introductory physics textbooks on the subject. Section 4 deals with (generally non-holomorphic) Type 0 CFTs in the language of vertex operator superalgebra (SVOA). Formulations of crossing symmetry and intertwining superoperators [resp. CFT vertex superoperators] of states in the Ramond (SVOA-twisted) sector [resp. the Ramond--Ramond sector] are presented by using two auxiliary parameters. The formulation allows systematic generalization to the class of simple current extensions characterized by four conditions: (1) the vanishing third cohomology, (2) skew-symmetric bimultiplicative form, (3) fixed-point-free action of simple currents, and (4) one additional condition described in the main text.