Legendre's Relation and the Quantum Equivalence osp(4|4)_(1) = osp(2|2)_(-2) + su(2)_(0)

Abstract

Using explicit results for the four-point correlation functions of the Wess-Zumino-Novikov-Witten (WZNW) model we discuss the conformal embedding osp(4|4)_(1) = osp(2|2)_(-2) + su(2)_(0). This embedding has emerged in Bernard and LeClair's recent paper [1]. Given that the osp(4|4)_(1) WZNW model is a free theory with power law correlation functions, whereas the su(2)_(0) and osp(2|2)_(-2) models are CFTs with logarithmic correlation functions, one immediately wonders whether or not it is possible to combine these logarithms and obtain simple power laws. Indeed, this very concern has been raised in a revised version of [1]. In this paper we demonstrate how one may recover the free field behaviour from a braiding of the solutions of the su(2)_(0) and osp(2|2)_(-2) Knizhnik-Zamolodchikov equations. We do this by implementing a procedure analogous to the conformal bootstrap programme [2]. Our ability to recover such simple behaviour relies on a remarkable identity in the theory of elliptic integrals known as Legendre's relation.