Graded Lie Derivatives and Short Distance Expansions in Two Dimensions
The exploration of algebraic structures in two-dimensional field theories underwent a rapid development in recent years, particularly through the investigation of W-algebras. It was demonstrated by Zamolodchikov that there exist algebras which close nonlinearly into finitely many (quasi-)primary fields1. For recent work on these W N -algebras see ref. 2–5 and references therein. A certain N → ∞ limit of these algebras, denoted meanwhile commonly as w 1+∞, has been studied by Bakas6 (see also ref. 7 for studies of homology complexes of a subalgebra of w 1+∞), and Pope, Romans and Shen succeeded8 in the construction of a unique W ∞-algebra with the property to admit central terms for all integer weights ≥ 2. For a discussion of field theoretic realisations of these algebras and their relations see ref. 9 and references therein. Both w 1+∞ and the related algebra w ∞ are subalgebras of the algebra of area-preserving diffeomorphisms in two dimensions, and it has been pointed out by Witten that these algebras are realized as symmetries of the ground ring of two-dimensional string theory10, implying in particular the existence of operator representations of these algebras in the framework of Liouville theory coupled to c = 1 matter fields. A very nice account of this has also been given by Eguchi, who studied the equivalence of Liouville theory coupled to matter fields and SU(2)/U(1) coset theory11.
KeywordsCentral Charge Central Extension Conformal Weight Primary Field Liouville Theory
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