Local Charge Algebras in Quantum Chiral Models and Gauge Theories

Abstract

Affine and loop algebras are analogous to local current algebras that occur in many physical field theories. Sugawara’s formulation of the chiral model is a particularly interesting case. If this model is extended by the addition of the Wess-Zumino term, then the quantum algebra of the local charge densities $[J_0^a(x),J_0^b(y)]$ becomes anomalous and has a central extension. This is the first example of a theory with such an anomaly. In 2 dimensions, the central extension is a c-number, just as for affine algebras. Gauge theories can be reformulated in terms of string variables, i.e. ordered line integrals of the form $(\exp i\int {_SA - dx)} {}_ +$. With a particular choice of rigid strings {S}, the theory can be rewritten in terms of the “strings” which are group elements rather than the gauge potentials A_μ which are in the algebra. In this formulation the canonical quantization of the theory again is in terms of a local non-Abelian charge algebra analogous to the chiral theory, or affine type algebras in higher dimensions.

Partially supported by the National Science Foundation through the Mathematical Sciences Research Institute.