WessZuminoWitten

Wess-Zumino-Witten, often abbreviated as WZW or WZW model, is a two-dimensional conformal field theory in which the field takes values in a compact Lie group G. It combines a nonlinear sigma model term with a topological Wess-Zumino term. On a worldsheet Σ, the field is a map g: Σ → G, extended to a three-manifold B with boundary ∂B = Σ. The action is S[g] = (k/8π) ∫_Σ Tr(g^{-1} ∂_μ g g^{-1} ∂^μ g) d^2x + (k/12π) ∫_B Tr(g^{-1} dg)^3, where k ∈ Z is the level. The Wess-Zumino term requires the extension to B and is quantized so that the path integral is well defined.

The model exhibits left and right current symmetries, generating holomorphic and antiholomorphic conserved currents that form affine Kac-Moody algebras at level k. Quantum mechanically the theory is conformally invariant for all k, with central charge c = (k dim G)/(k + h^∨), where h^∨ is the dual Coxeter number of G. Correlation functions decompose into chiral blocks determined by these current algebras, making the WZW model exactly solvable.

History and applications: the construction is attributed to Wess, Zumino, and Witten in the 1980s, and the WZW model serves as a fundamental building block of two-dimensional conformal field theories. In string theory, it describes strings propagating on group manifolds (for example, SU(2) at level k yields strings on S^3 with H-flux). In condensed matter physics, related coset models and gauged WZW theories describe critical spin systems and edge states. The framework also extends to coset constructions G_k/H_k, broadening its applicability.