In theoretical physics and mathematics, a Wess–Zumino–Witten (WZW) model, also called a Wess–Zumino–Novikov–Witten model, is a type of two-dimensional conformal field theory named after Julius Wess, Bruno Zumino, Sergei Novikov and Edward Witten.
provided the level obeys Integer values of the level also play an important role in the representation theory of the model's symmetry algebra, which is an affine Lie algebra.
If the level is a positive integer, the affine Lie algebra has unitary highest weight representations with highest weights that are dominant integral.
Such representations decompose into finite-dimensional subrepresentations with respect to the subalgebras spanned by each simple root, the corresponding negative root and their commutator, which is a Cartan generator.
The structure constants are completely anti-symmetric, and thus they define a 3-form on the group manifold of G. Thus, the integrand above is just the pullback of the harmonic 3-form to the ball
one then has This form leads directly to a topological analysis of the WZ term.
Geometrically, this term describes the torsion of the respective manifold.
[7] The presence of this torsion compels teleparallelism of the manifold, and thus trivialization of the torsionful curvature tensor; and hence arrest of the renormalization flow, an infrared fixed point of the renormalization group, a phenomenon termed geometrostasis.
The Wess–Zumino–Witten model is not only symmetric under global transformations by a group element in
an orthonormal basis (with respect to the Killing form) of the Lie algebra of
A second copy of the same affine Lie algebra is associated to the right-moving currents
The existence of the embedding shows that WZW models are conformal field theories.
The Sugawara construction is most concisely written at the level of the currents:
The central charge of the Virasoro algebra is given in terms of the level
of the affine Lie algebra by At the level of the generators of the affine Lie algebra, the Sugawara construction reads where the generators
is compact and simply connected, then the WZW model is rational and diagonal: rational because the spectrum is built from a (level-dependent) finite set of irreducible representations of the affine Lie algebra called the integrable highest weight representations, and diagonal because a representation of the left-moving algebra is coupled with the same representation of the right-moving algebra.
is compact but not simply connected, the WZW model is rational but not necessarily diagonal.
, and its spectrum is a non-diagonal combination of finitely many integrable highest weight representations.
WZW model is built from highest weight representations, plus their images under the spectral flow automorphisms of the affine Lie algebra.
[10] Non-factorizable representations are responsible for the fact that the corresponding WZW models are logarithmic conformal field theories.
The known conformal field theories based on affine Lie algebras are not limited to WZW models.
WZW model, modular invariant torus partition functions obey an ADE classification, where the
The conformal dimension of the affine primary field is given in terms of the quadratic Casimir
WZW model, the conformal dimension of a primary field of spin
is compact, the spectrum of the WZW model is made of highest weight representations, and all correlation functions can be deduced from correlation functions of affine primary fields via Ward identities.
is the Riemann sphere, correlation functions of affine primary fields obey Knizhnik–Zamolodchikov equations.
On Riemann surfaces of higher genus, correlation functions obey Knizhnik–Zamolodchikov–Bernard equations, which involve derivatives not only of the fields' positions, but also of the surface's moduli.
has been used by Juan Maldacena and Hirosi Ooguri to describe bosonic string theory on the three-dimensional anti-de Sitter space
[14][10] WZW models and their deformations have been proposed for describing the plateau transition in the integer quantum Hall effect.
gauged WZW model has an interpretation in string theory as Witten's two-dimensional Euclidean black hole.