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. A WZW model is associated to a Lie group (or supergroup), and its symmetry algebra is the affine Lie algebra built from the corresponding Lie algebra (or Lie superalgebra). By extension, the name WZW model is sometimes used for any conformal field theory whose symmetry algebra is an affine Lie algebra.

Action

Definition

a Riemann surface a Lie group a (generally complex) number, let us define the . The model is a nonlinear sigma model whose action:

is equipped with a flat Euclidean metric is the partial derivative is the Killing form on the Lie algebra. The Wess–Zumino term of the action is

is the completely anti-symmetric tensor is the Lie bracket.

Topological properties of the Wess–Zumino term

. This requires the homotopy group.

, and only depends on its homotopy class.

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 weightss 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.

In the case of the noncompact simple Lie group \mathrm{SL}(2,\R) is trivial, and the level is not constrained to be an integer.

Geometrical interpretation of the Wess–Zumino term

If ea are the basis vectors for the Lie algebra are the structure constants of the Lie algebra. 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 Denoting the harmonic 3-form by c 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. 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.

Symmetry algebra

Generalised group symmetry

symmetry., the following symmetry holds:

-valued fields:

group.

Affine Lie algebra

an orthonormal basis (with respect to the Killing form. We have the following operator product expansion:

is expanded in modes

then the current algebra is the affine Lie algebra of the WZW model.. The commutation relations of the affine Lie algebra are

of that second copy are antiholomorphic. The full symmetry algebra of the WZW model is the product of the two copies of the affine Lie algebra.

Sugawara construction

The Sugawara construction is an embedding of the Virasoro algebra into the universal enveloping algebra of the affine Lie algebra. The existence of the embedding shows that WZW models are conformal field theories. Moreover, it leads to Knizhnik–Zamolodchikov equations for correlation functions.

for the affine Lie algebra, and the energy-momentum tensor for the Virasoro algebra:

is the dual Coxeter number. By using the OPE of the currents and a version of Wick's theorem with itself is given by

of the affine Lie algebra by

At the level of the generators of the affine Lie algebra, the Sugawara construction reads

.

Spectrum

WZW models with compact, simply connected groups

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

such that

.

WZW models with other types of groups

, 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.

is a supergroup,. Non-factorizable representations are responsible for the fact that the corresponding WZW models are logarithmic conformal field theories.

Other theories based on affine Lie algebras

WZW model accounts for the A series only. WZW model, and the E series does not correspond to any WZW model.

is not a group, but a coset.

Fields and correlation functions

Fields

Given a simple representation, an affine primary field, such that

An affine primary field is also a primary field (i.e. the eigenvalue of the quadratic Casimir element of the Killing form) by

WZW model, the conformal dimension of a primary field of spin is

By the state-field correspondence, affine primary fields correspond to affine primary states, which are the highest weight states of highest weight representations of the affine Lie algebra.

Correlation functions

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.

Gauged WZW models

gauged WZW model (or coset model for the adjoint action WZW models, and whose central charge is the difference of their central charges.

Applications

The WZW model whose Lie group is the universal cover has been used by Juan Maldacena and Hirosi Ooguri to describe bosonic string theory on the three-dimensional anti-de Sitter space., or a deformation thereof if Ramond-Ramond flux is turned on.

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. The same model also describes certain two-dimensional statistical systems at criticality, such as the critical antiferromagnetic Potts model.

References