Wess-Zumino-Witten model

In theoretical physics and mathematics, the Wess-Zumino-Witten (WZW) model, also called the Wess-Zumino-Novikov-Witten model, is a simple model of conformal field theory whose solutions are realized by affine Kac-Moody algebras. It is named after Julius Wess, Bruno Zumino, Sergei P. Novikov and Edward Witten.

Action

Let G denote a compact simply-connected Lie group and g its simple Lie algebra. Suppose that γ is a G-valued field on the complex plane. More precisely, we want γ to be defined on the Riemann sphere , which is the complex plane compactified by adding a point at infinity.

The WZW model is then a nonlinear sigma model defined by γ with the action given by
:.
Here, is the partial derivative and the usual summation convention over indices is used, with a Euclidean metric. Here, is the Killing form on g, and thus the first term is the standard kinetic term of quantum field theory.

The term S^WZ is called the Wess-Zumino term and can be written as
:
where [,] is the commutator, is the completely anti-symmetric tensor, and the integration coordinates for i=1,2,3 range over the unit ball .
In this integral, the field γ has been extended so that it is defined on the interior of the unit ball. This extension can always be done because the homotopy group always vanishes for any compact, simply-connected Lie group, and we originally defined γ on the 2-sphere .

Pullback

Note that if are the basis vectors for the Lie algebra, then are the structure constants of the Lie algebra. Note also that 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 . Denoting the harmonic 3-form by c and the pullback by , one then has
:
This form leads directly to a topological analysis of the WZ term.

Topological obstructions

The extension of the field to the interior of the ball is not unique; the need to have the physics be independent of the extension imposes a quanitization condition on the coupling constant k. Consider two different extensions of γ to the interior of the ball. They are maps from flat 3-space into the Lie group G. Consider now glueing these two balls together at their boundary . The result of the glueing is a topological 3-sphere; each ball is a hemisphere of . The two different extensions of γ on each ball now becomes a map . However, the homotopy group
for any compact, connected simple Lie group G. Thus we have
:
where γ and γ' denote the two different extensions onto the ball, and n, an integer, is the winding number of the glued-together map. The physics that this model leads to will stay the same if

Thus, topological considerations leads one to conclude that that coupling constant k must be an integer.

Generalizations

Although in the above, the WZW model is defined on the Riemann sphere, it can be generalized so that the field γ lives on a compact Riemann surface.

Current algebra

The current algebra of the WZW model is a Virasoro algebra.

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