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 $S^2$, 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:$S\_k(gamma)=\; -\; ,\; frac\; \{k\}\{8pi\}\; int\_\{S^2\}\; d^2x,\; mathcal\{K\}\; (gamma^\{-1\}\; partial^mu\; gamma\; ,\; ,\; ,\; gamma^\{-1\}\; partial\_mu\; gamma)\; +\; 2pi\; k,\; S^\{mathrm\; WZ\}(gamma)$.Here, $partial\_mu\; =\; partial\; /\; partial\; x^mu$ is the partial derivative and the usual summation convention over indices is used, with a Euclidean metric. Here, $mathcal\{K\}$ 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:$S^\{mathrm\; WZ\}(gamma)\; =\; -\; ,\; frac\{1\}\{48pi^2\}\; int\_\{B^3\}\; d^3y,\; epsilon^\{ijk\}\; mathcal\{K\}\; left(\; gamma^\{-1\}\; ,\; frac\; \{partial\; gamma\}\; \{partial\; y^i\}\; ,\; ,\; ,\; left\; [gamma^\{-1\}\; ,\; frac\; \{partial\; gamma\}\; \{partial\; y^j\}\; ,\; ,\; ,gamma^\{-1\}\; ,\; frac\; \{partial\; gamma\}\; \{partial\; y^k\}\; ight]\; ight)$where [,] is the commutator, $epsilon^\{ijk\}$ is the completely anti-symmetric tensor, and the integration coordinates $y^i$ for "i"=1,2,3 range over the unit ball $B^3$.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 $pi\_2(G)$ always vanishes for any compact, simply-connected Lie group, and we originally defined γ on the 2-sphere $S^2=partial\; B^3$.

Pullback

Note that if $e\_a$ are the basis vectors for the Lie algebra, then $mathcal\{K\}\; (e\_a,\; [e\_b,\; e\_c]\; )$ 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 $B^3$. Denoting the harmonic 3-form by "c" and the pullback by $gamma^\{*\}$, one then has :$S^\{mathrm\; WZ\}(gamma)\; =\; int\_\{B^3\}\; gamma^\{*\}\; c$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 $S^2$. The result of the gluing is a topological 3-sphere; each ball $B^3$ is a hemisphere of $S^3$. The two different extensions of γ on each ball now becomes a map $S^3\; ightarrow\; G$. However, the homotopy group $pi\_3(G)=mathbb\{Z\}$ for any compact, connected simple Lie group "G". Thus we have:$S^\{mathrm\; WZ\}(gamma)\; =\; S^\{mathrm\; WZ\}(gamma\text{'})+n$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

:$exp\; left(i2pi\; k\; S^\{mathrm\; WZ\}(gamma)\; ight)=\; exp\; left(\; i2pi\; k\; S^\{mathrm\; WZ\}(gamma\text{'})\; ight)$

Thus, topological considerations leads one to conclude that coupling constant "k" must be an integer when "G" is a connected, compact, simple Lie group. For a semisimple or disconnected compact Lie group the level consists of an integer for each connected, simple component.

This topological obstruction can also be seen in the representation theory of the affine Lie algebra symmetry of the theory. When each level is a positive integer the affine Lie algebra has unitary highest weight representations with highest weights that are dominant integral. Such representations are easier to work with as they decompose into finite-dimensional subalgebras with respect to the subalgebras spanned by each simple root, the corresponding negative root and their commutator, which is a Cartan generator.

Often one is interested in a WZW model with a noncompact simple Lie group G, such as SL(2,R) which has been used by Juan Maldacena and Hirosi Ooguri to describe string theory on a three-dimensional anti de Sitter space, which is the universal cover of the group SL(2,R). In this case, as π₃(SL(2,R))=0, there is no topological obstruction and the level need not be integral. Correspondingly, the representation theory of such noncompact Lie groups is much richer than that of their compact counterparts.

Generalizations

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

Current algebra

The current algebra of the WZW model is a Kac-Moody algebra.

References

* J. Wess, B. Zumino, "Consequences of anomalous Ward identities", "Physics Letters B", 37 (1971) pp. 95-97.
* E. Witten, "Global aspects of current algebra", "Nuclear Physics B" 223 (1983) pp. 422-432.