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 Novikov and Edward Witten. ^{[1] [2] [3] [4]}

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 amounts to the complex plane compactified by adding a point at infinity.

The WZW model is then a nonlinear sigma model defined by γ with an action given by

$S_k(\gamma)= - \, \frac {k}{8\pi} \int_{S^2} d^2x\, \mathcal{K} (\gamma^{-1} \partial^\mu \gamma \, , \, \gamma^{-1} \partial_\mu \gamma) + 2\pi k\, S^{\mathrm WZ}(\gamma).$

Here, ∂_μ = ∂/∂x^μ 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

$S^{\mathrm WZ}(\gamma) = - \, \frac{1}{48\pi^2} \int_{B^3} d^3y\, \epsilon^{ijk} \mathcal{K} \left( \gamma^{-1} \, \frac {\partial \gamma} {\partial y^i} \, , \, \left[ \gamma^{-1} \, \frac {\partial \gamma} {\partial y^j} \, , \, \gamma^{-1} \, \frac {\partial \gamma} {\partial y^k} \right] \right)$

where [,] is the commutator, ε^ijk is the completely anti-symmetric tensor, and the integration coordinates yⁱ 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 π₂ (G) always vanishes for any compact, simply-connected Lie group, and we originally defined γ on the 2-sphere S ² = ∂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 γ^*, one then has

$S^{\mathrm WZ}(\gamma) = \int_{B^3} \gamma^{*} c.$

This form leads directly to a topological analysis of the WZ term.

Geometrically, this term describes the torsion of the respective manifold.^[5] 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.

Topological obstructions

The extension of the field to the interior of the ball is not unique; the need that the physics be independent of the extension imposes a quanitization condition on the coupling parameter k, the level. 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 S². The result of the gluing is a topological 3-sphere; each ball B³ is a hemisphere of S³. The two different extensions of γ on each ball now becomes a map S³ → G. However, the homotopy group π₃(G) = ℤ for any compact, connected simple Lie group G.

Thus, one has

$S^{\mathrm WZ}(\gamma) = S^{\mathrm WZ}(\gamma')+n ~,$

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

$\exp \left(i2\pi k S^{\mathrm WZ}(\gamma) \right)= \exp \left( i2\pi k S^{\mathrm WZ}(\gamma')\right)~.$

Thus, topological considerations lead one to conclude that the level 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,^[6] 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 γ lives on a compact Riemann surface.

Current algebra

The current algebra of the WZW model is a Kac–Moody algebra. The stress energy tensor is given by the Sugawara construction.

Coset construction

Taking the quotient of two WZW models gives a new conformal field theory whose central charge is the difference of the two original ones.

References

1. Wess, J.; Zumino, B. (1971). "Consequences of anomalous ward identities". Physics Letters B 37: 95. Bibcode:1971PhLB...37...95W. doi:10.1016/0370-2693(71)90582-X.
2. Witten, E. (1983). "Global aspects of current algebra". Nuclear Physics B 223 (2): 422–421. Bibcode:1983NuPhB.223..422W. doi:10.1016/0550-3213(83)90063-9.
3. Witten, E. (1984). "Non-abelian bosonization in two dimensions". Communications in Mathematical Physics 92 (4): 455–472. Bibcode:1984CMaPh..92..455W. doi:10.1007/BF01215276.
4. Novikov, S.P. (1981). "Multivalued functions and functionals. An analogue of the Morse theory". Sov. Math., Dokl. 24: 222–226.; Novikov, S. P. (1982). "The Hamiltonian formalism and a many-valued analogue of Morse theory". Russian Mathematical Surveys 37 (5): 1–9. Bibcode:1982RuMaS..37....1N. doi:10.1070/RM1982v037n05ABEH004020.
5. Braaten, E.; Curtright, T. L.; Zachos, C. K. (1985). "Torsion and geometrostasis in nonlinear sigma models". Nuclear Physics B 260 (3–4): 630. Bibcode:1985NuPhB.260..630B. doi:10.1016/0550-3213(85)90053-7.
6. Maldacena, J.; Ooguri, H. (2001). "Strings in AdS₃ and the SL(2,R) WZW model. I: The spectrum". Journal of Mathematical Physics 42 (7): 2929. arXiv:hep-th/0001053. Bibcode:2001JMP....42.2929M. doi:10.1063/1.1377273.