Wess–Zumino–Witten model

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, and its symmetry algebra is the affine Lie algebra built from the corresponding Lie algebra. By extension, the name WZW model is sometimes used for any conformal field theory whose symmetry algebra is an affine Lie algebra. 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, and its symmetry algebra is the affine Lie algebra built from the corresponding Lie algebra. By extension, the name WZW model is sometimes used for any conformal field theory whose symmetry algebra is an affine Lie algebra. For Σ {displaystyle Sigma } a Riemann surface, G {displaystyle G} a Lie group, and k {displaystyle k} a (generally complex) number, let us define the G {displaystyle G} -WZW model on Σ {displaystyle Sigma } at the level k {displaystyle k} . The model is a nonlinear sigma model whose action is a functional of a field γ : Σ → G {displaystyle gamma :Sigma o G} : Here, Σ {displaystyle Sigma } is equipped with a flat Euclidean metric, ∂ μ {displaystyle partial _{mu }} is the partial derivative, and K {displaystyle {mathcal {K}}} is the Killing form on the Lie algebra of G {displaystyle G} . The Wess–Zumino term of the action is Here ϵ i j k {displaystyle epsilon ^{ijk}} is the completely anti-symmetric tensor, and [ . , . ] {displaystyle } is the Lie bracket. The Wess–Zumino term is an integral over a three-dimensional manifold B 3 {displaystyle mathbf {B} ^{3}} whose boundary is ∂ B 3 = Σ {displaystyle partial mathbf {B} ^{3}=Sigma } . For the Wess–Zumino term to make sense, we need the field γ {displaystyle gamma } to have an extension to B 3 {displaystyle mathbf {B} ^{3}} . This requires the homotopy group π 2 ( G ) {displaystyle pi _{2}(G)} to be trivial, which is the case in particular for any compact Lie group G {displaystyle G} . The extension of a given γ : Σ → G {displaystyle gamma :Sigma o G} to B 3 {displaystyle mathbf {B} ^{3}} is in general not unique. For the WZW model to be well-defined, e i S k ( γ ) {displaystyle e^{iS_{k}(gamma )}} should not depend on the choice of the extension. The Wess–Zumino term is invariant under small deformations of γ {displaystyle gamma } , and only depends on its homotopy class. Possible homotopy classes are controlled by the homotopy group π 3 ( G ) {displaystyle pi _{3}(G)} . For any compact, connected simple Lie group G {displaystyle G} , we have π 3 ( G ) = Z {displaystyle pi _{3}(G)=mathbb {Z} } , and different extensions of γ {displaystyle gamma } lead to values of S W Z ( γ ) {displaystyle S^{mathrm {W} Z}(gamma )} that differ by integers. Therefore, they lead to the same value of e i S k ( γ ) {displaystyle e^{iS_{k}(gamma )}} 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. In the case of the noncompact simple Lie group S L ( 2 , R ) {displaystyle mathrm {SL} (2,mathbb {R} )} ,the homotopy group π 3 ( S L ( 2 , R ) ) {displaystyle pi _{3}(mathrm {SL} (2,mathbb {R} ))} is trivial, and the level is not constrained to be an integer.