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.[1][2][3][4] 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.[5]

ActionEdit

DefinitionEdit

For   a Riemann surface,   a Lie group, and   a (generally complex) number, let us define the  -WZW model on   at the level  . The model is a nonlinear sigma model whose action is a functional of a field  :

 

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

 

Here   is the completely anti-symmetric tensor, and   is the Lie bracket. The Wess–Zumino term is an integral over a three-dimensional manifold   whose boundary is  .

Topological properties of the Wess–Zumino termEdit

For the Wess–Zumino term to make sense, we need the field   to have an extension to  . This requires the homotopy group   to be trivial, which is the case in particular for any compact Lie group  .

The extension of a given   to   is in general not unique. For the WZW model to be well-defined,   should not depend on the choice of the extension. The Wess–Zumino term is invariant under small deformations of  , and only depends on its homotopy class. Possible homotopy classes are controlled by the homotopy group  .

For any compact, connected simple Lie group  , we have  , and different extensions of   lead to values of   that differ by integers. Therefore, they lead to the same value of   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  , the homotopy group   is trivial, and the level is not constrained to be an integer.[6]

Geometrical interpretation of the Wess–Zumino termEdit

Note that if ea 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.

Geometrically, this term describes the torsion of the respective manifold.[7] 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 algebraEdit

Affine Lie algebraEdit

Let   be a local complex coordinate on  ,   an orthonormal basis (with respect to the Killing form) of the Lie algebra of  , and   the quantisation of the field  . We have the following operator product expansion:

 

where   are the coefficients such that  . Equivalently, if   is expanded in modes

 

then the current algebra generated by   is the affine Lie algebra associated to the Lie algebra of  , with a level that coincides with the level   of the WZW model.[5] If  , the notation for the affine Lie algebra is  . The commutation relations of the affine Lie algebra are

 

This affine Lie algebra is the chiral symmetry algebra associated to the left-moving currents  . A second copy of the same affine Lie algebra is associated to the right-moving currents  . The generators   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 constructionEdit

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.

The Sugawara construction is most concisely written at the level of the currents:   for the affine Lie algebra, and the energy-momentum tensor   for the Virasoro algebra:

 

where the   denotes normal ordering, and   is the dual Coxeter number. By using the OPE of the currents and a version of Wick's theorem one may deduce that the OPE of   with itself is given by[5]

 

which is equivalent to the Virasoro algebra's commutation relations. The central charge of the Virasoro algebra is given in terms of the level   of the affine Lie algebra by

 

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

 
 

where the generators   of the Virasoro algebra are the modes of the energy-momentum tensor,  .

SpectrumEdit

WZW models with compact, simply connected groupsEdit

If the Lie group   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.[5]

For example, the spectrum of the   WZW model at level   is

 

where   is the affine highest weight representation of spin  : a representation generated by a state   such that

 

where   is the current that corresponds to a generator   of the Lie algebra of  .

WZW models with other types of groupsEdit

If the group   is compact but not simply connected, the WZW model is rational but not necessarily diagonal. For example, the   WZW model exists for even integer levels  , and its spectrum is a non-diagonal combination of finitely many integrable highest weight representations.[5]

If the group   is not compact, the WZW model is non-rational. Moreover, its spectrum may include non highest weight representations. For example, the spectrum of the   WZW model is built from highest weight representations, plus their images under the spectral flow automorphisms of the affine Lie algebra.[6]

If   is a supergroup, the spectrum may involve representations that do not factorize as tensor products of representations of the left- and right-moving symmetry algebras. This occurs for example in the case  ,[8] and also in more complicated supergroups such as  .[9] Non-factorizable representations are responsible for the fact that the corresponding WZW models are logarithmic conformal field theories.

Other theories based on affine Lie algebrasEdit

The known conformal field theories based on affine Lie algebras are not limited to WZW models. For example, in the case of the affine Lie algebra of the   WZW model, modular invariant torus partition functions obey an ADE classification, where the   WZW model accounts for the A series only.[10] The D series corresponds to the   WZW model, and the E series does not correspond to any WZW model.

Another example is the   model. This model is based on the same symmetry algebra as the   WZW model, to which it is related by Wick rotation. However, the   is not strictly speaking a WZW model, as   is not a group, but a coset.[11]

Fields and correlation functionsEdit

FieldsEdit

Given a simple representation   of the Lie algebra of  , an affine primary field   is a field that takes values in the representation space of  , such that

 

An affine primary field is also a primary field for the Virasoro algebra that results from the Sugawara construction. The conformal dimension of the affine primary field is given in terms of the quadratic Casimir   of the representation   (i.e. the eigenvalue of the quadratic Casimir element   where   is the inverse of the matrix   of the Killing form) by

 

For example, in the   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 functionsEdit

If the group   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.

If the Riemann surface   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.[12]

Gauged WZW modelsEdit

Given a Lie subgroup  , the   gauged WZW model (or coset model) is a nonlinear sigma model whose target space is the quotient   for the adjoint action of   on  . This gauged WZW model is a conformal field theory, whose symmetry algebra is a quotient of the two affine Lie algebras of the   and   WZW models, and whose central charge is the difference of their central charges.

ApplicationsEdit

The WZW model whose Lie group is the universal cover of the group   has been used by Juan Maldacena and Hirosi Ooguri to describe bosonic string theory on the three-dimensional anti-de Sitter space  .[6] Superstrings on   are described by the WZW model on the supergroup  , or a deformation thereof if Ramond-Ramond flux is turned on.[13][9]

WZW models and their deformations have been proposed for describing the plateau transition in the integer quantum Hall effect.[14]

The   gauged WZW model has an interpretation in string theory as Witten's two-dimensional Euclidean black hole.[15] The same model also describes certain two-dimensional statistical systems at criticality, such as the critical antiferromagnetic Potts model.[16]

ReferencesEdit

  1. ^ Wess, J.; Zumino, B. (1971). "Consequences of anomalous ward identities" (PDF). 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): 422–432. 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. ^ a b c d e Di Francesco, P.; Mathieu, P.; Sénéchal, D. (1997), Conformal Field Theory, Springer-Verlag, ISBN 0-387-94785-X
  6. ^ a b c 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.
  7. ^ 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.
  8. ^ V. Schomerus, H. Saleur, "The GL(1|1) WZW model: From supergeometry to logarithmic CFT", arxiv:hep-th/0510032
  9. ^ a b G. Gotz, T. Quella, V. Schomerus, "The WZNW model on PSU(1,1|2)", arxiv:hep-th/0610070
  10. ^ Andrea Cappelli and Jean-Bernard Zuber (2010), "A-D-E Classification of Conformal Field Theories", Scholarpedia 5(4):10314.
  11. ^ K. Gawedzki, "Non-Compact WZW Conformal Field Theories", arxiv:hep-th/9110076
  12. ^ G. Felder, C. Wieczerkowski, "Conformal blocks on elliptic curves and the Knizhnik--Zamolodchikov--Bernard equations", arxiv:hep-th/9411004
  13. ^ N. Berkovits, C. Vafa, E. Witten, "Conformal Field Theory of AdS Background with Ramond-Ramond Flux", arxiv:hep-th/9902098
  14. ^ M. Zirnbauer, "The integer quantum Hall plateau transition is a current algebra after all", arXiv:1805.12555
  15. ^ Witten, Edward (1991). "String theory and black holes". Physical Review D. 44 (2): 314–324. doi:10.1103/PhysRevD.44.314. ISSN 0556-2821.
  16. ^ N. Robertson, J. Jacobsen, H. Saleur, "Conformally invariant boundary conditions in the antiferromagnetic Potts model and the SL(2,ℝ)/U(1) sigma model", arXiv:1906.07565