Wess–Zumino–Witten model
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Wess–Zumino–Witten model
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]
Action
Definition
ForaRiemann surface,aLie group, anda (generally complex) number, let us define the **-WZW model onat the level**. The model is anonlinear sigma modelwhoseactionis a functional of a field:
Here,is equipped with a flatEuclidean metric,is thepartial derivative, andis theKilling formon theLie algebraof. The Wess–Zumino term of the action is
![{\displaystyle S^{\mathrm {W} Z}(\gamma )=-{\frac {1}{48\pi ^{2}}}\int _{\mathbf {B} ^{3}}d^{3}y\,\epsilon ^{ijk}{\mathcal {K}}\left(\gamma ^{-1}\partial _{i}\gamma ,\left[\gamma ^{-1}\partial _{j}\gamma ,\gamma ^{-1}\partial _{k}\gamma \right]\right).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/00f3879460f9f931086e0ca0a76edf41fc3cd37f)
Hereis thecompletely anti-symmetric tensor, andis theLie bracket.
The Wess–Zumino term is an integral over a three-dimensional manifoldwhose boundary is.
![{\displaystyle [.,.]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/740650be18c169f2d8f2d47ff04a721f726da613)
Topological properties of the Wess–Zumino term
For the Wess–Zumino term to make sense, we need the fieldto have an extension to. This requires thehomotopy groupto be trivial, which is the case in particular for any compact Lie group.
The extension of a giventois 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 itshomotopy class.
Possible homotopy classes are controlled by the homotopy group.
For any compact, connected simple Lie group, we have, and different extensions oflead to values ofthat differ by integers. Therefore, they lead to the same value ofprovided 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 groupis trivial, and the level is not constrained to be an integer.[6]
Geometrical interpretation of the Wess–Zumino term
Note that if *ea
- are the basis vectors for the
![{\mathcal {K}}(e_{a},[e_{b},e_{c}])](https://wikimedia.org/api/rest_v1/media/math/render/svg/5d62bc8d07365a71cd38c8888e48b6b37dfaa946)
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 algebra
Affine Lie algebra
Letbe a local complex coordinate on,an orthonormal basis (with respect to theKilling form) of the Lie algebra of, andthe quantisation of the field. We have the followingoperator product expansion:
whereare the coefficients such that.
Equivalently, ifis expanded in modes
![{\displaystyle [t^{a},t^{b}]=f_{c}^{ab}t^{c}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/93454dbc35780a77205abbd421248a7a4700f20c)
then thecurrent algebragenerated byis theaffine Lie algebraassociated to the Lie algebra of, with a level that coincides with the levelof the WZW model.[5]
If, the notation for the affine Lie algebra is.
The commutation relations of the affine Lie algebra are
![{\displaystyle [J_{n}^{a},J_{m}^{b}]=f_{c}^{ab}J_{m+n}^{c}+kn\delta ^{ab}\delta _{n+m,0}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/766ddc1931a1b60f9030d2a4f2937229b0c5cc65)
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 generatorsof 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 construction
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 theenergy-momentum tensorfor the Virasoro algebra:
where thedenotes normal ordering, andis thedual Coxeter number. By using theOPEof the currents and a version ofWick's theoremone may deduce that the OPE ofwith 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 levelof the affine Lie algebra by
Spectrum
WZW models with compact, simply connected groups
If the Lie groupis 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 integrablehighest 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 theWZW model at levelis
whereis the affine highest weight representation of spin: a representation generated by a statesuch that
whereis the current that corresponds to a generatorof the Lie algebra of.
WZW models with other types of groups
If the groupis compact but not simply connected, the WZW model is rational but not necessarily diagonal. For example, theWZW model exists for even integer levels, and its spectrum is a non-diagonal combination of finitely many integrable highest weight representations.[5]
If the groupis not compact, the WZW model is non-rational. Moreover, its spectrum may include non highest weight representations. For example, the spectrum of theWZW model is built from highest weight representations, plus their images under the spectral flow automorphisms of the affine Lie algebra.[6]
Ifis asupergroup, 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 arelogarithmic conformal field theories.
Other theories based on affine Lie algebras
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 theWZW model, modular invariant torus partition functions obey an ADE classification, where theWZW model accounts for the A series only.[10] The D series corresponds to theWZW model, and the E series does not correspond to any WZW model.
Another example is themodel. This model is based on the same symmetry algebra as theWZW model, to which it is related by Wick rotation. However, theis not strictly speaking a WZW model, asis not a group, but a coset.[11]
Fields and correlation functions
Fields
Given a simplerepresentationof the Lie algebra of, an affine primary fieldis a field that takes values in the representation space of, such that
An affine primary field is also aprimary fieldfor the Virasoro algebra that results from the Sugawara construction. The conformal dimension of the affine primary field is given in terms of the quadratic Casimirof the representation(i.e. the eigenvalue of the quadraticCasimir elementwhereis the inverse of the matrixof the Killing form) by
For example, in theWZW model, the conformal dimension of a primary field ofspinis
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 functions
If the groupis 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 viaWard identities.
If the Riemann surfaceis the Riemann sphere, correlation functions of affine primary fields obeyKnizhnik-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 models
Given a Lie subgroup, thegauged WZW model (or coset model) is a nonlinear sigma model whose target space is the quotientfor theadjoint actionofon. This gauged WZW model is a conformal field theory, whose symmetry algebra is a quotient of the two affine Lie algebras of theandWZW models, and whose central charge is the difference of their central charges.
Applications
The WZW model whose Lie group is theuniversal coverof the grouphas been used byJuan MaldacenaandHirosi Oogurito describe bosonicstring theoryon the three-dimensionalanti-de Sitter space.[6] Superstrings onare 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]
References
[1]
Citation Link//doi.org/10.1016%2F0370-2693%2871%2990582-XWess, 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.
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[2]
Citation Link//doi.org/10.1016%2F0550-3213%2883%2990063-9Witten, 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.
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[3]
Citation Link//doi.org/10.1007%2FBF01215276Witten, E. (1984). "Non-abelian bosonization in two dimensions". Communications in Mathematical Physics. 92 (4): 455–472. Bibcode:1984CMaPh..92..455W. doi:10.1007/BF01215276.
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Citation Link//doi.org/10.1070%2FRM1982v037n05ABEH004020Novikov, 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.
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Citation Linkopenlibrary.orgDi Francesco, P.; Mathieu, P.; Sénéchal, D. (1997), Conformal Field Theory*, Springer-Verlag, ISBN 0-387-94785-X***
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[6]
Citation Link//doi.org/10.1063%2F1.1377273Maldacena, 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.
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Citation Link//doi.org/10.1016%2F0550-3213%2885%2990053-7Braaten, 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.
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Citation Linkwww.scholarpedia.orgAndrea Cappelli and Jean-Bernard Zuber (2010), "A-D-E Classification of Conformal Field Theories", Scholarpedia 5(4):10314.
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[12]
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[14]
Citation Linkarxiv.orgM. Zirnbauer, "The integer quantum Hall plateau transition is a current algebra after all", arXiv:1805.12555
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[15]
Citation Linkportal.issn.orgWitten, Edward (1991). "String theory and black holes". Physical Review D. 44 (2): 314–324. doi:10.1103/PhysRevD.44.314. ISSN 0556-2821.
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