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 (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.
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 .
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.
If ea are the basis vectors for the Lie algebra, then are the structure constants of the Lie algebra. 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.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.
The Wess-Zumino-Witten model is not only symmetric under global transformations by a group element in , but also has a much richer symmetry. This symmetry is often called the symmetry. Namely, given any holomorphic -valued function , and any other (completely independent of ) antiholomorphic -valued function , where we have identified and in terms of the Euclidean space coordinates , the following symmetry holds:
One way to prove the existence of this symmetry is through repeated application of the Polyakov-Wiegmann identity regarding products of -valued fields:
The holomorphic and anti-holomorphic currents and are the conserved currents associated with this symmetry. The singular behaviour of the products of these currents with other quantum fields determine how those fields transform under infintessimal actions of the group.
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. 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.
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
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, .
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.
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 .
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.
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.
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 , and also in more complicated supergroups such as . Non-factorizable representations are responsible for the fact that the corresponding WZW models are logarithmic conformal field theories.
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. 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.
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.
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.
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.
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 . Superstrings on are described by the WZW model on the supergroup , or a deformation thereof if Ramond-Ramond flux is turned on.
WZW models and their deformations have been proposed for describing the plateau transition in the integer quantum Hall effect.
The gauged WZW model has an interpretation in string theory as Witten's two-dimensional Euclidean black hole. The same model also describes certain two-dimensional statistical systems at criticality, such as the critical antiferromagnetic Potts model.
In mathematics, an affine algebraic plane curve is the zero set of a polynomial in two variables. A projective algebraic plane curve is the zero set in a projective plane of a homogeneous polynomial in three variables. An affine algebraic plane curve can be completed in a projective algebraic plane curve by homogenizing its defining polynomial. Conversely, a projective algebraic plane curve of homogeneous equation h(x, y, t) = 0 can be restricted to the affine algebraic plane curve of equation h(x, y, 1) = 0. These two operations are each inverse to the other; therefore, the phrase algebraic plane curve is often used without specifying explicitly whether it is the affine or the projective case that is considered.
In differential geometry, the Lie derivative, named after Sophus Lie by Władysław Ślebodziński, evaluates the change of a tensor field, along the flow defined by another vector field. This change is coordinate invariant and therefore the Lie derivative is defined on any differentiable manifold.
In mathematics, the Virasoro algebra is a complex Lie algebra, the unique central extension of the Witt algebra. It is widely used in two-dimensional conformal field theory and in string theory.
The Chern–Simons theory is a 3-dimensional topological quantum field theory of Schwarz type developed by Edward Witten. It was discovered firstly by a mathematical physicist Albert Schwarz. It is named after mathematicians Shiing-Shen Chern and James Harris Simons who introduced the Chern–Simons 3-form. In the Chern–Simons theory, the action is proportional to the integral of the Chern–Simons 3-form.
Bosonic string theory is the original version of string theory, developed in the late 1960s. It is so called because it only contains bosons in the spectrum.
In mathematics, a Killing vector field, named after Wilhelm Killing, is a vector field on a Riemannian manifold that preserves the metric. Killing fields are the infinitesimal generators of isometries; that is, flows generated by Killing fields are continuous isometries of the manifold. More simply, the flow generates a symmetry, in the sense that moving each point on an object the same distance in the direction of the Killing vector will not distort distances on the object.
In mathematics, a vertex operator algebra (VOA) is an algebraic structure that plays an important role in two-dimensional conformal field theory and string theory. In addition to physical applications, vertex operator algebras have proven useful in purely mathematical contexts such as monstrous moonshine and the geometric Langlands correspondence.
In mathematics, an affine Lie algebra is an infinite-dimensional Lie algebra that is constructed in a canonical fashion out of a finite-dimensional simple Lie algebra. It is a Kac–Moody algebra for which the generalized Cartan matrix is positive semi-definite and has corank 1. From a purely mathematical point of view, affine Lie algebras are interesting because their representation theory, like representation theory of finite dimensional, semisimple Lie algebras is much better understood than that of general Kac–Moody algebras. As observed by Victor Kac, the character formula for representations of affine Lie algebras implies certain combinatorial identities, the Macdonald identities.
The gauge covariant derivative is a variation of the covariant derivative used in general relativity. If a theory has gauge transformations, it means that some physical properties of certain equations are preserved under those transformations. Likewise, the gauge covariant derivative is the ordinary derivative modified in such a way as to make it behave like a true vector operator, so that equations written using the covariant derivative preserve their physical properties under gauge transformations.
In mathematics, in particular in algebraic geometry and differential geometry, Dolbeault cohomology is an analog of de Rham cohomology for complex manifolds. Let M be a complex manifold. Then the Dolbeault cohomology groups depend on a pair of integers p and q and are realized as a subquotient of the space of complex differential forms of degree (p,q).
In algebraic geometry, a morphism of schemes generalizes a morphism of algebraic varieties just as a scheme generalizes an algebraic variety. It is, by definition, a morphism in the category of schemes.
In mathematical physics the Knizhnik–Zamolodchikov equations, or KZ equations, are linear differential equations satisfied by the correlation functions of two-dimensional conformal field theories associated with an affine Lie algebra at a fixed level. They form a system of complex partial differential equations with regular singular points satisfied by the N-point functions of affine primary fields and can be derived using either the formalism of Lie algebras or that of vertex algebras.
Towards the end of the nineteenth century, Sophus Lie introduced the notion of Lie group in order to study the solutions of ordinary differential equations (ODEs). He showed the following main property: the order of an ordinary differential equation can be reduced by one if it is invariant under one-parameter Lie group of point transformations. This observation unified and extended the available integration techniques. Lie devoted the remainder of his mathematical career to developing these continuous groups that have now an impact on many areas of mathematically based sciences. The applications of Lie groups to differential systems were mainly established by Lie and Emmy Noether, and then advocated by Élie Cartan.
In physics, Liouville field theory is a two-dimensional conformal field theory whose classical equation of motion is a generalization of Liouville's equation.
Projective space plays a central role in algebraic geometry. The aim of this article is to define the notion in terms of abstract algebraic geometry and to describe some basic uses of projective space.
In mathematics, the Kodaira–Spencer map, introduced by Kunihiko Kodaira and Donald C. Spencer, is a map associated to a deformation of a scheme or complex manifold X, taking a tangent space of a point of the deformation space to the first cohomology group of the sheaf of vector fields on X.
In algebraic geometry, a derived scheme is a pair consisting of a topological space X and a sheaf of commutative ring spectra on X such that (1) the pair is a scheme and (2) is a quasi-coherent -module. The notion gives a homotopy-theoretic generalization of a scheme.
In the theory of Lie groups, Lie algebras and their representation theory, a Lie algebra extensione is an enlargement of a given Lie algebra g by another Lie algebra h. Extensions arise in several ways. There is the trivial extension obtained by taking a direct sum of two Lie algebras. Other types are the split extension and the central extension. Extensions may arise naturally, for instance, when forming a Lie algebra from projective group representations. Such a Lie algebra will contain central charges.
A two-dimensional conformal field theory is a quantum field theory on a Euclidean two-dimensional space, that is invariant under local conformal transformations.
In two-dimensional conformal field theory, Virasoro conformal blocks are special functions that serve as building blocks of correlation functions. On a given punctured Riemann surface, Virasoro conformal blocks form a particular basis of the space of solutions of the conformal Ward identites. Zero-point blocks on the torus are characters of representations of the Virasoro algebra; four-point blocks on the sphere reduce to hypergeometric functions in special cases, but are in general much more complicated. In two dimensions as in other dimensions, conformal blocks play an essential role in the conformal bootstrap approach to conformal field theory.