Gauged WZNW Models Research Articles

Gauged non-compact WZNW models, revisited

The purpose of the present paper is to investigate the necessary conditions for unitarity of the spectrum of non-compact gauged WZNW models to some depth. In particular, we would like to investigate the necessity of integer weights and level. We will learn that the problem is very complex and we have not found any simple and general way to formulate the necessary conditions. Instead one must resort to studying the problem almost case by case. The only nearly complete conditions that we will find, is for the case g = su ( n , 1 ) . Furthermore, the horizontal part of the case g = su ( p , q ) is nearly completed as well. In other cases, we will find conditions associated with certain subalgebras and nodes in the Dynkin diagram close to the one corresponding to the non-compact root. In these examples we can give conditions for the horizontal part of the algebra. As a by-product of our investigation we will prove some nice formulae of character identities and explicit branching functions for these representations.

Super-WZNW with reductions to supersymmetric and fermionic integrable models

A systematic construction for an action describing a class of supersymmetric integrable models as well as for pure fermionic theories is discussed in terms of the gauged WZNW model associated to half integer graded affine Kac–Moody algebras. Explicit examples of the N = 1 , 2 super-sinh(sine)-Gordon models are discussed in detail. Pure fermionic theories arises for cosets sl ( p , 1 ) / sl ( p ) ⊗ u ( 1 ) when a maximal kernel condition is fulfilled. The integrability condition for such models is discussed and it is shown that the simplest example when p = 2 leads to the constrained Bukhvostov–Lipatov, Thirring, scalar massive and pseudo-scalar massless Gross–Neveu models.

Exact geometry of a Kerr-Taub-NUT solution of string theory

In this paper we study a solution of heterotic string theory corresponding to a rotating Kerr-Taub-NUT spacetime. It has an exact CFT description as a heterotic coset model, and a Lagrangian formulation as a gauged WZNW model. It is a generalisation of a recently discussed stringy Taub-NUT solution, and is interesting as another laboratory for studying the fate of closed timelike curves and cosmological singularities in string theory. We extend the computation of the exact metric and dilaton to this rotating case, and then discuss some properties of the metric, with particular emphasis on the curvature singularities.

On the Covariant Quantization of Type II Superstrings

In a series of papers Grassi, Policastro, Porrati and van Nieuwenhuizen have introduced a new method to covariantly quantize the GS-superstring by constructing a resolution of the pure spinor constraint of Berkovits' approach. Their latest version is based on a gauged WZNW model and a definition of physical states in terms of relative cohomology groups. We first put the off-shell formulation of the type II version of their ideas into a chirally split form and directly construct the free action of the gauged WZNW model, thus circumventing some complications of the super group manifold approach to type II. Then we discuss the BRST charges that define the relative cohomology and the N=2 superconformal algebra. A surprising result is that nilpotency of the BRST charge requires the introduction of another quartet of ghosts.

The quantum superstring as a WZNW model with N=2 superconformal symmetry

We present a new development in our approach to the covariant quantization of superstrings in 10 dimensions which is based on a gauged WZNW model. To incorporate worldsheet diffeomorphisms we need the quartet of ghosts ( b zz , c z , β zz , γ z ) for topological gravity. The currents of this combined system form an N=2 superconformal algebra. The model has vanishing central charge and contains two anticommuting BRST charges, Q S = Q W +∮ γ z b zz and Q V =∮ c z ( T W zz +(1/2) T top zz )+ γ z ( B W zz +(1/2) B top zz −(1/2) c z F zzz −(1/4) γ z Φ zzz ). Physical states form the cohomology of Q S + Q V +∮ η z , where η z is obtained by the usual fermionization of β zz , γ z , have non-negative grading, and are annihilated by b 0 and β 0. We no longer introduce any ghosts by hand, and the formalism is completely Lorentz covariant.

Asymmetrically gauged WZNW models

AbstractMany interesting exactly solvable backgrounds can be obtained by gauging WZNW models asymmetrically. These include the base of the conifold and the time dependent Nappi‐Witten background in which a 3‐dimensional universe passes through a series of big‐bang big‐crunch singularities. In this short note we review recent results on the conformal field theory description of asymmetric cosets. In particular, we present formulas for their bulk modular invariant partition functions and for a large number of D‐brane boundary states.

GAUGING OF GEOMETRIC ACTIONS AND INTEGRABLE HIERARCHIES OF THE KP TYPE

This work consists of two interrelated parts. First, we derive massive gauge-invariant generalizations of geometric actions on coadjoint orbits of arbitrary (infinite-dimensional) groups G with central extensions, with gauge group H being certain (infinite-dimensional) subgroup of G. We show that there exist generalized "zero-curvature" representations of the pertinent equations of motion on the coadjoint orbit. Second, in the special case of G being the Kac–Moody group, the equations of motion of the underlying gauged WZNW geometric action are identified as additional-symmetry flows of generalized Drinfeld–Sokolov integrable hierarchies based on the loop algebra [Formula: see text]. For [Formula: see text] the latter hierarchies are equivalent to the class c KP R,M of constrained (reduced) KP hierarchies. We describe in some detail the loop algebras of additional (nonisospectral) symmetries of c KP R,M hierarchies. Apart from gauged WZNW models, certain higher-dimensional nonlinear systems such as Davey–Stewartson and N-wave resonant systems are also identified as additional symmetry flows of c KP R,M hierarchies. Also we present the explicit derivation of the general Darboux–Bäcklund solutions of c KP R,M hierarchies preserving their additional (nonisospectral) symmetries, which for R=1 contain among themselves solutions to the gauged SL (M+1)/ U (1)× SL (M) WZNW field equations.

Canonical approach to 2D WZNW model, non-Abelian bosonization, and anomalies

The gauged WZNW model has been derived as an effective action, whose Poisson brackets algebra of the constraints is isomorphic to the commutator algebra of operators in quantized fermionic theory. As a consequence, the Hamiltonian as well as the usual Lagrangian non-Abelian bosonization rules have been obtained for the chiral currents and chiral densities. The expression for the anomaly has been obtained as a function of the Schwinger term, using canonical methods.

Integration of the SL(2, [formula omitted])/U(1) gauged WZNW model with periodic boundary conditions

Gauged WZNW models are integrable conformal field theories. We integrate the classical SL(2, R )/U(1) theory with periodic boundary conditions, which describes closed strings moving in a curved target-space geometry. We calculate its Poisson bracket structure by solving an initial state problem. The results differ from previous field-theoretic calculations due to zero-modes. For a future exact canonical quantization the physical fields are (non-locally) transformed onto canonical free fields.

The Formulae of Kontsevich and Verlinde¶from the Perspective of the Drinfeld Double

A two dimensional gauge theory is canonically associated to every Drinfeld double. For particular doubles, the theory turns out to be e.g. the ordinary Yang-Mills theory, the G/G gauged WZNW model or the Poisson $\sigma$-model that underlies the Kontsevich quantization formula. We calculate the arbitrary genus partition function of the latter. The result is the $q$-deformation of the ordinary Yang-Mills partition function in the sense that the series over the weights is replaced by the same series over the $q$-weights. For $q$ equal to a root of unity the series acquires the affine Weyl symmetry and its truncation to the alcove coincides with the Verlinde formula.

Supersymmetric gauged WZNW models as dressing cosets

The domain of applicability of the Poisson-Lie T-duality is enlarged to include the gauged WZW models.

The Complete Solution of the Classical SL(2,ℝ/U(1) Gauged WZNW Field Theory

We prove that any gauged WZNW model has a Lax pair representation, and give explicitly the general solution of the classical equations of motion of the SL(2,R)/U(1) theory. We calculate the symplectic structure of this solution by solving a differential equation of the Gelfand-Dikii type with initial state conditions at infinity, and transform the canonical physical fields non-locally onto canonical free fields. The results will, finally, be collected in a local B\"acklund transformation. These calculations prepare the theory for an exact canonical quantization.

The Semi-Infinite Cohomology of Affine Lie Algebras

We study in detail the semi-infinite or BRST cohomology of general affine Lie algebras. This cohomology is relevant in the BRST approach to gauged WZNW models. We prove the existence of an infinite sequence of elements in the cohomology for non-zero ghost numbers. This will imply that the BRST approach to topological WZNW model admits many more states than a conventional coset construction. This conclusion also applies to some non-topological models. Our work will also contain results on the structure of Verma modules over affine Lie algebras. In particular, we generalize the results of Verma and Bernstein-Gel'fand-Gel'fand,for finite dimensional Lie algebras, on the structure and multiplicities of Verma modules.

Gauged Knizhnik-Zamolodchikov equation

Correlation functions of gauged WZNW models are shown to satisfy a differential equation, which is a gauge generalization of the Knizhnik-Zamolodchikov equation.

Gauge Dressing of 2D Field Theories

By using the gauge Ward identities, we study correlation functions of gauged WZNW models. We show that the gauge dressing of the correlation functions can be taken into account as a solution of the Knizhnik–Zamolodchikov equation. Our method is analogous to the analysis of the gravitational dressing of 2D field theories.

Two-dimensional instantons with bosonization and physics of adjoint two-dimensional QCD.

We evaluate partition functions ${Z}_{I}$ in topologically nontrivial (instanton) gauge sectors in the bosonized version of the Schwinger model and in a gauged WZNW model corresponding to two-dimensional QCD (QC${\mathrm{D}}_{2}$) with adjoint fermions. We show that the bosonized model is equivalent to the fermion model only if a particular form of the WZNW action with a gauge-invariant integrand is chosen. For the exact correspondence, it is necessary to integrate over the ways the gauge group $\frac{\mathrm{SU}(N)}{{Z}_{N}}$ is embedded into the full $\mathrm{O}({N}^{2}\ensuremath{-}1)$ group for the bosonized matter field. For even $N$, one should also take into account the contributions of both disconnected components in $\mathrm{O}({N}^{2}\ensuremath{-}1)$. In that case, ${Z}_{I}\ensuremath{\propto}{m}^{{n}_{0}}$ for small fermion masses where $2{n}_{0}$ coincides with the number of fermion zero modes in a particular instanton background. The Taylor expansion of $\frac{{Z}_{I}}{{m}^{{n}_{0}}}$ in mass involves only even powers of $m$, as it should. The physics of adjoint QC${\mathrm{D}}_{2}$ is discussed. We argue that, for odd $N$, the discrete chiral symmetry ${Z}_{2}\ensuremath{\bigotimes}{Z}_{2}$ present in the action is broken spontaneously down to ${Z}_{2}$ and the fermion condensate ${〈\overline{\ensuremath{\lambda}}\ensuremath{\lambda}〉}_{0}$ is formed. The system undergoes a first order phase transition at ${T}_{c}=0$ so that the condensate is zero at an arbitrary small temperature. It is not yet quite clear what happens for even $N>~4$.

WZNW models and gauged WZNW models based on a family of solvable Lie algebras

A family of solvable self-dual Lie algebras that are not double extensions of Abelian algebras and, therefore, cannot be obtained through a Wigner contraction, is presented. We construct WZNW and gauged WZNW models based on the first two algebras in this family. We also analyze some general phenomena arising in such models.

A duality-like transformation in WZNW models inspired by dual riemannian globally symmetric spaces

We investigate transformations on the group manifold element and gauged fields on two different kinds of gauged WZNW models and thus obtain a duality-like transformation between chiral- and vector-gauged WZNW models with null gauged subgroups that exactly converts the chiral-gauged WZNW action to vector-gauged WZNW action and vice versa. These duality-like transformations correspond to the duality in Riemannian globally symmetric spaces.

Construction of BRST invariant states in [formula omitted] WZNW models

We study the cohomology arising in the BRST formulation of G H gauged WZNW models. We will derive for a general simple group H with arbitrary representations and levels, conditions for which the cohomology is non-trivial. We show, by introducing a small perturbation due to Jantzen, in the highest weights of the representations, how states in the cohomology, “singlet pairs”, arise from unphysical states, “Kugo-Ojima quartets”, as the perturbation is set to zero. This will enable us to identify and construct states in the cohomology. The ghost numbers that will occur are ± p, with p uniquely determined by the representations of the algebras involved. Our construction is given in terms of the current modes and relies on the explicit form of highest weight null-states given by Malikov, Feigen and Fuchs.

VECTOR-CHIRAL EQUIVALENCE IN NULL GAUGED WZNW THEORY

We consider the standard vector and chiral gauged WZNW models gauged by their maximal null subgroups and show that they can be mapped to each other by a special transformation. We give an explicit expression for the map in the case of the classical Lie groups AN, BN, CN, DN, and note its connection with the duality map for the Riemannian globally symmetric spaces.