Non-abelian Toda Theories Research Articles

Supersymmetric Liouville theory in AdS2 and AdS/CFT

In a series of recent papers, a special kind of AdS2/CFT1 duality was observed: the boundary correlators of elementary fields that appear in the Lagrangian of a 2d conformal theory in rigid AdS2 background are the same as the correlators of the corresponding primary operators in the chiral half of that 2d CFT in flat space restricted to the real line. The examples considered were: (i) the Liouville theory where the operator dual to the Liouville scalar in AdS2 is the stress tensor; (ii) the abelian Toda theory where the operators dual to the Toda scalars are the mathcal{W} -algebra generators; (iii) the non-abelian Toda theory where the Liouville field is dual to the stress tensor while the extra gauged WZW theory scalars are dual to non-abelian parafermionic operators. By direct Witten diagram com- putations in AdS2 one can check that the structure of the boundary correlators is indeed consistent with the Virasoro (or higher) symmetry. Here we consider a supersymmetric generalization: the mathcal{N} = 1 superconformal Liouville theory in AdS2. We start with the super Liouville theory coupled to 2d supergravity and show that a consistent restriction to rigid AdS2 background requires a non-zero value of the supergravity auxiliary field and thus a modification of the Liouville potential from its familiar flat-space form. We show that the Liouville scalar and its fermionic partner are dual to the chiral half of the stress tensor and the supercurrent of the super Liouville theory on the plane. We perform tests supporting the duality by explicitly computing AdS2 Witten diagrams with bosonic and fermionic loops.

Non-abelian Toda theory on AdS2 and {mathrm{AdS}}_2/{mathrm{CFT}}_2^{raisebox{1ex}{1}!left/ !raisebox{-1ex}{2}right.} duality

It was recently observed that boundary correlators of the elementary scalar field of the Liouville theory on AdS2 background are the same (up to a non-trivial proportionality coefficient) as the correlators of the chiral stress tensor of the Liouville CFT on the complex plane restricted to the real line. The same relation generalizes to the conformal abelian Toda theory: boundary correlators of Toda scalars on AdS2 are directly related to the correlation functions of the chiral mathcal{W} -symmetry generators in the Toda CFT and thus are essentially controlled by the underlying infinite-dimensional symmetry. These may be viewed as examples of AdS2/CFT1 duality where the CFT1 is the chiral half of a 2d CFT; we shall to this as {mathrm{AdS}}_2/{mathrm{CFT}}_2^{raisebox{1ex}{1}!left/ !raisebox{-1ex}{2}right.} . In this paper we demonstrate that this duality applies also to the non-abelian Toda theory containing a Liouville scalar coupled to a 2d σ-model originating from the SL(2, ℝ)/U(1) gauged WZW model. Here the Liouville scalar is again dual to the chiral stress tensor T while the other two scalars are dual to the parafermionic operators V± of the non-abelian Toda CFT. We explicitly check the duality at the next-to-leading order in the large central charge expansion by matching the chiral CFT correlators of (T, V+, V−) (computed using a free field representation) with the boundary correlators of the three Toda scalars given by the tree-level and one-loop Witten diagrams in AdS2.

Nonabelian Toda Theories from Parafermionic Reductions of the WZW Model

We investigate a class of conformal nonabelian-Toda models representing noncompact SL(2, R)/U(1) parafermions (PF) interacting with specific abelian Toda theories and having a global U(1) symmetry. A systematic derivation of the conserved currents, their algebras, and the exact solution of these models are presented. An important property of this class of models is the affine SL(2, R)q algebra spanned by charges of the chiral and antichiral nonlocal currents and the U(1) charge. The classical (Poisson brackets) algebras of symmetries VGn of these models appear to be of mixed PF-WGn type. They contain together with the local quadratic terms specific for the Wn-algebras the nonlocal terms similar to the ones of the classical PF-algebra. The renormalization of the spins of the nonlocal currents is the main new feature of the quantum VAn-algebras. The quantum VA2-algebra and its degenerate representations are studied in detail

On the classification of real forms of non-Abelian Toda theories and W-algebras

We consider conformal non-Abelian Toda theories obtained by Hamiltonian reduction from Wess-Zumino-Witten models based on general real Lie groups. We study in detail the possible choices of reality conditions which can be imposed on the WZW or Toda fields and prove correspondences with sl(2, R) embeddings into real Lie algebras and with the possible real forms of the associated W -algebras. We devise a method for finding all real embeddings which can be obtained from a given embeddings of sl(2, C) into a complex Lie algebra. We then apply this to give a complete classification of real embeddings which are principal in some simple regular subalgebra of a classical Lie algebra.

SU(2, R) q symmetries of Non-Abelian Toda theories

The classical and quantum algebras of a class of conformal NA-Toda models are studied. It is shown that the SL(2, R) q Poisson brackets algebra generated by certain chiral and antichiral charges of the nonlocal currents and the global U(1) charge appears as an algebra of the symmetries of these models.

Exact solution of the SL(2, [formula omitted])/U(1) WZNW model

It is shown that the exact solution of the gauged SL(2, R )/U(1) Wess-Zumino-Novikov-Witten (WZNW) black hole model is asymptotically related to the solution of an integrable non-abelian Toda theory. A second-order differential equation characterizes the structure of the theory, and its fundamental solutions describe the general solution of the WZNW model as well. As in Liouville theory, physical and free fields are non-locally related transforming the energy-momentum tensor into a canonical free field form. The black hole model is expected to be canonically quantizable.

Gravitational dressing of massive soliton theories

The massive soliton theories describe integrable perturbations of WZW cosets as generalized multi-component sine-Gordon models. We study their coupling to 2-dim gravity in the conformal gauge and show that the resulting models can be interpreted as conformal non-Abelian Toda theories when a certain algebraic condition is satisfied. These models, however, do not provide quantum mechanically consistent string backgrounds in the case the underlying WZW constraints are first solved classically.

Real forms of non-abelian Toda theories and their W-algebras

We consider real forms of Lie Algebras and embeddings of sl(2) which are consistent with the construction of integrable models via Hamiltonian reduction. In other words: we examine possible non-standard reality conditions for non-abelian Toda theories. We point out in particular that the usual restriction to the maximally non-compact form of the algebra is unnecessary, and we show how relaxing this condition can lead to new real forms of the resulting w-algebras. Previous results for abelian Toda theories are recovered as special cases. The construction can be extended straightforwardly to deal with osp(1|2) embeddings in Lie superalgebras. Two examples are worked out in detail, one based on a bosonic Lie algebra, the other based on a Lie superalgebra leading to an action which realizes the N = 4 superconformal algebra.

Consistent string backgrounds and completely integrable 2D field theories

After reviewing the β-function equations for consistent string backgrounds in the σ-model approach, including metric and antisymmetric tensor, dilaton and tachyon potential, we apply this formalism to WZW models. We particularly emphasize the case where the WZW model is perturbed by an integrable marginal tachyon potential operator leading to the non-abelian Toda theories. Already in the simplest such theory, there is a large non-linear and non-local chiral algebra that extends the Virasoro algebra. This theory is shown to have two formulations, one being a classical reduction of the other. Only the non-reduced theory is shown to satisfy the β-function equations.

Higher grading generalisations of the Toda systems

In the present paper we obtain some integrable generalisations of the Toda system generated by flat connection forms taking values in higher Z-grading subspaces of a simple Lie algebra, and construct their general solutions. One may think of our systems as describing some new fields of the matter type coupled to the standard Toda systems. This is of special interest in nonabelian Toda theories where the latter involve black hole target space metrics. We also give a derivation of our conformal system on the base of the Hamiltonian reduction of the WZNW model; and discuss a relation between abelian and nonabelian systems generated by a gauge transformation that maps the first grading description to the second. The latter involves grades larger than one.

Non-abelian Toda theory: A completely integrable model for strings on a black hole background

The present paper studies a completely integrable conformally invariant model in 1 + 1 dimensions that corresponds to string propagation on the two-dimensional black hole background (semi-infinite cigar). Besides the two space-time string fields there is a third (internal) field with a very specific Liouville-type interaction leading to the complete integrability. This system is known as the non-abelian Toda theory. I give the general explicit classical solution. It realizes a rather involved transformation expressing the interacting string fields in terms of (three plus three) functions ϕ j(u) and ϕ j(v) of one light-cone variable only. The latter are shown to lead to standard harmonic oscillator (free-field) Poisson brackets thus paving the way towards quantization. There are three left-moving and three right-moving conserved quantities. The right(left)-moving conserved quantities form a new closed non-linear, non-local Poisson bracket algebra. This algebra is a Virasoro algebra extended by two conformal dimension-two primaries.

Hamiltonian reductions of self-dual Yang-Mills theory

The Hamiltonian reductions of self-dual Yang-Mills theory associated with left-right dual regular constant constraints are analysed. It is shown that the reduced theory is precisely the four dimensional analogue of the nonabelian Toda theory. In the special case of principal gradation with constraints of the first degree this theory becomes the usual Toda theory in four dimensions. The effective action and the linear systems are all obtained from the reduction procedure.

Black holes from non-abelian Toda theories

Non-abelian Toda theories are shown to provide exactly solvable conformal systems in the presence of a black hole. They correspond to gauged WZNW models where the gauge group is nilpotent, and are thus basically different from the ones currently considered, following Witten. The non-abelian Toda potential gives a cosmological term which may be integrated exactly at the classical level.

Towards a classification of W algebras arising from non-abelian Toda theories

We establish a correspondence between (integral and half-integral) gradings of a simple Lie algebra G specifying Toda models and SL(2)×U(1) subalgebras of G . The U(1)factor is submitted to special conditions which insure in each case that the theory is not degenerate. The conformal spin content of the W algebra underlying each Toda model can easily be computed in this approach.

Higher order corrections for quasiparticle RPA: Liang Zhao and Andrew Sustich. National Superconducting Cyclotron Laboratory and Department of Physics and Astronomy, Michigan State University, East Lansing, Michigan 48824

We consider real forms of Lie Algebras and embeddings of sl(2) which are consistent with the construction of integrable models via Hamiltonian reduction. In other words: we examine possible non-standard reality conditions for non-abelian Toda theories. We point out in particular that the usual restriction to the maximally non-compact form of the algebra is unnecessary, and we show how relaxing this condition can lead to new real forms of the resulting w-algebras. Previous results for abelian Toda theories are recovered as special cases. The construction can be extended straightforwardly to deal with osp(1|2) embeddings in Lie superalgebras. Two examples are worked out in detail, one based on a bosonic Lie algebra, the other based on a Lie superalgebra leading to an action which realizes the N = 4 superconformal algebra.

Anomaly cancellation for super- [formula omitted] -gravity

We generalise the description of minimal superconformal models coupled to supergravity, due to Distler, Hlousek and Kawaii, to super- W -gravity. When the chiral algebra is the generalisation of the W-algebra associated to any contragredient Lie superalgebra the total central charge vanishes as a result of Lie superalgebra identities. When the algebra has only fermionic simple roots there is N = 1 superconformal invariance and for this case we describe the Lax operators and construct gravitationally dressed primary superfields of weight zero. We also prove the anomaly cancellation associated with the generalised non-abelian Toda theories.