Gauged WZW Model Research Articles

On a class of conformal mathcal{E} -models and their chiral Poisson algebras

In this paper, we study conformal points among the class of mathcal{E} -models. The latter are σ-models formulated in terms of a current Poisson algebra, whose Lie-theoretic definition allows for a purely algebraic description of their dynamics and their 1-loop RG-flow. We use these results to formulate a simple algebraic condition on the defining data of such a model which ensures its 1-loop conformal invariance and the decoupling of its observables into two chiral Poisson algebras, describing the classical left- and right-moving fields of the theory. In the case of so-called non-degenerate mathcal{E} -models, these chiral sectors form two current algebras and the model takes the form of a WZW theory once realised as a σ-model. The case of degenerate mathcal{E} -models, in which a subalgebra of the current algebra is gauged, is more involved: the conformal condition yields a wider class of theories, which includes gauged WZW models but also other examples, seemingly different, which however sometimes turn out to be related to gauged WZW models based on other Lie algebras. For this class, we build non-local chiral fields of parafermionic-type as well as higher-spin local ones, forming classical mathcal{W} -algebras. In particular, we find an explicit and efficient algorithm to build these local chiral fields. These results (and their potential generalisations discussed at the end of the paper) open the way for the quantisation of a large class of conformal mathcal{E} -models using the standard operator formalism of two-dimensional CFT.

Webs of integrable theories

We present an intuitive diagrammatic representation of a new class of integrable σ-models. It is shown that to any given diagram corresponds an integrable theory that couples N WZW models with a certain number of each of the following four fundamental integrable models, the PCM, the YB model, both based on a group G, the isotropic σ-model on the symmetric space G/H and the YB model on the symmetric space G/H. To each vertex of a diagram we assign the matrix of one of the aforementioned fundamental integrable theories. Any two vertices may be connected with a number of lines having an orientation and carrying an integer level ki. Each of these lines is associated with an asymmetrically gauged WZW model at an arbitrary level ki. Gauge invariance of the full action is translated to level conservation at the vertices. We also show how to immediately read from the diagrams the corresponding σ-model actions. The most generic of these models depends on at least n2+1 parameters, where n is the total number of vertices/fundamental integrable models. Finally, we discuss the case where the level conservation at the vertices is relaxed and the case where the deformation matrix is not diagonal in the space of integrable models.

Little strings, long strings, and fuzzballs

At high energy densities, fivebranes are populated by a Hagedorn phase of so- called little strings, whose statistical mechanics underlies black fivebrane thermodynamics. A particular limit of this phase yields BTZ black holes in AdS3, leading us to the idea that in this context fuzzballs and highly excited little strings are one and the same. We explore these ideas through an analysis of D-brane probes of fivebrane supertube backgrounds. String theory dynamics on these backgrounds is described by an exactly solvable null- gauged WZW model. We develop the formalism of null gauging on worldsheets with boundaries, and find that D-branes wrapping topology at the bottom of the supertube throat are avatars of the “long string” structure that dominates the thermodynamics of the black hole regime, appearing here as excitations of supertubes lying near but slightly outside the black hole regime.

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.

Integrable asymmetric \u03bb-deformations

We construct integrable deformations of the λ-type for asymmetrically gauged WZW models. This is achieved by a modification of the Sfetsos gauging procedure to account for a possible automorphism that is allowed in G/G models. We verify classical integrability, derive the one-loop beta function for the deformation parameter and give the construction of integrable D-brane configurations in these models. As an application, we detail the case of the λ-deformation of the cigar geometry corresponding to the axial gauged SL(2, R)/U(1) theory at large k. Here we also exhibit a range of both A-type and B-type integrability preserving D-brane configurations.

Effective theories for 2+1 dimensional non-Abelian topological spin liquids

In this work we propose an effective low-energy theory for a large class of 2+1 dimensional non-Abelian topological spin liquids whose edge states are conformal degrees of freedom with central charges corresponding to the coset structure su(2)k ⊕ su(2)k′ /su(2)k+k′ . For particular values of k′ it furnishes the series for unitary minimal and superconformal models. These gapped phases were recently suggested to be obtained from an array of one-dimensional coupled quantum wires. In doing so we provide an explicit relationship between two distinct approaches: quantum wires and Chern-Simons bulk theory. We firstly make a direct connection between the interacting quantum wires and the corresponding conformal field theory at the edges, which turns out to be given in terms of chiral gauged WZW models. Relying on the bulk-edge correspondence we are able to construct the underlying non-Abelian Chern-Simons effective field theory.

S-matrices and quantum group symmetry of k-deformed sigma models

Recently, two kinds of integrable deformations of the string world sheet theory in the gauge/gravity correspondence have been constructed (Delduc et al 2014 Phys. Rev. Lett. 112 051601; Hollowood et al 2014 J. Phys. A: Math. Theor. 47 495402). One class of these, the k deformations associated to the more general q deformations but with a root of unity, has been shown to be related to a particular discrete deformation of the principal chiral models and (semi-)symmetric space sigma models involving a gauged WZW model. We conjecture a form for the exact S-matrices of the bosonic integrable field theories of this type. The S-matrices imply that the theories have a hidden infinite dimensional affine quantum group symmetry. We provide some evidence, via quantum inverse scattering techniques, that the theories do indeed possess the finite-dimensional part of this quantum group symmetry.

The continuum limit of [formula omitted] spin chains

Building on our previous work for a2(2) and a3(2) we explore systematically the continuum limit of gapless aN−1(2) vertex models and spin chains. We find the existence of three possible regimes. Regimes I and II for a2n−1(2) are related with a2n−1(2) Toda, and described by n compact bosons. Regime I for a2n(2) is related with a2n(2) Toda and involves n compact bosons, while regime II is related instead with B(1)(0,n) super Toda, and involves in addition a single Majorana fermion. The most interesting is regime III, where non-compact degrees of freedom appear, generalising the emergence of the Euclidean black hole CFT in the a2(2) case. For a2n(2) we find a continuum limit made of n compact and n non-compact bosons, while for a2n−1(2) we find n compact and n−1 non-compact bosons. We also find deep relations between aN−1(2) in regime III and the gauged WZW models SO(N)/SO(N−1).

Supergravity background of λ-deformed model for AdS2 × S2 supercoset

Starting with the Fˆ/G supercoset model corresponding to the AdSn×Sn superstring one can define the λ-model of arXiv:1409.1538 either as a deformation of the Fˆ/Fˆ gauged WZW model or as an integrable one-parameter generalisation of the non-abelian T-dual of the AdSn×Sn superstring sigma model with respect to the whole supergroup Fˆ. Here we consider the case of n=2 and find the explicit form of the 4d target space background for the λ-model for the PSU(1,1|2)/SO(1,1)×SO(2) supercoset. We show that this background represents a solution of type IIB 10d supergravity compactified on a 6-torus with only metric, dilaton Φ and the RR 5-form (represented by a 2-form F in 4d) being non-trivial. This implies that the λ-model is Weyl invariant at the quantum level and thus defines a consistent superstring sigma model. The supergravity solution we find is different from the one in arXiv:1410.1886 which should correspond to a version of the λ-model where only the bosonic subgroup of Fˆ is gauged. Still, the two solutions have equivalent scaling limit of arXiv:1504.07213 leading to the isometric background for the metric and eΦF which is related to the η-deformed AdS2×S2 sigma model of arXiv:1309.5850. Similar results are expected in the AdS3×S3 and AdS5×S5 cases.

Beta function of k deformed AdS5 × S 5 string theory

We calculate the one loop beta function for the would-be marginal coupling on the world sheet of the k deformed sigma models associated to a quantum group with q=exp(i pi/k). This includes the bosonic principal chiral models and symmetric space sigma models but also the k deformed semi-symmetric space sigma model describing strings in a deformation of AdS_5 x S^5. The world sheet sigma model is a current-current deformation of the gauged WZW model for the supergroup PSU(2,2|4) with level k. In the string theory context the beta function is shown to vanish because of the vanishing of the Killing form of PSU(2,2|4) which is another piece of evidence that the k deformed theories define consistent string theories.

Type IIB supergravity solution for the T-dual of the η-deformed AdS 5 × S 5 superstring

We find an exact type IIB supergravity solution that represents a one-parameter deformation of the T-dual of the AdS 5 × S 5 background (with T-duality applied in all 6 abelian bosonic isometric directions). The non-trivial fields are the metric, dilaton and RR 5-form only. The latter has remarkably simple “undeformed” form when written in terms of a “deformation-rotated” vielbein basis. An unusual feature of this solution is that the dilaton contains a linear dependence on the isometric coordinates of the metric precluding a straightforward reversal of T-duality. If we still formally dualize back, we find exactly the metric, B-field and product of dilaton with RR field strengths as recently extracted from the η-deformed AdS 5 × S 5 superstring action in arXiv:1507.04239 . We also discuss similar solutions for deformed AdS n × S n backgrounds with n = 2, 3. In the η → i limit we demonstrate that all these backgrounds can be interpreted as special limits of gauged WZW models and are also related to (a limit of) the Pohlmeyer-reduced models of the AdS n × S n superstrings.

Defects in Chern-Simons theory, gauged WZW models on the brane, and level-rank duality

We consider Hanany-Witten setups of 3- and 5-branes in type IIB string theory that realize $$ \mathcal{N}=\left(1,0\right) $$ , (2, 0) and (1, 1) gauged WZW models in 1 + 1 dimensions. The gauged WZW models arise as theories residing on the boundary of D3 branes ending on D5 branes. From the point of view of low energy dynamics the D5 branes play the role of half-BPS co-dimension-1 defects (domain walls) in 3d $$ \mathcal{N}=1 $$ or $$ \mathcal{N}=2 $$ Chern-Simons theories. Extending the analysis of previous works on the subject of boundary conditions in (supersymmetric) Chern-Simons theory, we discuss in detail the field theory construction of a large class of Chern-Simons domain wall theories and its embedding in open string dynamics. Finally, we exhibit how standard brane moves that result to 3d Seiberg duality, translate in our setup to a generalized level-rank duality in gauged-WZW models.

Integrable λ-deformations: squashing coset CFTs and AdS5 × S5

We examine integrable λ-deformations of SO(n + 1)/SO(n) coset CFTs and their analytic continuations. We provide an interpretation of the deformation as a squashing of the corresponding coset σ-model’s target space. We realise the λ-deformation for n = 5 case as a solution to supergravity supported by non-vanishing five-form and dilaton. This interpolates between the coset CFT SO(4, 2)/SO(4, 1) × SO(6)/SO(5) constructed as a gauged WZW model and the non-Abelian T-dual of the AdS5 × S5 spacetime.

On integrable deformations of superstring sigma models related to [formula omitted] supercosets

We consider two integrable deformations of 2d sigma models on supercosets associated with AdSn×Sn. The first, the “η-deformation” (based on the Yang–Baxter sigma model), is a one-parameter generalization of the standard superstring action on AdSn×Sn, while the second, the “λ-deformation” (based on the deformed gauged WZW model), is a generalization of the non-abelian T-dual of the AdSn×Sn superstring. We show that the η-deformed model may be obtained from the λ-deformed one by a special scaling limit and analytic continuation in coordinates combined with a particular identification of the parameters of the two models. The relation between the couplings and deformation parameters is consistent with the interpretation of the first model as a real quantum deformation and the second as a root of unity quantum deformation. For the AdS2×S2 case we then explore the effect of this limit on the supergravity background associated with the λ-deformed model. We also suggest that the two models may form a dual Poisson–Lie pair and provide direct evidence for this in the case of the integrable deformations of the coset associated with S2.

Spacetimes for λ-deformations

We examine a recently proposed class of integrable deformations to two-dimensional conformal field theories. These {\lambda}-deformations interpolate between a WZW model and the non-Abelian T-dual of a Principal Chiral Model on a group G or, between a G/H gauged WZW model and the non-Abelian T-dual of the geometric coset G/H. {\lambda}-deformations have been conjectured to represent quantum group q-deformations for the case where the deformation parameter is a root of unity. In this work we show how such deformations can be given an embedding as full string backgrounds whose target spaces satisfy the equations of type-II supergravity. One illustrative example is a deformation of the Sl(2,R)/U(1) black-hole CFT. A further example interpolates between the $\frac{SU(2)\times SU(2)}{SU(2)}\times\frac{SL(2,R)\times SL(2,R)}{SL(2,R)} \times U(1)^4$ gauged WZW model and the non-Abelian T-dual of $AdS_3\times S^3\times T^4$ supported with Ramond flux.

An integrable deformation of the AdS5 × S5 superstring

The S-matrix on the world-sheet theory of the string in AdS has previously been shown to admit a deformation where the symmetry algebra is replaced by the associated quantum group. The case where q is real has been identified as a particular deformation of the Green–Schwarz sigma model. An interpretation of the case with q a root of unity has, until now, been lacking. We show that the Green–Schwarz sigma model admits a discrete deformation which can be viewed as a rather simple deformation of the gauged WZW model, where . The deformation parameter q is then a kth root of unity where k is the level. The deformed theory has the same equations-of-motion as the Green–Schwarz sigma model but has a different symplectic structure. We show that the resulting theory is integrable and has just the right amount of kappa-symmetries that appear as a remnant of the fermionic part of the original gauge symmetry. This points to the existence of a fully consistent deformed string background.

Global Gauge Anomalies in Coset Models of Conformal Field Theory

We study the occurrence of global gauge anomalies in the coset models of two-dimensional conformal field theory that are based on gauged WZW models. A complete classification of the non-anomalous theories for a wide family of gauged rigid adjoint or twisted-adjoint symmetries of WZW models is achieved with the help of Dynkin's classification of Lie subalgebras of simple Lie algebras.

G/G gauged WZW-matter model, Bethe Ansatz for q-boson model and Commutative Frobenius algebra

We investigate the correspondence between two dimensional topological gauge theories and quantum integrable systems discovered by Moore, Nekrasov, Shatashvili. This correspondence means that the hidden quantum integrable structure exists in the topological gauge theories. We showed the correspondence between the G/G gauged WZW model and the phase model in JHEP 11 (2012) 146 (arXiv:1209.3800). In this paper, we study a one-parameter deformation for this correspondence and show that the G/G gauged WZW model coupled to additional matters corresponds to the q-boson model. Furthermore, we investigate this correspondence from a viewpoint of the commutative Frobenius algebra, the axiom of the two dimensional topological quantum field theory.

Integrable interpolations: From exact CFTs to non-Abelian T-duals

We derive two new classes of integrable theories interpolating between exact CFT WZW or gauged WZW models and non-Abelian T-duals of principal chiral models or geometric coset models. They are naturally constructed by gauging symmetries of integrable models. Our analysis implies that non-Abelian T-duality preserves integrability and suggests a novel way to understand the global properties of the corresponding backgrounds.

Pohlmeyer reduction for superstrings in AdS space

The Pohlmeyer reduced equations for strings moving only in the AdS subspace of AdS5 × S5 have been used recently in the study of classical Euclidean minimal surfaces for Wilson loops and some semiclassical three-point correlation functions. We find an action that leads to these reduced superstring equations. For example, for a bosonic string in AdSn such an action contains a Liouville scalar part plus a K/K gauged WZW model for the group K = SO(n − 2) coupled to another term depending on two additional fields transforming as vectors under K. Solving for the latter fields gives a non-Abelian Toda model coupled to the Liouville theory. For n = 5 we generalize this bosonic action to include the S5 contribution and fermionic terms. The corresponding reduced model for the AdS2 × S2 truncation of the full AdS5 × S5 superstring turns out to be equivalent to super Liouville theory. Our construction is based on taking a limit of the previously found reduced theory actions for bosonic strings in AdSn × S1 and superstrings in AdS5 × S5. This new action may be useful as a starting point for possible quantum generalizations or deformations of the classical Pohlmeyer-reduced theory. We give examples of simple extrema of this reduced superstring action which represent strings moving in the AdS5 part of the space. Expanding near these backgrounds we compute the corresponding fluctuation spectra and show that they match the spectra found in the original superstring theory.