Jia Tian, Jue Hou, Bin Chen
We study the integrable asymmetric $λ$-deformations of the $SO(n+1)/SO(n)$ coset models, following the prescription proposed in \cite{AsyLambda}. We construct all corresponding deformed geometries in an inductive way. Remarkably we find a $Z_2$ transformation which maps the asymmetric $λ$--deformed models to the symmetric $λ$--deformed models.
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We study dynamics of scalar fields on a large class of geometries described by integrable sigma models. Although equations of motion are not separable due to absence of isometries and Killing tensors, we completely determine the spectra using algebraic and group-theoretic methods.
A general Lagrangian formulation of integrably deformed G/H-coset models is given. We consider the G/H-coset model in terms of the gauged Wess-Zumino-Witten action and obtain an integrable deformation by adding a potential energy term $Tr(gTg^{-1}\Tb )$, where algebra elements $T, \Tb $ belong to the center of the algebra {\bf h} associated with the subgroup H. We show that the classical equation of motion of the deformed coset model can be identified with the integrability condition of certain linear equations which makes the use of the inverse scattering method possible. Using the linear equ…
Type-II backgrounds from deformed coset CFTs
通过可积变形余代CFT构造含AdS因子的Type-II超引力解，提升特定λ-变形模型，实性条件约束参数，部分解无T-对偶极限。
We study the $SU(2)_k/U(1)$-parafermion model perturbed by its first thermal operator. By formulating the theory in terms of a (perturbed) fermionic coset model we show that the model is equivalent to interacting WZW fields modulo free fields. In this scheme, the order and disorder operators of the $Z_k$ parafermion theory are constructed as gauge invariant composites. We find that the theory presents a duality symmetry that interchanges the roles of the spin and dual spin operators. For two particular values of the coupling constant we find that the theory recovers conformal invariance and th…
The relationship between the nonlinear Schrodinger hierarchy and the parafermion and SL(2,R)/U(1) coset models, analogous to the relationship between the KdV hierarchy and the minimal models, is explained. To do this I first present an in depth study of a series of integrable hierarchies related to NLS, and write down an action from which any of these hierarchies, and the associated second Poisson bracket structures, can be obtained. In quantizing the free part of this action we find many features in common with the bosonized parafermion and SL(2,R)/U(1) models, and particularly it is clear th…
We present the construction of the $λ$-deformation of $AdS_5\times S^5$ superstring from the four dimensional Chern-Simons-type gauge theory. The procedure is applicable to all the semi-symmetric coset models and generalizes the previous construction of $λ$-deformation of the symmetric coset models.
We continue our study of $λ$-deformed $σ$-models by setting up a $1/k$ perturbative expansion around the free field point for cosets, in particular for the $λ$-deformed $SU(2)/U(1)$ coset CFT. We construct an interacting field theory in which all deformation effects are manifestly encoded in the interaction vertices. Using this we reproduce the known $β$-function and the anomalous dimension of the composite operator perturbing away from the conformal point. We introduce the $λ$-dressed parafermions which have an essential Wilson-like phase in their expressions. Subsequently, we compute their a…
We study the fuzzy structure of the general Kaehler coset space G/S\otimes{U(1)}^k deformed by the Fedosov formalism. It is shown that the Killing potentials satisfy the fuzzy algebrae working in the Darboux coordinates.
We obtain classical solutions of $ł$-deformed $\s$-models based on $SL(2,\mathbb{R})/U(1)$ and $SU(2)/U(1)$ coset manifolds. Using two different sets of coordinates, we derive two distinct classes of solutions. The first class is expressed in terms of hyperbolic and trigonometric functions, whereas the second one in terms of elliptic functions. We analyze their properties along with the boundary conditions and discuss string systems that they describe. It turns out that there is an apparent similarity between the solutions of the second class and the motion of a pendulum.
We study the $λ$--deformation of symmetric coset models from the viewpoint of a four dimensional Chern-Simons theory \cite{CY3}. In addition, by applying the "dual" boundary conditions of the ones used in the construction $η$--deformed PCM in the trigonometric description \cite{TRIGYB} we construct a $λ$--deformation type model.
The aim of this work is to present a general theory of coset models G/H in which different left and right actions of H on G are gauged. Our main results include a formula for their modular invariant partition function, the construction of a large set of boundary states and a general description of the corresponding brane geometries. The paper concludes with some explicit applications to the base of the conifold and to the time-dependent Nappi-Witten background.
We consider $λ$-deformed current algebra CFTs at level $k$, interpolating between an exact CFT in the UV and a PCM in the IR. By employing gravitational techniques, we derive the two-loop, in the large $k$ expansion, $β$-function. We find that this is covariant under a remarkable exact symmetry involving the coupling $λ$, the level $k$ and the adjoint quadratic Casimir of the group. Using this symmetry and CFT techniques, we are able to compute the Zamolodchikov metric, the anomalous dimension of the bilinear operator and the Zamolodchikov $C$-function at two-loops in the large $k$ expansion, …
We analyze asymmetric marginal deformations of SU(2)_k and SL(2,R)_k WZW models. These appear in heterotic string backgrounds with non-vanishing Neveu--Schwarz three-forms plus electric or magnetic fields, depending on whether the deformation is elliptic, hyperbolic or parabolic. Asymmetric deformations create new families of exact string vacua. The geometries which are generated in this way, deformed S^3 or AdS_3, include in particular geometric cosets such as S^2, AdS_2 or H_2. Hence, the latter are consistent, exact conformal sigma models, with electric or magnetic backgrounds. We discuss v…
I review some marginal deformations of SU(2) and SL(2,R) Wess-Zumino-Witten models, which are relevant for the investigation of the moduli space of NS5/F1 brane configurations. Particular emphasis is given to the asymmetric deformations, triggered by electric or magnetic fluxes. These exhibit critical values, where the target spaces become exact geometric cosets such as S2 = SU(2)/U(1) or AdS2 = SL(2,R)/U(1)-space. I comment about further generalizations towards the appearance of flag spaces as exact string solutions.
We study the large N anomalous dimensions of operators in a Leigh-Strassler deformation of N=4 super Yang-Mills theory. The operators that we study have a bare dimension of order N (so that the large N limit is not captured by planar diagrams) and are AdS/CFT dual to giant gravitons. The diagonalization of the dilatation operator factorizes into two problems. One of these problems is solved using a double coset ansatz. The second problem is equivalent to a set of decoupled harmonic oscillators.
Antoniadis et al proposed a relation between the Omega-deformation and refined correlation functions of the topological string theory. We investigate the proposal for the deformed conifold geometry from a non-compact Gepner model approach. The topological string theory on the deformed conifold has a dual description in terms of the c=1 non-critical string theory at the self-dual radius, and the Omega-deformation yields the radius deformation. We show that the refined correlation functions computed from the twisted SL(2,R)/U(1) Kazama-Suzuki coset model at level k=1 have direct c=1 non-critical…
By employing CFT techniques, we show how to compute in the context of λ-deformations of current algebras and coset CFTs the exact in the deformation parameters C-function for a wide class of integrable theories that interpolate between a UV and an IR point. We explicitly consider RG flows for integrable deformations of left-right asymmetric current algebras and coset CFTs. In all cases, the derived exact C-functions obey all the properties asserted by Zamolodchikov's c-theorem in two-dimensions.
A novel class of integrable $σ$-models interpolating between exact coset conformal field theories in the IR and hyperbolic spaces in the UV is constructed. We demonstrate the relation to the asymptotic limit of $λ$-deformed models for cosets of non-compact groups. An integrable model interpolating between two spacetimes with cosmological and black hole interpretations and exact conformal field theory descriptions is also provided. In the process of our work, a new zoom-in limit, distinct from the well known non-Abelian T-duality limit, is found.
We construct the solution of type IIB supergravity describing the integrable lambda-deformation of the AdS_3 x S^3 supercoset. While the geometry corresponding to the deformation of the bosonic coset has been found in the past, our background is more natural for studying superstrings, and several interesting features distinguish our solution from its bosonic counterpart. We also report progress towards constructing the lambda-deformation of the AdS_5 x S^5 supercoset.
Kerr-Schild perturbations in General Relativity provide a fruitful way of constructing new exact solutions starting from known ones, elucidating also the structure of the spacetimes. We initiate such a study in the context of string theory and supergravity. Specifically, we explicitly construct Kerr-Schild perturbations of coset CFTs based on low dimensionality orthogonal groups. We show that these give rise to scale, but not Weyl, invariant integrable $σ$-models. We explicitly demonstrate that these models can also be derived from a particular limiting procedure of $λ$-deformed coset CFTs bas…
We show that O(2,2) transformation of SU(2) WZNW model gives rise to marginal deformation of this model by the operator $\int d^2 z J(z)\bar J(\bar z)$ where $J$, $\bar J$ are U(1) currents in the Cartan subalgebra. Generalization of this result to other WZNW theories is discussed. We also consider O(3,3) transformation of the product of an SU(2) WZNW model and a gauged SU(2) WZNW model. The three parameter set of models obtained after the transformation is shown to be the result of first deforming the product of two SU(2) WZNW theories by marginal operators of the form $\sum_{i,j=1}^2 C_{ij} …
We study the notion of strong integrability for classically integrable $λ$-deformed CFTs and coset CFTs. To achieve this goal we employ the Poisson brackets of the spatial Lax matrix which we prove that it assumes the Maillet $r/s$-matrix algebra. As a consequence the system in question are integrable in the strong sense. Furthermore, we show that the derived Maillet $r/s$-matrix algebras can be realized in terms of twist functions, at the poles of which we recover the underlying symmetry algebras.
A procedure is developed for constructing deformations of integrable sigma-models which are themselves classically integrable. When applied to the principal chiral model on any compact Lie group F, one recovers the Yang-Baxter sigma-model introduced a few years ago by C. Klimcik. In the case of the symmetric space sigma-model on F/G we obtain a new one-parameter family of integrable sigma-models. The actions of these models correspond to a deformation of the target space geometry and include a torsion term. An interesting feature of the construction is the q-deformation of the symmetry corresp…
We study some properties of kink solutions of the model with non-polynomial potential obtained by deforming the well-known $φ^6$ field model. We consider the excitation spectrum of the kink. We also discuss the properties of the `kink+antikink' system as a whole that are not inherent to a solitary kink or antikink.
We consider two integrable deformations of 2d sigma models on supercosets associated with AdS_n x S^n. The first, the "eta-deformation" (based on the Yang-Baxter sigma model), is a one-parameter generalization of the standard superstring action on AdS_n x S^n, while the second, the "lambda-deformation" (based on the deformed gauged WZW model), is a generalization of the non-abelian T-dual of the AdS_n x S^n superstring. We show that the eta-deformed model may be obtained from the lambda-deformed one by a special scaling limit and analytic continuation in coordinates combined with a particular …
We calculate the all-loop anomalous dimensions of current operators in $λ$-deformed $σ$-models. For the isotropic integrable deformation and for a semi-simple group $G$ we compute the anomalous dimensions using two different methods. In the first we use the all-loop effective action and in the second we employ perturbation theory along with the Callan-Symanzik equation and in conjunction with a duality-type symmetry shared by these models. Furthermore, using CFT techniques we compute the all-loop anomalous dimensions of bilinear currents for the isotropic deformation case and a general $G$. Fi…
$T\overline{T}$ Deformation of $\mathcal{N}=(1,1)$ Off-Shell Supersymmetry and Partially Broken Supersymmetry
本文构造了二维自由N=(1,1)超对称模型的TTbar形变超作用量，证明离壳超对称在形变下被修改，并将其解释为N=(2,2)部分破缺至N=(1,1)的Goldstone超场有效作用量，同时通过陪集超空间和约束超场方法重现该结果。
Integrable $λ$-deformed $σ$-models are characterized by an underlying current algebra/coset model CFT deformed, at the infinitesimal level, by current/parafermion bilinears. We promote the deformation parameters to dynamical functions of time introduced as an extra coordinate. It is conceivable that by appropriately constraining them, the beta-functions vanish and consequently the $σ$-model stays conformal. Remarkably, we explicitly materialize this scenario in several cases having a single and even multiple deformation parameters. These generically obey a system of non-linear second-order ord…
We compute the anomalous dimension of the single current operator in the case of single and doubly deformed asymmetric $λ$-models with a general deformation matrix. Our method uses the underlying geometry of the coupling space, as well as an auxiliary group interaction, which completely decouples from the asymmetric model in a specific limit, consistent with the Renormalization Group flow. Our results are valid to all orders in the deformation parameters and leading order to the levels of the underlying current algebras. We specialize our general result to several models of particular interest…
We compute the all-loop anomalous dimensions of current and primary field operators in deformed current algebra theories based on a general semi-simple group, but with different (large) levels for the left and right sectors. These theories, unlike their equal level counterparts, possess a new non-trivial fixed point in the IR. By computing the exact in $λ$ two- and three-point functions for these operators we deduce their OPEs and their equal-time commutators. Using these we argue on the nature of the CFT at the IR fixed point. The associated to the currents Poisson brackets are a two-paramete…
Integrable models based on non-semi-simple groups and plane wave target spacetimes
本文首次构造基于非半单群 E_2^c 的可积 λ-变形 WZW 模型，得到两个不等价但对称代数相同的 Nappi-Witten 平面波变形，并进一步从陪集 CFT 的 Penrose 极限构造平面波，同时简要讨论 Yang-Baxter σ-模型收缩。
We construct a new class of integrable $σ$-models based on current algebra theories for a general semisimple group $G$ by utilizing a left-right asymmetric gauging. Their action can be thought of as the all-loop effective action of two independent WZW models for $G$ both at level $k$, perturbed by current bilinears mixing the different WZW models. A non-perturbative symmetry in the couplings parametric space is revealed. We perform the Hamiltonian analysis of the action and demonstrate integrability in several cases. We extend our construction to deformations of $G/H$ CFTs and show integrabili…
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