We study the worldsheet scattering theory of the $η$ deformation of the AdS$_\mathsf{5} \times $S$^\mathsf{5}$ superstring corresponding to the purely fermionic Dynkin diagram. This theory is a Weyl-invariant integrable deformation of the AdS$_\mathsf{5} \times $S$^\mathsf{5}$ superstring, with trigonometric quantum-deformed symmetry. We compute the two-body worldsheet S matrix of this string in the light-cone gauge at tree level to quadratic order in fermions. The result factorizes into two elementary blocks, and solves the classical Yang-Baxter equation. We also determine the corresponding exact factorized S matrix, and show that its perturbative expansion matches our tree-level results, once we correctly identify the deformed light-cone symmetry algebra of the string. Finally, we briefly revisit the computation of the corresponding S matrix for the $η$ deformation based on the distinguished Dynkin diagram, finding a tree-level S matrix that factorizes and solves the classical Yang-Baxter equation, in contrast to previous results.
We compute perturbative worldsheet S-matrix of beta-deformed AdS/CFT in the strong and weak `t Hooft coupling limit to compare with exact S-matrix. For the purpose we take near BMN limit of TsT-transformed AdS_5 x S^5 with the twisted boundary condition and compute the S-matrix on worldsheet using light-cone gauge fixed Lagrangian. For the weak coupling side, we compute the S-matrix in the SU(3) sector by applying coordinate Bethe ansatz method to the one-loop dilatation operator obtained from the deformed super Yang-Mills theory. These analysis support the conjectured exact S-matrix in the le…
Liouville strings in AdS$_3$: the worldsheet story
Center for Cosmology and Particle Physics Department of Physics, New York University（纽约大学宇宙学与粒子物理中心物理系）；Departamento de Física, Universidad de Buenos Aires, FCEN-UBA Instituto de Física de Buenos Aires, IFIBA-CONICET（布宜诺斯艾利斯大学物理系、理学院、布宜诺斯艾利斯物理研究所）
Dipartimento di Fisica, Università degli Studi di Milano–Bicocca（米兰比可卡大学物理系）；INFN, Sezione di Milano–Bicocca（意大利国家核物理研究所米兰比可卡分部）；School of Mathematics, University of Birmingham（伯明翰大学数学学院）；INFN, Sezione di Padova（意大利国家核物理研究所帕多瓦分部）
研究超对称 $\lambda$-形变 AdS3 × S3 × T4 超弦的微扰玻色子树级光锥规范世界片 S 矩阵，发现非弹性散射过程抵消保持可积性，但 $\lambda \to 1$ 时 S 矩阵病态，表明该极限未捕捉到 T-对偶理论的完整世界片动力学。
We investigate the worldsheet S matrix of string theory in $η$-deformed $AdS_5\times S^5$. By computing the six-point tree-level S matrix we explicitly show that there is no particle production at this level, as required by the classical integrability of the theory. At one and two loops we show that integrability requires that the classical two-particle states be redefined in a non-local and $η$-dependent way. This is a significant departure from the undeformed theory which is probably related to the quantum group symmetry of the worldsheet theory. We use generalized unitarity to carry out the…
We consider various integrable two-parameter deformations of the $AdS_3 \times S^3 \times T^4$ superstring with quantum group symmetry. Working on the string worldsheet in light-cone gauge and to quadratic order in fermions, we obtain their common massive tree-level two-body S matrix, which matches the expansion of the conjectured exact $q$-deformed S matrix. We then analyze the behavior of the exact S matrix under mirror transformation -- a double Wick rotation on the worldsheet -- and find that it satisfies a mirror duality relation analogous to the distinguished $q$-deformed $AdS_5 \times S…
We investigate the scattering matrix in mass-deformed N>=4 Chern-Simons models including as special cases the BLG and ABJM theories of multiple M2 branes. Curiously the structure of this scattering matrix in three spacetime dimensions is equivalent to (a) the two-dimensional worldsheet matrix found in the context of AdS/CFT integrability and (b) the R-matrix of the one-dimensional Hubbard model. The underlying reason is that all three models are based on an extension of the psu(2|2) superalgebra which constrains the matrix completely. We also compute scattering amplitudes in one-loop field the…
We study $T\bar T$ deformations of 2d CFTs with periodic boundary conditions. We relate these systems to string models on $\mathbb{R}\times {S}^1\times{\cal M}$, where $\cal M$ is the target space of a 2d CFT. The string model in the light cone gauge is identified with the corresponding 2d CFT and in the static gauge it reproduces its $T\bar T$ deformed system. This relates the deformed system and the initial one by a worldsheet coordinate transformation, which becomes a time dependent canonical map in the Hamiltonian treatment. The deformed Hamiltonian defines the string energy and we express…
We study a very special class of $T\bar{J}$ deformations of conformal field theories in two dimensions. While the deformations break the Lorentz symmetry, they preserve the twisted Lorentz symmetry. The resulting theory has right-moving Virasoro as well as left-moving translation and left-moving (chiral) scale symmetry without left-moving special conformal symmetry (nor left-moving Virasoro symmetry). As in the original $T\bar{J}$ deformations, they may be regarded as an operator dependent non-local change of coordinates. We show concrete examples based on worldsheet string theory and discuss …
We investigate unoriented strings and superstrings in two dimensions and their dual matrix quantum mechanics. Most of the models we study have a tachyon tadpole coming from the RP^2 worldsheet which needs to be cancelled by a renormalization of the worldsheet theory. We find evidence that the dual matrix models describe the renormalized theory. The singlet sector of the matrix models is integrable and can be formulated in terms of fermions moving in an external potential and interacting via the Calogero-Moser potential. We show that in the double-scaling limit the latter system exhibits partic…
Two distinct $η$-deformations of strings on AdS$_5\times$S$^5$ can be defined; both amount to integrable quantum deformations of the string non-linear sigma model, but only one is itself a superstring background. In this paper we compare their conjectured all-loop worldsheet S matrices and derive the corresponding Bethe equations. We find that, while the S matrices are apparently different, they lead to the same Bethe equations. Moreover, in either case the eigenvalues of the transfer matrix, which encode the conserved charges of each system, also coincide. We conclude that the integrable stru…
As was observed by Grigoriev and Tseytlin, the Pohlmeyer-reduced AdS_2 x S^2 superstring theory possesses N=(2,2) worldsheet supersymmetry. We show, at the classical level, that the AdS_3 x S^3 and AdS_5 x S^5 superstring theories in the Pohlmeyer-reduced form reveal hidden N=(4,4) and N=(8,8) worldsheet supersymmetries. Our consideration is based on the modified mass-deformed gauged WZW action for the superstring equations. We present the explicit form of the supersymmetry transformations for both the off-shell action and the superstring equations. The characteristic feature of these transfor…
String theory on NS-NS AdS_3 x S^3 admits an exactly marginal deformation which breaks the SL(2,R)_R x SL(2,R)_L isometry of AdS_3 down to SL(2,R)_R x U(1)_L. The holographic dual is an exotic and only partially understood type of two-dimensional CFT with a reduced unbroken global conformal symmetry group. In this paper we study the deformed theory on the string worldsheet. It is found to be related by a spectral flow which is nonlocal in spacetime to the undeformed worldsheet theory. An exact formula for the spectrum of massive strings is presented.
We revisit $T\bar T$ deformations of $d=2$ theories with fermions with a view toward the quantization. As a simple illustration, we compute the deformed Dirac bracket for a Majorana doublet and confirm the known eigenvalue flows perturbatively. We mostly consider those $T\bar T$ theories that can be reconstructed from string-like theories upon integrating out the worldsheet metric. After a quick overview of how this works when we add NSR-like or GS-like fermions, we obtain a known non-supersymmetric $T\bar T$ deformation of a $\cN=(1,1)$ theory from the latter, based on the Noether energy-mome…
We study a toy model of the Kerr/CFT correspondence using string theory on AdS$_3 \times S^3$. We propose a single trace irrelevant deformation of the dual CFT generated by a vertex operator with spacetime dimensions (2,1). This operator shares the same quantum numbers as the integrable $T\bar{J}$ deformation of two-dimensional CFTs where $\bar{J}$ is a chiral $U(1)$ current. We show that the deformation is marginal on the worldsheet and that the target spacetime is deformed to null warped AdS$_3$ upon dimensional reduction. We also calculate the spectrum of the deformed theory on the cylinder…
Effects of mass deformations on 1/2 Bogomol'nyi-Prasad-Sommerfield (BPS) non-Abelian vortices are studied in 4d N=2 supersymmetric U(1) \times SO(2n) and U(1) \times USp(2n) gauge theories, with Nf=2n quark multiplets. The 2d N=(2,2) effective worldsheet sigma models on the Hermitian symmetric spaces SO(2n)/U(n) and USp(2n)/U(n) found recently which describe the low-energy excitations of the orientational moduli of the vortices, are generalized to the respective massive sigma models. The continuous vortex moduli spaces are replaced by a finite number (2^{n-1} or 2^{n}) of vortex solutions. The…
In this paper we prove the equivalence among (i) the weakly coupled worldsheet string theory described by the coset sigma model $\frac{SL(2,\mathbb{R})_k\times U(1)}{U(1)}\times S^3 \times T^4$ with $SL(2,\mathbb{R})$ WZW level $k\geq 2$, (ii) the full near horizon theory of the NS5 branes with $k$ NS5 branes wrapping $T^4\times S^1$, $p\gg1$ F1 strings wrapping $S^1$ and $n$ units of momentum along the $S^1$ and (iii) the single trace $T\bar{T}$ deformation of string theory in $AdS_3\times S^3\times T^4$. As a check we compute the spectrum of the spacetime theory by performing BRST quantizati…
Sine-square deformation, a recently found modulation of the coupling strength in certain statistical models, is discussed in the context of two-dimensional conformal field theories, with particular attention to open/closed string duality. This deformation is shown to be non-trivial and leads to a divergence in the worldsheet metric. The structure of the vacua of the deformed theory is also investigated. The approach advocated here may provide an understanding of string duality through the worldsheet dynamics.
We study thermodynamic aspects of a tractable toy model of holography for extremal Kerr black holes proposed in [arXiv:1806.10127]. On the gravity side, the theory can be described by the worldsheet action of string theory on a warped AdS$_3$ background supported by NS-NS flux. Once we turn on temperature, the deformed background is described by a black string solution of type IIB supergravity that features a locally warped AdS$_3$ factor. The dual field theory is conjectured to be a single-trace version of a $J\bar{T}$-deformed CFT at finite temperature. As evidence for the correspondence we …
We construct highest-weight modules and a Yangian extension of the centrally extended sl(1|1)^2 superalgebra, that is a symmetry of the worldsheet scattering associated with the AdS3/CFT3 duality. We demonstrate that the R-matrix intertwining atypical modules is of a trigonometric type. We also consider a quantum deformation of this superalgebra, its modules, and obtain a quantum affine extension of the Drinfeld-Jimbo type that describes a deformed worldsheet scattering.
In this PhD thesis we review some aspects of integrable models related to string backgrounds or their deformations. In the first part we develop methods to obtain exact results in the AdS3/CFT2 correspondence. We consider the AdS_3 x S^3 x T^4 background with pure Ramond-Ramond flux and we find the all-loop worldsheet S-matrix by exploiting the symmetries of the model in light-cone gauge. As we naturally include the massless modes on the worldsheet, we derive the full set of Bethe-Yang equations. In the massive sector we give also a spin-chain description and we write down solutions compatible…
We investigate a deformed matrix model of type 0A theory related to supersymmetric Witten's black hole in two-dimensions, generalization of bosonic model suggested by Kazakov et. al. We find a free field realization of the partition function of the matrix model, which includes Ramond-Ramond perturbations in the type 0A theory. In a simple case, the partition function is factorized into two determinants, which are given by $τ$ function of an integrable system. We work out the genus expansion of the partition function. Holographic relation with the supersymmetric Witten's black hole is checked b…
String theory on AdS$_3$ with NS-NS fluxes admits a solvable irrelevant deformation which is close to the $T\bar{T}$ deformation of the dual CFT$_2$. This consists of deforming the worldsheet action, namely the action of the $SL(2,\mathbb{R})$ WZW model, by adding to it the operator $J^-\bar{J}^-$, constructed with two Kac-Moody currents. The geometrical interpretation of the resulting theory is that of strings on a conformally flat background that interpolates between AdS$_3$ in the IR and a flat linear dilaton spacetime with Hagedorn spectrum in the UV, having passed through a transition reg…
We consider the worldsheet boundary scattering and the corresponding boundary algebras for the Z=0 giant graviton and the Z=0 D7-brane in the AdS/CFT correspondence. We consider two approaches to the boundary scattering, the usual one governed by the (generalized) twisted Yangians and the q-deformed model of these boundaries governed by the quantum affine coideal subalgebras. We show that the q-deformed approach leads to boundary algebras that are of a more compact form than the corresponding twisted Yangians, and thus are favourable to use for explicit calculations. We obtain the q-deformed r…
We consider non-Abelian BPS-saturated flux tubes (strings) in N=2 supersymmetric QCD deformed by superpotential terms of a special type breaking N=2 supersymmetry down to N=1. Previously it was believed that worldsheet supersymmetry is "accidentally" enhanced due to the facts that N =(1,1) SUSY is automatically elevated up to N=(2,2) on CP(N-1) and, at the same time, there are no N=(0,2) generalizations of the bosonic CP(N-1) model. Edalati and Tong noted that the target space is in fact CP(N-1) x C rather than CP(N-1). This allowed them to suggest a "heterotic" N=(0,2) sigma model, with the C…
We continue the study of a recently proposed solvable irrelevant deformation of an AdS$_3$/CFT$_2$ correspondence that leads in the UV to a theory with Hagedorn spectrum. This can be thought of as a single trace analog of the $T\bar{T}$-deformation of the dual CFT$_2$. Here we focus on the deformed worldsheet theory in presence of a conformal boundary. First, we compute the expectation value of a bulk primary operator on the disc geometry. We give a closed expression for such observable, from which we obtain the anomalous conformal dimension induced by the deformation. We compare the result wi…
We continue explorations of non-Abelian strings, focusing on the solution of a heterotic deformation of the CP(N-1) model with an extra right-handed fermion field and N=(0,2) supersymmetry. This model emerges as a low-energy theory on the worldsheet of the BPS-saturated flux tubes (strings) in N=2 supersymmetric QCD deformed by a superpotential of a special type breaking N=2 supersymmetry down to N=1. Using large-N expansion we solve this model to the leading order in 1/N. Our solution exhibits spontaneous supersymmetry breaking for all values of the deformation parameter. We identify the Gold…
$AdS_3$ string theory in the stringy regime $k=(R_{AdS}/\ell_{str})^2 < 1$ provides a laboratory for the study of holography in which both sides of AdS/CFT duality are under fairly good control. Worldsheet string theory is solvable, and for closed strings the dual spacetime CFT is a deformation of a symmetric product orbifold. Here we extend this construction to include open strings by adding a probe D-string, described semi-classically by an $AdS_2$ D-brane in $AdS_3$. The dual defect or boundary conformal field theory (BCFT) is again a deformed symmetric product, which now describes the Fock…
Knizhnik-Zamolodchikov equations and the analytic structure of strings in AdS$_3$
Instituto de Astronomía y Física del Espacio (IAFE), CONICET - Universidad de Buenos Aires, Facultad de Ciencias Exactas y Naturales（空间天体物理学与物理研究所，阿根廷国家科学研究委员会-布宜诺斯艾利斯大学，理学院）；Universidad Abierta Interamericana, Facultad de Arquitectura（泛美开放大学，建筑学院）；Instituto de Física de La Plata - CONICET（拉普拉塔物理研究所，阿根廷国家科学研究委员会）；Departamento de Física, Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires（布宜诺斯艾利斯大学理学院物理系）；Instituto de Investigaciones Matemáticas Luis A. Santaló (IMAS), CONICET - Universidad de Buenos Aires, Facultad de Ciencias Exactas y Naturales（路易斯·A·桑塔洛数学研究所，阿根廷国家科学研究委员会-布宜诺斯艾利斯大学，理学院）
We propose a new $AdS_3/CFT_2$ duality, in which the bulk string theory has a target spacetime $AdS_3$ times a squashed three-sphere $S^3_\flat$, and the dual $CFT_2$ is a symmetric product of sigma models on $R_ϕ\times S^3_\flat$, deformed by a $ϕ$-dependent $Z_2$ twist operator. The duality maps the asymptotic region of $AdS_3$ to the region $ϕ\to\infty$, where the twist interaction in the $CFT_2$ turns off. The $AdS_3$ backgrounds in question have $R_{AdS}<\ell_s$, and so lie on the string side of the string/black hole correspondence transition. As a consequence, the high energy density of …
Doing a double Wick rotation in the worldsheet theory of the light cone $\mathrm{AdS}_5 \times \mathrm{S}^5$ superstring results in an inequivalent, so-called mirror theory that plays a central role in the field of integrability in AdS/CFT. We show that this mirror theory can be interpreted as the light cone theory of a free string on a different background. This background is related to $\mathrm{dS}_5 \times \mathrm{H}^5$ by a double T duality, and has hidden supersymmetry. The geometry can also be extracted from an integrable deformation of the $\mathrm{AdS}_5 \times \mathrm{S}^5$ sigma mode…
We study a Jordanian deformation of the $AdS_5 \times S^5$ superstring that preserves 12 superisometries. It is an example of homogeneous Yang-Baxter deformations, a class that generalises TsT deformations to the non-abelian case. Many of the attractive features of TsT carry over to this more general class, from the possibility of generating new supergravity solutions to the preservation of worldsheet integrability. In this paper, we exploit the fact that the deformed $σ$-model with periodic boundary conditions can be reformulated as an undeformed one with twisted boundary conditions, to discu…
We propose a worldsheet formula for tree-level correlation functions describing a scalar field with arbitrary mass and quartic self-interaction in de Sitter space, which is a simple model for inflationary cosmology. The correlation functions are located on the future boundary of the spacetime and are Fourier-transformed to momentum space. Our formula is supported on mass-deformed scattering equations involving conformal generators in momentum space and reduces to the CHY formula for $ϕ^4$ amplitudes in the flat space limit. Using the global residue theorem, we verify that it reproduces the Wit…
A solvable irrelevant deformation of AdS$_3$/CFT$_2$ correspondence leading to a theory with Hagedorn spectrum at high energy has been recently proposed. It consists of a single trace deformation of the boundary theory, which is inspired by the recent work on solvable $T\bar{T}$ deformations of two-dimensional CFTs. Thought of as a worldsheet $σ$-model, the interpretation of the deformed theory from the bulk viewpoint is that of string theory on a background that interpolates between AdS$_3$ in the IR and a linear dilaton vacuum of little string theory in the UV. The insertion of the operator …
We consider the exact $R$-matrix of $AdS_3/CFT_2$, which is the building block for describing the scattering of worldsheet excitations of the light-cone gauge-fixed backgrounds $AdS_3 \times S^3 \times T^4$ and $AdS_3 \times S^3 \times S^3 \times S^1$ with pure Ramond-Ramond fluxes. We show that $R$ is invariant under a "deformed boost" symmetry, for which we write an explicit exact coproduct, i.e. its action on 2-particle states. When we include the boost, the symmetries of the $R$-matrix close into a $q$-Poincaré superalgebra. Our findings suggest that the recently discovered boost invarianc…
We discuss the spectrum of a string propagating on eta-deformed AdS_5 x S^5 by treating its world-sheet theory as an integrable quantum field theory. The exact S-matrix of this field theory is given by a q-deformation of the AdS_5 x S^5 world-sheet S-matrix with real deformation parameter. By considering mirror (double Wick-rotated) versions of these world-sheet theories we give the Thermodynamic Bethe Ansatz description of their exact finite size spectra. Interestingly, this class of models maps onto itself under the mirror transformation. At the level of the string this appears to say that t…
The homogeneous Yang-Baxter deformation is part of a larger web of integrable deformations and dualities that recently have been studied with motivations in integrable $σ$-models, solution-generating techniques in supergravity and Double Field Theory, and possible generalisations of the AdS/CFT correspondence. The $σ$-models obtained by the homogeneous Yang-Baxter deformation with periodic boundary conditions on the worldsheet are on-shell equivalent to undeformed models, yet with twisted boundary conditions. While this has been known for some time, the expression provided so far for the twist…
We consider the four-point function of operators in the stress tensor multiplet of the $U(N)_k\times U(N)_{-k}$ ABJM theory, in the limit where $N$ is taken to infinity while $N/k^{5}$ is held fixed. In this limit, ABJM theory is holographically dual to type IIA string theory on $AdS_4\times \mathbb{CP}^3$ at finite string coupling $g_s \sim (N/k^5)^{1/4}$. While at leading order in $1/N$, the stress tensor multiplet four-point function can be computed from type IIA supergravity, in this work we focus on the first subleading correction, which comes from tree level Witten diagrams with an $R^4$…
In this work we explore the complexity path integral optimization process for the case of warped $\text{AdS}_3$/warped $\text{CFT}_2$ correspondence. We first present the specific renormalization flow equations and analyze the differences with the case of CFT. We discuss how the "chiral Liouville action" could replace the Liouville action as the suitable cost function for this case. Starting from the other side of the story, we also show how the deformed Liouville actions could be derived from the spacelike, timelike and null warped metrics and how the behaviors of boundary topological terms c…
