We examine the behavior of the Ramond ground states in the D1-D5 CFT after a deformation of the free-orbifold sigma model on target space $({\mathbb T}^4)^N / S_N$ by a marginal interaction operator. These states are compositions of Ramond ground states of the twisted and untwisted sectors. They are characterized by a conjugacy class of $S_N$ and by the set of their "spins", including both R-charge and "internal" SU(2) charge. We compute the four-point functions of an arbitrary Ramond ground state with its conjugate and two interaction operators, for genus-zero covering surfaces representing the leading orders in the large-$N$ expansion. We examine short distance limits of these four-point functions, shedding light on the dynamics of the interacting theory. We find the OPEs and a collection of structure constants of the ground states with the interaction operators and a set of resulting non-BPS twisted operators. We also calculate the integrals of the four-point functions over the positions of the interaction operators and show that they vanish. This provides an explicit demonstration that the Ramond ground states remain protected against deformations away of the free orbifold point, as expected from algebraic considerations using the spectral flow of the ${\mathcal N} = (4,4)$ superconformal algebra with central charge $c = 6N$.
To model the process of absorption and emission of quanta by an extremal D1-D5 black hole in the dual CFT, we consider transitions between different Ramond vacua via absorption and emission of chiral primaries. We compute the probabilities to reach different CFT states starting with a special Ramond vacuum, using techniques of the orbifold CFT. It is found that the processes involving the change of angular momentum by $k$ units are suppressed as 1/N^k
We investigate gauge/gravity duals with flavour for which pure-gauge Kalb-Ramond B fields are turned on in the background, into which a D7 brane probe is embedded. First we consider the case of a magnetic field in two of the spatial boundary directions. We show that at finite temperature, i.e. in the AdS-Schwarzschild background, the B field has a stabilizing effect on the mesons and chiral symmetry breaking occurs for a sufficiently large value of the B field. Then we turn to the electric case of a B field in the temporal direction and one spatial boundary direction. In this case, there is a …
The free orbifold point of the D1-D5 CFT must be deformed with a scalar marginal operator driving it to the region in moduli space where the holographic supergravity description of fuzzball microstates becomes available. We discuss the effects of the deformation operator on the twisted Ramond ground states of the CFT by computing four-point functions. One can thus extract the OPEs of the deformation operator with these Ramond fields to find the conformal dimensions of intermediate non-BPS states and the relevant structure constants. We also compute the anomalous dimensions at second order in p…
The correspondence between the 't Hooft limit of N=4 super Yang-Mills theory and tree-level IIB superstring theory on AdS(5)xS(5) in a Ramond-Ramond background at values of lambda=g^2 N ranging from infinity to zero is examined in the context of unitarity. A squaring relation for the imaginary part of the holographic scattering of identical string fields in the two-particle channels is found, and a mismatch between weak and strong 't Hooft coupling is pointed out within the correspondence. Several interpretations and implications are proposed.
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 use AGT correspondence between N=2 SUSY Yang-Mills theory on ${\mathbb R}^4/{\mathbb Z}_2$ and two-dimensional CFT model with the algebra $ {\cal H} \oplus \hat{sl}(2)_2 \oplus \text{NSR}$ to obtain the explicit expressions for 4-point NSR conformal blocks including Ramond fields in terms of Nekrasov partition functions and correlation functions of $\hat{sl}(2)_2$ WZW model.
As we have shown in the previous work, using the formalism of matrix and eigenvalue models, to a given classical algebraic curve one can associate an infinite family of quantum curves, which are in one-to-one correspondence with singular vectors of a certain (e.g. Virasoro or super-Virasoro) underlying algebra. In this paper we reformulate this problem in the language of conformal field theory. Such a reformulation has several advantages: it leads to the identification of quantum curves more efficiently, it proves in full generality that they indeed have the structure of singular vectors, it e…
The D1D5 CFT gives a holographic dual description of a near-extremal black hole in string theory. The interaction in this theory is given by a marginal deformation operator, which is composed of supercharges acting on a twist operator. The twist operator links together different copies of a free CFT. We study the effect of this deformation operator when it links together CFT copies with winding numbers M and N to produce a copy with winding M+N, populated with excitations of a particular form. We compute the effect of the deformation operator in the full supersymmetric theory, firstly on a Ram…
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 utilize the integrality conjecture to show that the torus partition function of a fermionic rational conformal theory in the Ramond-Ramond sector becomes a constant when the bound $h^R \ge \frac{c}{24}$ is satisfied, where $h^R$ denote the conformal weights of Ramond states and $c$ is the central charge. The constant-valued Ramond-Ramond partition function strongly suggests the presence of supersymmetry unless a given theory has free fermions. The lower bound $h^R \ge \frac{c}{24}$ can then be identified with the unitarity bound of $\mathcal{N}=1$ supersymmetry. We thus propose that, for ra…
We propose an integrable bootstrap framework for the computation of correlation functions for superstrings in $AdS_3\times S^3\times T^4$ backgrounds supported by an arbitrary mixture or Ramond-Ramond and Neveu-Schwarz-Neveu-Schwarz fluxes. The framework extends the "hexagon tessellation" approach which was originally proposed for $AdS_5\times S^5$ and for the first time it demonstrates its applicability to other (less supersymmetric) setups. We work out the hexagon form factor for two-particle states, including its dressing factors which follow from those of the spectral problem, and we show …
The D1D5 brane bound state is believed to have an `orbifold point' in its moduli space which is the analogue of the free Yang Mills theory for the D3 brane bound state. The supergravity geometry generated by D1 and D5 branes is described by a different point in moduli space, and in moving towards this point we have to deform the CFT by a marginal operator: the `twist' which links together two copies of the CFT. In this paper we find the effect of this deformation operator on the simplest physical state of the CFT -- the Ramond vacuum. The twist deformation leads to a final state that is popula…
We compute correlators of two heavy and two light operators in the strong coupling and large $c$ limit of the D1D5 CFT which is dual to weakly coupled AdS$_3$ gravity. The light operators have dimension two and are scalar descendants of the chiral primaries considered in arXiv:1705.09250, while the heavy operators belong to an ensemble of Ramond-Ramond ground states. We derive a general expression for these correlators when the heavy states in the ensemble are close to the maximally spinning ground state. For a particular family of heavy states we also provide a result valid for any value of t…
We study multipoint correlators of protected scalars on the Maldacena-Wilson line in $\mathcal{N}=4$ SYM. Working at weak coupling in the planar limit, we derive an explicit recursion relation that captures next-to-leading order correlators with an arbitrary number of insertions of the fundamental scalar field. By pinching fundamental scalars together, we can build composite protected operators with higher values of the R-charge. Our result then encompasses arbitrary $n$-point correlators of protected operators with arbitrary weight. As a demonstration of our method, we give explicit formulae …
B-type D-branes are constructed on two different K3-fibrations over IP_1 using boundary conformal field theory at the rational Gepner points of these models. The microscopic CFT charges are compared with the Ramond charges of D-branes wrapped on holomorphic cycles of the corresponding Calabi-Yau manifold. We study in particular D4-branes and bundles localized on the K3 fibers, and find from CFT that each irreducible component of a bundle on K3 gains one modulus upon fibration over IP_1. This is in agreement with expectations and so provides a further test of the boundary CFT.
We compute correlation functions in the AdS/CFT correspondence to study the emergence of effective spacetime geometries describing complex underlying microstates. The basic argument is that almost all microstates of fixed charges lie close to certain "typical" configurations. These give a universal response to generic probes, which is captured by an emergent geometry. The details of the microstates can only be observed by atypical probes. We compute two point functions in typical ground states of the Ramond sector of the D1-D5 CFT, and compare with bulk two-point functions computed in asymptot…
The D1D5P system has a large set of BPS states at its orbifold point. Perturbing away from this 'free' point leads to some states joining up into long supermultiplets and lifting, while other states remain BPS. We consider the simplest orbifold which exhibits this lift: that with $N=2$ copies of the free $c=6$ CFT. We write down the number of lifted and unlifted states implied by the index at all levels upto $6$. We work to second order in the perturbation strength $λ$. For levels upto $4$, we find the wavefunctions of the lifted states, their supermultiplet structure and the value of the lift…
Near-extremal black holes are obtained by exciting the Ramond sector of the D1-D5 CFT, where the ground state is highly degenerate. We find that the dual geometries for these ground states have throats that end in a way that is characterized by the CFT state. Below the black hole threshold we find a detailed agreement between propagation in the throat and excitations of the CFT. We study the breakdown of the semiclassical approximation and relate the results to the proposal of gr-qc/0007011 for resolving the information paradox: semiclassical evolution breaks down if hypersurfaces stretch too …
We propose a diagnostic tool for detecting non-trivial symmetry protected topological (SPT) phases protected by a symmetry group $G$ in 2+1 dimensions. Our method is based on directly studying the 1+1-dimensional anomalous edge conformal field theory (CFT) of SPT phases. We claim that if the CFT is the edge theory of an SPT phase, then there must be an obstruction to cutting it open. This obstruction manifests in the in-existence of boundary states that preserves both the conformal symmetry and the global symmetry $G$. We discuss the relation between edgeability, the ability to find a consiste…
A particularly rich class of integrable systems arises from the AdS/CFT duality. There, the two-dimensional quantum field theory living on the string worldsheet may be understood in terms of a non-relativistic factorized S matrix, and the energy spectrum may be derived by techniques such as the mirror thermodynamic Bethe ansatz or the quantum spectral curve. In the case of AdS$_3$/CFT$_2$ without Ramond-Ramond fluxes, the worldhseet theory is a Wess-Zumino-Witten model with continous and discrete representations which, for the lowest allowed level, is dual to the symmetric product orbifold of …
We prove inductively that every k-trace operator of SO(6)_R irrep with Young tableau partition {r_1,r_2,r_3}, constructed out of k chiral primaries in the twenty dimensional SO(6)_R irrep, leads to a quasi primary field with protected conformal dimension. Our argument is based on perturbative evaluations of certain four point functions up to order 1/N^2.
We study two families of composite twisted Ramond fields (made by products of two operators) in the $\cal {N}=(4,4)$ supersymmetric D1-D5 SCFT$_2$ deformed by a marginal modulus operator away from its $(T^4)^N/ S_N$ free orbifold point. We construct the large-$N$ contributions to the four-point functions with two composite operators and two deformation fields. These functions allow us to derive short-distance OPE limits and to calculate the anomalous dimensions of the composite operators. We demonstrate that one can distinguish two sets of composite Ramond states with twists $m_1$ and $m_2$: p…
We describe the effect of the marginal deformation of the $\cal N = (4, 4)$ superconformal $(T^4)^N /S_N$ orbifold theory on a doublet of R-neutral twisted Ramond fields, in the large-$N$ approximation. Our analysis of their dynamics explores the explicit analytic form of the genus-zero four-point function involving two R-neutral Ramond fields and two deformation operators. We compute this correlation function with two different approaches: the Lunin-Mathur path-integral technique and the stress-tensor method. From its short distance limits, we extract the OPE structure constants and the scali…
We derive universal constraints on $(1+1)d$ rational conformal field theories (CFTs) that can arise as transitions between topological theories protected by a global symmetry. The deformation away from criticality to the trivially gapped phase is driven by a symmetry preserving relevant deformation and under renormalization group flow defines a conformal boundary condition of the CFT. When a CFT can make a transition between distinct trivially gapped phases the spectrum of the CFT quantized on an interval with the associated boundary conditions has degeneracies at each energy level. Using tech…
We study the moduli dependence of the chiral ring in N = (4,4) superconformal field theories, with special emphasis on those CFTs that are dual to type IIB string theory on AdS3xS3xX4. The chiral primary operators are sections of vector bundles, whose connection describes the operator mixing under motion on the moduli space. This connection can be exactly computed using the constraints from N = (4,4) supersymmetry. Its curvature can be determined using the tt* equations, for which we give a derivation in the physical theory which does not rely on the topological twisting. We show that for N = …
We study $S_N$-invariant four-point functions with two generic multi-cycle fields and two twist-2 fields, at the free orbifold point of the D1-D5 CFT. We derive the explicit factorization of these functions following from the action of the symmetric group on the composite multi-cycle fields. Apart from non-trivial symmetry factors that we compute, the function with multi-cycle operators is reduced to a sum of connected correlators in which the composite fields have, at most, two cycles. The correlators with two double-cycle and two single-cycle fields give the leading order contribution in the…
In the previous paper, the authors pointed out correspondence between a supersymmetric double-well matrix model and two-dimensional type IIA superstring theory on a Ramond-Ramond background from the viewpoint of symmetry and spectrum. This was confirmed by agreement between planar correlation functions in the matrix model and tree-level amplitudes in the superstring theory. In order to investigate the correspondence further, in this paper we compute correlation functions to all order of genus expansion in the double scaling limit of the matrix model. One-point functions of operators protected …
We study tree level one-point functions of non-protected scalar operators in the defect CFT, based on N=4 SYM, which is dual to the SO(5) symmetric D3-D7 probe brane system with non-vanishing instanton number. Whereas symmetries prevent operators from the SU(2) and SU(3) sub-sectors from having non-vanishing one-point functions, more general scalar conformal operators, which in particular constitute Bethe eigenstates of the integrable SO(6) spin chain, are allowed to have non-trivial one-point functions. For a series of operators with a small number of excitations we find closed expressions in…
Recently, Freyhult, Rej and Staudacher (FRS) proposed an integral equation determining the leading logarithmic term of the anomalous dimension of sl(2) twist-operators in N=4 SYM for large Lorentz spin M and twist L at fixed j = L/log(M). We discuss the large j limit of the FRS equation. This limit can be matched with the {\em fast long string} limit of AdS_5 X S^5 superstring perturbation theory at all couplings. In particular, a certain part of the classical and one-loop string result is known to be protected and can be computed in the weakly coupled large-j limit of the FRS equation. We pre…
We construct the gravity dual of a field theory with flavour in which there are Fayet-Iliopoulos (FI) terms present. For this purpose we turn on a constant Kalb-Ramond B field in four internal space directions of AdS_5 x S^5 together with a D7 brane probe wrapping AdS_5 x S^3. The B field induces noncommutativity on the four internal directions of the D7 brane probe perpendicular to the AdS boundary. We argue on general grounds that the moduli space of the Higgs part of mixed Coulomb-Higgs states in the dual field theory at the boundary is described by the ADHM equations for noncommutative ins…
The AdS/CFT correspondence provides a rich testing ground for many important topics in theoretical physics. The earliest and most striking example of the correspondence is the conjectured duality between the energy spectrum of type IIB superstring theory on AdS_5 x S^5 and the operator anomalous dimensions of N=4 supersymmetric Yang-Mills theory in four dimensions. While there is a substantial amount of evidence in support of this conjecture, direct tests have been elusive. The difficulty of quantizing superstring theory in a curved Ramond-Ramond background is compounded by the problem of comp…
We show that, although the correlator of four stress-tensor multiplets in N=4 SYM is known to have radiative corrections, certain linear combinations of its components are protected from perturbative renormalisation and remain at their free-field values. This result is valid for weak as well as for strong coupling and for any gauge group. Our argument uses Intriligator's insertion formula, and includes a proof that the possible contact term contributions cannot change the form of the amplitudes. Combining this new non-renormalisation theorem with Maldacena's conjecture allows us to make a pred…
We study correlation functions of local operator insertions on the 1/2-BPS Wilson line in ${\cal N}=4$ super Yang-Mills theory. These correlation functions are constrained by the 1d superconformal symmetry preserved by the 1/2-BPS Wilson line and define a defect CFT$_1$ living on the line. At strong coupling, a set of elementary operator insertions with protected scaling dimensions correspond to fluctuations of the dual fundamental string in AdS$_5 \times S^5$ ending on the line at the boundary and can be thought of as light fields propagating on the AdS$_2$ worldsheet. We use AdS/CFT techniqu…
We apply recently developed tools from exceptional field theory to calculate the full Kaluza-Klein spectrum of the AdS$_5$ Pilch-Warner solution of type IIB supergravity. Through the AdS/CFT correspondence this yields detailed information about the spectrum of protected and unprotected operators of the four-dimensional $\mathcal{N}=1$ Leigh-Strassler SCFT, in the planar limit. We also calculate explicitly the superconformal index of the SCFT in this limit and show that it agrees precisely with the spectrum of protected operators in the supergravity calculation.
We compute a set of correlation functions of operator insertions on the 1/8 BPS Wilson loop in $\mathcal{N}=4$ SYM by employing supersymmetric localization, OPE and the Gram-Schmidt orthogonalization. These correlators exhibit a simple determinant structure, are position-independent and form a topological subsector, but depend nontrivially on the 't Hooft coupling and the rank of the gauge group. When applied to the 1/2 BPS circular (or straight) Wilson loop, our results provide an infinite family of exact defect CFT data, including the structure constants of protected defect primaries of arbi…
We study a class of non-protected local composite operators which occur in the R symmetry singlet channel of the OPE of two stress-tensor multiplets in {\cal N}=4 SYM. At tree level these are quadrilinear scalar dimension four operators, two single-traces and two double-traces. In the presence of interaction, due to a non-trivial mixing under renormalization, they split into linear combinations of conformally covariant operators. We resolve the mixing by computing the one-loop two-point functions of all the operators in an {\cal N}=1 setup, then diagonalizing the anomalous dimension matrix and…
Four-point functions with fractional R-symmetry excitations in the D1-D5 CFT
Universidade Federal de Pernambuco（伯南布哥联邦大学）；Universidade Federal do Espírito Santo（圣埃斯皮里图联邦大学）；Institute for Nuclear Research and Nuclear Energy, Bulgarian Academy of Sciences（保加利亚科学院核能研究所）
We consider the 1d CFT defined by the half-BPS Wilson line in planar $\mathcal{N}=4$ super Yang-Mills. Using analytic bootstrap methods we derive the four-point function of the super-displacement operator at fourth order in a strong coupling expansion. Via AdS/CFT, this corresponds to the first three-loop correlator in AdS ever computed. To do so we address the operator mixing problem by considering a family of auxiliary correlators. We further extract the anomalous dimension of the lightest non-protected operator and find agreement with the integrability-based numerical result of Grabner, Gro…
We calculate planar tree level one-point functions of non-protected operators in the defect conformal field theory dual to the D3-D5 brane system with k units of the world volume flux. Working in the operator basis of Bethe eigenstates of the Heisenberg XXX_{1/2} spin chain we express the one-point functions as overlaps of these eigenstates with a matrix product state. For k=2 we obtain a closed expression of determinant form for any number of excitations, and in the case of half-filling we find a relation with the Néel state. In addition, we present a number of results for the limiting case o…
Entanglement renormalization is a real-space renormalization group (RG) transformation for quantum many-body systems. It generates the multi-scale entanglement renormalization ansatz (MERA), a tensor network capable of efficiently describing a large class of many-body ground states, including those of systems at a quantum critical point or with topological order. The MERA has also been proposed to be a discrete realization of the holographic principle of string theory. In this paper we propose the use of symmetric tensors as a mechanism to build a symmetry protected RG flow, and discuss two im…
One of the predictions of the AdS/CFT correspondence is the matching of protected operators between a superconformal field theory and its holographic dual. We review the spectrum of protected operators in quiver gauge theories that flow to superconformal field theories at low energies. The spectrum is determined by the cyclic homology of an algebra associated to the quiver gauge theory. Identifying the spectrum of operators with cyclic homology allows us to apply the Hochschild-Kostant-Rosenberg theorem to relate the cyclic homology groups to deRham cohomology groups. The map from cyclic homol…
The non-renormalization of the 3-point functions $tr X^{k_1} tr X^{k_2} tr X^{k_3}$ of chiral primary operators in N=4 super-Yang-Mills theory is one of the most striking facts to emerge from the AdS/CFT correspondence. A two-fold puzzle appears in the extremal case, e.g. k_1 = k_2 + k_3. First, the supergravity calculation involves analytic continuation in the k_i variables to define the product of a vanishing bulk coupling and an infinite integral over AdS. Second, extremal correlators are uniquely sensitive to mixing of the single-trace operators $tr X^k$ with protected multi-trace operator…
Surface operators in the 6d (2,0) theory at large $N$ have a holographic description in terms of M2 branes probing the AdS$_7 \times S^4$ M-theory background. The most symmetric, 1/2-BPS, operator is defined over a planar or spherical surface, and it preserves a 2d superconformal group. This includes, in particular, an $SO(2,2)$ subgroup of 2d conformal transformations, so that the surface operator may be viewed as a conformal defect in the 6d theory. The dual M2 brane has an AdS$_3$ induced geometry, reflecting the 2d conformal symmetry. Here we use the holographic description to extract the …
We perform exact computations of correlation functions of 1/2-BPS local operators and protected operator insertions on the 1/8-BPS Wilson loop in $\mathcal{N}=4$ SYM. This generalizes the results of our previous paper arXiv:1802.05201, which employs supersymmetric localization, OPE and the Gram-Schmidt process. In particular, we conduct a detailed analysis for the 1/2-BPS circular (or straight) Wilson loop in the planar limit, which defines an interesting nontrivial defect CFT. We compute its bulk-defect structure constants at finite 't Hooft coupling, and present simple integral expressions i…
We demonstrate by explicit multi-loop calculation that γ-deformed planar N=4 SYM, supplemented with a set of double-trace counter-terms, has two nontrivial fixed points in the recently proposed double scaling limit, combining vanishing 't Hooft coupling and large imaginary deformation parameter. We provide evidence that, at the fixed points, the theory is described by an integrable non-unitary four-dimensional CFT. We find a closed expression for the four-point correlation function of the simplest protected operators and use it to compute the exact conformal data of operators with arbitrary Lo…
We probe the conformal block structure of a scalar four-point function in $d\geq2$ conformal field theories by including higher-order derivative terms in a bulk gravitational action. We consider a heavy-light four-point function as the boundary correlator at large central charge. Such a four-point function can be computed, on the gravity side, as a two-point function of the light operator in a black hole geometry created by the heavy operator. We consider analytically solving the corresponding scalar field equation in a near-boundary expansion and find that the multi-stress tensor conformal bl…
In this dissertation we explore various aspects of the AdS/CFT correspondence. As string quantization on general backgrounds with fluxes is very difficult, one often uses the duality at the level of canonical fields of supergravity and the corresponding half-BPS operators in SYM, since they both belong to the shortest multiplets of the superconformal group SU(2,2|4). In addition to half-BPS operators, there are others with non-renormalization properties. One such class of operators is the quarter-BPS operators, which are dual to threshold bound states of elementary supergravity excitations. Th…
4D Lorentzian conformal field theory (CFT) is mapped into 5D anti-de Sitter spacetime (AdS), from the viewpoint of "geometrizing" conformal current algebra. A large-N expansion of the CFT is shown to lead to (infinitely many) weakly coupled AdS particles, in one-to-one correspondence with minimal-color-singlet CFT primary operators. If all but a finite number of "protected" primary operators have very large scaling dimensions, it is shown that there exists a low-AdS-curvature effective field theory regime for the corresponding finite set of AdS particles. Effective 5D gauge theory and General …
