We study a certain class of nonextremal D1D5 geometries and their ergoregion emission. Using a detailed CFT computation and the formalism developed in arXiv:0906.2015 [hep-th], we compute the full spectrum and rate of emission from the geometries and find exact agreement with the gravity answer. Previously, only part of the spectrum had been reproduced using a CFT description. We close with a discussion of the context and significance of the calculation.
In the D1D5 CFT the twist operator of order 2 can twist together two copies in the untwisted sector into a single joined copy in the twisted sector. Traditionally, this effect is computed by using the covering map method. Recently, a new method was developed using the Bogoliubov ansatz and conformal symmetry to compute this effect in a toy model of one free boson. In this paper, we use this method with superconformal symmetry to compute the effect of the twist operator in the D1D5 CFT. This may provide more effective tools for computing correlation functions of twist operators in this system.
The deformation operator of the D1D5 orbifold CFT, a twist 2 operator, drives the CFT towards the black hole dual and its physics is key to understanding thermalization in the D1D5 system. To further study this deformation, we extend previous work on the effect of twist 2 operators to a method that works for higher orders, in the continuum limit. Our component twist method works by building higher twist operators out of twist 2 operators together with knowledge of Bogoliubov transformations. Consequently, this method sidesteps limitations in Lunin-Mathur technology by avoiding lifts to the cov…
We consider general 2D orbifold CFTs of the form M^N/S_N, with M a target space manifold and S_N the symmetric group, and generalize the Lunin-Mathur covering space technique in two ways. First, we consider excitations of twist operators by modes of fields that are not twisted by that operator, and show how to account for these excitations when computing correlation functions in the covering space. Second, we consider non-twist sector operators and show how to include the effects of these insertions in the covering space. We work two examples, one using a simple bosonic CFT, and one using the …
Thermalization in the D1D5 CFT should occur via interactions caused by the twist operator, which deforms the theory off its free orbifold point. Earlier studies investigating this deformation at first order did not show any definite evidence of thermalization. In this paper we study the deformation to second order, where we do expect to see the effects that should give thermalization. We compute the effect of two twist operators on an initial vacuum state, which generates a squeezed state analogous to the case for a single twist. We obtain expressions for the Bogoliubov coefficients in this 2-…
The D1D5 CFT is a holographic dual of a near-extremal black hole in string theory. The interaction in this theory involves a twist operator which joins together different copies of a free CFT. Given a large number of D1 and D5 branes, the effective length of the circle on which the CFT lives is very large. We develop a technique to study the effect of the twist operator in the limit where the wavelengths of excitations are short compared to this effective length, which we call the 'continuum limit'. The method uses Bogoliubov coefficients to compute the effect of the twist operator in this lim…
The D1D5 CFT has been very useful in the study of black holes. The interaction in this theory involves a twist operator, which links together different copies of a free CFT. For the bosonic fields, we examine the action of this twist when it links together CFT copies with winding numbers M and N to produce a copy with winding M+N. Starting with the vacuum state generates a squeezed state, which we compute. Starting with an initial excitation on one of the copies gives a linear combination of excitations on the final state, which we also compute. These results generalize earlier computations wh…
We consider a brane universe in a Reissner-Nordstrom-de Sitter background spacetime of arbitrary dimensionality. It is shown that the brane evolution is described by generalized Friedmann equations for radiative matter along with a stiff-matter contribution. On the basis of the (conjectured) dS/CFT correspondence, we identify various thermodynamic properties of the brane. It is then demonstrated that, when the brane crosses the de Sitter cosmological horizon, the CFT thermodynamics and Friedmann-like equations coincide. Moreover, the CFT entropy is shown to be expressible in a generalized Card…
We are interested in thermalization in the D1D5 CFT, since this process is expected to be dual to black hole formation. We expect that the lowest order process where thermalization occurs will be at second order in the perturbation that moves us away from the orbifold point. The operator governing the deformation off of the orbifold point consists of a twist operator combined with a supercharge operator acting on this twist. In a previous paper we computed the action of two twist operators on an arbitrary state of the CFT. In the present work we compute the action of the supercharges on these …
We review the computation of the anomalous dimension of twist operators in the planar limit of N=4 SYM using the asymptotic Bethe ansatz and demonstrate how this quantity is obtained at weak, strong and intermediate values of the coupling constant. The anomalous dimension of twist operators in the limit of large Lorentz spin played a major role in the construction as well as in many tests of the asymptotic Bethe equations, this aspect of the story is emphasised.
A large class of the recently found unimodular nonabelian homogeneous Yang-Baxter deformations of the AdS_5 x S^5 superstring can be realized as sequences of noncommuting TsT transformations. I show that many of them are duals to various noncommutative versions of supersymmetric Yang-Mills theory, structurally determined directly in terms of the associated r matrices, in line with previous expectations in the literature.
We consider a brane universe in an asymptotically de Sitter background spacetime of arbitrary dimensionality. In particular, the bulk spacetime is described by a ``topological de Sitter'' solution, which has recently been investigated by Cai, Myung and Zhang. In the current study, we begin by showing that the brane evolution is described by Friedmann-like equations for radiative matter. Next, on the basis of the dS/CFT correspondence, we identify the thermodynamic properties of the brane universe. We then demonstrate that many (if not all) of the holographic aspects of analogous AdS-bulk scena…
We stress that the dS/CFT correspondence should be formulated using unitary principal series representations of the de Sitter isometry group/conformal group, rather than highest-weight representations as originally proposed. These representations, however, are infinite-dimensional, and so do not account for the finite gravitational entropy of de Sitter space in a natural way. We then propose to replace the classical isometry group by a q-deformed version. This is carried out in detail for two-dimensional de Sitter and we find that the unitary principal series representations deform to finite-d…
We consider the issue of thermalization in the D1D5 CFT. Thermalization is expected to correspond to the formation of a black hole in the dual gravity theory. We start from the orbifold point, where the theory is essentially free, and does not thermalize. In earlier work it was noted that there was no clear thermalization effect when the theory was deformed off the orbifold point to first order in the relevant twist perturbation. In this paper we consider the deformation to second order in the twist, where we do find effects that can cause thermalization of an initial perturbation. We consider…
We compute 1-loop correction E_1 to the energy of folded string in AdS_5 x S^5 (carrying spin S in AdS_5 and momentum J in S^5) using ``long string'' approximation in which S >> J >> 1. According to AdS/CFT E_1 should represent the first subleading correction to strong coupling expansion of anomalous dimension of higher twist SL(2) sector operators of the form Tr D^S Z^J. We show that E_1 smoothly interpolates between the ln S regime (previously found in the J=0 case) and the λ/J^2 ln^3 (S/J) regime (which is the leading correction to the thermodynamic limit on the spin chain side). This suppo…
We explore the OPE of certain twist operators in symmetric product ($S_N$) orbifold CFTs, extending our previous work arXiv:1804.01562 to the case of $\mathcal{N}=(4,4)$ supersymmetry. We consider a class of twist operators related to the chiral primaries by spectral flow parallel to the twist. We conjecture that at large $N$, the OPE of two such operators contains only fields in this class, along with excitations by fractional modes of the superconformal currents. We provide evidence for this by studying the coincidence limits of two 4-point functions to several non-trivial orders. We show ho…
The D1-D5 system has an orbifold point in its moduli space, at which it may be described by an ${\cal N} = (4,4)$ supersymmetric sigma model with target space $M^N/S(N)$ where $M$ is $\mathbb{T}^4$ or $K3$. In this paper we consider correlation functions involving chiral operators constructed from twist fields: we find explicit expressions for processes involving a twist $n$ operator joining $n$ twist operators of arbitrary twist. These expressions are universal, in that they are independent of the choice of $M$, and the final results can be expressed in a compact form. We explain how these re…
We study the operator product expansion (OPE) of two identical scalar primary operators in the lightcone limit in a conformal field theory where a scalar is the operator with lowest twist. We see that in CFTs where both the stress tensor and a scalar are the lowest twist operators, the stress tensor contributes at the leading order in the lightcone OPE and the scalar contributes at the subleading order. We also see that there does not exist a scalar analogue of the average null energy condition (ANEC) for a CFT where a scalar is the lowest twist operator.
We consider the set of partition functions that result from the insertion of twist operators compatible with conformal invariance in a given 2D Conformal Field Theory (CFT). A consistency equation, which gives a classification of twists, is written and solved in particular cases. This generalises old results on twisted torus boundary conditions, gives a physical interpretation of Ocneanu's algebraic construction, and might offer a new route to the study of properties of CFT.
We give an AdS/CFT interpretation to homogeneous Yang-Baxter deformations of the AdS_5 x S^5 superstring as noncommutative deformations of the dual gauge theory, going well beyond the canonical noncommutative case. These homogeneous Yang-Baxter deformations can be of so-called abelian or jordanian type. While abelian deformations have a clear interpretation in string theory and many already had well understood gauge theory duals, jordanian deformations appear novel on both counts. We discuss the symmetry structure of the deformed string from the uniformizing perspective of Drinfeld twists and …
We consider correlation functions in symmetric product ($S_N$) orbifold CFTs at large $N$ with arbitrary seed CFT, expanding on our earlier work arXiv:2211.04633 . Using covering space techniques, we calculate descent relations using fractional Virasoro generators in correlators, writing correlators of descendants in terms of correlators of ancestors. We first consider the case three-point functions of the form ($m$-cycle)-($n$-cycle)-($q$-cycle) which lift to arbitrary primaries on the cover, and descendants thereof. In these examples we show that the final descent relations do not depend on …
The D1D5 CFT has a large set of states that are supersymmetric at the `free' orbifold point in moduli space. When we perturb away from this point, some of these states join into long multiplets and lift in energy, while others remain supersymmetric. The count of unlifted states can be bounded below by an index, but the index does not yield the pattern of lifting; i.e., which states join into a long multiplet and how much this multiplet lifts. In this paper we consider the simple case of the D1D5 CFT where the orbifold CFT is a sigma model with targets space $(T^4)^2/S_2$ and consider states at…
A class of marginal deformations of four-dimensional N=4 super Yang-Mills theory has been found to correspond to a set of smooth, multiparameter deformations of the S^5 target subspace in the holographic dual on AdS_5 x S^5. We present here an analogous set of deformations that act on global toroidal isometries in the AdS_5 subspace. Remarkably, certain sectors of the string theory remain classically integrable in this larger class of so-called gamma-deformed AdS_5 x S^5 backgrounds. Relying on studies of deformed su(2)_gamma models, we formulate a local sl(2)_gamma Lax representation that adm…
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 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…
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 …
We take an analytic approach to the CFT bootstrap, studying the 4-pt correlators of d > 2 dimensional CFTs in an Eikonal-type limit, where the conformal cross ratios satisfy |u| << |v| < 1. We prove that every CFT with a scalar operator ϕmust contain infinite sequences of operators O_{τ,l} with twist approaching τ-> 2Δ_ϕ+ 2n for each integer n as l -> infinity. We show how the rate of approach is controlled by the twist and OPE coefficient of the leading twist operator in the ϕx ϕOPE, and we discuss SCFTs and the 3d Ising Model as examples. Additionally, we show that the OPE coefficients of ot…
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…
It is believed that the D1D5 brane system is described by an 'orbifold CFT' at a special point in moduli space. We first develop a general formulation relating amplitudes in a d-dimensional CFT to absorption/emission of quanta from flat infinity. We then construct the D1D5 vertex operators for minimally coupled scalars in supergravity, and use these to compute the CFT amplitude for emission from a state carrying a single excitation. Using spectral flow we relate this process to one where we have emission from a highly excited initial state. In each case the radiation rate is found to agree wit…
The role of local higher-twist ($τ> 3$) spin-1/2 fermionic operators of the strongly coupled ${\cal {N}}=4$ supersymmetric Yang-Mills theory on the symmetric and antisymmetric deep inelastic scattering (DIS) structure functions is investigated. The calculations are carried out in terms of the duality between ${\cal {N}}=4$ SYM theory and type IIB supergravity on AdS$_5 \times S^5$. Particularly, we explicitly obtain the structure functions for single-trace spin-1/2 fermionic operators in the 20$^*$ and 60$^*$ irreducible representations of $SU(4)_R$, corresponding to twists 4 and 5, respective…
We elaborate on various aspects of the conformal field theory of the symmetric orbifold. We collect various results that have appeared in the literature, and we present a coherent picture of the operator content of this CFT, relying on the orbifold extension of the Virasoro algebra. We then focus on the large $N$-limit of this theory, discuss the OPE of two twist operators, and find various selection rules. We review how to calculate four-point functions of twist operators, and we write down the most general four-point function in the covering space for large $N$. We show that it depends on so…
We study twist operators in higher dimensional CFT's. In particular, we express their conformal dimension in terms of the energy density for the CFT in a particular thermal ensemble. We construct an expansion of the conformal dimension in power series around n=1, with n being replica parameter. We show that the coefficients in this expansion are determined by higher point correlations of the energy-momentum tensor. In particular, the first and second terms, i.e. the first and second derivatives of the scaling dimension, have a simple universal form. We test these results using holography and f…
Correlators of unitary quantum field theories in Lorentzian signature obey certain analyticity and positivity properties. For interacting unitary CFTs in more than two dimensions, we show that these properties impose general constraints on families of minimal twist operators that appear in the OPEs of primary operators. In particular, we rederive and extend the convexity theorem which states that for the family of minimal twist operators with even spins appearing in the reflection-symmetric OPE of any scalar primary, twist must be a monotonically increasing convex function of the spin. Our arg…
Gravitational shockwaves are insensitive to higher-curvature corrections in the action. Recent work found that the OPE coefficients of lowest-twist multi-stress-tensor operators, computed holographically in a planar black hole background, are insensitive as well. In this paper, we analyze the relation between these two limits. We explicitly evaluate the two-point function on a shockwave background to all orders in a large central charge expansion. In the geodesic limit, we find that the ANEC exponentiates in the multi-stress-tensor sector. To compare with the black hole limit, we obtain a recu…
We give a unified treatment of dispersive sum rules for four-point correlators in conformal field theory. We call a sum rule dispersive if it has double zeros at all double-twist operators above a fixed twist gap. Dispersive sum rules have their conceptual origin in Lorentzian kinematics and absorptive physics (the notion of double discontinuity). They have been discussed using three seemingly different methods: analytic functionals dual to double-twist operators, dispersion relations in position space, and dispersion relations in Mellin space. We show that these three approaches can be mapped…
We study a refined large spin limit for twist operators in the sl(2) sector of AdS/CFT. We derive a novel non-perturbative equation for the generalized two-parameter scaling function associated to this limit, and analyze it at weak coupling. It is expected to smoothly interpolate between weakly coupled gauge theory and string theory at strong coupling.
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…
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 define a holographic dual to the Donaldson-Witten topological twist of $\mathcal{N}=2$ gauge theories on a Riemannian four-manifold. This is described by a class of asymptotically locally hyperbolic solutions to $\mathcal{N}=4$ gauged supergravity in five dimensions, with the four-manifold as conformal boundary. Under AdS/CFT, minus the logarithm of the partition function of the gauge theory is identified with the holographically renormalized supergravity action. We show that the latter is independent of the metric on the boundary four-manifold, as required for a topological theory. Supersy…
In this dissertation, we review work presented in arXiv:0906.2015, arXiv:0907.1663, arXiv:1002.3132, arXiv:1003.2746, and arXiv:1007.2202 on the D1D5 system. We begin with some motivational material for black holes in string theory. In Chapter 2, we review the D1D5 system, including the gravity and CFT descriptions. In Chapter 3, we show how to perturbatively relax the decoupling limit in a general AdS-CFT setting. This allows one to compute the emission out of the AdS/CFT into the asymptotic flat space. In Chapter 4, we apply that formalism to some particular geometries, and exactly reproduce…
We study the emission rates of scalar, spinor and vector particles from a 5 dimensional black hole for arbitrary partial waves. The solution is lifted to 6 dimensions, and the near horizon $ BTZ \times S^3$ geometry of the black hole solution is probed to determine the greybody factors. We show that the exact decay rates can be reproduced from a $(1+1)$-dimensional conformal field theory which lies on the boundary of the near horizon geometry. The AdS/CFT correspondence is used to determine the dimension of the CFT operators corresponding to the bulk fields. These operators couple to plane wav…
The D1D5 CFT has provided a useful microscopic model for studying black holes. The coupling in this theory is a twist deformation whose action on the vacuum generates a squeezed state. We give a new derivation of the expression for this squeezed state using the conformal Ward identity; this derivation provides an insight into several features of the state. We also examine the squeezed state in a continuum limit where we describe it in terms of position space correlations created by the twist.
In the AdS/CFT correspondence a chiral primary is described by a supergravity solution with mass equaling angular momentum. For AdS_3 X S^3 we are led to consider three special families of metrics with this property: metrics with conical defects, Aichelburg-Sexl type metrics generated by rotating particles, and metrics generated by giant gravitons. We find that the first two of these are special cases of the complete family of chiral primary metrics which can be written down using the general solution in hep-th/0109154, but they correspond to two extreme limits - the conical defect metrics map…
We study the Feynman graph structure and compute certain exact four-point correlation functions in chiral CFT$_4$ proposed by Ö.Gürdoğan and one of the authors as a double scaling limit of $γ$-deformed $\mathcal{N}=4$ SYM theory. We give full description of bulk behavior of large Feynman graphs: it shows a generalized "dynamical fishnet" structure, with a dynamical exchange of bosonic and Yukawa couplings. We compute certain four-point correlators in the full chiral CFT$_4$, generalizing recent results for a particular one-coupling version of this theory -- the bi-scalar "fishnet" CFT. We sum …
Prompted by the Mathieu Moonshine observation, we identify a pair of 45-dimensional vector spaces of states that account for the first order term in the massive sector of the elliptic genus of K3 in every Z2-orbifold CFT on K3. These generic states are uniquely characterized by the fact that the action of every geometric symmetry group of a Z2-orbifold CFT yields a well-defined faithful representation on them. Moreover, each such representation is obtained by restriction of the 45-dimensional irreducible representation of the Mathieu group M24 constructed by Margolin. Thus we provide a piece o…
We analytically study the lightcone limit of the conformal bootstrap equations for 4-point functions containing global symmetry currents and the stress tensor in 3d CFTs. We show that the contribution of the stress tensor to the anomalous dimensions of large spin double-twist states is negative if and only if the conformal collider physics bounds are satisfied. In the context of AdS/CFT these results indicate a relation between the attractiveness of AdS gravity and positivity of the CFT energy flux. We also study the contribution of non-Abelian conserved currents to the anomalous dimensions of…
Groenewold-Moyal twists, integrable spin-chains and AdS/CFT
Instituto Galego de Física de Altas Enerxías (IGFAE), and Departamento de Física de Partículas, Universidade de Santiago de Compostela（加利西亚高等物理研究所（IGFAE），圣地亚哥-德孔波斯特拉大学粒子物理系）
We develop a method for computing correlation functions of twist operators in the bosonic 2-d CFT arising from orbifolds M^N/S_N, where M is an arbitrary manifold. The path integral with twist operators is replaced by a path integral on a covering space with no operator insertions. Thus, even though the CFT is defined on the sphere, the correlators are expressed in terms of partition functions on Riemann surfaces with a finite range of genus g. For large N, this genus expansion coincides with a 1/N expansion. The contribution from the covering space of genus zero is `universal' in the sense th…
We study the spectrum and OPE coefficients of the three-dimensional critical O(2) model, using four-point functions of the leading scalars with charges 0, 1, and 2 ($s$, $ϕ$, and $t$). We obtain numerical predictions for low-twist OPE data in several charge sectors using the extremal functional method. We compare the results to analytical estimates using the Lorentzian inversion formula and a small amount of numerical input. We find agreement between the analytic and numerical predictions. We also give evidence that certain scalar operators lie on double-twist Regge trajectories and obtain est…
We consider high spin, $s$, long twist, $L$, planar operators (asymptotic Bethe Ansatz) of strong ${\cal N}=4$ SYM. Precisely, we compute the minimal anomalous dimensions for large 't Hooft coupling $λ$ to the lowest order of the (string) scaling variable $\ell \sim L/ (\ln \mathcal{S} \sqrtλ)$ with GKP string size $\sim\ln \mathcal{S}\equiv 2 \ln (s/\sqrtλ)$. At the leading order $(\ln \mathcal{S}) \cdot \ell ^2 $, we can confirm the O(6) non-linear sigma model description for this bulk term, without boundary term $(\ln \mathcal{S})^0$. Going further, we derive, extending the O(6) regime, the…
We compute numerically the dimensions and OPE coefficients of several operators in the 3d Ising CFT, and then try to reverse-engineer the solution to crossing symmetry analytically. Our key tool is a set of new techniques for computing infinite sums of SL(2,R) conformal blocks. Using these techniques, we solve the lightcone bootstrap to all orders in an asymptotic expansion in large spin, and suggest a strategy for going beyond the large spin limit. We carry out the first steps of this strategy for the 3d Ising CFT, deriving analytic approximations for the dimensions and OPE coefficients of se…
