General CFT Research Articles

Affine Yangian of mathfrak{gl} (2) and integrable structures of superconformal field theory

This paper is devoted to study of integrable structures in superconformal field theory and more general coset CFT’s related to the affine Yangian Y( hat{mathfrak{gl}} (2)). We derive the relation between the RLL and current realizations and prove Bethe anzatz equations for the spectrum of Integrals of Motion.

Entanglement in descendants

We study the single interval entanglement and relative entropies of conformal descendants in 2d CFT. Descendants contain non-trivial entanglement, though the entanglement entropy of the canonical primary in the free boson CFT contains no additional entanglement compared to the vacuum, we show that the entanglement entropy of the state created by its level one descendant is non-trivial and is identical to that of the U(1) current in this theory. We determine the first sub-leading corrections to the short interval expansion of the entanglement entropy of descendants in a general CFT from their four point function on the n-sheeted plane. We show that these corrections are determined by multiplying squares of appropriate dressing factors to the corresponding corrections of the primary. Relative entropy between descendants of the same primary is proportional to the square of the difference of their dressing factors. We apply our results to a class of descendants of generalized free fields and descendants of the vacuum and show that their dressing factors are universal.

Interplay between reflection positivity and crossing symmetry in the bootstrap approach to CFT

Crossing symmetry (CS) is the main tool in the bootstrap program applied to CFT. This consists in an equality which imposes restrictions on the CFT data of a model, i.e., the OPE coefficients and the conformal dimensions. Reflection positivity (RP) has also played a role in this program, since this condition is what leads to the unitary bound and reality of the OPE coefficients. In this paper, we show that RP can still reveal more information, explaining how RP itself can capture an important part of the restrictions imposed by the full CS equality. In order to do that, we use a connection used by us in a previous work between RP and positive definiteness of a function of a single variable. This allows us to write constraints on the OPE coefficients in a concise way. These constraints are encoded in the conditions that certain functions of the cross-ratio will be positive defined and in particular completely monotonic. We will consider how the bounding of scalar conformal dimensions and OPE coefficients arise in this RP based approach. We will illustrate the conceptual and practical value of this view trough examples of general CFT models in d-dimensions.

Dispersion relations and exact bounds on CFT correlators

We derive new crossing-symmetric dispersion formulae for CFT correlators restricted to the line. The formulae are equivalent to the sum rules implied by what we call master functionals, which are analytic extremal functionals which act on the crossing equation. The dispersion relations provide an equivalent formulation of the constraints of the Polyakov bootstrap and hence of crossing symmetry on the line. The built in positivity properties imply simple and exact lower and upper bounds on the values of general CFT correlators on the Euclidean section, which are saturated by generalized free fields. Besides bounds on correlators, we apply this technology to determine new universal constraints on the Regge limit of arbitrary CFTs and obtain very simple and accurate representations of the 3d Ising spin correlator.

Four-point functions in momentum space: conformal ward identities in the scalar/tensor case

We derive and analyze the conformal Ward identities (CWI’s) of a tensor 4-point function of a generic CFT in momentum space. The correlator involves the stress–energy tensor T and three scalar operators O (TOOO). We extend the reconstruction method for tensor correlators from 3- to 4-point functions, starting from the transverse traceless sector of the TOOO. We derive the structure of the corresponding CWI’s in two different sets of variables, relevant for the analysis of the 1–3 (1 graviton rightarrow 3 scalars) and 2–2 (graviton + scalar rightarrow two scalars) scattering processes. The equations are all expressed in terms of a single form factor. In both cases we discuss the structure of the equations and their possible behaviors in various asymptotic limits of the external invariants. A comparative analysis of the systems of equations for the TOOO and those for the OOOO, both in the general (conformal) and dual-conformal/conformal (dcc) cases, is presented. We show that in all the cases the Lauricella functions are homogeneous solutions of such systems of equations, also described as parametric 4K integrals of modified Bessel functions.

Structural Performance of Welded Built-Up Mega Composite Columns Subject to Compressive Force and Moment

The width thickness ratio of a concrete-filled steel tube column is limited in order to prevent the local buckling of the tube. Determined by column width and steel tube thickness, width-thickness ratio decides whether the cross-section is compact, non-compact or slender. Different compressive strength formulas are applied depending on the type of the cross-section. This study suggests the shape of binding frames which are placed at a constant distance inside the steel tube of a mega column having large cross-sections to improve the composite effect and confinement effect. Structural tests are conducted on 9 welded built-up and generic CFT columns with different steel thicknesses to evaluate their structural performance under compressive force and moment applied simultaneously. The maximum load of the welded built-up specimens with compact, non-compact and slender cross-sections is approximately 38%, 30% and 20% higher when compared with their generic CFT counterparts. While the effective stiffness of the members of concrete-filled composite columns is calculated by one formula, the difference among the specimens is greater than 10% depending on the width-thickness ratio. Therefore, it is deemed that effective stiffness needs to be categorized depending on the width-thickness ratio as in the case of compressive strength and bending strength.

Bulk metric reconstruction from boundary entanglement

Most of the literature in the \emph{bulk reconstruction program} in holography focuses on recovering local bulk operators propagating on a quasilocal bulk geometry and the knowledge of the bulk geometry is always assumed or guessed. The fundamental problem of the bulk reconstruction program, which is \emph{recovering the bulk background geometry (metric)} from the boundary CFT state is still outstanding. In this work, we formulate a recipe to extract the bulk metric itself from the boundary state, specifically, the modular Hamiltonian information of spherical subregions in the boundary. Our recipe exploits the recent construction of Kabat and Lifschytz \cite{Kabat:2017mun} to first compute the bulk two point function of scalar fields directly in the CFT without knowledge of the bulk metric or the equations of motion, and then to take a large scaling dimension limit (WKB) to extract the geodesic distance between two close points in the bulk i.e. the metric. As a proof of principle, we consider three dimensional bulk and selected CFT states such as the vacuum and the thermofield double states. We show that they indeed reproduce the pure AdS and the regions outside the Rindler wedge and the BTZ black hole \emph{up to a rigid conformal factor}. Since our approach does not rely on symmetry properties of the CFT state, it can be applied to reconstruct asymptotically AdS geometries dual to arbitrary general CFT states provided the modular Hamiltonian is available. We discuss several obvious extensions to the case of higher spacetime dimensions as well as some future applications, in particular, for constructing metric beyond the causal wedge of a boundary region. In the process, we also extend the construction of \cite{Kabat:2017mun} to incorporate the first order perturbative locality for AdS scalars.

Bulk phase shift, CFT Regge limit and Einstein gravity

The bulk phase shift, related to a CFT four-point function, describes two-to-two scattering at fixed impact parameter in the dual AdS spacetime. We describe its properties for a generic CFT and then focus on large N CFTs with classical bulk duals. We compute the bulk phase shift for vector operators using Regge theory. We use causality and unitarity to put bounds on the bulk phase shift. The resulting constraints bound three-point functions of two vector operators and the stress tensor in terms of the gap o the theory. Similar bounds should hold for any spinning operator in a CFT. Holographically this implies that in a classical gravitational theory any non-minimal coupling to the graviton, as well as any other particle with spin greater than or equal to two, is suppressed by the mass of higher spin particles.

Shear sum rule in higher derivative gravity theories

We study holographic shear sum rules in Einstein gravity with curvature squared corrections. Sum rules relate weighted integral over spectral densities of retarded correlators in the shear channel to the one point functions of the CFTs. The proportionality constant can be written in terms of the data of three point functions of the stress tensors of the CFT (t2 and t4). For CFTs dual to two derivative Einstein gravity, this proportionality constant is just frac{d}{2left(d+1right)} . This has been verified by a direct holographic computation of the retarded correlator for Einstein gravity in AdSd+1 black hole background. We compute corrections to the holographic shear sum rule in presence of higher derivative corrections to the Einstein-Hilbert action. We find agreement between the sum rule obtained from a general CFT analysis and holographic computation for Gauss Bonnet theories in AdS5 black hole background. We then generalize the sum rule for arbitrary curvature squared corrections to Einstein-Hilbert action in d ≥ 4. Evaluating the parameters t2 and t4 for the possible dual CFT in presence of such curvature corrections, we find an agreement with the general field theory derivation to leading order in coupling constants of the higher derivative terms.

Semiclassics, Goldstone bosons and CFT data

Hellerman et al. (arXiv:1505.01537) have shown that in a generic CFT the spectrum of operators carrying a large U(1) charge can be analyzed semiclassically in an expansion in inverse powers of the charge. The key is the operator state correspondence by which such operators are associated with a finite density superfluid phase for the theory quantized on the cylinder. The dynamics is dominated by the corresponding Goldstone hydrodynamic mode and the derivative expansion coincides with the inverse charge expansion. We illustrate and further clarify this situation by first considering simple quantum mechanical analogues. We then systematize the approach by employing the coset construction for non-linearly realized space-time symmetries. Focussing on CFT3 we illustrate the case of higher rank and non-abelian groups and the computation of higher point functions. Three point function coefficients turn out to satisfy universal scaling laws and correlations as the charge and spin are varied.

Exploring soft constraints on effective actions

We study effective actions for simultaneous breaking of space-time and internal symmetries. Novel features arise due to the mixing of Goldstone modes under the broken symmetries which, in contrast to the usual Adler's zero, leads to non-vanishing soft limits. Such scenarios are common for spontaneously broken SCFT's. We explicitly test these soft theorems for $\mathcal{N}=4$ sYM in the Coulomb branch both perturbatively and non-perturbatively. We explore the soft constraints systematically utilizing recursion relations. In the pure dilaton sector of a general CFT, we show that all amplitudes up to order $s^{n} \sim \partial^{2n}$ are completely determined in terms of the $k$-point amplitudes at order $s^k$ with $k \leq n$. Terms with at most one derivative acting on each dilaton insertion are completely fixed and coincide with those appearing in the conformal DBI, i.e. DBI in AdS. With maximal supersymmetry, the effective actions are further constrained, leading to new non-renormalization theorems. In particular, the effective action is fixed up to eight derivatives in terms of just one unknown four-point coefficient and one more coefficient for ten-derivative terms. Finally, we also study the interplay between scale and conformal invariance in this context.

Large interval limit of Rényi entropy at high temperature

In this paper, we propose a novel expansion to compute the large interval limit of the R\'enyi entropy of 2D CFT at high temperature. Via the replica trick, the single interval R\'enyi entropy of 2D CFT at finite temperature could be read from the partition function on $n$-sheeted torus connected with each other along a branch cut. We calculate the partition function by inserting a complete basis across the branch cut. Because of the monodromy condition across the branch cut in the large interval limit, the basis of the states should be the ones in the twist sector. We study the twist sector of a general module of CFT and find that there is an one-to-one correspondence between the twist sector states and the normal sector states. As an application, we revisit the non-compact free scalar theory and discuss the large interval limit of the R\'enyi entropy of this theory by using our proposal. We find complete agreement in the leading and next-leading orders with direct expansion of the exact partition function. Moreover, we prove the relation (\ref{th}) between thermal entropy and the entanglement entropy for a generic CFT with discrete spectrum.

Rényi entropy, stationarity, and entanglement of the conformal scalar

We extend previous work on the perturbative expansion of the Renyi entropy, $S_q$, around $q=1$ for a spherical entangling surface in a general CFT. Applied to conformal scalar fields in various spacetime dimensions, the results appear to conflict with the known conformal scalar Renyi entropies. On the other hand, the perturbative results agree with known Renyi entropies in a variety of other theories, including theories of free fermions and vector fields and theories with Einstein gravity duals. We propose a resolution stemming from a careful consideration of boundary conditions near the entangling surface. This is equivalent to a proper treatment of total-derivative terms in the definition of the modular Hamiltonian. As a corollary, we are able to resolve an outstanding puzzle in the literature regarding the Renyi entropy of ${\cal N}=4$ super-Yang-Mills near $q=1$. A related puzzle regards the question of stationarity of the renormalized entanglement entropy (REE) across a circle for a (2+1)-dimensional massive scalar field. We point out that the boundary contributions to the modular Hamiltonian shed light on the previously-observed non-stationarity. Moreover, IR divergences appear in perturbation theory about the massless fixed point that inhibit our ability to reliably calculate the REE at small non-zero mass.

Novel charges in CFT’s

In this paper we construct two infinite sets of self-adjoint commuting charges for a quite general CFT. They come out naturally by considering an infinite embedding chain of Lie algebras, an underlying structure that share all theories with gauge groups U(N), SO(N) and Sp(N). The generality of the construction allows us to carry all gauge groups at the same time in a unified framework, and so to understand the similarities among them. The eigenstates of these charges are restricted Schur polynomials and their eigenvalues encode the value of the correlators of two restricted Schurs. The existence of these charges singles out restricted Schur polynomials among the number of bases of orthogonal gauge invariant operators that are available in the literature.

A strongly coupled zig-zag transition

Abstract The zig-zag symmetry transition is a phase transition in 1D quantum wires, in which a Wigner lattice of electrons transitions to two staggered lattices. Previous studies model this transition as a Luttinger liquid coupled to a Majorana fermion. The model exhibits interesting RG flows, involving quenching of velocities in subsectors of the theory. We suggest an extension of the model which replaces the Majorana fermion by a more general CFT; this includes an experimentally realizable case with two Majorana fermions. We analyse the RG flow both in field theory and using AdS/CFT techniques in the large central charge limit of the CFT. The model has a rich phase structure with new qualitative features, already in the two Majorana fermion case. The AdS/CFT calculation involves considering back reaction in space-time to capture subleading effects.

Holographic calculations of Rényi entropy

We extend the approach of Casini, Huerta and Myers to a new calculation of the Renyi entropy of a general CFT in d dimensions with a spherical entangling surface, in terms of certain thermal partition functions. We apply this approach to calculate the Renyi entropy in various holographic models. Our results indicate that in general, the Renyi entropy will be a complicated nonlinear function of the central charges and other parameters which characterize the CFT. We also exhibit the relation between this new thermal calculation and a conventional calculation of the Renyi entropy where a twist operator is inserted on the spherical entangling surface. The latter insight also allows us to calculate the scaling dimension of the twist operators in the holographic models.

Novel CFT duals for extreme black holes

In this paper, we study the CFT duals for extreme black holes in the stretched horizon formalism. We consider the extremal RN, Kerr–Newman–AdS–dS, as well as the higher dimensional Kerr–AdS–dS black holes. In all these cases, we reproduce the well-established CFT duals. Actually we show that for stationary extreme black holes, the stretched horizon formalism always gives rise to the same dual CFT pictures as the ones suggested by ASG of corresponding near horizon geometries. Furthermore, we propose new CFT duals for 4D Kerr–Newman–AdS–dS and higher dimensional Kerr–AdS–dS black holes. We find that every dual CFT is defined with respect to a rotation in certain angular direction, along which the translation defines a U ( 1 ) Killing symmetry. In the presence of two sets of U ( 1 ) symmetry, the novel CFT duals are generated by the modular group SL ( 2 , Z ) , and for n sets of U ( 1 ) symmetry there are general CFT duals generated by T-duality group SL ( n , Z ) .

General hidden conformal symmetry of 4D Kerr-Newman and 5D Kerr black holes

There are two known CFT duals, namely the J-picture and the Q-picture, for a four-dimensional Kerr-Newman black hole, corresponding to the angular momentum J and the electric charge Q respectively. In our recent study we found a one-parameter class of CFT duals for extremal Kerr-Newman black hole, connecting these two pictures. In this paper we study these novel CFT duals for the generic non-extremal Kerr-Newman black hole. We investigate the hidden conformal symmetry in the low frequency scattering off Kerr-Newman black hole, from which the dual temperatures could be read. We find that there still exists a hidden conformal symmetry for a general CFT dual. We reproduce the correct Bekenstein-Hawking entropy from the Cardy formula, assuming the form of the central charge being invariant. Moreover we compute the retarded Green’s function in the general CFT dual picture and find it is in good match with the CFT prediction. Furthermore we discuss the hidden conformal symmetries of the five dimensional Kerr black hole and obtain the similar evidence to support the general dual CFT pictures.

Entanglement Rényi entropies in holographic theories

Ryu and Takayanagi conjectured a formula for the entanglement (von Neumann) entropy of an arbitrary spatial region in an arbitrary holographic field theory. The von Neumann entropy is a special case of a more general class of entropies called Renyi entropies. Using Euclidean gravity, Fursaev computed the entanglement Renyi entropies (EREs) of an arbitrary spatial region in an arbitrary holographic field theory, and thereby derived the RT formula. We point out, however, that his EREs are incorrect, since his putative saddle points do not in fact solve the Einstein equation. We remedy this situation in the case of two-dimensional CFTs, considering regions consisting of one or two intervals. For a single interval, the EREs are known for a general CFT; we reproduce them using gravity. For two intervals, the RT formula predicts a phase transition in the entanglement entropy as a function of their separation, and that the mutual information between the intervals vanishes for separations larger than the phase transition point. By computing EREs using gravity and CFT techniques, we find evidence supporting both predictions. We also find evidence that large-$N$ symmetric-product theories have the same EREs as holographic ones.

Radiation from the non-extremal fuzzball

The fuzzball proposal says that the information of the black hole state is distributed throughout the interior of the horizon in a ‘quantum fuzz’. There are special microstates where in the dual CFT we have ‘many excitations in the same state’; these are described by regular classical geometries without horizons. Jejjala et al (2005 Phys. Rev. D 71 124030) constructed non-extremal regular geometries of this type. Cardoso et al (2006 Phys. Rev. D 73 064031, 2007 Phys. Rev. D 76 105015) then found that these geometries had a classical instability. In this paper, we show that the energy radiated through the unstable modes is exactly the Hawking radiation for these microstates. We do this by (i) starting with the semiclassical Hawking radiation rate, (ii) using it to find the emission vertex in the CFT, (iii) replacing the Boltzman distributions of the generic CFT state with the ones describing the microstate of interest, (iv) observing that the emission now reproduces the classical instability. Because the CFT has ‘many excitations in the same state’ we get the physics of a Bose–Einstein condensate rather than a thermal gas, and the usually slow Hawking emission increases, by Bose enhancement, to a classically radiated field. This system therefore provides a complete gravity description of information-carrying radiation from a special microstate of the non-extremal hole.