2d CFTs Research Articles

Universal spectrum of 2d conformal field theory in the large c limit

Two-dimensional conformal field theories exhibit a universal free energy in the high temperature limit $T \to \infty$, and a universal spectrum in the Cardy regime, $\Delta \to \infty$. We show that a much stronger form of universality holds in theories with a large central charge $c$ and a sparse light spectrum. In these theories, the free energy is universal at all values of the temperature, and the microscopic spectrum matches the Cardy entropy for all $\Delta \geq c/6$. The same is true of three-dimensional quantum gravity; therefore our results provide simple necessary and sufficient criteria for 2d CFTs to behave holographically in terms of the leading spectrum and thermodynamics. We also discuss several applications to CFT and gravity, including operator dimension bounds derived from the modular bootstrap, universality in symmetric orbifolds, and the role of non-universal `enigma' saddlepoints in the thermodynamics of 3d gravity.

Universality of long-distance AdS physics from the CFT bootstrap

We begin by explicating a recent proof of the cluster decomposition principle in AdS_{d+1} from the CFT_d bootstrap in d > 2. The CFT argument also computes the leading interactions between distant objects in AdS, and we confirm the universal agreement between the CFT bootstrap and AdS gravity in the semi-classical limit. We proceed to study the generalization to 2d CFTs, which requires knowledge of the Virasoro conformal blocks in a lightcone OPE limit. We compute these blocks in a semiclassical, large central charge approximation, and use them to prove a suitably modified theorem. In particular, from the 2d bootstrap we prove the existence of large spin operators with fixed 'anomalous dimensions' indicative of the presence of deficit angles in AdS_3. As we approach the threshold for the BTZ black hole, interpreted as a CFT scaling dimension, the twist spectrum of large spin operators becomes dense. Due to the exchange of the Virasoro identity block, primary states above the BTZ threshold mimic a thermal background for light operators. We derive the BTZ quasi-normal modes, and we use the bootstrap equation to prove that the twist spectrum is dense. Corrections to thermality could be obtained from a more refined computation of the Virasoro conformal blocks.

Bootstrapping the O(N ) vector models

We study the conformal bootstrap for 3D CFTs with O(N) global symmetry. We obtain rigorous upper bounds on the scaling dimensions of the first O(N) singlet and symmetric tensor operators appearing in the $\phi_i \times \phi_j$ OPE, where $\phi_i$ is a fundamental of O(N). Comparing these bounds to previous determinations of critical exponents in the O(N) vector models, we find strong numerical evidence that the O(N) vector models saturate the bootstrap constraints at all values of N. We also compute general lower bounds on the central charge, giving numerical predictions for the values realized in the O(N) vector models. We compare our predictions to previous computations in the 1/N expansion, finding precise agreement at large values of N.

Probing higher spin black holes from CFT

In a class of 2D CFTs with higher spin symmetry, we compute thermal two-point functions of certain scalar primary operators in the presence of nonzero chemical potential for higher spin charge. These are shown to agree with the same quantity calculated holographically using scalar fields propagating in a charged black hole background of 3D higher spin gravity. This match serves as further evidence for the duality between W_N minimal models at large central charge and 3D higher spin gravity. It also supports a recent prescription for computing boundary correlators of multi-trace scalar primary operators in higher spin theories.

Large N techniques for Nekrasov partition functions and AGT conjecture

The AGT conjecture relates $ \mathcal{N} $ = 2 4d SUSY gauge theories to 2d CFTs. Matrix model techniques can be used to investigate both sides of this relation. The large N limit refers here to the size of Young tableaux in the expression of the gauge theory partition function. It corresponds to the vanishing of Ω-background equivariant deformation parameters, and should not be confused with the t’Hooft expansion at large number of colors. In this paper, a saddle point approach is employed to study the Nekrasov-Shatashvili limit of the gauge theory, leading to define β-deformed, or quantized, Seiberg-Witten curve and differential form. Then this formalism is compared to the large N limit of the Dijkgraaf-Vafa β-ensemble. A transformation law relating the wave functions appearing at both sides of the conjecture is proposed. It implies a transformation of the Seiberg-Witten 1-form in agreement with the definition specified earlier. As a side result, a remarkable property of $ \mathcal{N} $ = 2 theories emerged: the instanton contribution to the partition function can be determined from the perturbative term analysis.

$ \mathcal{N} = 2 $ superconformal blocks and instanton partition functions

We consider the problem of computing (irregular) conformal blocks in 2d CFTs whose chiral symmetry algebra is the N=2 superconformal algebra. Our construction uses two ingredients: (i) the relation between the representation theories of the N=2 superconformal algebra and the affine sl(2) algebra, extended to the level of the conformal blocks, and (ii) the relation between affine sl(2) conformal blocks and instanton partition functions in the 4d N=2 SU(2) gauge theory with a surface defect. By combining these two facts we derive combinatorial expressions for the N=2 superconformal blocks in the Gaiotto limit.

Carving out the space of 4D CFTs

We introduce a new numerical algorithm based on semidefinite programming to efficiently compute bounds on operator dimensions, central charges, and OPE coefficients in 4D conformal and N=1 superconformal field theories. Using our algorithm, we dramatically improve previous bounds on a number of CFT quantities, particularly for theories with global symmetries. In the case of SO(4) or SU(2) symmetry, our bounds severely constrain models of conformal technicolor. In N=1 superconformal theories, we place strong bounds on dim(Phi*Phi), where Phi is a chiral operator. These bounds asymptote to the line dim(Phi*Phi) <= 2 dim(Phi) near dim(Phi) ~ 1, forbidding positive anomalous dimensions in this region. We also place novel upper and lower bounds on OPE coefficients of protected operators in the Phi x Phi OPE. Finally, we find examples of lower bounds on central charges and flavor current two-point functions that scale with the size of global symmetry representations. In the case of N=1 theories with an SU(N) flavor symmetry, our bounds on current two-point functions lie within an O(1) factor of the values realized in supersymmetric QCD in the conformal window.

Emergent IR dual 2d CFTs in chargedAdS5black holes

We study the possible dynamical emergence of IR conformal invariance describing the low energy excitations of near-extremal $R$-charged global ${\mathrm{AdS}}_{5}$ black holes. We find interesting behavior especially when we tune parameters in such a way that the relevant extremal black holes have classically vanishing horizon area, i.e. no classical ground-state entropy, and when we combine the low energy limit with a large $N$ limit of the dual gauge theory. We consider both near-BPS and non-BPS regimes and their near horizon limits, emphasize the differences between the local ${\mathrm{AdS}}_{3}$ throats emerging in either case, and discuss potential dual IR 2d CFTs for each case. We compare our results with the predictions obtained from the Kerr/CFT correspondence, and obtain a natural quantization for the central charge of the near-BPS emergent IR CFT which we interpret in terms of the open strings stretched between giant gravitons.

Current OPEs in superconformal theories

It's well known that in conformal theories the two- and three-point functions of a subset of the local operators-the conformal primaries-suffice, via the operator product expansion (OPE), to determine all local correlation functions of operators. It's less well known that, in superconformal theories, the OPE of superdescendants is generally undetermined from those of the superprimaries, and there is no universal notion of superconformal blocks. We recall these and related aspects of 4d (S)CFTs, and then we focus on the super operator product expansion (sOPE) of conserved currents in 4d N=1 SCFTs. The current-current OPE J(x)J(0) has applications to general gauge mediation. We show how superconformal symmetry, when combined with current conservation, determines the OPE coefficients of superconformal descendants in terms of those of the superconformal primaries. We show that only integer-spin real superconformal primary operators of vanishing R-charge, and their descendants, appear in the sOPE. We also discuss superconformal blocks for four-point functions of the conserved currents.

The virtue of defects in 4D gauge theories and 2D CFTs

We advance a correspondence between the topological defect operators in Liouville and Toda conformal field theories - which we construct - and loop operators and domain wall operators in four dimensional N=2 supersymmetric gauge theories on S^4. Our computation of the correlation functions in Liouville/Toda theory in the presence of topological defect operators, which are supported on curves on the Riemann surface, yields the exact answer for the partition function of four dimensional gauge theories in the presence of various walls and loop operators; results which we can quantitatively substantiate with an independent gauge theory analysis. As an interesting outcome of this work for two dimensional conformal field theories, we prove that topological defect operators and the Verlinde loop operators are different descriptions of the same operators.

Large-N limits of 2d CFTs, quivers and AdS3 duals

We explore the large-N limits of 2d CFTs, focusing mostly on WZW models and their cosets. The $SU(N)_k$ theory is parametrized in this limit by a 't Hooft-like coupling. We show a duality between strong coupling, where the theory is described by almost free fermions, and weak coupling where the theory is described by bosonic fields by an analysis of spectra and correlators. The AdS$_3$ dual is described, and several quantitative checks are performed. Besides the more standard states that should correspond to bulk black holes we find ground states with large degeneracy that can dominate the standard Cardy entropy at weak coupling and are expected to correspond to regular horizonless semiclassical bulk solutions.

Microscopic realization of the Kerr/CFT correspondence

Supersymmetric M/string compactifications to five dimensions contain BPS black string solutions with magnetic graviphoton charge P and near-horizon geometries which are quotients of AdS_3 x S^2. The holographic duals are typically known 2D CFTs with central charges c_L=c_R=6P^3 for large P. These same 5D compactifications also contain non-BPS but extreme Kerr-Newman black hole solutions with SU(2)_L spin J_L and electric graviphoton charge Q obeying Q^3 \leq J_L^2. It is shown that in the maximally charged limit Q^3 -> J_L^2, the near-horizon geometry coincides precisely with the right-moving temperature T_R=0 limit of the black string with magnetic charge P=J_L^{1/3}. The known dual of the latter is identified as the c_L=c_R=6J_L CFT predicted by the Kerr/CFT correspondence. Moreover, at linear order away from maximality, one finds a T_R \neq 0 quotient of the AdS_3 factor of the black string solution and the associated thermal CFT entropy reproduces the linearly sub-maximal Kerr-Newman entropy. Beyond linear order, for general Q^3<J_L^2, one has a finite-temperature quotient of a warped deformation of the magnetic string geometry. The corresponding dual deformation of the magnetic string CFT potentially supplies, for the general case, the c_L=c_R=6J_L CFT predicted by Kerr/CFT.

Self-dual warpedAdS3black holes

We study a new class of solutions of three-dimensional topological massive gravity. These solutions can be taken as non-extremal black holes, with their extremal counterparts being discrete quotients of spacelike warped AdS$_3$ along the $U(1)_L$ isometry. We study the thermodynamics of these black holes and show that the first law is satisfied. We also show that for consistent boundary conditions, the asymptotic symmetry generators form only one copy of the Virasoro algebra with central charge $c_L = \frac{4\nu\ell}{G(\nu^2+3)}$, with which the Cardy formula reproduces the black hole entropy. We compute the real-time correlators of scalar perturbations and find a perfect match with the dual CFT predictions. Our study provides a novel example of warped AdS/CFT correspondence: the self-dual warped AdS$_3$ black hole is dual to a CFT with nonvanishing left central charge. Moreover our investigation suggests that the quantum topological massive gravity asymptotic to the same spacelike warped AdS$_3$ in different consistent ways may be dual to different 2D CFTs.

Entanglement Entropy of AdS Black Holes

We review recent progress in understanding the entanglement entropy of gravitational configurations for anti-de Sitter gravity in two and three spacetime dimensions using the AdS/CFT correspondence. We derive simple expressions for the entanglement entropy of two- and three-dimensional black holes. In both cases, the leading term of the entanglement entropy in the large black hole mass expansion reproduces exactly the Bekenstein-Hawking entropy, whereas the subleading term behaves logarithmically. In particular, for the BTZ black hole the leading term of the entanglement entropy can be obtained from the large temperature expansion of the partition function of a broad class of 2D CFTs on the torus.

Twofold hidden conformal symmetries of the Kerr-Newman black hole

In this paper, we suggest that there are two different individual 2D CFTs holographically dual to the Kerr-Newman black hole, coming from the corresponding two possible limits --- the Kerr/CFT and Reissner-Nordstr\"om/CFT correspondences, namely there exist the Kerr-Newman/CFTs dualities. A probe scalar field at low frequencies turns out can exhibit two different 2D conformal symmetries (named by $J$- and $Q$-pictures, respectively) in its equation of motion when the associated parameters are suitably specified. These twofold dualities are supported by the matchings of entropies, absorption cross sections and real time correlators computed from both the gravity and the CFT sides. Our results lead to a fascinating "microscopic no hair conjecture" --- for each macroscopic hair parameter, in additional to the mass of a black hole in the Einstein-Maxwell theory, there should exist an associated holographic CFT$_2$ description.

GCA in 2d

We make a detailed study of the infinite dimensional Galilean Conformal Algebra (GCA) in the case of two spacetime dimensions. Classically, this algebra is precisely obtained from a contraction of the generators of the relativistic conformal symmetry in 2d. Here we find quantum mechanical realisations of the (centrally extended) GCA by considering scaling limits of certain 2d CFTs. These parent CFTs are non-unitary and have their left and right central charges become large in magnitude and opposite in sign. We therefore develop, in parallel to the usual machinery for 2d CFT, many of the tools for the analysis of the quantum mechanical GCA. These include the representation theory based on GCA primaries, Ward identities for their correlation functions and a nonrelativistic Kac table. In particular, the null vectors of the GCA lead to differential equations for the four point function. The solution to these equations in the simplest case is explicitly obtained and checked to be consistent with various requirements.

Holographic Entanglement Entropy of the BTZ Black Hole

We investigate quantum entanglement of gravitational configurations in 3D AdS gravity using the AdS/CFT correspondence. We derive explicit formulas for the holographic entanglement entropy (EE) of the BTZ black hole, conical singularities and regularized AdS$_{3}$. The leading term in the large temperature expansion of the holographic EE of the BTZ black hole reproduces exactly its Bekenstein-Hawking entropy S_BH, whereas the subleading term behaves as ln S_BH. We also show that the leading term of the holographic EE for the BTZ black hole can be obtained from the large temperature expansion of the partition function of a broad class of 2D CFTs on the torus. This result indicates that black hole EE is not a fundamental feature of the underlying theory of quantum gravity but emerges when the semiclassical notion of spacetime geometry is used to describe the black hole.

Higher-spin fields in braneworlds

The dynamics of higher-spin fields in braneworlds is discussed. In particular, we study fermionic and bosonic higher-spin fields in AdS 5 and their localization on branes. We find that four-dimensional zero modes exist only for spin-one fields, if there are no couplings to the boundaries. If boundary couplings are allowed, as in the case of the bulk graviton, all bosons acquire a zero mode irrespective of their spin. We show that there are boundary conditions for fermions, which generate chiral zero modes in the four-dimensional spectrum. We also propose a gauge invariant on-shell action with cubic interactions by adding non-minimal couplings, which depend on the Weyl tensor. In addition, consistent couplings between higher-spin fields and matter on the brane are presented. Finally, in the AdS/CFT correspondence, where bulk 5D theories on AdS are related to 4D CFTs, we explicitly discuss the holographic picture of higher-spin theories in AdS 5 with and without boundaries.

Holography for fermions

The holographic interpretation is a useful tool to describe 5D field theories in a 4D language. In particular it allows one to relate 5D AdS theories with 4D CFTs. We elaborate on the 5D/4D dictionary for the case of fermions in AdS5 with boundaries. This dictionary is quite useful to address phenomenological issues in a very simple manner, as we show by giving some examples.

Monopole Operators and Mirror Symmetry in Three Dimensions

We study vortex-creating, or monopole, operators in 3d CFTs which are the infrared limit of N=2 and N=4 supersymmetric QEDs in three dimensions. Using large-Nf expansion, we construct monopole operators which are primaries of short representations of the superconformal algebra. Mirror symmetry in three dimensions makes a number of predictions about such operators, and our results confirm these predictions. Furthermore, we argue that some of our large-Nf results are exact. This implies, in particular, that certain monopole operators in N=4 d=3 SQED with Nf=1 are free fields. This amounts to a proof of 3d mirror symmetry in a special case.