Four-dimensional Conformal Field Theories Research Articles

Three-point functions of a fermionic higher-spin current in 4D conformal field theory

We investigate the properties of a four-dimensional conformal field theory possessing a fermionic higher-spin current $Q_{\alpha(2k) \dot{\alpha}}$. Using a computational approach, we examine the number of independent tensor structures contained in the three-point correlation functions of two fermionic higher-spin currents with the conserved vector current $V_{m}$, and with the energy-momentum tensor $T_{m n}$. In particular, the $k = 1$ case corresponds to a "supersymmetry-like" current, that is, a fermionic conserved current with identical properties to the supersymmetry current which appears in $\mathcal{N} = 1$ superconformal field theories. However, we show that in general, the three-point correlation functions $\langle QQV \rangle$ and $\langle QQT \rangle$ are not consistent with $\mathcal{N} = 1$ supersymmetry.

Renormalized holographic entanglement entropy for quadratic curvature gravity

We derive a covariant expression for the renormalized holographic entanglement entropy for conformal field theories (CFTs) dual to quadratic curvature gravity in arbitrary dimensions. This expression is written as the sum of the bare entanglement entropy functional obtained using standard conical defect techniques, and a counterterm defined at the boundary of the extremal surface of the functional. The latter corresponds to the cod-2 self-replicating part of the extrinsic counterterms when evaluated on the replica orbifold. This renormalization method isolates the universal terms of the holographic entanglement entropy functional. We use it to compute the standard $C$-function candidate for CFTs of arbitrary dimension, and the type-B anomaly coefficient $c$ for four-dimensional CFTs.

Strongly γ-Deformed N=4 Supersymmetric Yang-Mills Theory as an Integrable Conformal Field Theory.

We demonstrate by explicit multiloop calculation that γ-deformed planar N=4 supersymmetric Yang-Mills (SYM) theory, supplemented with a set of double-trace counterterms, 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 nonunitary four-dimensional conformal field theory. 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 Lorentz spin. We conjecture that both conformal symmetry and integrability should survive in γ-deformed planar N=4 SYM theory for arbitrary values of the deformation parameters.

Canceling the Weyl anomaly from a position-dependent coupling

Once we put a quantum field theory on a curved manifold, it is natural to further assume that coupling constants are position-dependent. The position-dependent coupling constants then provide an extra contribution to the Weyl anomaly so that we may attempt to cancel the entire Weyl anomaly on the curved manifold. We show that such a cancellation is possible for constant Weyl transformation or infinitesimal but generic Weyl transformation in two- and four-dimensional conformal field theories with exactly marginal deformations. When the Weyl scaling factor is annihilated by conformal powers of Laplacian (e.g., by the Fradkin-Tseytlin-Riegert-Paneitz operator in four dimensions), the cancellation persists even at the finite order thanks to a nice mathematical property of the $Q$ curvature under the Weyl transformation.

Universal bounds on operator dimensions from the average null energy condition

We show that the average null energy condition implies novel lower bounds on the scaling dimensions of highly-chiral primary operators in four-dimensional conformal field theories. Denoting the spin of an operator by a pair of integers left(k,overline{k}right) specifying the transformations under chiral mathfrak{s}mathfrak{u}(2) rotations, we explicitly demonstrate these new bounds for operators transforming in (k, 0) and (k, 1) representations for sufficiently large k. Based on these calculations, along with intuition from free field theory, we conjecture that in any unitary conformal field theory, primary local operators of spin left(k,overline{k}right) and scaling dimension Δ satisfy varDelta ge max left{k,overline{k}right} . If left|k-overline{k}right|>4 , this is stronger than the unitarity bound.

Displacement Operators and Constraints on Boundary Central Charges.

Boundary conformal field theories have several additional terms in the trace anomaly of the stress tensor associated purely with the boundary. We constrain the corresponding boundary central charges in three- and four-dimensional conformal field theories in terms of two- and three-point correlation functions of the displacement operator. We provide a general derivation by comparing the trace anomaly with scale dependent contact terms in the correlation functions. We conjecture a relation between the a-type boundary charge in three dimensions and the stress tensor two-point function near the boundary. We check our results for several free theories.

Einstein gravity 3-point functions from conformal field theory

We study stress tensor correlation functions in four-dimensional conformal field theories with large N and a sparse spectrum. Theories in this class are expected to have local holographic duals, so effective field theory in anti-de Sitter suggests that the stress tensor sector should exhibit universal, gravity-like behavior. At the linearized level, the hallmark of locality in the emergent geometry is that stress tensor three-point functions 〈T T T 〉, normally specified by three constants, should approach a universal structure controlled by a single parameter as the gap to higher spin operators is increased. We demonstrate this phenomenon by a direct CFT calculation. Stress tensor exchange, by itself, violates causality and unitarity unless the three-point functions are carefully tuned, and the unique consistent choice exactly matches the prediction of Einstein gravity. Under some assumptions about the other potential contributions, we conclude that this structure is universal, and in particular, that the anomaly coefficients satisfy a ≈ c as conjectured by Camanho et al. The argument is based on causality of a four-point function, with kinematics designed to probe bulk locality, and invokes the chaos bound of Maldacena, Shenker, and Stanford.

Universality of small black hole instability in AdS/CFT

[Formula: see text] Type IIb supergravity compactifications on five-dimensional Einstein manifolds [Formula: see text] realize holographic duals to four-dimensional conformal field theories. Black holes in such geometries are dual to thermal states in these CFTs. When black holes become sufficiently small in (global) [Formula: see text], they are expected to suffer Gregory–Laflamme instability with respect to localization on [Formula: see text]. Previously, the instability was demonstrated for gravitational dual of [Formula: see text] SYM, where [Formula: see text]. We extend stability analysis to arbitrary [Formula: see text]. We point out that the quasinormal mode equation governing the instabilities is universal. The precise onset of the instability is [Formula: see text]-sensitive, as it is governed by the lowest nonvanishing eigenvalue [Formula: see text] of its Laplacian.

Shape Dependence of Holographic Rényi Entropy in Conformal Field Theories.

We develop a framework for studying the well-known universal term in the Rényi entropy for an arbitrary entangling region in four-dimensional conformal field theories that are holographically dual to gravitational theories. The shape dependence of the Rényi entropy S_{n} is described by two coefficients: f_{b}(n) for traceless extrinsic curvature deformations and f_{c}(n) for Weyl tensor deformations. We provide the first calculation of the coefficient f_{b}(n) in interacting theories by relating it to the stress tensor one-point function in a deformed hyperboloid background. The latter is then determined by a straightforward holographic calculation. Our results show that a previous conjecture f_{b}(n)=f_{c}(n), motivated by surprising evidence from a variety of free field theories and studies of conical defects, fails holographically.

Upper bound on the mass anomalous dimension in many-flavor gauge theories: a conformal bootstrap approach

We study four-dimensional conformal field theories with an $SU(N)$ global symmetry by employing the numerical conformal bootstrap. We consider the crossing relation associated with a four-point function of a spin~$0$ operator~$\phi_i^{\Bar{k}}$ which belongs to the adjoint representation of $SU(N)$. For~$N=12$ for example, we found that the theory contains a spin~$0$ $SU(12)$-breaking relevant operator when the scaling dimension of~$\phi_i^{\Bar{k}}$, $\Delta_{\phi_i^{\Bar{k}}}$, is smaller than~$1.71$. Considering the lattice simulation of many-flavor quantum chromodynamics with $12$~flavors on the basis of the staggered fermion, the above $SU(12)$-breaking relevant operator, if it exists, would be induced by the flavor-breaking effect of the staggered fermion and prevent an approach to an infrared fixed point. Actual lattice simulations do not show such signs. Thus, assuming the absence of the above $SU(12)$-breaking relevant operator, we have an upper bound on the mass anomalous dimension at the fixed point~$\gamma_m^*\leq1.29$ from the relation~$\gamma_m^*=3-\Delta_{\phi_i^{\Bar{k}}}$. Our upper bound is not so strong practically but it is strict within the numerical accuracy. We also find a kink-like behavior in the boundary curve for the scaling dimension of another $SU(12)$-breaking operator.

Lie algebra extensions of current algebras on S3

An affine Kac–Moody algebra is a central extension of the Lie algebra of smooth mappings from S1 to the complexification of a Lie algebra. In this paper, we shall introduce a central extension of the Lie algebra of smooth mappings from S3 to the quaternization of a Lie algebra and investigate its root space decomposition. We think this extension of current algebra might give a mathematical tool for four-dimensional conformal field theory as Kac–Moody algebras give it for two-dimensional conformal field theory.

Infinite Chiral Symmetry in Four Dimensions

We describe a new correspondence between four-dimensional conformal field theories with extended supersymmetry and two-dimensional chiral algebras. The meromorphic correlators of the chiral algebra compute correlators in a protected sector of the four-dimensional theory. Infinite chiral symmetry has far-reaching consequences for the spectral data, correlation functions, and central charges of any four-dimensional theory with \({\mathcal{N}=2}\) superconformal symmetry.

On conformal field theories with extremal $ \frac{a}{c} $ values

Unitary conformal field theories (CFTs) are believed to have positive (non-negative) energy correlators. Energy correlators are universal observables in higher-dimensional CFTs built out of integrated Wightman functions of the stress-energy tensor. We analyze energy correlators in parity invariant four-dimensional CFTs. The goal is to use the positivity of energy correlators to further constrain unitary CFTs. It is known that the positivity of the simplest one-point energy correlator implies that $ \frac{1}{3}\leq \frac{a}{c}\leq \frac{31 }{18 } $ where a and c are the Weyl anomaly coefficients. We use the positivity of higher point energy correlators to show that CFTs with extremal values of $ \frac{a}{c} $ have trivial scattering observables. More precisely, for $ \frac{a}{c}=\frac{1}{3} $ and $ \frac{a}{c}=\frac{31 }{18 } $ all energy correlators are fixed to be the ones of the free boson and the free vector theory correspondingly. Similarly, we show that the positivity and finiteness of energy correlators together imply that the three-point function of the stress tensor in a CFT cannot be proportional to the one in the theory of free boson, free fermion or free vector field.

Six-dimensional methods for four-dimensional conformal field theories. II. Irreducible fields

This note supplements an earlier paper on conformal field theories. There it was shown how to construct tensor, spinor, and spinor-tensor primary fields in four dimensions from their counterparts in six dimensions, where conformal transformations act simply as $SO(4,2)$ Lorentz transformations. Here we show how to constrain fields in six dimensions so that the corresponding primary fields in four dimensions transform according to irreducible representations of the four-dimensional Lorentz group, even when the irreducibility conditions on these representations involve the four-component Levi-Civita tensor ${ϵ}_{\ensuremath{\mu}\ensuremath{\nu}\ensuremath{\rho}\ensuremath{\sigma}}$.

Gravity and Large Black Holes in Randall-Sundrum II Braneworlds

We show how to construct low energy solutions to the Randall-Sundrum II (RSII) model by using an associated five-dimensional anti-de Sitter space (AdS(5)) and/or four-dimensional conformal field theory (CFT(4)) problem. The RSII solution is given as a perturbation of the AdS(5)-CFT(4) solution, with the perturbation parameter being the radius of curvature of the brane metric compared to the AdS length ℓ. The brane metric is then a specific perturbation of the AdS(5)-CFT(4) boundary metric. For low curvatures the RSII solution reproduces 4D general relativity on the brane. Recently, AdS(5)-CFT(4) solutions with a 4D Schwarzschild boundary metric were numerically constructed. We modify the boundary conditions to numerically construct large RSII static black holes with radius up to ~20ℓ. For a large radius, the RSII solutions are indeed close to the associated AdS(5)-CFT(4) solution.

A-MAXIMIZATION IN ${\mathcal N}=1$ SUPERSYMMETRIC Spin(10) GAUGE THEORIES

A summary is reported on our previous publications about four-dimensional [Formula: see text] supersymmetric Spin(10) gauge theory with chiral superfields in the spinor and vector representations in the non-Abelian Coulomb phase. Carrying out the method of a-maximization, we studied decoupling operators in the infrared and the renormalization flow of the theory. We also give a brief review on the non-Abelian Coulomb phase of the theory after recalling the unitarity bound and the a-maximization procedure in four-dimensional conformal field theory.

Six-dimensional methods for four-dimensional conformal field theories

The calculation of both spinor and tensor Green's functions in four-dimensional conformally invariant field theories can be greatly simplified by six-dimensional methods. For this purpose, four-dimensional fields are constructed as projections of fields on the hypercone in six-dimensional projective space, satisfying certain transversality conditions. In this way some Green's functions in conformal field theories are shown to have structures more general than those commonly found by use of the inversion operator. These methods fit in well with the assumption of AdS/CFT duality. In particular, it is transparent that if fields on AdS$_5$ approach finite limits on the boundary of AdS$_5$, then in the conformal field theory on this boundary these limits transform with conformal dimensionality zero if they are tensors (of any rank), but with conformal dimension 1/2 if they are spinors or spinor-tensors.

Gravitational Lorentz violation and superluminality via AdS/CFT duality

A weak quantum mechanical coupling is constructed permitting superluminal communication within a preferred region of a gravitating ${\mathrm{AdS}}_{5}$ spacetime. This is achieved by adding a spatially nonlocal perturbation of a special kind to the Hamiltonian of a four-dimensional conformal field theory with a weakly coupled ${\mathrm{AdS}}_{5}$ dual, such as maximally supersymmetric Yang-Mills theory. In particular, two issues are given careful treatment: (1) the UV-completeness of our deformed conformal field theory (CFT), guaranteeing the existence of a ``deformed string theory'' AdS dual and (2) the demonstration that superluminal effects can take place in AdS, both on its boundary as well as in the bulk. Exotic Lorentz-violating properties such as these may have implications for tests of general relativity, addressing the cosmological constant problem, or probing behind horizons. Our construction may give insight into the interpretation of wormhole solutions in Euclidean AdS gravity.

Entanglement entropy, trace anomalies and holography

The holographic representation of the entanglement entropy of four-dimensional conformal field theories is studied. By generalizing the replica trick the anomalous terms in the entanglement entropy are evaluated. The same terms in the holographic representation are calculated by a method which does not require the solution of the equations of motion or a cut-off. The two calculations disagree for rather generic geometries. The reasons for the disagreement are analyzed.

Non-critical holography and four-dimensional CFT's with fundamentals

We find non-critical string backgrounds in five and eight dimensions, holographically related to four-dimensional conformal field theories with N=0 and N=1 supersymmetries. In the five-dimensional case we find an AdS_5 background metric for a string model related to non-supersymmetric, conformal QCD with large number of colors and flavors and discuss the conjectured existence of a conformal window from the point of view of our solution. In the eight-dimensional string theory, we build a family of solutions of the form AdS_5 x \tilde{S}^3 with \tilde{S}^3 a squashed three-sphere. For a special value of the ratio N_f/N_c, the background can be interpreted as the supersymmetric near-horizon limit of a system of color and flavor branes on R^{1,3} times a known four-dimensional generalization of the cigar. The N=1 dual theory with fundamental matter should have an IR fixed point only for a fixed ratio N_f/N_c. General features of the string/gauge theory correspondence for theories with fundamental flavors are also addressed.