Three-dimensional CFTs Research Articles

CFT correlators from shape deformations in Cubic Curvature Gravity

We find a covariant expression for the universal part of the holographic entanglement entropy which is valid for CFTs dual to generic higher curvature gravities in up to five bulk dimensions. We use this functional to compute universal coefficients of stress-tensor correlators in three-dimensional CFTs dual to Cubic Curvature Gravity. Using gauge/gravity duality, we work out an expression for the entanglement entropy of deformed entangling regions and read the coefficients from the power expansion of the entropy in the deformation parameter. In particular, we obtain the t4 coefficient of the 3-point function and exhibit a difference between the results obtained using the entanglement entropy functional for minimal and non-minimal splittings. We compare the obtained expressions for t4 derived considering both splittings with results obtained through other holographic methods which are splitting-independent. We find agreement with the result obtained from the non-minimal splitting, whereas the result derived from the minimal splitting is inconsistent and it is therefore ruled out.

Disks globally maximize the entanglement entropy in 2 + 1 dimensions

The entanglement entropy corresponding to a smooth region in general three-dimensional CFTs contains a constant universal term, −F ⊂ SEE. For a disk region, F|disk ≡ F0 coincides with the free energy on \U0001d54a3 and provides an RG-monotone for general theories. As opposed to the analogous quantity in four dimensions, the value of F generally depends in a complicated (and non-local) way on the geometry of the region and the theory under consideration. For small geometric deformations of the disk in general CFTs as well as for arbitrary regions in holographic theories, it has been argued that F is precisely minimized by disks. Here, we argue that F is globally minimized by disks with respect to arbitrary regions and for general theories. The proof makes use of the strong subadditivity of entanglement entropy and the geometric fact that one can always place an osculating circle within a given smooth entangling region. For topologically non-trivial entangling regions with nB boundaries, the general bound can be improved to F ≥ nBF0. In addition, we provide accurate approximations to F valid for general CFTs in the case of elliptic regions for arbitrary values of the eccentricity which we check against lattice calculations for free fields. We also evaluate F numerically for more general shapes in the so-called “Extensive Mutual Information model”, verifying the general bound.

OPE meets semiclassics

We show that the correlator of three large charge operators with minimal scaling dimension can be computed semiclassically in CFTs with a $U(1)$ symmetry for arbitrary fixed values of the ratios of their charges. We obtain explicitly the OPE coefficient from the numerical solution of a nonlinear boundary value problem in the conformal superfluid EFT in $3d$. The result applies in all three-dimensional CFTs with a $U(1)$ symmetry whose large charge sector is a superfluid.

Shape dependence of renormalized holographic entanglement entropy

We study the holographic entanglement entropy of deformed entangling regions in three-dimensional CFTs dual to Einstein-AdS gravity, using a renormalization scheme based on the addition of extrinsic counterterms. In this prescription, when even- dimensional manifolds are considered, the universal contribution to the entanglement entropy is identified as the renormalized volume of the Ryu-Takayanagi hypersurface, which is written as the sum of a topological and a curvature term. It is shown that the change in the renormalized entanglement entropy due to the deformation of the entangling surface is encoded purely in the curvature contribution. In turn, as the topological part is given by the Euler characteristic of the Ryu-Takayanagi surface, it remains shape independent. Exploiting the covariant character of the extrinsic counterterms, we apply the renormalization scheme for the case of deformed entangling regions in AdS4/CFT3, recovering the results found in the literature. Finally, we provide a derivation of the relation between renormalized entanglement entropy and Willmore energy. The presence of a lower bound of the latter makes manifest the relation between the AdS curvature of the Ryu-Takayanagi surface and the strong subadditivity property.

Partition functions on slightly squashed spheres and flux parameters

We argue that the conjectural relation between the subleading term in the small-squashing expansion of the free energy of general three-dimensional CFTs on squashed spheres and the stress-tensor three-point charge t4 proposed in arXiv:1808.02052: {F}_{{mathbbm{S}}_{varepsilon}^3}^{(3)}(0)=frac{1}{630}{pi}^4{C}_T{t}_4 , holds for an infinite family of holographic higher-curvature theories. Using holographic calculations for quartic and quintic Generalized Quasi-topological gravities and general-order Quasi-topological gravities, we identify an analogous analytic relation between such term and the charges t2 and t4 valid for five-dimensional theories: {F}_{{mathbbm{S}}_{varepsilon}^5}^{(3)}(0)=frac{2}{15}{pi}^6{C}_Tleft[1+frac{3}{40}{t}_2+frac{23}{630}{t}_4right] . We test both conjectures using new analytic and numerical results for conformally-coupled scalars and free fermions, finding perfect agreement.

An AdS/EFT correspondence at large charge

Considering theories in sectors of large global charge Q results in a semiclassical effective field theory (eft) description for some strongly-coupled conformal field theories (cfts) with continuous global symmetries. Hence, when studying dualities at large charge, we can have control over the strongly coupled side of the duality and gain perturbative access to both dual pairs.In this work we discuss the AdS/CFT correspondence in the regime Q≫CT≫1 where both the eft and gravity descriptions are valid and stable (CT being the central charge). We present the observation that the ground state energy as a function of the Abelian charge Q for a simple eft in some three-dimensional cfts coincides with the expression for the mass of an anti-de Sitter–Reissner–Nordström black hole as a function of its charge. Using this observation we propose a tentative dictionary relating cft, eft and holographic descriptions. We also find agreement for the higher-derivative corrections on both sides, suggesting a large-CT expansion on the eft side.

Renormalization of entanglement entropy from topological terms

We propose a renormalization scheme for entanglement entropy of three-dimensional CFTs with a four-dimensional asymptotically AdS gravity dual in the context of the gauge/gravity correspondence. The procedure consists in adding the Chern form as a boundary term to the area functional of the Ryu-Takayanagi minimal surface. We provide an explicit prescription for the renormalized entanglement entropy, which is derived via the replica trick. This is achieved by considering a Euclidean gravitational action renormalized by the addition of the Chern form at the spacetime boundary, evaluated in the conically-singular replica manifold. We show that the addition of this boundary term cancels the divergent part of the entanglement entropy, recovering the results obtained by Taylor and Woodhead. We comment on how this prescription for renormalizing the entanglement entropy is in line with the general program of topological renormalization in asymptotically AdS gravity.

Comments on squashed-sphere partition functions

We study the partition function of odd-dimensional conformal field theories placed on spheres with a squashed metric. We establish that the round sphere provides a local extremum for the free energy which, in general, is not a global extremum. In addition, we show that the leading quadratic correction to the free energy around this extremum is proportional to the coefficient, CT , determining the two-point function of the energy-momentum tensor in the CFT. For three-dimensional CFTs, we compute explicitly this proportionality constant for a class of squashing deformations which preserve an SU(2) × U(1) isometry group on the sphere. In addition, we evaluate the free energy as a function of the squashing parameter for theories of free bosons, free fermions, as well as CFTs holographically dual to Einstein gravity with a negative cosmological constant. We observe that, after suitable normalization, the dependence of the free energy on the squashing parameter for all these theories is nearly universal for a large region of parameter space and is well approximated by a simple quadratic function arising from holography. We generalize our results to five-dimensional CFTs and, in this context, we also study theories holographically dual to six-dimensional Gauss-Bonnet gravity.

Critical solutions of topologically gauged $ \mathcal{N} $ = 8 CFTs in three dimensions

In this paper we discuss some special (critical) background solutions that arise in topological gauged ${\mathcal N}=8$ three-dimensional CFTs with SO(N) gauge group. These solutions solve the TMG equations (containing the parameters $\mu$ and $l$) for a certain set of values of $\mu l$ obtained by varying the number of scalar fields with a VEV. Apart from Minkowski, chiral round $AdS_3$ and null-warped $AdS_3$ (or Schr\"odinger(z=2)) we identify also a more exotic solution recently found in $TMG$ by Ertl, Grumiller and Johansson. We also discuss the spectrum, symmetry breaking pattern and the supermultiplet structure in the various backgrounds and argue that some properties are due to their common origin in a conformal phase. Some of the scalar fields, including all higgsed ones, turn out to satisfy three-dimensional singleton field equations. Finally, we note that topologically gauged ${\mathcal N}=6$ ABJ(M) theories have a similar, but more restricted, set of background solutions.

Holographic striped phases

We discuss new types of instabilities of D=4 electrically charged AdS-Reissner-Nordstr\"om black branes that involve neutral pseudo-scalars. The instabilities spontaneously break translational invariance and are associated with the dual three-dimensional CFTs, at finite temperature and fixed chemical potential with respect to a global abelian symmetry, acquiring striped phases. We show that such instabilities are present for the infinite class of skew-whiffed $AdS_4\times SE_7$ solutions of D=11 supergravity, albeit at a lower temperature than the known superfluid instabilities.

SL(2,Bbb Z) action on three-dimensional CFTs and holography

We show that there is a natural action of SL(2,Z) on the two-point functions of the energy momentum tensor and of higher-spin conserved currents in three-dimensional CFTs. The dynamics behind the S-operation of SL(2,Z) is that of an irrelevant current-current deformation and we point out its similarity to the dynamics of a wide class of three-dimensional CFTs. The holographic interpretation of our results raises the possibility that many three-dimensional CFTs have duals on AdS4 with SL(2,Z) duality properties at the linearized level.

Three-dimensional CFTs and RG flow from squashing M2-brane horizon

Utilizing AdS/CFT correspondence in M-theory, an example of interacting d=3 conformal field theories and renormalization group flow between them is presented. Near-horizon geometry of N coincident M2-branes located on a conical singularity on eight-dimensional hyper-Kähler manifold or manifold with Spin(7) holonomy is, in large- N limit, AdS 4× X 7, where X 7 is seven-sphere with squashing. Deformation from round S 7 to squashed one is known to lead to spontaneous breaking of N=8 local supersymmetry in gauged AdS 4 supergravity to N=1,0 . Via AdS/CFT correspondence, it is interpreted as renormalization group flow from SO(5)× SO(3) symmetric UV fixed point to SO(8) symmetric IR fixed point. Evidence for the interpretation is found both from the supergravity scalar potential and from the existence of interpolating static domain-wall thereof, and from the conformal dimensions of the relevant chiral primary operator at each fixed point.

On the free-energy of three-dimensional CFTs and polylogarithms

We study the O( N) vector model and the U( N) Gross-Neveu model with fixed total fermion number, in three dimensions. Using non-trivial polylogarithmic identities, we calculate the large- N renormalized free-energy density of these models, at their conformal points in a “slab” geometry with one finite dimension of length L. We comment on the possible implications of our results.