Conformal Dimension Research Articles

Relative entropy and the RG flow

We consider the relative entropy between vacuum states of two different theories: a conformal field theory (CFT), and the CFT perturbed by a relevant operator. By restricting both states to the null Cauchy surface in the causal domain of a sphere, we make the relative entropy equal to the difference of entanglement entropies. As a result, this difference has the positivity and monotonicity properties of relative entropy. From this it follows a simple alternative proof of the c-theorem in d = 2 space-time dimensions and, for d > 2, the proof that the coefficient of the area term in the entanglement entropy decreases along the renormalization group (RG) flow between fixed points. We comment on the regimes of convergence of relative entropy, depending on the space-time dimensions and the conformal dimension Δ of the perturbation that triggers the RG flow.

Rigidity of Quasisymmetric Mappings on Self-affine Carpets

We show that the class of quasisymmetric maps between horizontal self-affine carpets is rigid. Such maps can only exist when the dimensions of the carpets coincide, and in this case, the quasisymmetric maps are quasi-Lipschitz. We also show that horizontal self-affine carpets are minimal for the conformal Assouad dimension.

On classification of extremal non-holomorphic conformal field theories

Rational chiral conformal field theories are organized according to their genus, which consists of a modular tensor category and a central charge c. A long-term goal is to classify unitary rational conformal field theories based on a classification of unitary modular tensor categories. We conjecture that for any unitary modular tensor category , there exists a unitary chiral conformal field theory so that its modular tensor category is . In this paper, we initiate a mathematical program in and around this conjecture. We define a class of extremal vertex operator algebras with minimal conformal dimensions as large as possible for their central charge, and non-trivial representation theory. We show that there are finitely many different characters of extremal vertex operator algebras possessing at most three different irreducible modules. Moreover, we list all of the possible characters for such vertex operator algebras with .

Model of a polymer chain twisted inside a statistical segment. Poly(3‐hexylthiophene) in chloroform solutions

AbstractA set of hydrodynamic methods in extremely dilute solutions (determination of intrinsic viscosity [η], translational diffusion D and flotation in a centrifugal field s) was used in the studies of conformational properties, dimensions and equilibrium rigidity of individual molecules of a series of samples of poly(3‐hexylthiophene). Absolute molecular masses MsD were determined. The Mark–Kuhn–Houwink equations for the range of MsD from 2240 to 15 300 g mol−1 are given. For the first time, an analysis of conformational properties and equilibrium flexibility of poly(3‐hexylthiophene) molecules was carried out using the model of a statistical segment containing a twisted polymer chain. © 2017 Society of Chemical Industry

Exactly marginal deformations from exceptional generalised geometry

We apply exceptional generalised geometry to the study of exactly marginal deformations of mathcal{N} = 1 SCFTs that are dual to generic AdS5 flux backgrounds in type IIB or eleven-dimensional supergravity. In the gauge theory, marginal deformations are parametrised by the space of chiral primary operators of conformal dimension three, while exactly marginal deformations correspond to quotienting this space by the complexified global symmetry group. We show how the supergravity analysis gives a geometric interpretation of the gauge theory results. The marginal deformations arise from deformations of generalised structures that solve moment maps for the generalised diffeomorphism group and have the correct charge under the generalised Reeb vector, generating the R-symmetry. If this is the only symmetry of the background, all marginal deformations are exactly marginal. If the background possesses extra isometries, there are obstructions that come from fixed points of the moment maps. The exactly marginal deformations are then given by a further quotient by these extra isometries.Our analysis holds for any mathcal{N} = 2 AdS5 flux background. Focussing on the particular case of type IIB Sasaki-Einstein backgrounds we recover the result that marginal deformations correspond to perturbing the solution by three-form flux at first order. In various explicit examples, we show that our expression for the three-form flux matches those in the literature and the obstruction conditions match the one-loop beta functions of the dual SCFT.

Emergent Conformal Symmetry and Geometric Transport Properties of Quantum Hall States on Singular Surfaces.

We study quantum Hall states on surfaces with conical singularities. We show that the electronic fluid at the cone tip possesses an intrinsic angular momentum, which is due solely to the gravitational anomaly. We also show that quantum Hall states behave as conformal primaries near singular points, with a conformal dimension equal to the angular momentum. Finally, we argue that the gravitational anomaly and conformal dimension determine the fine structure of the electronic density at the conical point. The singularities emerge as quasiparticles with spin and exchange statistics arising from adiabatically braiding conical singularities. Thus, the gravitational anomaly, which appears as a finite size correction on smooth surfaces, dominates geometric transport on singular surfaces.

Two-dimensional conformal field theories with matrix-valued level

We introduce a new class of two dimensional conformal field theories by extending Wess-Zumino-Witten (WZW) models to chiral algebras with matrix-valued levels. The new CFTs are based on holomorphic currents with an operator product expansion characterized by a positive integer-valued matrix $K_{AB}$. We use the Sugawara construction to compute the energy-momentum tensor, the central charge, and the spectrum of conformal dimensions of the CFTs based on this algebra. We also construct a set of genus-$1$ characters and show that they fulfil a representation of the modular group $\text{SL}(2,\mathbb{Z})$ up to a modular anomaly.

Localized AdS_{5}×S^{5} Black Holes.

According to heuristic arguments, global AdS_{5}×S^{5} black holes are expected to undergo a phase transition in the microcanonical ensemble. At high energies, one expects black holes that respect the symmetries of the S^{5}; at low energies, one expects "localized" black holes that appear pointlike on the S^{5}. According to anti-de Sitter/conformal field theory correspondence, N=4 supersymmetric Yang-Mills (SYM) theory on a 3-sphere should therefore exhibit spontaneous R-symmetry breaking at strong coupling. In this Letter, we numerically construct these localized black holes. We extrapolate the location of this phase transition, and compute the expectation value of the broken scalar operator with lowest conformal dimension. Via the correspondence, these results offer quantitative predictions for N=4 SYM theory.

Non-equilibrium phase and entanglement entropy in 2D holographic superconductors via gauge–string duality

An alternative method of developing the theory of non-equilibrium two-dimensional holographic superconductor is to start from the definition of a time-dependent AdS3 background. As originally proposed, many of these formulae were cast in exponential form, but the adoption of the numeric method of expression throughout the bulk serves to show more clearly the relationship between the various parameters. The time dependence behavior of the scalar condensation and Maxwell fields are fitted numerically. A usual value for Maxwell field on AdS horizon is exp(–bt), and the exponential log ratio is therefore 10−8 s−1. The coefficient b of the time in the exponential term exp(–bt) can be interpreted as a tool to measure the degree of dynamical instability; its reciprocal 1/b is the time in which the disturbance is multiplied in the ratio. A discussion of some of the exponential formulae is given by the scalar field ψ(z, t) near the AdS boundary. It may be possible that a long interval would elapse in the system, which tends to the equilibrium state, where the normal mass and conformal dimensions emerged. A somewhat curious calculation has been made to illustrate the holographic entanglement entropy for this system. The foundation of all this calculation is, of course, a knowledge of multiple (connected and disconnected) extremal surfaces. There are several cases in which exact and approximate solutions are jointly used; a variable numerical quantity is represented by a graph, and the principles of approximation are then applied to determine related numerical quantities. In the case of the disconnected phase with a finite extremal area, we find a discontinuity in the first derivative of the entanglement entropy as the conserved charge J is increased.

The Weak Gravity Conjecture in three dimensions

We study weakly coupled $U(1)$ theories in $AdS_3$, their associated charged BTZ solutions, and their charged spectra. We find that modular invariance of the holographic dual two-dimensional CFT and compactness of the gauge group together imply the existence of charged operators with conformal dimension significantly below the black hole threshold. We regard this as a form of the Weak Gravity Conjecture (WGC) in three dimensions. We also explore the constraints posed by modular invariance on a particular discrete $\mathbb{Z}_N$ symmetry which arises in our discussion. In this case, modular invariance does not guarantee the existence of light $\mathbb{Z}_N$-charged states. We also highlight the differences between our discussion and the usual heuristic arguments for the WGC based on black hole remnants.

Embeddings of locally compact hyperbolic groups into Lp-spaces

In the last years, there has been a large amount of research on embeddability properties of finitely generated hyperbolic groups. In this paper, we elaborate on the more general class of locally compact hyperbolic groups. We compute the equivariant Lp-compression in a number of locally compact examples, such as the groups SO(n,1). Next, we show that although there are locally compact, non-discrete hyperbolic groups G with Kazhdan's property (T), it is true that any locally compact hyperbolic group admits a proper affine isometric action on an Lp-space for p larger than the Ahlfors regular conformal dimension of ∂G. This answers a question asked by Yves de Cornulier. Finally, we elaborate on the locally compact version of property A and show that, as in the discrete case, a locally compact second countable group has property A if its non-equivariant compression is greater than 1/2.

Symbolic Consumption in Upscale Cafés

The growth in premium coffee consumption in South Korea along with increased incomes and lifestyle changes has created important opportunities for marketers to target customers in the international market. To take advantage of these opportunities, it is critical to understand the underlying factors that motivate Gen Y coffee consumers in Korea in order to provide guidance for international food and beverage businesses. This study investigates three possible mechanisms in terms of the individual, social, and functional evaluations of customers with respect to coffee consumption in upscale cafés. The results of this study provide empirical evidence that the basic motivational drivers of Korean Gen Y consumers’ premium coffee consumption in cafés are similar to dimensions of luxury value such as materialism, conformity, conspicuous tendencies, and functional dimensions. Moreover, this study discovered the moderating effect of income source according to the context in which it is obtained (gifted money or earned money) in terms of young consumers spending habits.

Coulomb branches for rank 2 gauge groups in 3d N = 4 $$ \mathcal{N}=4 $$ gauge theories

The Coulomb branch of 3-dimensional N=4 gauge theories is the space of bare and dressed BPS monopole operators. We utilise the conformal dimension to define a fan which, upon intersection with the weight lattice of a GNO-dual group, gives rise to a collection of semi-groups. It turns out that the unique Hilbert bases of these semi-groups are a sufficient, finite set of monopole operators which generate the entire chiral ring. Moreover, the knowledge of the properties of the minimal generators is enough to compute the Hilbert series explicitly. The techniques of this paper allow an efficient evaluation of the Hilbert series for general rank gauge groups. As an application, we provide various examples for all rank two gauge groups to demonstrate the novel interpretation.

Universal corrections to entanglement entropy of local quantum quenches

We study the time evolution of single interval Renyi and entanglement entropies following local quantum quenches in two dimensional conformal field theories at finite temperature for which the locally excited states have a finite temporal width, \epsilon. We show that, for local quenches produced by the action of a conformal primary field, the time dependence of Renyi and entanglement entropies at order \epsilon^2 is universal. It is determined by the expectation value of the stress tensor in the replica geometry and proportional to the conformal dimension of the primary field generating the local excitation. We also show that in CFTs with a gravity dual, the \epsilon^2 correction to the holographic entanglement entropy following a local quench precisely agrees with the CFT prediction. We then consider CFTs admitting a higher spin symmetry and turn on a higher spin chemical potential \mu. We calculate the time dependence of the order \epsilon^2 correction to the entanglement entropy for small \mu, and show that the contribution at order \mu^2 is universal. We verify our arguments against exact results for minimal models and the free fermion theory.

Role of nonlocal probes of thermalization for a strongly interacting non-Abelian plasma

We use a holographic method to investigate thermalization of a boost-invariant strongly interacting non-Abelian plasma. Boundary sourcing, a distorsion of the boundary metric, is employed to drive the system far from equilibrium. Thermalization is analyzed through nonlocal probes: the equal-time two-point correlation function of large conformal dimension operators in the boundary theory, and Wilson loops of different shapes. We study the dependence of the thermalization time on the size of the probes, and compare the results to the ones obtained using local observables: the onset of thermalization is first observed at short distances.

Relative entropy of excited states in two dimensional conformal field theories

We study the relative entropy and the trace square distance, both of which measure the distance between reduced density matrices of two excited states in two dimensional conformal field theories. We find a general formula for the relative entropy between two primary states with the same conformal dimension in the limit of a single small interval and find that in this case the relative entropy is proportional to the trace square distance. We check our general formulae by calculating the relative entropy between two generalized free fields and the trace square distance between the spin and disorder operators of the critical Ising model. We also give the leading term of the relative entropy in the small interval expansion when the two operators have different conformal dimensions. This turns out to be universal when the CFT has no primaires lighter than the stress tensor. The result reproduces the previously known special cases.

Quasisymmetric Maps on Kakeya Sets

I show that $L^{p}-L^{q}$ estimates for the Kakeya maximal function yield lower bounds for the conformal dimension of Kakeya sets, and upper bounds for how much quasisymmetries can increase the Hausdorff dimension of line segments inside Kakeya sets. Combining the known $L^{p}-L^{q}$ estimates of Wolff and Katz-Tao with the main result of the paper, the conformal dimension of Kakeya sets in $\mathbb{R}^{n}$ is at least $\max\{(n + 2)/2,(4n + 3)/7\}$. Moreover, if $f$ is a quasisymmetry from a Kakeya set $K \subset \mathbb{R}^{n}$ onto any at most $n$-dimensional metric space, the $f$-image of a.e. line segment inside $K$ has dimension at most $\min\{2n/(n + 2),7n/(4n + 3)\}$. The Kakeya maximal function conjecture implies that the bounds can be improved to $n$ and $1$, respectively.

Long, partial-short, and special conformal fields

In the framework of metric-like approach, totally symmetric arbitrary spin bosonic conformal fields propagating in flat space-time are studied. Depending on the values of conformal dimension, spin, and dimension of space-time, we classify all conformal field as long, partial-short, short, and special conformal fields. An ordinary-derivative (second-derivative) Lagrangian formulation for such conformal fields is obtained. The ordinary-derivative Lagrangian formulation is realized by using double-traceless gauge fields, Stueckelberg fields, and auxiliary fields. Gauge-fixed Lagrangian invariant under global BRST transformations is obtained. The gauge-fixed BRST Lagrangian is used for the computation of partition functions for all conformal fields. Using the result for the partition functions, numbers of propagating D.o.F for the conformal fields are also found.

Holographic construction of excited CFT states

We present a systematic construction of bulk solutions that are dual to CFT excited states. The bulk solution is constructed perturbatively in bulk fields. The linearised solution is universal and depends only on the conformal dimension of the primary operator that is associated with the state via the operator-state correspondence, while higher order terms depend on detailed properties of the operator, such as its OPE with itself and generally involve many bulk fields. We illustrate the discussion with the holographic construction of the universal part of the solution for states of two dimensional CFTs, either on $R \times S^1$ or on $R^{1,1}$. We compute the 1-point function both in the CFT and in the bulk, finding exact agreement. We comment on the relation with other reconstruction approaches.

Entanglement entropy from corner transfer matrix in Forrester–Baxter non-unitary RSOS models

Using a corner transfer matrix approach, we compute the bipartite entanglement Rényi entropy in the off-critical perturbations of non-unitary conformal minimal models realised by lattice spin chains Hamiltonians related to the Forrester–Baxter RSOS models (Bianchini et al 2015 J. Stat. Mech. P03010) in regime III. This allows to show on a set of explicit examples that the Rényi entropies for non-unitary theories rescale near criticality as the logarithm of the correlation length with a coefficient proportional to the effective central charge. This complements a similar result, recently established for the size rescaling at the critical point (Bianchini et al 2015 J. Phys. A: Math. Theor. 48 04FT01), showing the expected agreement of the two behaviours. We also compute the first subleading unusual correction to the scaling behaviour, showing that it is expressible in terms of expansions of various fractional powers of the correlation length, related to the differences between the conformal dimensions of fields in the theory and the minimal conformal dimension. Finally, a few observations on the limit leading to the off-critical logarithmic minimal models of Pearce and Seaton (2012 J. Stat. Mech. P09014) are put forward.