Spectral sum rules for conformal field theories in arbitrary dimensions (original) (raw)

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 (t 2 and t 4). For CFTs dual to two derivative Einstein gravity, this proportionality constant is just d 2(d+1). This has been verified by a direct holographic computation of the retarded correlator for Einstein gravity in AdS d+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 holo-graphic computation for Gauss Bonnet theories in AdS 5 black hole background. We then generalize the sum rule for arbitrary curvature squared corrections to Einstein-Hilbert action in d ≥ 4. Evaluating the parameters t 2 and t 4 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.

Spectral flow between conformal field theories in 1 + 1 dimensions

Nuclear Physics B, 1992

Consider the q5 13-perturbed minimal unitary conformal field theories .,~ç,(with diagonal modular invariant partition function) of central charge c= 1 -(6/p(p + 1)), p = 3, 4,... It is believed that for one sign of the perturbing parameter they are massive scattering theories of kinks, whereas for the other they are massless but not scale-invariant quantum field theories interpolating between 4; in the UV and~in the JR limit. We propose integral equations for the exact finite-volume energies of the first and kth excited states in çb13-perturbed 4'2k+2 on the cylinder (of circumference R). These integral equations are similar to the thermodynamic Bethe Ansatz equations recently proposed by Al. Zamolodchikov for the ground-state energy in these theories. The behaviour of the conjectured expressions for the finite-volume energies at small and large R, which we investigate analytically and numerically, is in excellent agreement with predictions of conformal perturbation theory around the UV and JR fixed point, respectively, providing strong support for our proposal. In particular, we study in some detail the flow of the spin field (dimension d =~) of the tricritical Ising model 4'~to the spin field (d =~) of the critical Ising model .%~. For 4 13-perturbed 4~we also compare the small-R behaviour of the conjectured first energy-gap with results from the "truncated conformal-space approach", and discuss the limitations of the latter method when conformal perturbation theory has UV divergencies. * More precisely the r2 term is actually an "anti-bulk term", i.e. it has to be present in the small r expansion so that e(r)/r has a finite limitin particular no bulk r2 term corresponding to E(R) having a term linear in Rat large r. Similarly one can understand the r2 In r term. Cf. ref. [131for details, and also remarks in subsect. 2.2. * Operator product coefficients C4,,1,.4,5 of arbitrary conformal fields can be calculated using conformal invariance [11given the coefficients for the primary fields, which are known in all minimal CFTs [30].

Duality Constraints on Thermal Spectra of 3D Conformal Field Theories and 4D Quasinormal Modes

2024

Thermal spectra of correlation functions in holographic 3D large-N conformal field theories (CFTs) correspond to quasinormal modes of classical gravity and other fields in asymptotically anti-de Sitter black hole spacetimes. Using general properties of such spectra along with constraints imposed by the S duality (or the particle-vortex duality), we derive a spectral duality relation that all such spectra must obey. Its form is universal and relates infinite products over QNMs with bulk algebraically special frequencies. In the process, we also derive a new sum rule constraining products over QNMs. The spectral duality relation, which imposes an infinite set of constraints on the QNMs, is then investigated and a number of well-known holographic examples that demonstrate its validity are examined. Our results also allow us to understand several new aspects of the pole-skipping phenomenon.

Thermodynamics of conformal field theories and cosmology

We study the ratio of the entropy to the total energy in conformal field theories at finite temperature. For the free field realizations of N = 4 super Yang-Mills theory in D = 4 and the (2, 0) tensor multiplet in D = 6, the ratio is bounded from above. The corresponding bounds are less stringent than the recently proposed Verlinde bound. For strongly coupled CFTs with AdS duals, we show that the ratio obeys the Verlinde bound even in the presence of rotation. For such CFTs, we point out an intriguing resemblance in their thermodynamic formulas with the corresponding ones of two-dimensional CFTs. The discussion is based on hep-th/0101076 [1]

Entropy bounds in cosmology and conformal field theories

Relativistic viscous hydrodynamics, conformal invariance, and holography

Journal of High Energy Physics, 2008

We consider second-order viscous hydrodynamics in conformal field theories at finite temperature. We show that conformal invariance imposes powerful constraints on the form of the second-order corrections. By matching to the AdS/CFT calculations of correlators, and to recent results for Bjorken flow obtained by Heller and Janik, we find three (out of five) second-order transport coefficients in the strongly coupled N = 4 supersymmetric Yang-Mills theory. We also discuss how these new coefficents can arise within the kinetic theory of weakly coupled conformal plasmas. We point out that the Müller-Israel-Stewart theory, often used in numerical simulations, does not contain all allowed second-order terms and, frequently, terms required by conformal invariance.

Pressure and Compressibility of Conformal Field Theories from the AdS/CFT Correspondence

Entropy

The equation of state associated with N = 4 supersymmetric Yang-Mills in 4 dimensions, for SU(N) in the large N limit, is investigated using the AdS/CFT correspondence. An asymptotically AdS black-hole on the gravity side provides a thermal background for the Yang-Mills theory on the boundary in which the cosmological constant is equivalent to a volume. The thermodynamic variable conjugate to the cosmological constant is a pressure and the P − V diagram is studied. It is known that there is a critical point where the heat capacity diverges and this is reflected in the isothermal compressibility. Critical exponents are derived and found to be mean field in the large N limit. The same analysis applied to 3 and 6 dimensional conformal field theories again yields mean field exponents associated with the compressibility at the critical point. 1 DIAS-STP-16-05 1 Feature invited paper for special edition "Black hole thermodynamics II" in Entropy.

Stress Tensor Perturbations in Conformal Field Theory

International Journal of Modern Physics A, 1991

We reconsider the problem of deforming a conformal field theory to a neighboring theory which is again critical. An invariant formulation of this problem is important for understanding the underlying symmetry of string theory. We give a simple derivation of A. Sen’s recent formula for the change in the stress tensor and show that, when correctly interpreted, it is coordinate-invariant. We give the corresponding superconformal perturbation for superfield backgrounds and explain why it has no direct analog for spin-field backgrounds.

Published for SISSA by Springer Holographic two-point functions in conformal gravity

In this paper we compute the holographic two-point functions of four dimensional conformal gravity. Precisely we calculate the two-point functions for Energy-Momentum (EM) and Partially Massless Response (PMR) operators that have been identified as two response functions for two independent sources in the dual CFT. The correlation function of EM with PMR tensors turns out to be zero which is expected according to the conformal symmetry. The two-point function of EM is that of a transverse and traceless tensor, and the two-point function of PMR which is a traceless operator contains two distinct parts, one for a transverse-traceless tensor operator and another one for a vector field, both of which fulfill criteria of a CFT. We also discuss about the unitarity of the theory. A Linearization 30 B Asymptotic analysis 30 C Linearization in transverse-traceless gauge 31

Geometric and renormalized entropy in conformal field theory

Nuclear Physics B, 1994

In statistical physics, useful notions of entropy are defined with respect to some coarse graining procedure over a microscopic model. Here we consider some special problems that arise when the microscopic model is taken to be relativistic quantum field theory. These problems are associated with the existence of an infinite number of degrees of freedom per unit volume. Because of these the microscopic entropy can, and typically does, diverge for sharply localized states. However the difference in the entropy between two such states is better behaved, and for most purposes it is the useful quantity to consider. In particular, a renormalized entropy can be defined as the entropy relative to the ground state. We make these remarks quantitative and precise in a simple model situation: the states of a conformal quantum field theory excited by a moving mirror. From this work, we attempt to draw some lessons concerning the "information problem" in black hole physics.

Spectral sum rules for conformal field theories in arbitrary dimensions (original) (raw)

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