D-dimensional Conformal Field Theories Research Articles

Superintegrability of d-Dimensional Conformal Blocks.

We observe that conformal blocks of scalar four-point functions in a d-dimensional conformal field theory can be mapped to eigenfunctions of a two-particle hyperbolic Calogero-Sutherland Hamiltonian. The latter describes two coupled Pöschl-Teller particles. Their interaction, whose strength depends smoothly on the dimension d, is known to be superintegrable. Our observation enables us to exploit the rich mathematical literature on Calogero-Sutherland models in deriving various results for conformal field theory. These include an explicit construction of conformal blocks in terms of Heckman-Opdam hypergeometric functions. We conclude with a short outlook, in particular, on the consequences of integrability for the theory of conformal blocks.

Causality constraints in conformal field theory

Causality places nontrivial constraints on QFT in Lorentzian signature, for example fixing the signs of certain terms in the low energy Lagrangian. In d-dimensional conformal field theory, we show how such constraints are encoded in crossing symmetry of Euclidean correlators, and derive analogous constraints directly from the conformal bootstrap (analytically). The bootstrap setup is a Lorentzian four-point function corresponding to propagation through a shockwave. Crossing symmetry fixes the signs of certain log terms that appear in the conformal block expansion, which constrains the interactions of low-lying operators. As an application, we use the bootstrap to rederive the well known sign constraint on the $(\partial\phi)^4$ coupling in effective field theory, from a dual CFT. We also find constraints on theories with higher spin conserved currents. Our analysis is restricted to scalar correlators, but we argue that similar methods should also impose nontrivial constraints on the interactions of spinning operators.

The Casimir energy in curved space and its supersymmetric counterpart

We study d-dimensional Conformal Field Theories (CFTs) on the cylinder, $$ {S}^{d-1}\times \mathrm{\mathbb{R}} $$ , and its deformations. In d = 2 the Casimir energy (i.e. the vacuum energy) is universal and is related to the central charge c. In d = 4 the vacuum energy depends on the regularization scheme and has no intrinsic value. We show that this property extends to infinitesimally deformed cylinders and support this conclusion with a holographic check. However, for $$ \mathcal{N}=1 $$ supersymmetric CFTs, a natural analog of the Casimir energy turns out to be scheme independent and thus intrinsic. We give two proofs of this result. We compute the Casimir energy for such theories by reducing to a problem in supersymmetric quantum mechanics. For the round cylinder the vacuum energy is proportional to a + 3c. We also compute the dependence of the Casimir energy on the squashing parameter of the cylinder. Finally, we revisit the problem of supersymmetric regularization of the path integral on Hopf surfaces.

Sphere partition functions and the Zamolodchikov metric

We study the finite part of the sphere partition function of d-dimensional Conformal Field Theories (CFTs) as a function of exactly marginal couplings. In odd dimensions, this quantity is physical and independent of the exactly marginal couplings. In even dimensions, this object is generally regularization scheme dependent and thus unphysical. However, in the presence of additional symmetries, the partition function of even-dimensional CFTs can become physical. For two-dimensional N=(2,2) supersymmetric CFTs, the continuum partition function exists and computes the Kahler potential on the chiral and twisted chiral superconformal manifolds. We provide a new elementary proof of this result using Ward identities on the sphere. The Kahler transformation ambiguity is identified with a local term in the corresponding N=(2,2) supergravity theory. We derive an analogous, new, result in the case of four-dimensional N=2 supersymmetric CFTs: the S^4 partition function computes the Kahler potential on the superconformal manifold. Finally, we show that N=1 supersymmetry in four dimensions and N=(1,1) supersymmetry in two dimensions are not sufficient to make the corresponding sphere partition functions well-defined functions of the exactly marginal parameters.

An ansatz for one dimensional steady state configurations

.We conjecture a universal formula for the heat current of a steady state connecting two asymptotic equilibrium systems in d-dimensional conformal field theories. Our proposal is verified by comparing it to exact expressions in 1 + 1 dimensions and linear hydrodynamics as well as numerical simulations in an Israel–Stewart like theory of second order viscous hydrodynamics.

Holography for Schrödinger backgrounds

We discuss holography for Schrodinger solutions of both topologically massive gravity in three dimensions and massive vector theories in (d+1) dimensions. In both cases the dual field theory can be viewed as a d-dimensional conformal field theory (two dimensional in the case of TMG) deformed by certain operators that respect the Schrodinger symmetry. These operators are irrelevant from the viewpoint of the relativistic conformal group but they are exactly marginal with respect to the non-relativistic conformal group. The spectrum of linear fluctuations around the background solutions corresponds to operators that are labeled by their scaling dimension and the lightcone momentum k_v. We set up the holographic dictionary and compute 2-point functions of these operators both holographically and in field theory using conformal perturbation theory and find agreement. The counterterms needed for holographic renormalization are non-local in the v lightcone direction.

Holography for chiral scale-invariant models

Deformation of any d-dimensional conformal field theory by a constant null source for a vector operator of dimension (d + z -1) is exactly marginal with respect to anisotropic scale invariance, of dynamical exponent z. The holographic duals to such deformations are AdS plane waves, with z=2 being the Schrodinger geometry. In this paper we explore holography for such chiral scale-invariant models. The special case of z=0 can be realized with gravity coupled to a scalar, and is of particular interest since it is related to a Lifshitz theory with dynamical exponent two upon dimensional reduction. We show however that the corresponding reduction of the dual field theory is along a null circle, and thus the Lifshitz theory arises upon discrete light cone quantization of an anisotropic scale invariant field theory.

Renormalization group and infinite algebraic structure inD-dimensional conformal field theory

We consider scalar field theory in the D-dimensional space with nontrivial metric and local action functional of most general form. It is possible to construct for this model a generalization of renormalization procedure and RG-equations. In the fixed point the diffeomorphism and Weyl transformations generate an infinite algebraic structure of D-Dimensional conformal field theory models. The Wilson expansion and crossing symmetry enable to obtain sum rules for dimensions of composite operators and Wilson coefficients.

ON THE CONFORMAL ANOMALY FROM HIGHER DERIVATIVE GRAVITY IN AdS/CFT CORRESPONDENCE

We follow Witten's proposal1 in the calculation of conformal anomaly from (d + 1)-dimensional higher derivative gravity via AdS/CFT correspondence. It is assumed that some d-dimensional conformal field theories have a description in terms of above (d + 1)-dimensional higher derivative gravity which includes not only the Einstein term and cosmological constant but also curvature squared terms. The explicit expression for two-dimensional and four-dimensional anomalies is found, it contains higher derivative corrections. In particular, it is shown that not only Einstein gravity but also theory with the Lagrangian L =aR2 + bRμνRμν + Λ (even when a=0 or b=0) is five-dimensional bulk theory for [Formula: see text] super-Yang–Mills theory in AdS/CFT correspondence. Similarly, the d + 1 = 3 theory with (or without) Einstein term may describe d = 2 scalar or spinor CFT's. That gives new versions of bulk side which may be useful in different aspects. As application of our general formalism we find next-to-leading corrections to the conformal anomaly of [Formula: see text] supersymmetric theory from d = 5 AdS higher derivative gravity (low energy string effective action).

Holography and the Weyl anomaly

We review our calculation of the Weyl anomaly for d-dimensional conformal field theories that have a description in terms of a (d+1)-dimensional gravity theory.

Holography and the Weyl Anomaly

We review our calculation of the Weyl anomaly for d-dimensional conformal field theories that have a description in terms of a (d + 1)-dimensional gravity theory.

New Developments in D-dimensional conformal quantum field theory

A review of recent developments in conformal quantum field theory in D-dimensional space is presented. The conformally invariant solution of the Ward identities is studied. We demonstrate the existence of D-dimensional analogues of primary and secondary fields, the central charge, and the null vectors. The Hilbert space is shown to possess a specific model-independent structure defined by the 1 2 (D+1)(D+2) -dimensional symmetry and the Ward identities. In particular, there exists a sector H of the Hilbert space related to an infinite family of “secondary” fields which are generated by the currents and the energy-momentum tensor. The general solution of the Ward identities in D>2 defining the sector H necessarily includes the contribution of the gauge fields. We derive the conditions which single out the conformal theories of a direct (non-gauge) interaction. We examine the class of models satisfying these conditions. It is shown that the Green functions of the current and the energy-momentum tensor in these models are uniquely determined by the Ward identities for any D≥2. The anomalous Ward identities containing contributions of c-number and operator analogues of the central charge, are discussed. Closed sets of expressions for the Green functions of secondary fields are obtained in D-dimensional space. A family of exactly solvable conformal models in D≥2 is constructed. Each model is defined by the requirement of vanishing of a certain field Q s , s=1,2…. The fields Q s are constructed as definite superpositions of secondary fields. After that, one requires each field Q s to be primary. The latter is possible for specific values of scale dimensions of fundamental fields (a D-dimensional analogue of the Kac formula). The states Q s |0〉 are analogous to null vectors. One can derive closed sets of differential equations for higher Green functions in each of the models. These results are demonstrated on examples of several exactly solvable models in D>2. The approach developed here is based on the finite-dimensional conformal symmetry for any D≥2. However the family of models under consideration does have the structure identical to that of two-dimensional conformal theories. This analogy is discussed in detail. It is shown that when D=2, the above family coincides with the well-known family of models based on infinite-dimensional conformal symmetry. The analysis of this phenomenon indicates the possibility of existence of D-dimensional analogue of the Virasoro algebra.

Anomalous Ward identities in D-dimensional conformal theory.

Anomalous Ward identities for four-point Green functions containing conserved operators (currents and the energy-momentum tensor) in D-dimensional conformal field theory are obtained. They contain operator anomalous terms generalizing the central charge in two dimensions. \textcopyright{} 1996 The American Physical Society.

Infinite symmetry group in D-dimensional conformal quantum field theory

The diffeomorphism group of the D-dimensional space as the symmetry group for conformal quantum field theory models is proposed. The problem of calculation of critical exponents turns out to be reduced to the study of special diffeomorphism group representations. The general properties of these representations on the space of symmetric traceless tensors are investigated.