Arbitrary Conformal Field Theory Research Articles

Lorentzian Threads as Gatelines and Holographic Complexity.

The continuous min flow-max cut principle is used to reformulate the "complexity=volume" conjecture using Lorentzian flows-divergenceless norm-bounded timelike vector fields whose minimum flux through a boundary subregion is equal to the volume of the homologous maximal bulk Cauchy slice. The nesting property is used to show the rate of complexity is bounded below by "conditional complexity," describing a multistep optimization with intermediate and final target states. Conceptually, discretized Lorentzian flows are interpreted in terms of threads or gatelines such that complexity is equal to the minimum number of gatelines used to prepare a conformal field theory (CFT) state by an optimal tensor network (TN) discretizing the state. We propose a refined measure of complexity, capturing the role of suboptimal TNs, as an ensemble average. The bulk symplectic potential provides a "canonical" thread configuration characterizing perturbations around arbitrary CFT states. Its consistency requires the bulk to obey linearized Einstein's equations, which are shown to be equivalent to the holographic first law of complexity, thereby advocating a notion of "spacetime complexity."

A conformal dispersion relation: correlations from absorption

We introduce the analog of Kramers-Kronig dispersion relations for correlators of four scalar operators in an arbitrary conformal field theory. The correlator is expressed as an integral over its “absorptive part”, defined as a double discontinuity, times a theory-independent kernel which we compute explicitly. The kernel is found by resumming the data obtained by the Lorentzian inversion formula. For scalars of equal scaling dimensions, it is a remarkably simple function (elliptic integral function) of two pairs of cross-ratios. We perform various checks of the dispersion relation (generalized free fields, holographic theories at tree-level, 3D Ising model), and get perfect matching. Finally, we derive an integral relation that relates the “inverted” conformal block with the ordinary conformal block.

Almost linear Haldane pseudopotentials and emergent conformal block wave functions in a Landau level

We consider a two dimensional (2D) model of particles interacting in a Landau level. We work in a finite disk geometry and take the particles to interact with a linearly decreasing two-body Haldane pseudo-potential. We show that the ground state subspace of this model is spanned by the wave-functions that can be written as polynomial conformal blocks (of an arbitrary conformal field theory) consistent with the filling fraction (scaling dimension). To remove degeneracies, we then add a quadratic perturbation to the Hamiltonian and show that; 1. Conformal blocks constructed using the Moore-Read construction (e.g. Laughlin, Pfaffian, and Read-Rezayi states) remain exact eigenstates of this model in the thermodynamic limit and 2. By tuning an externally imposed single-body $-L_z^2$ potential we can enforce Moore-Read conformal blocks to become exact ground states of this model in the thermodynamic limit. We cannot rule out the possibility of residual degeneracies in this limit. This model has no filling dependence and is comprised only from two-body long-range interactions and external single-body potentials. Our results provide insight into how conformal block wave-functions can emerge in a Landau level.

Flow equation, conformal symmetry, and anti-de Sitter geometry

We argue that the Anti-de-Sitter (AdS) geometry in d+1 dimensions naturally emerges from an arbitrary conformal field theory in d dimensions using the free flow equation. We first show that an induced metric defined from the flowed field generally corresponds to the quantum information metric, called the Bures or Helstrom metric, if the flowed field is normalized appropriately. We next verify that the induced metric computed explicitly with the free flow equation always becomes the AdS metric when the theory is conformal. We finally prove that the conformal symmetry in d dimensions converts to the AdS isometry in d+1 dimensions after d dimensional quantum averaging. This guarantees the emergence of AdS geometry without explicit calculation.

Quantum corrections for spinning particles in de Sitter

We compute the one-loop quantum corrections to the gravitational potentials of a spinning point particle in a de Sitter background, due to the vacuum polarisation induced by conformal fields in an effective field theory approach. We consider arbitrary conformal field theories, assuming only that the theory contains a large number N of fields in order to separate their contribution from the one induced by virtual gravitons. The corrections are described in a gauge-invariant way, classifying the induced metric perturbations around the de Sitter background according to their behaviour under transformations on equal-time hypersurfaces. There are six gauge-invariant modes: two scalar Bardeen potentials, one transverse vector and one transverse traceless tensor, of which one scalar and the vector couple to the spinning particle. The quantum corrections consist of three different parts: a generalisation of the flat-space correction, which is only significant at distances of the order of the Planck length; a constant correction depending on the undetermined parameters of the renormalised effective action; and a term which grows logarithmically with the distance from the particle. This last term is the most interesting, and when resummed gives a modified power law, enhancing the gravitational force at large distances. As a check on the accuracy of our calculation, we recover the linearised Kerr-de Sitter metric in the classical limit and the flat-space quantum correction in the limit of vanishing Hubble constant.

Quantum corrections to holographic mutual information

We compute the leading contribution to the mutual information (MI) of two disjoint spheres in the large distance regime for arbitrary conformal field theories (CFT) in any dimension. This is achieved by refining the operator product expansion method introduced by Cardy \cite{Cardy:2013nua}. For CFTs with holographic duals the leading contribution to the MI at long distances comes from bulk quantum corrections to the Ryu-Takayanagi area formula. According to the FLM proposal\cite{Faulkner:2013ana} this equals the bulk MI between the two disjoint regions spanned by the boundary spheres and their corresponding minimal area surfaces. We compute this quantum correction and provide in this way a non-trivial check of the FLM proposal.

Weyl anomaly induced stress tensors in general manifolds

Considering arbitrary conformal field theories in general (non-conformally flat) backgrounds, we adopt a dimensional regularization approach to obtain stress tensors from Weyl anomalies. The results of Type A anomaly-induced stress tensors in four and six dimensions generalize the previous results calculated in a conformally flat background. On the other hand, regulators are needed to have well-defined Type B anomaly-induced stress tensors. We also discuss ambiguities related to Type D anomalies, Weyl invariants and order of limit issues.

Conformal boundary state for the rectangular geometry

We discuss conformal field theories (CFTs) in rectangular geometries, and develop a formalism that involves a conformal boundary state for the 1+1d open system. We focus on the case of homogeneous boundary conditions (no insertion of a boundary condition changing operator), for which we derive an explicit expression of the associated boundary state, valid for any arbitrary CFT. We check the validity of our solution, comparing it with known results for partition functions, numerical simulations of lattice discretizations, and coherent state expressions for free theories.

Branes: from free fields to general backgrounds

Motivated by recent developments in string theory, we study the structure of boundary conditions in arbitrary conformal field theories. A boundary condition is specified by two types of data: first, a consistent collection of reflection coefficients for bulk fields on the disk; and second, a choice of an automorphism ω of the fusion rules that preserves conformal weights and can be implemented on the spaces of chiral blocks. Non-trivial automorphisms ω correspond to D-brane configurations for arbitrary conformal field theories. The choice of the fusion rule automorphism ω amounts to fixing the dimension and certain global topological features of the D-brane world volume and the background gauge field on it. We present evidence that for fixed choice of ω the boundary conditions are classified as the irreducible representations of some commutative associative algebra, a generalization of the fusion rule algebra. Each of these irreducible representations corresponds to a choice of the moduli for the world volume of the D-brane and the moduli of the flat connection on it.

GLUING THEOREM, STAR PRODUCT AND INTEGRATION IN OPEN STRING FIELD THEORY IN ARBITRARY BACKGROUND FIELDS

We discuss how to define star product and integration in a string field theory formulated in a background provided by an arbitrary conformal field theory on the upper half-plane with central charge 26.

Canonical quantization of the induced 2D gravity

We present a bosonized version of an arbitrary conformal field theory on a fluctuating space-time. We develop the canonical formalism in the light-cone gauge. The SL(2, R ) current algebra structure is identified and the constraints are imposed by the BRST procedure. We quantize the system for space both a circle and a line. We recover for this model the renormalization of the central term of the current algebra and of the scaling dimensions.

NO GHOST THEOREM AND COHOMOLOGY THEOREM FOR STRINGS IN ARBITRARY STATIC BACKGROUNDS

We consider a string moving in an arbitrary time-independent background given by an arbitrary conformal field theory of appropriate central charge (e.g., c=25 for bosonic string) and one flat time-like dimension. We show that the physical subspace of the Hilbert space is positive semi-definite (no ghost theorem) and that the cohomology of the BRST operator is trivial except for the ghost number one (for open bosonic string) sector (cohomology theorem). Both the proofs are reductio ad absurdum proofs based on the corresponding theorems for the strings moving in flat background. In cases where there is an extra flat space-like dimension (besides the flat time-like one), the transverse subspace with positive-definite norm can be constructed.