Theorems for the lightcone bootstrap

We study the OPE coefficients cΔ, J for heavy-light scalar four-point functions, which can be obtained holographically from the two-point function of a light scalar of some non-integer conformal dimension ΔL in an AdS black hole. We verify that the OPE coefficient cd,0 = 0 for pure gravity black holes, consistent with the tracelessness of the holographic energy-momentum tensor. We then study the OPE coefficients from black holes involving matter fields. We first consider general charged AdS black holes and we give some explicit low-lying examples of the OPE coefficients. We also obtain the recursion formula for the lowest-twist OPE coefficients with at most two current operators. For integer ΔL, although the OPE coefficients are not fully determined, we set up a framework to read off the coefficients γΔ,J of the log(z overline{z} ) terms that are associated with the anomalous dimensions of the exchange operators and obtain a general formula for γΔ,J. We then consider charged AdS black holes in gauged supergravity STU models in D = 5 and D = 7, and their higher-dimensional generalizations. The scalar fields in the STU models are conformally massless, dual to light operators with ΔL = d − 2. We derive the linear perturbation of such a scalar in the STU charged AdS black holes and obtain the explicit OPE coefficient cd−2,0. Finally, we analyse the asymptotic properties of scalar hairy AdS black holes and show how cd,0 can be nonzero with exchanging scalar operators in these backgrounds.

In this paper, we use crossing symmetry and unitarity constraints to put a lower bound on the central charge of conformal field theories in large space-time dimensions D. Specifically, we work with the four-point function of identical scalars ϕ with scaling dimension ∆ϕ, and use a certain class of analytic functionals to show that the OPE coefficient squared {c}_{phi phi {T}^{mu nu}}^2 must be exponentially small in D. For this to hold, we need to make a certain assumption about the nature of the spectrum below 2∆ϕ. Our argument is robust and can be applied to any OPE coefficient squared {c}_{phi phi O}^2 with ∆O< 2∆ϕ. This suggests that conformal field theories in large dimensions (if they exist) must be exponentially close to generalized free field theories.

In d-dimensional CFTs with a large number of degrees of freedom an important set of operators consists of the stress tensor and its products, multi stress tensors. Thermalization of such operators, the equality between their expectation values in heavy states and at finite temperature, is equivalent to a universal behavior of their OPE coefficients with a pair of identical heavy operators. We verify this behavior in a number of examples which include holographic and free CFTs and provide a bootstrap argument for the general case. In a free CFT we check the thermalization of multi stress tensor operators directly and also confirm the equality between the contributions of multi stress tensors to heavy-heavy-light-light correlators and to the corresponding thermal light-light two-point functions by disentangling the contributions of other light operators. Unlike multi stress tensors, these light operators violate the Eigenstate Thermalization Hypothesis and do not thermalize.

We show that for a unitary modular invariant 2D CFT with central charge c>11$$\\end{document}]]> and having a nonzero twist gap in the spectrum of Virasoro primaries, for sufficiently large spin J, there always exist spin-J operators with twist falling in the interval (c-112-ε,c-112+ε) with ε=O(J-1/2logJ). We establish that the number of Virasoro primary operators in such a window has a Cardy-like, i.e., exp2π(c-1)J6 growth. A similar result is proven for a family of holographic CFTs with the twist gap growing linearly in c and a uniform boundedness condition, in the regime J≫c3≫1. From the perspective of near-extremal rotating BTZ black holes (without electric charge), our result is valid when the Hawking temperature is much lower than the “gap temperature.”

We compute numerically the dimensions and OPE coefficients of several operators in the 3d Ising CFT, and then try to reverse-engineer the solution to crossing symmetry analytically. Our key tool is a set of new techniques for computing infinite sums of SL(2, mathrm{mathbb{R}} ) conformal blocks. Using these techniques, we solve the lightcone bootstrap to all orders in an asymptotic expansion in large spin, and suggest a strategy for going beyond the large spin limit. We carry out the first steps of this strategy for the 3d Ising CFT, deriving analytic approximations for the dimensions and OPE coefficients of several infinite families of operators in terms of the initial data {Δσ, Δϵ, fσσϵ, fϵϵϵ, cT}. The analytic results agree with numerics to high precision for about 100 low-twist operators (correctly accounting for O(1) mixing effects between large-spin families). Plugging these results back into the crossing equations, we obtain approximate analytic constraints on the initial data.

We study the defect operator product expansion (OPE) of displacement operators in free and interacting conformal field theories using replica methods. We show that as n approaches 1 a contact term can emerge when the OPE contains defect operators of twist d − 2. For interacting theories and general states we give evidence that the only possibility is from the defect operator that becomes the stress tensor in the n → 1 limit. This implies that the quantum null energy condition (QNEC) is always saturated for CFTs with a twist gap. As a check, we show independently that in a large class of near vacuum states, the second variation of the entanglement entropy is given by a simple correlation function of averaged null energy operators as studied by Hofman and Maldacena. This suggests that sub-leading terms in the defect OPE are controlled by a defect version of the spin-3 non-local light ray operator and we speculate about the possible origin of such a defect operator. For free theories this contribution condenses to a contact term that leads to violations of QNEC saturation.

Using a recent proposal of circuit complexity in quantum field theories introduced by Jefferson and Myers, we compute the time evolution of the complexity following a smooth mass quench characterized by a time scale δt in a free scalar field theory. We show that the dynamics has two distinct phases, namely an early regime of approximately linear evolution followed by a saturation phase characterized by oscillations around a mean value. The behavior is similar to previous conjectures for the complexity growth in chaotic and holographic systems, although here we have found that the complexity may grow or decrease depending on whether the quench increases or decreases the mass, and also that the time scale for saturation of the complexity is of order δt (not parametrically larger).

We compute four-point functions with two maximal giant gravitons and two chiral primary operators at three-loop order in planar N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} = 4 Super-Yang-Mills theory. The Lagrangian insertion method, together with symmetries of the theory fix the integrand up to a few constants. Under the additional assumption of planarity, all these constants can be determined by lower loop data. The obtained result can be regarded as the planar diagram contribution to the full four-point function and is written in terms of the known three-loop conformal integrals. From the four-point function, we extract the OPE coefficients of two giant gravitons and one non-BPS twist-2 operator with arbitrary spin at three-loops, given in terms of harmonic sums. We observe an intriguingly simple relation between the giant graviton OPE coefficients and the OPE coefficients of three single-trace operators.

Free massless Boson fields are defined as manifestly covariant unitary representations of the Poincare group for zero mass and integer spin s . The fields are tensors which, in the simplest case, belong to the representationD(s,0) ⊕ D(0,s) of the Lorentz group. They are characterized by wave equations of two types: (i) The symmetry conditions, which impose the requirement that the tensors indeed carry the representation D(s,0) ⊕ D(0,s), and(ii) the unitarity conditions, which turn out to be of the form {im573-1}

Boundaries not only are fundamental elements in nearly all realistic physical systems, but also greatly enrich the structure of quantum field theories. In this paper, we demonstrate that conformal field theory (CFT) with a boundary, known as surface CFT in three dimensions, can be studied with the setup of fuzzy sphere. We consider the example of surface criticality of the 3d Ising CFT. We propose two schemes by cutting a boundary in the orbital space and the real space respectively to realise the ordinary and the normal surface CFTs on the fuzzy sphere. We obtain the operator spectra through state-operator correspondence. We observe integer spacing of the conformal multiplets, and thus provide direct evidence of conformal symmetry. We identify the ordinary surface primary o o , the displacement operator \mathrm{D} D and their conformal descendants and extract their scaling dimensions. We also study the one-point and two-point correlation functions and extract the bulk-to-surface OPE coefficients, some of which are reported for the first time. In addition, using the overlap of the bulk CFT state and the polarised state, we calculate the boundary central charges of the 3d Ising surface CFTs non-perturbatively. Other conformal data obtained in this way also agrees with prior methods. We also discuss future directions, such as special surface CFT, cone defects, and surfaces of other bulk CFTs.

The quantum null energy condition (QNEC) is a quantum generalization of the null energy condition, which gives a lower bound on the null energy in terms of the second derivative of the von Neumann entropy or entanglement entropy of some region with respect to a null direction. The QNEC states that $⟨{T}_{kk}{⟩}_{p}\ensuremath{\ge}{\mathrm{lim}}_{A\ensuremath{\rightarrow}0}(\frac{\ensuremath{\hbar}}{2\ensuremath{\pi}A}{S}_{\mathrm{out}}^{\ensuremath{'}\ensuremath{'}})$, where ${S}_{\mathrm{out}}$ is the entanglement entropy restricted to one side of a codimension-2 surface $\mathrm{\ensuremath{\Sigma}}$, which is deformed in the null direction about a neighborhood of point $p$ with area $A$. A proof of QNEC has been given before, which applies to free and super-renormalizable bosonic field theories, and to any points that lie on a stationary null surface. Using similar assumptions and methods, we prove the QNEC for fermionic field theories.

In 1914, Ramanujan unveiled 17 extraordinary infinite series for 1/π. In this Letter, we uncover their physics origin by relating them to 2D logarithmic conformal field theories (LCFTs), which emerge in diverse settings such as the fractional quantum Hall effect, percolation, polymers, and even holography. Through this LCFT connection, we reinterpret such infinite series in terms of fundamental CFT data-the operator spectrum and OPE coefficients. This perspective leads to novel physics-inspired approximations for 1/π. Drawing lessons from Ramanujan's formulas, we construct a new family of bases for expanding LCFT correlators that converge far more rapidly than the standard conformal block decomposition. This is achieved using recently developed stringy/parametric crossing-symmetric dispersion relations. Remarkably, when working with these new expansions, the action of a certain differential operator (which arises naturally from the Ramanujan connection) dramatically enhances convergence, with the entire contribution collapsing to that of the logarithmic identity operator. This striking simplification hints at a universal property of LCFTs. Finally, we discuss a new holographic interpretation of this unexpected mathematics-physics connection.

We review and systematize two (analytic) bootstrap techniques in two-dimensional conformal field theories using the S-modular transformation. The first one gives universal results in asymptotic regimes by relating extreme temperatures. Along with the presentation of known results, we use this technique to also derive asymptotic formulae for the Zamolodchikov recursion coefficients which match previous conjectures from numerics and from Regge asymptotic analysis. The second technique focuses on intermediate temperatures. We use it to sketch a methodology to derive a bound on off-diagonal squared OPE coefficients, as well as to improve existing bounds on the spectrum in case of non-negative diagonal OPE coefficients.

We consider the algebras generated by observables in quantum field theory localized in regions in the null plane. For a scalar free field theory, we show that the one-particle structure can be decomposed into a continuous direct integral of lightlike fibres and the modular operator decomposes accordingly. This implies that a certain form of QNEC is valid in free fields involving the causal completions of half-spaces on the null plane (null cuts). We also compute the relative entropy of null cut algebras with respect to the vacuum and some coherent states.

We set up the conventional conformal bootstrap equations in Mellin space and analyse the anomalous dimensions and OPE coefficients of large spin double trace operators. By decomposing the equations in terms of continuous Hahn polynomials, we derive explicit expressions as an asymptotic expansion in inverse conformal spin to any order, reproducing the contribution of any primary operator and its descendants in the crossed channel. The expressions are in terms of known mathematical functions and involve generalized Bernoulli (Nørlund) polynomials and the Mack polynomials and enable us to derive certain universal properties. Comparing with the recently introduced reformulated equations in terms of crossing symmetric tree level exchange Witten diagrams, we show that to leading order in anomalous dimension but to all orders in inverse conformal spin, the equations are the same as in the conventional formulation. At the next order, the polynomial ambiguity in the Witten diagram basis is needed for the equivalence and we derive the necessary constraints for the same.