Search

We study quantum field theory on a de Sitter spacetime dS$_{d+1}$ background. Our main tool is the Hilbert space decomposition in irreducible unitary representations of its isometry group $SO(d+1,1)$. As the first application of the Hilbert space formalism, we recover the K\"allen-Lehmann spectral decomposition of the scalar bulk two-point function. In the process, we exhibit a relation between poles in the corresponding spectral densities and the boundary CFT data. Moreover, we derive an inversion formula for the spectral density through analytical continuation from the sphere and use it to find the spectral decompisiton for a few examples. Next, we study the conformal partial wave decomposition of the four-point functions of boundary operators. These correlation functions are very similar to the ones of standard conformal field theory, but have different positivity properties that follow from unitarity in de Sitter. We conclude by proposing a non-perturbative conformal bootstrap approach to the study of these late-time four-point functions, and we illustrate our proposal with a concrete example for QFT in dS$_2$., Comment: 54 pages + appendices, 3 figures; v2 corrections, fixed typos, added references, JHEP version

Quantum Field Theories (QFTs) in Anti-de Sitter (AdS) spacetime are often strongly coupled when the radius of AdS is large, and few methods are available to study them. In this work, we develop a Hamiltonian truncation method to compute the energy spectrum of QFTs in two-dimensional AdS. The infinite volume of constant timeslices of AdS leads to divergences in the energy levels. We propose a simple prescription to obtain finite physical energies and test it with numerical diagonalization in several models: the free massive scalar field, $\phi^4$ theory, Lee-Yang and Ising field theory. Along the way, we discuss spontaneous symmetry breaking in AdS and derive a compact formula for perturbation theory in quantum mechanics at arbitrary order. Our results suggest that all conformal boundary conditions for a given theory are connected via bulk renormalization group flows in AdS., Comment: 63 pages + appendices, 44 figures

We study fixed points with N scalar fields in $4 - \varepsilon$ dimensions to leading order in $\varepsilon$ using a bottom-up approach. We do so by analyzing O(N) invariants of the quartic coupling $\lambda_{ijkl}$ that describes such CFTs. In particular, we show that $\lambda_{iijj}$ and $\lambda_{ijkl}^2$ are restricted to a specific domain, refining a result by Rychkov and Stergiou. We also study averages of one-loop anomalous dimensions of composite operators without gradients. In many cases, we are able to show that the O(N) fixed point maximizes such averages. In the final part of this work, we generalize our results to theories with N complex scalars and to bosonic QED. In particular we show that to leading order in $\varepsilon$, there are no bosonic QED fixed points with N < 183 flavors., Comment: 29 pages + appendices, 10 figures. v2: misprints corrected; v3: added comparison between complex and real bounds, minor edits, version to appear in JHEP

We describe a nonperturbative method to compute the partition function and correlation functions for scalar QFTs set on the $d$-dimensional sphere $S^d$. The method relies on a Hamiltonian picture, where the theory is quantized on $S^{d-1}$ and states evolve in time by means of a time-dependent Hamiltonian. Crucially, the Hilbert space on $S^{d-1}$ is truncated to a finite set of states below a cutoff. Throughout this work we focus on the $\phi^2$ and $i\phi^3$ flows in three dimensions. In the first part of this paper we analyze the cutoff-dependence of various observables, computing both divergent and RG-improvement counterterms to be added to the action. Next we present nonperturbative results for the massive scalar on $S^3$, finding good agreement in the strong-coupling regime between numerical data and the $F$-coefficient of the free scalar CFT. We also check that the renormalized $i \phi^3$ theory on $S^3$ is nonperturbatively UV-finite. The scheme in question breaks the $\mathrm{SO}(d+1)$ spacetime symmetry group of $S^d$ down to $\mathrm{SO}(d)$, and in an example we study how the full symmetry is restored in the continuum limit. The relation between our method and earlier work by Al. B. Zamolodchikov involving a specific RG flow on $S^2$ is explained as well., Comment: 27 pages + appendices, 6 figures. v2: typos, added refs, minor changes

This paper investigates d-dimensional CFTs in the presence of a codimension-one boundary and CFTs defined on real projective space RP^d. Our analysis expands on the alpha space method recently proposed for one-dimensional CFTs in arXiv:1702.08471. In this work we establish integral representations for scalar two-point functions in boundary and crosscap CFTs using plane-wave-normalizable eigenfunctions of different conformal Casimir operators. CFT consistency conditions imply integral equations for the spectral densities appearing in these decompositions, and we study the relevant integral kernels in detail. As a corollary, we find that both the boundary and crosscap kernels can be identified with special limits of the d=1 crossing kernel., Comment: 30 pages, 2 figures; v2: references added

We initiate the study of the conformal bootstrap using Sturm-Liouville theory, specializing to four-point functions in one-dimensional CFTs. We do so by decomposing conformal correlators using a basis of eigenfunctions of the Casimir which are labeled by a complex number alpha. This leads to a systematic method for computing conformal block decompositions. Analyzing bootstrap equations in alpha space turns crossing symmetry into an eigenvalue problem for an integral operator K. The operator K is closely related to the Wilson transform, and some of its eigenfunctions can be found in closed form., Comment: 34 pages, 3 figures. v2: some minor changes and added subsection 2.3 on convergence

Logarithmic conformal field theories have a vast range of applications, from critical percolation to systems with quenched disorder. In this paper we thoroughly examine the structure of these theories based on their symmetry properties. Our analysis is model-independent and holds for any spacetime dimension. Our results include a determination of the general form of correlation functions and conformal block decompositions, clearing the path for future bootstrap applications. Several examples are discussed in detail, including logarithmic generalized free fields, holographic models, self-avoiding random walks and critical percolation., Comment: 55 pages + appendices

We consider the dimensional reduction of a CFT, breaking multiplets of the d-dimensional conformal group SO(d+1,1) up into multiplets of SO(d,1). This leads to an expansion of d-dimensional conformal blocks in terms of blocks in d-1 dimensions. In particular, we obtain a formula for 3d conformal blocks as an infinite sum over 2F1 hypergeometric functions with closed-form coefficients., Comment: 12 pages, 1 figure

We consider the continuation of free and interacting scalar field theory to non-integer spacetime dimension d. We find that the correlation functions in these theories are necessarily incompatible with unitarity (or with reflection positivity in Euclidean signature). In particular, the theories contain negative norm states unless d is a positive integer. These negative norm states can be obtained via the OPE from simple positive norm operators, and are therefore an integral part of the theory. At the Wilson-Fisher fixed point the non-unitarity leads to the existence of complex anomalous dimensions. We demonstrate that they appear already at leading order in the epsilon expansion., Comment: 31 pages, 2 figures; v2: references added, version to appear in Phys. Rev. D

We show how to perform accurate, nonperturbative and controlled calculations in quantum field theory in d dimensions. We use the Truncated Conformal Space Approach (TCSA), a Hamiltonian method which exploits the conformal structure of the UV fixed point. The theory is regulated in the IR by putting it on a sphere of a large finite radius. The QFT Hamiltonian is expressed as a matrix in the Hilbert space of CFT states. After restricting ourselves to energies below a certain UV cutoff, an approximation to the spectrum is obtained by numerical diagonalization of the resulting finite-dimensional matrix. The cutoff dependence of the results can be computed and efficiently reduced via a renormalization procedure. We work out the details of the method for the phi^4 theory in d dimensions with d not necessarily integer. A numerical analysis is then performed for the specific case d = 2.5, a value chosen in the range where UV divergences are absent. By going from weak to intermediate to strong coupling, we are able to observe the symmetry-preserving, symmetry-breaking, and conformal phases of the theory, and perform rough measurements of masses and critical exponents. As a byproduct of our investigations we find that both the free and the interacting theories in non integral d are not unitary, which however does not seem to cause much effect at low energies., Comment: 56 pages, 22 figures; V2: defective figure file replaced; V3: minor corrections; refs added; V4: refs updated

Conformal blocks in any number of dimensions depend on two variables z, zbar. Here we study their restrictions to the special "diagonal" kinematics z = zbar, previously found useful as a starting point for the conformal bootstrap analysis. We show that conformal blocks on the diagonal satisfy ordinary differential equations, third-order for spin zero and fourth-order for the general case. These ODEs determine the blocks uniquely and lead to an efficient numerical evaluation algorithm. For equal external operator dimensions, we find closed-form solutions in terms of finite sums of 3F2 functions., Comment: 21 pages; v2: misprints in Eqs. 2.17 and 4.9 corrected; v3: misprints in A.5 and A.6 corrected

We develop the theory of conformal blocks in CFT_d expressing them as power series with Gegenbauer polynomial coefficients. Such series have a clear physical meaning when the conformal block is analyzed in radial quantization: individual terms describe contributions of descendants of a given spin. Convergence of these series can be optimized by a judicious choice of the radial quantization origin. We argue that the best choice is to insert the operators symmetrically. We analyze in detail the resulting "rho-series" and show that it converges much more rapidly than for the commonly used variable z. We discuss how these conformal block representations can be used in the conformal bootstrap. In particular, we use them to derive analytically some bootstrap bounds whose existence was previously found numerically., Comment: 27 pages, 9 figures; v2: misprints corrected

Worm methods to simulate the Ising model in the Aizenman random current representation including a low noise estimator for the connected four point function are extended to allow for antiperiodic boundary conditions. In this setup several finite size renormalization schemes are formulated and studied with regard to the triviality of \phi^4 theory in four dimensions. With antiperiodicity eliminating the zero momentum Fourier mode a closer agreement with perturbation theory is found compared to the periodic torus., Comment: 20 pages, 3 double-figures, 6 tables