Distributions in CFT. Part II. Minkowski space

CFTs in Euclidean signature satisfy well-accepted rules, such as the convergent Euclidean OPE. It is nowadays common to assume that CFT correlators exist and have various properties also in Lorentzian signature. Some of these properties may represent extra assumptions, and it is an open question if they hold for familiar statistical-physics CFTs such as the critical 3d Ising model. Here we consider Wightman 4-point functions of scalar primaries in Lorentzian signature. We derive a minimal set of their properties solely from the Euclidean unitary CFT axioms, without using extra assumptions. We establish all Wightman axioms (temperedness, spectral property, local commutativity, clustering), Lorentzian conformal invariance, and distributional convergence of the s-channel Lorentzian OPE. This is done constructively, by analytically continuing the 4-point functions using the s-channel OPE expansion in the radial cross-ratios ρ, $$ \overline{\rho} $$ ρ ¯ . We prove a key fact that |ρ|, $$ \left|\overline{\rho}\right| $$ ρ ¯ < 1 inside the forward tube, and set bounds on how fast |ρ|, $$ \left|\overline{\rho}\right| $$ ρ ¯ may tend to 1 when approaching the Minkowski space.We also provide a guide to the axiomatic QFT literature for the modern CFT audience. We review the Wightman and Osterwalder-Schrader (OS) axioms for Lorentzian and Euclidean QFTs, and the celebrated OS theorem connecting them. We also review a classic result of Mack about the distributional OPE convergence. Some of the classic arguments turn out useful in our setup. Others fall short of our needs due to Lorentzian assumptions (Mack) or unverifiable Euclidean assumptions (OS theorem).

Classical solutions and their double copy in split signature

The three-point amplitude is the key building block in the on-shell approach to scattering amplitudes. We show that the classical objects computed by massive three-point amplitudes in gauge theory and gravity are Newman-Penrose scalars in a split-signature spacetime, where three-point amplitudes can be defined for real kinematics. In fact, the quantum state set up by the particle is a coherent state fully determined by the three-point amplitude due to an eikonal-type exponentiation. Having identified this simplest classical solution from the perspective of scattering amplitudes, we explore the double copy of the Newman-Penrose scalars induced by the traditional double copy of amplitudes, and find that it coincides with the Weyl version of the classical double copy. We also exploit the Kerr-Schild version of the classical double copy to determine the exact spacetime metric in the gravitational case. Finally, we discuss the direct implication of these results for Lorentzian signature via analytic continuation.

Replica wormholes for an evaporating 2D black hole

Quantum extremal islands reproduce the unitary Page curve of an evaporating black hole. This has been derived by including replica wormholes in the gravitational path integral, but for the transient, evaporating black holes most relevant to Hawking’s paradox, these wormholes have not been analyzed in any detail. In this paper we study replica wormholes for black holes formed by gravitational collapse in Jackiw-Teitelboim gravity, and confirm that they lead to the island rule for the entropy. The main technical challenge is that replica wormholes rely on a Euclidean path integral, while the quantum extremal islands of an evaporating black hole exist only in Lorentzian signature. Furthermore, the Euclidean equations for the Schwarzian mode are non-local, so it is unclear how to connect to the local, Lorentzian dynamics of an evaporating black hole. We address these issues with Schwinger-Keldysh techniques and show how the non-local equations reduce to the local ‘boundary particle’ description in special cases.

Local Nets of Von Neumann Algebras in the Sine–Gordon Model

The Haag–Kastler net of local von Neumann algebras is constructed in the ultraviolet finite regime of the Sine–Gordon model, and its equivalence with the massive Thirring model is proved. In contrast to other authors, we do not add an auxiliary mass term, and we work completely in Lorentzian signature. The construction is based on the functional formalism for perturbative Algebraic Quantum Field Theory together with estimates originally derived within Constructive Quantum Field Theory and adapted to Lorentzian signature. The paper extends previous work by two of us.

A note on the $$ \mathcal{N} $$ = 2 super-$$ {\mathcal{W}}_3 $$ holographic dictionary

This is a long-overdue companion paper to [1]. We study the relation between sl(3|2) Chern-Simons supergravity on AdS3 and two-dimensional CFT’s with $$ \mathcal{N} $$ N = 2 super-$$ {\mathcal{W}}_3 $$ W 3 symmetry. Specifically, we carry out a complete analysis of asymptotic symmetries in a basis that makes the superconformal structure transparent, allowing us to establish the precise dictionary between currents and transformation parameters in the bulk and their boundary counterparts. We also discuss the incorporation of sources and display in full detail the corresponding holographic Ward identities. By imposing suitable hermiticity conditions on the CFT currents, we identify the superalgebra su(2, 1|1, 1) as the appropriate real form of sl(3|2) in Lorentzian signature. We take the opportunity to review some of the properties of the $$ \mathcal{N} $$ N = 2 super-$$ {\mathcal{W}}_3 $$ W 3 conformal algebra, including its multiplet structure, OPE’s and spectral flow invariance, correcting some minor typos present in the literature.

On the flow of states under $T\overline{T}$

We study the TT deformation of two dimensional quantum field theories from a Hamiltonian point of view, focusing on aspects of the theory in Lorentzian signature. Our starting point is a simple rewriting of the spatial integral of the TT operator, which directly implies the deformed energy spectrum of the theory. Using this rewriting, we then derive flow equations for various quantities in the deformed theory, such as energy eigenstates, operators, and correlation functions. On the plane, we find that the deformation merely has the effect of implementing successive canonical/Bogoliubov transformations along the flow. This leads us to define a class of non-local, “dressed” operators (including a dressed stress tensor) which satisfy the same commutation relations as in the undeformed theory. This further implies that on the plane, the deformed theory retains its symmetry algebra, including conformal symmetry, if the original theory is a CFT. On the cylinder the TT deformation is much more non-trivial, but even so, correlation functions of certain dressed operators are integral transforms of the original ones. Finally, we propose a tensor network interpretation of our results in the context of AdS/CFT.

Spherically symmetric static wormhole models in the Einsteinian cubic gravity

Our aim is to discuss spherically symmetric static wormholes with the Lorentzian signature in the Einsteinian cubic gravity for two different models of pressure sources. First, we calculate the modified fields equations for the Einsteinian cubic gravity for the wormhole geometry under the anisotropic matter. Then we investigate the shape-function for two different models, which can be taken as a part of the general relation, namely, [Formula: see text]. We further study the energy conditions for both the models in the background of the Einsteinian cubic gravity. We show that our obtained shape-functions satisfy all the necessary conditions for the existence of wormhole solutions in the Einsteinian cubic gravity for some particular values of the different involved parameters. We also discuss the behavior of the energy conditions especially the null and the weak energy conditions for the wormhole models in the Einsteinian cubic gravity.

A generalized Nachtmann theorem in CFT

Correlators of unitary quantum field theories in Lorentzian signature obey certain analyticity and positivity properties. For interacting unitary CFTs in more than two dimensions, we show that these properties impose general constraints on families of minimal twist operators that appear in the OPEs of primary operators. In particular, we rederive and extend the convexity theorem which states that for the family of minimal twist operators with even spins appearing in the reflection-symmetric OPE of any scalar primary, twist must be a monotonically increasing convex function of the spin. Our argument is completely non-perturbative and it also applies to the OPE of nonidentical scalar primaries in unitary CFTs, constraining the twist of spinning operators appearing in the OPE. Finally, we argue that the same methods also impose constraints on the Regge behavior of certain CFT correlators.