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Register a new userBounds on the density of states and the spectral gap in CFT$_{2}$
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We improve the recently discovered upper and lower bounds on the $O(1)$ correction to the Cardy formula for the density of states integrated over an energy window (of width $2delta$), centered at high energy in 2 dimensional conformal field theory. We prove optimality of the lower bound for $deltato 1^{-}$. We prove a conjectured upper bound on the asymptotic gap between two consecutive Virasoro primaries for a central charge greater than $1,$ demonstrating it to be $1.$ Furthermore, a systematic method is provided to establish a limit on how tight the bound on the $O(1)$ correction to the Cardy formula can be made using bandlimited functions. The techniques and the functions used here are of generic importance whenever the Tauberian theorems are used to estimate some physical quantities.
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We introduce a new approach to find the Tomita-Takesaki modular flow for multi-component regions in general chiral conformal field theory. Our method is based on locality and analyticity of primary fields as well as the so-called Kubo-Martin-Schwinger (KMS) condition. These features can be used to transform the problem to a Riemann-Hilbert problem on a covering of the complex plane cut along the regions, which is equivalent to an integral equation for the matrix elements of the modular Hamiltonian. Examples are considered.
In this paper the relativistic quantum dynamics of a spin-1/2 neutral particle with a magnetic moment $mu$ in the cosmic string spacetime is reexamined by applying the von Neumann theory of self--adjoint extensions. Contrary to previous studies where the interaction between the spin and the line of charge is neglected, here we consider its effects. This interaction gives rise to a point interaction: $boldsymbol{ abla} cdot mathbf{E}= (2lambda/alpha)delta(r)/r$. Due to the presence of the Dirac delta function, by applying an appropriated boundary condition provided by the theory of self--adjoint extensions, irregular solutions for the Hamiltonian are allowed. We address the scattering problem obtaining the phase shift, S-matrix and the scattering amplitude. The scattering amplitude obtained shows a dependency with energy which stems from the fact that the helicity is not conserved in this system. Examining the poles of the S-matrix we obtain an expression for the bound states. The presence of bound states for this system has not been discussed before in the literature.
In this paper we study in detail the deformations introduced in [1] of the integrable structures of the AdS$_{2,3}$ integrable models. We do this by embedding the corresponding scattering matrices into the most general solutions of the Yang-Baxter equation. We show that there are several non-trivial embeddings and corresponding deformations. We work out crossing symmetry for these models and study their symmetry algebras and representations. In particular, we identify a new elliptic deformation of the $rm AdS_3 times S^3 times M^4$ string sigma model.
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 $rho, bar{rho}$. We prove a key fact that $|rho|, |bar{rho}| < 1$ inside the forward tube, and set bounds on how fast $|rho|, |bar{rho}|$ 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).
We show that the four-point functions in conformal field theory are defined as distributions on the boundary of the region of convergence of the conformal block expansion. The conformal block expansion converges in the sense of distributions on this boundary, i.e. it can be integrated term by term against appropriate test functions. This can be interpreted as a giving a new class of functionals that satisfy the swapping property when applied to the crossing equation, and we comment on the relation of our construction to other types of functionals. Our language is useful in all considerations involving the boundary of the region of convergence, e.g. for deriving the dispersion relations. We establish our results by elementary methods, relying only on crossing symmetry and the standard convergence properties of the conformal block expansion. This is the first in a series of papers on distributional properties of correlation functions in conformal field theory.
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