78 research outputs found
Convexity and Liberation at Large Spin
We consider several aspects of unitary higher-dimensional conformal field
theories (CFTs). We first study massive deformations that trigger a flow to a
gapped phase. Deep inelastic scattering in the gapped phase leads to a
convexity property of dimensions of spinning operators of the original CFT. We
further investigate the dimensions of spinning operators via the crossing
equations in the light-cone limit. We find that, in a sense, CFTs become free
at large spin and 1/s is a weak coupling parameter. The spectrum of CFTs enjoys
additivity: if two twists tau_1, tau_2 appear in the spectrum, there are
operators whose twists are arbitrarily close to tau_1+tau_2. We characterize
how tau_1+tau_2 is approached at large spin by solving the crossing equations
analytically. We find the precise form of the leading correction, including the
prefactor. We compare with examples where these observables were computed in
perturbation theory, or via gauge-gravity duality, and find complete agreement.
The crossing equations show that certain operators have a convex spectrum in
twist space. We also observe a connection between convexity and the ratio of
dimension to charge. Applications include the 3d Ising model, theories with a
gravity dual, SCFTs, and patterns of higher spin symmetry breaking.Comment: 61 pages, 13 figures. v2: added reference and minor correctio
Modular Invariance, Tauberian Theorems, and Microcanonical Entropy
We analyze modular invariance drawing inspiration from tauberian theorems.
Given a modular invariant partition function with a positive spectral density,
we derive lower and upper bounds on the number of operators within a given
energy interval. They are most revealing at high energies. In this limit we
rigorously derive the Cardy formula for the microcanonical entropy together
with optimal error estimates for various widths of the averaging energy shell.
We identify a new universal contribution to the microcanonical entropy
controlled by the central charge and the width of the shell. We derive an upper
bound on the spacings between Virasoro primaries. Analogous results are
obtained in holographic 2d CFTs. We also study partition functions with a UV
cutoff. Control over error estimates allows us to probe operators beyond the
unity in the modularity condition. We check our results in the 2d Ising model
and the Monster CFT and find perfect agreement.Comment: 39 pages, 9 figure
On Fine Structure of Strings: The Universal Correction to the Veneziano Amplitude
We consider theories of weakly interacting higher spin particles in flat
spacetime. We focus on the four-point scattering amplitude at high energies and
imaginary scattering angles. The leading asymptotic of the amplitude in this
regime is universal and equal to the corresponding limit of the Veneziano
amplitude. In this paper, we find that the first sub-leading correction to this
asymptotic is universal as well. We compute the correction using a model of
relativistic strings with massive endpoints. We argue that it is unique using
holography, effective theory of long strings and bootstrap techniques.Comment: 54 pages, 5 figure
Conformal Bootstrap With Slightly Broken Higher Spin Symmetry
We consider conformal field theories with slightly broken higher spin
symmetry in arbitrary spacetime dimensions. We analyze the crossing equation in
the double light-cone limit and solve for the anomalous dimensions of higher
spin currents with large spin . The result depends on the
symmetries and the spectrum of the unperturbed conformal field theory. We
reproduce all known results and make further predictions. In particular we make
a prediction for the anomalous dimensions of higher spin currents in the 3d
Ising model.Comment: 41 pages, 2 figures, %\draftmod
On Conformal Field Theories With Extremal a/c Values
Unitary conformal field theories (CFTs) are believed to have positive
(non-negative) energy correlators. Energy correlators are universal observables
in higher-dimensional CFTs built out of integrated Wightman functions of the
stress-energy tensor. We analyze energy correlators in parity invariant
four-dimensional CFTs. The goal is to use the positivity of energy correlators
to further constrain unitary CFTs. It is known that the positivity of the
simplest one-point energy correlator implies that 1/3 <= a/c <= 31/18 where a
and c are the Weyl anomaly coefficients. We use the positivity of higher point
energy correlators to show that CFTs with extremal values of a/c have trivial
scattering observables. More precisely, for a/c=1/3 and a/c=31/18 all energy
correlators are fixed to be the ones of the free boson and the free vector
theory correspondingly. Similarly, we show that the positivity and finiteness
of energy correlators together imply that the three-point function of the
stress tensor in a CFT cannot be proportional to the one in the theory of free
boson, free fermion or free vector field.Comment: 24 pages, 3 figure
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