The existence of a
positive linear functional acting on the space of (differences between)
conformal blocks has been shown to rule out regions in the parameter space of
conformal field theories (CFTs). We argue that at the boundary of the allowed
region the extremal functional contains, in principle, enough information to
determine the dimensions and OPE coefficients of an infinite number of
operators appearing in the correlator under analysis. Based on this idea we
develop the Extremal Functional Method (EFM), a numerical procedure for
deriving the spectrum and OPE coefficients of CFTs lying on the boundary (of
solution space). We test the EFM by using it to rederive the low lying spectrum
and OPE coefficients of the 2d Ising model based solely on the dimension of a
single scalar quasi-primary -- no Virasoro algebra required. Our work serves as
a benchmark for applications to more interesting, less known CFTs in the near
future, such as the 3d Ising model.
The stress-energy tensor in a conformal field theory has
zero trace.
Hence AdS boundary stress-tensors are traceless by
construction, to match this property of the dual CFT. An elegant (aka nifty)
construction based on the conformal isometry of AdS will be presented which
shows that in an asymptotically AdS spacetime, the sum of the ADM mass and the
ADM tensions is zero. This result follows strictly from the gravitational point
of view- that is, the Einstein equations and the definitions of the ADM
charges. Further, it turns out that perturbative stress-energy sources in an asymptotically
AdS spacetime must satisfy a local version of this constraint, namely that the
sum of the energy density minus the pressures equals zero. The situation with
positive cosmological constant is both similar and distinct in interesting
ways, which will be briefly discussed. The analogous (analytically continued)
conformal isometry in dS is the root of the
``k^4 “ power spectrum for causal cosmological
perturbations. Work in progress (speculations) will be presented about a
corresponding sum-rule for gravitational charges defined at future infinity in
a spacetime that approaches dS at late times.
Holographic cosmology maps cosmological time evolution to
the inverse RG flow of a dual three-dimensional QFT. In cases where this RG flow
connects two closely separated fixed points, QFT correlators may be calculated
perturbatively in terms of the conformal field theory associated with one of the
fixed points, even when the dual QFT is at strong coupling.
Realising slow-roll inflation in these terms, we show how to derive
standard slow-roll inflationary power spectra and non-Gaussianities through
purely holographic calculations. The form of slow-roll inflationary correlators
is seen to be determined by the perturbative breaking of conformal symmetry
away from the fixed point.