Six-dimensional Methods for Four-dimensional Conformal Field Theories

By S. Weinberg. 2010

240 citations recorded by the source corpus.

The calculation of both spinor and tensor Green's functions in four-dimensional conformally-invariant field theories can be greatly simplified by six-dimensional methods. For this purpose, four-dimensional fields are constructed as projections of fields on the hypercone in six-dimensional projective space, satisfying certain transversality conditions. In this way some Green's functions in conformal field theories are shown to have structures more general than those commonly found by use of the inversion operator. These methods fit in well with the assumption of AdS/CFT duality. In particular, it is transparent that if fields on AdS{sub 5} approach finite limits on the boundary of AdS{sub 5}, then in the conformal field theory on this boundary these limits transform with conformal dimensionality zero if they are tensors (of any rank), but with conformal dimension 1/2 if they are spinors or spinor-tensors.

Subfields: Astronomy and Astrophysics, Nuclear and High Energy Physics

View Paper: https://www.semanticscholar.org/paper/31d7a8054e566f8e4740bf520bb1f64e6210b28e

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Citations in the corpus

On graviton non-gaussianities during inflation (2011, background): “An elegant and simple way to write correlation functions is to go to the embedding space formalism [45–47].”
Superconformal symmetry and two-loop amplitudes in planar N = 4 super Yang-Mills (2011, methodology): “(2.3) 2Our fermions are related simply to those of Weinberg [34], which satisfy the constraint Xψ̃Weinberg = 0: ψ̃Weinberg = Xψus.”
Bootstrapping 3D fermions (2015, methodology): “To find the embedding of fields in 5D spacetime we follow the approach of [26].”
Projectors, shadows, and conformal blocks (2012, background): “As shown in [33], spinors lift to twistors.”
Analyticity and the holographic S-matrix (2011, unknown): “It will be useful to represent the kinematics in d + 2 dimensional embedding space coordinates P α [51].”
EPFL Lectures on Conformal Field Theory in D ≥ 3 Dimensions (2016, background): “It was then discussed by Weinberg [18] (who apparently rediscovered it independently of all the previous literature).”
Modular Hamiltonians on the null plane and the Markov property of the vacuum state (2017, methodology): “Instead, the CFT is naturally defined on a cylinder, obtained by compactifying Minkowski space [29, 30].”
Defects in conformal field theory (2016, background): “In fact, it is possible to embed the former space into the latter in such a way that the generators of SO(d+ 1, 1) act on the embedded Rd precisely as conformal transformations [2, 3, 4].”
Unitarity and the holographic S-Matrix (2011, methodology): “For this purpose, it will be simpler and more elegant to use the embedding formalism [23, 24], where we can use the machinery that was developed and nicely explained in [25].”
The cosmological bootstrap: weight-shifting operators and scalar seeds (2019, methodology): “The latter are defined most naturally in embedding space [65–67].”
Spinning conformal blocks (2011, methodology): “This approach to CFT computations goes back to the work of Dirac [27], and has been used on and oﬀ since the 1970’s [1, 28, 29] (see also [30, 31] for recent work).”
A natural language for AdS/CFT correlators (2011, methodology): “We will be discussing CFT correlation functions, so it is natural to use variables [40, 41] that are acted on linearly by the Euclidean conformal group SO(1, d+ 1).”