Projectors, shadows, and conformal blocks

By David Simmons-Duffin. 2012

304 citations recorded by the source corpus.

A bstractWe introduce a method for computing conformal blocks of operators in arbitrary Lorentz representations in any spacetime dimension, making it possible to apply bootstrap techniques to operators with spin. The key idea is to implement the “shadow formalism” of Ferrara, Gatto, Grillo, and Parisi in a setting where conformal invariance is manifest. Conformal blocks in d-dimensions can be expressed as integrals over the projective null-cone in the “embedding space” $ \mathbb{R} $d+1,1. Taking care with their analytic structure, these integrals can be evaluated in great generality, reducing the computation of conformal blocks to a bookkeeping exercise. To facilitate calculations in four-dimensional CFTs, we introduce techniques for writing down conformally-invariant correlators using auxiliary twistor variables, and demonstrate their use in some simple examples.

Subfields: Nuclear and High Energy Physics, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/678ddbe81ea621f7d3b0b9903fbbeac29f540b37

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Citations in the corpus

Conformal Regge theory (2012, background): “The second conformal block is usually called the shadow of the first (see for example [21] for details).”
Bootstrapping mixed correlators in the 3D Ising model (2014, methodology): “2) Other recent studies of global conformal blocks can be found in [11, 41, 42, 45, 46, 49–55], with connections to Mellin amplitudes in [56–65].”
Witten diagrams revisited: the AdS geometry of conformal blocks (2015, background): “One can take this second-order differential equation, plus the corresponding one with 1, 2↔ 3, 4, supplemented with appropriate boundary conditions, as one’s definition of W∆,` [74].”
Spinning AdS propagators (2014, methodology): “The integral over the boundary point P is conformal and can be done using the equality [13] ∫”
Radial Coordinates for Conformal Blocks (2013, background): “Conformal blocks for such correlators have been studied recently in [22, 23](20), but the results were not yet put to concrete use.”
Bootstrapping the O(N ) vector models (2013, methodology): “Although much progress has been made computing conformal blocks for these types of operators using series manipulations [40, 41, 43, 13, 42], derivative relations [44, 45], and conformal integrals [46], we believe the residues ci are the most convenient and directly useful data for numerical bootstr”
Bootstrapping 3D fermions (2015, methodology): “As described in [27], the conformal block of O can be obtained from the integral”
Celestial amplitudes and conformal soft theorems (2019, background): “13) where tilde denotes the shadow transform (see [59]).”
Conformal basis for flat space amplitudes (2017, background): “To study the shadow transform of our conformal primary wavefunction, let us first note a useful identity (see, for example, [34]) ∫ d~z 1 |~z − ~ w|2(d−∆) 1 (−q(~z) ·X)∆ = π d 2 Γ(∆− d 2 ) Γ(∆) (−X(2)) d2−∆ (−q(~ w) ·X)d−∆ .”