Recursion relations for conformal blocks

151

We compute in closed analytical form the minimal set of "seed" conformal blocks associated to the exchange of generic mixed symmetry spinor/tensor operators in an arbitrary representation ( ,¯ ) of the Lorentz group in four dimensional conformal field theories. These blocks arise from 4-point functions involving two scalars, one (0, | −¯ |) and one (| −¯ |, 0) spinors or tensors. We directly solve the set of Casimir equations, that can elegantly be written in a compact form for any ( ,¯ ), by using an educated ansatz and reducing the problem to an algebraic linear system. Various details on the form of the ansatz have been deduced by using the so called shadow formalism. The complexity of the conformal blocks depends on the value of p = | −¯ | and grows with p, in analogy to what happens to scalar conformal blocks in d even space-time dimensions as d increases. These results open the way to bootstrap 4-point functions involving arbitrary spinor/tensor operators in four dimensional conformal field theories.

“…[12] and the further developments in ref. [13], CBs associated to the exchange of fermion operators in 3D have recently been computed [14]. In ref.…”

Section: Jhep02(2016)183mentioning

confidence: 99%

Seed conformal blocks in 4D CFT

Echeverri

Elkhidir

Karateev

et al. 2016

151

“…We cast our discussion in terms of the embedding space formalism [11,16,[45][46][47] throughout. Other recent work on spinning conformal blocks can be found in [31,32,[48][49][50][51][52][53][54][55][56][57][58][59].…”

Section: Jhep11(2017)060mentioning

confidence: 99%

Spinning geodesic Witten diagrams

Dyer¹,

Freedman²,

Sully³

2017

We present an expression for the four-point conformal blocks of symmetric traceless operators of arbitrary spin as an integral over a pair of geodesics in Anti-de Sitter space, generalizing the geodesic Witten diagram formalism of Hijano et al.[1] to arbitrary spin. As an intermediate step in the derivation, we identify a convenient basis of bulk threepoint interaction vertices which give rise to all possible boundary three point structures. We highlight a direct connection between the representation of the conformal block as geodesic Witten diagram and the shadow operator formalism.

“…It will be interesting to relate the results in this section (or the zero-modes) to the rational representations of conformal blocks [33][34][35].…”

Section: Jhep10(2016)060mentioning

confidence: 99%

Partition functions with spin in AdS2 via quasinormal mode methods

Keeler¹,

Lisbão²,

Ng³

2016

Abstract:We extend the results of [1], computing one loop partition functions for massive fields with spin half in AdS 2 using the quasinormal mode method proposed by Denef, Hartnoll, and Sachdev [2]. We find the finite representations of SO(2, 1) for spin zero and spin half, consisting of a highest weight state |h and descendants with non-unitary values of h. These finite representations capture the poles and zeroes of the one loop determinants. Together with the asymptotic behavior of the partition functions (which can be easily computed using a large mass heat kernel expansion), these are sufficient to determine the full answer for the one loop determinants. We also discuss extensions to higher dimensional AdS 2n and higher spins.