Scalar-vector bootstrap

Abstract

Abstract:We work out all of the details required for implementation of the conformal bootstrap program applied to the four-point function of two scalars and two vectors in an abstract conformal field theory in arbitrary dimension. This includes a review of which tensor structures make appearances, a construction of the projectors onto the required mixed symmetry representations, and a computation of the conformal blocks for all possible operators which can be exchanged. These blocks are presented as differential operators acting upon the previously known scalar conformal blocks. Finally, we set up the bootstrap equations which implement crossing symmetry. Special attention is given to the case of conserved vectors, where several simplifications occur.

Conformal two-point correlation functions from the operator product expansion

“…Using the definition of the three-point tensorial function (2.5) and the general result of [1] for the action of the differential operator, we find that there exists a simple conformal substitution rule for (2.12), analogous to the one in the three-point case [19]. It can be explicitly and concisely given as 13) where onlyP N 24 andP N 34 are expressed in terms of the homogeneized three-point embedding space coordinates (2.8) for the three-point conformal substitution (2.5). After the three-point conformal substitution has been implemented but before the four-point one is performed, all the embedding space coordinates are re-expressed in terms of the homogeneized four-point embedding space coordinates (2.10) and the conformal cross-ratios (2.11), with the homogeneized three-point quantities (2.8) given byη…”

“…With the revival of interest in the conformal bootstrap, several new results for conformal blocks were developed more recently [12][13][14] using a variety of different methods.…”

Seed conformal blocks in 4D CFT

“…Applications of this method for some specific correlators appeared in ref. [21]. Some other limits of the known CBs have been discussed in refs.…”

Spinning geodesic Witten diagrams

“…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].…”