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.
Highlights
The data of an abstract unitary conformal field theory (CFT) in D dimensions is encoded by the spectrum of primary operators and their operator product expansion (OPE), which are in turn specified by a finite number of real constants for each triplet of primary operators
Using the shadow formalism and the results from appendix A, appendix B, and appendix D, we compute the required integrals and perform the monodromy projection to obtain the conformal blocks
The contribution of a primary operator, either O or A, and all its descendants to the four-point function is captured by a conformal block gprs, where the indices r and s refer to possible tensor structures in either the three-point function SV O or SV A, while p refers to SV SV
Summary
The data of an abstract unitary conformal field theory (CFT) in D dimensions is encoded by the spectrum of primary operators and their OPEs, which are in turn specified by a finite number of real constants for each triplet of primary operators From this information, we can in principle compute any correlation function by iteratively performing OPEs to reduce the correlator to a two-point function. The step in this program is to use the results of this paper to obtain bounds (numerical or analytical) on the data of a general class of CFTs, and in particular for a CFT with a conserved primary vector operator and an associated continuous global symmetry. The real prize would be to implement the bootstrap with four conserved stress-energy tensors, gleaning extremely general information about the space of consistent unitary CFTs
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