Abstract
In the context of conformal field theories in general space-time dimension, we find all the possible singularities of the conformal blocks as functions of the scaling dimension $\Delta$ of the exchanged operator. In particular, we argue, using representation theory of parabolic Verma modules, that in odd spacetime dimension the singularities are only simple poles. We discuss how to use this information to write recursion relations that determine the conformal blocks. We first recover the recursion relation introduced in 1307.6856 for conformal blocks of external scalar operators. We then generalize this recursion relation for the conformal blocks associated to the four point function of three scalar and one vector operator. Finally we specialize to the case in which the vector operator is a conserved current.
Highlights
In the context of conformal field theories in general space-time dimension, we find all the possible singularities of the conformal blocks as functions of the scaling dimension ∆ of the exchanged operator
We first recover the recursion relation introduced in [1] for conformal blocks of external scalar operators. We generalize this recursion relation for the conformal blocks associated to the four point function of three scalar and one vector operator
The conformal bootstrap program [2, 3] is a non-perturbative approach to the construction of interacting conformal field theories (CFT), which has made remarkable progress in recent years since the pioneering work of [4]
Summary
The conformal bootstrap program [2, 3] is a non-perturbative approach to the construction of interacting conformal field theories (CFT), which has made remarkable progress in recent years since the pioneering work of [4]. Conformal blocks (CB) are the basic ingredients needed to set up the conformal bootstrap equations They encode the contribution to a four point function from the exchange of a primary operator and all its descendants. The case of the vector operator illustrates a new feature that arises when there is more than one CB for a given set of labels of the external and exchanged operators. In this case, the residues in the ∆ complex plane become linear combinations of several CBs. In section 6, we give a detailed discussion of the structure of general conformal families in any spacetime dimension. We conclude with some remarks about the residues of these poles in general spacetime dimension but leave their explicit computation for future work
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