Abstract
Many CFT problems, e.g. ones with global symmetries, have correlation functions with a crossing antisymmetric sector. We show that such a crossing antisymmetric function can be expanded in terms of manifestly crossing antisymmetric objects, which we call the ‘+ type Polyakov blocks’. These blocks are built from AdSd+1 Witten diagrams. In 1d they encode the ‘+ type’ analytic functionals which act on crossing antisymmetric functions. In general d we establish this Witten diagram basis from a crossing antisymmetric dispersion relation in Mellin space. Analogous to the crossing symmetric case, the dispersion relation imposes a set of independent ‘locality constraints’ in addition to the usual CFT sum rules given by the ‘Polyakov conditions’. We use the Polyakov blocks to simplify more general analytic functionals in d > 1 and global symmetry functionals.
Highlights
One of the most important tools available to theoretical physicists in CFT problems is crossing symmetry or the Conformal Bootstrap
The latter is an idea that a crossing symmetric correlator can be expanded in terms of a manifestly crossing symmetric sum of Witten diagrams which must be equal to the usual conformal block expansion [11,12,13]
For this we show that a crossing antisymmetric function can be obtained from a dispersion relation in Mellin space
Summary
The analytic functionals are perhaps most well understood in 1d, where they are related to another formulation of crossing symmetry: the Polyakov bootstrap [10] The latter is an idea that a crossing symmetric correlator can be expanded in terms of a manifestly crossing symmetric sum of Witten diagrams (a Polyakov block) which must be equal to the usual conformal block expansion [11,12,13]. The idea of a + type Polyakov block is extended to general dimensions, which requires an infinite number of exchange Witten diagrams and crossing antisymmetric contact diagrams For this we show that a crossing antisymmetric function can be obtained from a dispersion relation in Mellin space. This, rather counter-intuitive, notation is for consistency with the analytic functional literature
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