Crossing antisymmetric Polyakov blocks plus dispersion relation

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 AdS(d+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.

Automated summary

This study shows that correlation functions in many CFT problems, including ones with global symmetries, have a crossing antisymmetric sector. It is demonstrated that this crossing antisymmetric function can be expanded using '+ type Polyakov blocks'. These blocks are constructed from AdS(d+1) Witten diagrams and play a crucial role in simplifying analytic functionals and global symmetry functionals.