Crossing antisymmetric Polyakov blocks + Dispersion relation

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 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.

arXiv comment32 pages,3 figures; v2: typos corrected

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• Positivity and Geometric Function Theory Constraints on Pion Scattering
• Locality and Analyticity of the Crossing Symmetric Dispersion Relation
• Mixed Correlator Dispersive CFT Sum Rules
• Positivity, low twist dominance and CSDR for CFTs
• A Celestial Route to AdS Bulk Locality
• Bootstrap Method in Theoretical Physics