Crossing antisymmetric Polyakov blocks + 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 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.

Symmetry and Antisymmetry of the CMB Anisotropy Pattern

Given an arbitrary function, we may construct symmetric and antisymmetric functions under a certain operation. Since statistical isotropy and homogeneity of our Universe has been a fundamental assumption of modern cosmology, we do not expect any particular symmetry or antisymmetry in our Universe. Besides fundamental properties of our Universe, we may also figure our contamination and improve the quality of the CMB data products, by matching the unusual symmetries and antisymmetries of the CMB data with known contaminantions. If we let the operation to be a coordinate inversion, the symmetric and antisymmetric functions have even and odd-parity respectively. The investigation on the parity of the recent CMB data shows a large-scale odd-parity preference, which is very unlikely in the statistical isotropic and homogeneous Universe. We investigated the association of the WMAP systematics with the anomaly, but did not find a definite non-cosmological cause. Besides the parity anomaly, there is anomalous lack of large-scale correlation in CMB data. We show that the odd-parity preference at low multipoles is, in fact, phenomenologically identical with the lack of large-angle correlation.