Aspects of CFTs on real projective space

Source abstract

Abstract We present an analytic study of conformal field theories on the real projective space R P d , focusing on the two-point functions of scalar operators. Due to the partially broken conformal symmetry, these are non-trivial functions of a conformal cross ratio and are constrained to obey a crossing equation. After reviewing basic facts about the structure of correlators on R P d , we study a simple holographic setup which captures the essential features of boundary correlators on R P d . The analysis is based on calculations of Witten diagrams on the quotient space A d S d + 1 / Z 2 , and leads to an analytic approach to two-point functions. In particular, we argue that the structure of the conformal block decomposition of the exchange Witten diagrams suggests a natural basis of analytic functionals, whose action on the conformal blocks turns the crossing equation into certain sum rules. We test this approach in the canonical example of ϕ 4 theory in dimension d = 4 − ϵ , extracting the CFT data to order ϵ 2 . We also check our results by standard field theory methods, both in the large N and ϵ expansions. Finally, we briefly discuss the relation of our analysis to the problem of construction of local bulk operators in AdS/CFT.