Aspects of CFTs on Real Projective Space

Abstract

We present an analytic study of conformal field theories on the real projective space $\mathbb{RP}^d$, 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 $\mathbb{RP}^d$, we study a simple holographic setup which captures the essential features of boundary correlators on $\mathbb{RP}^d$. The analysis is based on calculations of Witten diagrams on the quotient space $AdS_{d+1}/\mathbb{Z}_2$, 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 $\phi^4$ theory in dimension $d=4-\epsilon$, extracting the CFT data to order $\epsilon^2$. We also check our results by standard field theory methods, both in the large $N$ and $\epsilon$ expansions. Finally, we briefly discuss the relation of our analysis to the problem of construction of local bulk operators in AdS/CFT.