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 $Ad{S}_{d+1}/{\mathbb{Z}}_2$, 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 ${\varphi }^4$ 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.

v1: 54 pages, 8 figures; v2: minor improvements, references added, to appear on J. Phys. A; v3: typos corrected in Sec. 6.3

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