Aspects of CFTs on real projective space

Abstract

<jats:title>Abstract</jats:title> <jats:p>We present an analytic study of conformal field theories on the real projective space <jats:inline-formula> <jats:tex-math><?CDATA $\mathbb{R}{\mathbb{P}}^{d}$?></jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:mi mathvariant="double-struck">R</mml:mi> <mml:msup> <mml:mrow> <mml:mi mathvariant="double-struck">P</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>d</mml:mi> </mml:mrow> </mml:msup> </mml:math> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="aabcf59ieqn7.gif" xlink:type="simple" /> </jats:inline-formula>, 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 <jats:inline-formula> <jats:tex-math><?CDATA $\mathbb{R}{\mathbb{P}}^{d}$?></jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:mi mathvariant="double-struck">R</mml:mi> <mml:msup> <mml:mrow> <mml:mi mathvariant="double-struck">P</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>d</mml:mi> </mml:mrow> </mml:msup> </mml:math> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="aabcf59ieqn8.gif" xlink:type="simple" /> </jats:inline-formula>, we study a simple holographic setup which captures the essential features of boundary correlators on <jats:inline-formula> <jats:tex-math><?CDATA $\mathbb{R}{\mathbb{P}}^{d}$?></jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:mi mathvariant="double-struck">R</mml:mi> <mml:msup> <mml:mrow> <mml:mi mathvariant="double-struck">P</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>d</mml:mi> </mml:mrow> </mml:msup> </mml:math> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="aabcf59ieqn9.gif" xlink:type="simple" /> </jats:inline-formula>. The analysis is based on calculations of Witten diagrams on the quotient space <jats:inline-formula> <jats:tex-math><?CDATA ${\mathrm{A}\mathrm{d}\mathrm{S}}_{d+1}/{\mathbb{Z}}_{2}$?></jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:msub> <mml:mrow> <mml:mi mathvariant="normal">A</mml:mi> <mml:mi mathvariant="normal">d</mml:mi> <mml:mi mathvariant="normal">S</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>d</mml:mi> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msub> <mml:mo>/</mml:mo> <mml:msub> <mml:mrow> <mml:mi mathvariant="double-struck">Z</mml:mi> </mml:mrow> <mml:mrow> <mml:mn>2</mml:mn> </mml:mrow> </mml:msub> </mml:math> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="aabcf59ieqn10.gif" xlink:type="simple" /> </jats:inline-formula>, 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 <jats:italic>ϕ</jats:italic> <jats:sup>4</jats:sup> theory in dimension <jats:italic>d</jats:italic> = 4 − <jats:italic>ϵ</jats:italic>, extracting the CFT data to order <jats:italic>ϵ</jats:italic> <jats:sup>2</jats:sup>. We also check our results by standard field theory methods, both in the large <jats:italic>N</jats:italic> and <jats:italic>ϵ</jats:italic> expansions. Finally, we briefly discuss the relation of our analysis to the problem of construction of local bulk operators in AdS/CFT.</jats:p>