论文标题
CFT在实际投影空间上的各个方面
Aspects of CFTs on Real Projective Space
论文作者
论文摘要
我们介绍了对真实投影空间$ \ mathbb {rp}^d $的共形场理论的分析研究，重点是标量运算符的两点函数。由于部分断裂的形式对称性，这些是共形交叉比例的非平凡功能，并且被限制以遵守交叉方程。在回顾了有关相关器在$ \ mathbb {rp}^d $上的结构的基本事实之后，我们研究了一个简单的全息设置，该设置捕获了$ \ mathbb {rp}^d $上的边界相关器的基本特征。该分析基于对商空间上的Witten图的计算$ ads_ {D+1}/\ Mathbb {Z} _2 $，并导致对两点函数的分析方法。特别是，我们认为交换图的保形块分解的结构表明了分析功能的自然基础，其在保形块上的作用将交叉方程式转化为某些总和规则。我们在$ ϕ^4 $理论的规范示例中测试了这种方法$ d =4-ε$，将CFT数据提取到订单$ε^2 $。我们还通过标准现场理论方法检查结果，无论是在大型$ n $还是$ε$扩展中。最后，我们简要讨论了我们的分析与广告/CFT中本地散装运营商的构建问题的关系。
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 $ϕ^4$ 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.
