arXiv 2009.032902020-09hep-th

实射影空间上的CFTs问题

Aspects of CFTs on Real Projective Space

Simone Giombi, Himanshu Khanchandani, Xinan Zhou

备注：: v1: 54 pages, 8 figures; v2: minor improvements, references added, to appear on J. Phys. A; v3: typos corrected in Sec. 6.3
DOI：: 10.1088/1751-8121/abcf59

摘要：

本文对实射影空间$\mathbb{RP}^d$上的共形场论进行了分析研究，重点研究了标量算子的两点函数。由于部分破缺的共形对称性，它们是共形交比的非平凡函数，并且被约束为服从交叉方程。在回顾了关于$\mathbb{RP}^d$上的边界元结构的基本事实之后，我们研究了一个简单的全息装置，该装置捕获了$\mathbb{RP}^d$上的边界元的基本特征。分析的基础上计算Witten图的商空间$AdS_{d+1}/\mathbb{Z}_2$，并导致两点函数的分析方法。特别是，我们认为，交换威滕图的共形块分解的结构表明解析泛函的自然基础，其作用在共形块上的交叉方程变成一定的求和规则。我们测试这种方法在规范的例子中的$ϕ^4$理论在尺寸$d=4-ε$，提取CFT数据，以订购$ε^2$。我们还检查我们的结果，由标准的场论方法，无论是在大$N$和$ε$扩展。最后，我们简要地讨论了我们的分析的AdS/CFT中的局部散装运营商的建设问题的关系。

英文摘要：

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.

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