# CFTs on curved spaces

## TL;DR

This paper explores the structure of conformal groups and correlation functions of conformal field theories on various curved and unorientable manifolds, revealing symmetry enhancements and proposing candidates for conformal manifolds without supersymmetry.

## Contribution

It systematically analyzes conformal groups on curved spaces, including boundaries and unorientable manifolds, and introduces new insights into conformal symmetries on tori and their implications for CFTs.

## Key findings

- Conformal groups on certain curved spaces are explicitly computed.
- Symmetry enhancement occurs on $ ext{S}^1_l imes ext{H}^2_r$ with specific radius relations.
- The conformal group on $ ext{T}^d$ is isomorphic to $U(1)^d$ for $d o 2$ and higher.

## Abstract

We study conformal field theories (CFTs) on curved spaces including both orientable and unorientable manifolds possibly with boundaries. We first review conformal transformations on curved manifolds. We then compute the identity components of conformal groups acting on various metric spaces using a simple fact; given local coordinate systems be single-valued. Boundary conditions thus obtained which must be satisfied by conformal Killing vectors (CKVs) correctly reproduce known conformal groups. As a byproduct, on $\mathbb S^1_l\times\mathbb H^2_r$, by setting their radii $l=Nr$ with $N\in\mathbb N^\times$, we find (the identity component of) the conformal group enhances, whose persistence in higher dimensions is also argued. We also discuss forms of correlation functions on these spaces using the symmetries. Finally, we study a $d$-torus $\mathbb T^d$ in detail, and show the identity component of the conformal group acting on the manifold in general is given by $\text{Conf}_0(\mathbb T^d)\simeq U(1)^d$ when $d\ge2$. Using the fact, we suggest some candidates of conformal manifolds of CFTs on $\mathbb T^d$ without assuming the presence of supersymmetry (SUSY). In order to clarify which parts of correlation functions are physical, we also discuss renormalization group (RG) and local counterterms on curved spaces.