CFTs on curved spaces

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.