The author claims that conformal symmetry is useful but not powerful enough to give rise to the large machinery of conformal field theory (CFT) and its multitude of nontrivial results. In order to do so, one must consider some locality principle of quantum field theory.
Motivated by this argument, the groupoid \({\mathcal C}= \{(g,A)\}\), \((g, A)(g', A')= (g\circ g', A')\) providing \(A= g'(A')\) is introduced. Here \(A\) is any simply connected domain of the Riemann sphere \(\widehat{\mathbb{C}}\) and \(g: A\to\widehat{\mathbb{C}}\) is a univalent conformal map (\(A\) is often assumed to be the hyperbolic type). The topology of \({\mathcal C}\) is defined similarly to the topology of a sheaf (§2.2). So it is not Hausdorff and the tangent space becomes a Fréchet space. But based on Hadamard derivation, the \(A\)-differentiation is defined on a suitable class of functions defined on a neighborhood of a point \(\Sigma\) of \({\mathcal C}\) by \[ \lim_{\eta\to 0} {f(g_\eta\cdot\Sigma)- f(\Sigma)\over\eta}= \nabla^A f(\Sigma)/h,\quad h=\partial(g_\eta; \eta> 0) \] (Definition 3.4). The definiton of the path \((g_\eta)\) is given in §3.1. \(\nabla^Af(\Sigma)\) is a linear functional on \(H^{<}(A)\), the space of holomorphic vector fields on \(A\) and the fiber of the tangent bundle of \({\mathcal C}\). The conformal derivative of \(f\) at \(\Sigma\) is defined to be \(\nabla^A(\Sigma)\). By the structure of the dual of \(H^{<}(A)\) (see Lemma 3.1 in [W. Rudin, Functional analysis. New York, NY: McGraw-Hill Book Comp. (1973; Zbl 0253.46001)]), there is a class \(\Delta^Af(\Sigma)= \{\gamma+u: u\in H^{<}(A)\}\), where \(\gamma\) is a holomorphic vector field on an annular neighborhood of \(\partial A\). This class is referred to as the holomorphic \(A\)-class of \(f\) at \(\Sigma\). For any \(a\in\widehat{\mathbb{C}}\setminus A\), there is a unique member of this class given by \[ \{z\to \Delta^A_{a;z} f(\Sigma)\in H^{<}_a(\widehat{\mathbb{C}}\setminus A)\}. \] These are called “holomorphic \(A\)-derivative of \(f\) at \(\Sigma\)”. Then it is shown \(\Delta^A_{a;z}f(\Sigma)\) to be independent of \(a\), so denote by \(\Delta^{[A]}f(\Sigma)\) (Theorem 3.1). If the \(B\)-derivative of \(f\) at \(\Sigma\) is zero, it is shown \[ \Delta^{[A]}_wf(\Sigma)=(\partial g(w))^2 \Delta^{[A]}_{g(w)} f'(\Sigma') \] (Corollary 3.11). This general theory is applied to CFT, taking conformal derivatives of correlation functions with conjugation of a global holomorphic derivative anda it reproduces the conformal Ward identity and the boundary condition of Theorem 4.1 in [J. Cardy, “Conformal invariance and surface critical behavior”, Nucl. Phys. B 240, 514 (1984)].
The topology of \({\mathcal C}\) is defined taking \[ N_{r,K}= \{(g',A'): \max\{d(g(z), g'(z));\, z\in K\}< r,\, K\subset A'\} \] as the neighborhood of \((g, A)\). So it is not Hausdorff and the usual results on Lie groupoid can not be used. In §2, fundamental geometric notions on \({\mathcal C}\), such as vector bundle over \({\mathcal C}\) and their local sections, trajectories, Lie derivatives and so on are presented. Definitions and properties of conformal differentiability are given in §3, which is the main part of this paper. Some technical parts used in this section, such as determination of the dual of \(H^{<}(A)\) (used in the proof of Lemma 3), factorization of conformal maps on annular domain (used in the proof of Theorem 3.3, which derives Corollary 3.11), etc. are given in appendices. In §4, after explaining the conformal Ward identity, extended Ward identities (the conformal Ward identity and the boundary conditions) are derived from conformal derivatives (Theorem 4.1). It is also shown that the one-point average of the stress-energy tensor is \[ (T(w))_{{\mathcal C}}= \Delta^{[\widehat{\mathbb{C}}_w]}_{w[\partial C\cup\partial D]}\log Z(C|D), \] where \(\overline D\subset C\) (§4.4).
The author claims that the main idea of this paper comes from Shramm-Loewner evolution (SLE) [J. Cardy, Ann. Phys. 318, No. 1, 81–118 (2005; Zbl 1073.81068)] and conformal loop ensembles (CLE) [S. Sheffield, Duke Math. J. 147, 79 (2009; Zbl 1170.60008)]. They lead to consider CFT as a statistical field theory. Hence the theorems of §3 have no immediate equivalent in the context of the universal Teichmüller space or of Diff\((S^1)\). These are explained in the introduction together with remarks on relations with some previous works [B. Doyon et al., Commun. Math. Phys. 268, No. 3, 687–716 (2006; Zbl 1121.81108)].