Calculus on manifolds of conformal maps and CFT
type: Item
instance of: scholarly article
title: Calculus on manifolds of conformal maps and CFT
zbMATH Open document ID: 1258.81076
DOI: 10.1088/1751-8113/45/31/315202
author: Benjamin Doyon
published in: Journal of Physics A: Mathematical and Theoretical
publication date: 2012-09-05T00:00:00Z
full work available at URL: https://arxiv.org/abs/1004.0138
Mathematics Subject Classification ID: 81T40
Mathematics Subject Classification ID: 58H05
zbMATH DE Number: 6077993
zbMATH Keywords: conformal field theory
zbMATH Keywords: conformal loop ensemble
zbMATH Keywords: groupoid of conformal maps
zbMATH Keywords: Fréchet space
zbMATH Keywords: Ward identity
zbMATH Keywords: Schramm-Loewner evolution
MaRDI profile type: Publication
OpenAlex ID: W2044414053
arXiv ID: 1004.0138
review text: 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.NEWLINENEWLINE 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 (\S2.2). 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 NEWLINE\[NEWLINE\lim_{\eta\to 0} {f(g_\eta\cdot\Sigma)- f(\Sigma)\over\eta}= \nabla^A f(\Sigma)/h,\quad h=\partial(g_\eta; \eta> 0)NEWLINE\]NEWLINE (Definition 3.4). The definiton of the path \((g_\eta)\) is given in \S3.1. \(\nabla^Af(\Sigma)\) 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 [\textit{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 NEWLINE\[NEWLINE\{z\to \Delta^A_{a;z} f(\Sig…
reviewed by: Akira Asada
description: scientific article; zbMATH DE number 6077993