Calculus on manifolds of conformal maps and CFT

Abstract

In conformal field theory (CFT) on simply connected domains of the Riemann sphere, the natural conformal symmetries under self-maps are extended, in a certain way, to local symmetries under general conformal maps, and this is at the basis of the powerful techniques of CFT. Conformal maps of simply connected domains naturally have the structure of an infinite-dimensional groupoid, which generalizes the finite-dimensional group of self-maps. We put a topological structure on the space of conformal maps on simply connected domains, which makes it into a topological groupoid. Further, we (almost) extend this to a local manifold structure based on the infinite-dimensional Fréchet topological vector space of holomorphic functions on a given domain A. From this, we develop the notion of conformal A-differentiability at the identity. Our main conclusion is that quadratic differentials characterizing cotangent elements on the local manifold enjoy properties similar to those of the holomorphic stress–energy tensor of CFT; these properties underpin the local symmetries of CFT. Applying the general formalism to CFT correlation functions, we show that the stress–energy tensor is exactly such a quadratic differential. This is at the basis of constructing the stress–energy tensor in conformal loop ensembles. It also clarifies the relation between Cardy's boundary conditions for CFT on simply connected domains, and the expression of the stress–energy tensor in terms of metric variations.