Abstract
Conformal nets provide a mathematical formalism for conformal field theory. Associated to a conformal net with finite index, we give a construction of the ‘bundle of conformal blocks’, a representation of the mapping class groupoid of closed topological surfaces into the category of finite-dimensional projective Hilbert spaces. We also construct infinite-dimensional spaces of conformal blocks for topological surfaces with smooth boundary. We prove that the conformal blocks satisfy a factorization formula for gluing surfaces along circles, and an analogous formula for gluing surfaces along intervals. We use this interval factorization property to give a new proof of the modularity of the category of representations of a conformal net.
Highlights
Given a 2d conformal field theory (CFT), its partition function Z assigns a number to every Riemann surface
Conformal blocks were first introduced in the famous paper of Belavin, Polyakov, and Zamolodchikov [11] for the special case of the Virasoro conformal field theory
They are certain holomorphic functions Fa that enter in a formula for the correlation and partition functions
Summary
It is expected that the spaces of conformal blocks form a finite-dimensional vector bundle with hermitian inner product and projectively flat unitary connection over the moduli space of Riemann surfaces. If is a compact oriented topological surface with boundary, and ∂ is equipped with a smooth structure, our construction produces an infinite dimensional Hilbert-space-well-defined-up-to-phase V ( ), equipped with an action of the von Neumann algebra A(∂ ). As a consequence of the functoriality of the construction → V ( ), for every surface and every conformal net A with finite index, there is a projective (anti)unitary representations of the following infinite dimensional topological group: G( ) := f ∈ Homeo+( ) ∪ Homeo−( ) : f |∂ is smooth isotopy rel ∂.
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