Conformal nets II: conformal blocks

TL;DR

This paper constructs and analyzes the bundle of conformal blocks associated with conformal nets, demonstrating their factorization properties and modularity, thus advancing the mathematical understanding of conformal field theory.

Contribution

It introduces a new construction of conformal blocks for conformal nets, including boundary cases, and proves their factorization and modularity properties.

Findings

Constructed the bundle of conformal blocks for conformal nets.

Proved factorization formulas for gluing surfaces.

Established the modularity of the representation category.

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.