Towards fully-local 2d chiral CFTs from conformal nets: bicommutant categories and fusion of their modules

Thesis

We give a new definition of a bicommutant category, a categorified analogue of a von Neumann algebra. Our definition is independent of a choice of faithful representation: as a bi-involutive W∗ -tensor category, it is required to act on its own absorbing ideal, with the left and right actions being each other’s commutants. The absorbing ideal thus plays the role that the standard form plays for a von Neumann algebra. We develop a string calculus that makes the coherences of this definition manifest. We define modules over bicommutant categories and the ‘categorified’ Connes fusion of such modules, and we propose a definition of the 3-category that bicommutant categories form.

We introduce left- and right-localised endomorphisms of the algebras that a conformal net A assigns to multi-intervals, generalising the Doplicher–Haag–Roberts endomorphisms, and prove that these form a monoidal category equivalent to the category Sol(A) of solitonic representations of the net. Using this equivalence, we prove that Sol(A) is a bicommutant category for every conformal net, assuming neither finite-index nor strong additivity. As a corollary, its Drinfeld centre is the representation category Rep(A).

We partially develop a fully-local Segal-style functorial framework for two-dimensional chiral conformal field theories arising from conformal nets. We construct the dimension-0 and dimension-1 components, assigning a bicommutant category to a point with a one-dimensional collar and appropriate bimodule W∗ -categories to 1-dimensional cobordisms, and we establish compatibility of the fusion of module categories with the gluing of 1-dimensional cobordisms. The circle is assigned Rep(A) abs, which agrees with the Drinfeld centre of the value on the point up to passing to absorbing ideal.

Authors: Nivedita

• DOI: 10.5287/ora-prgbb5jk9
• Type of award: DPhil
• Level of award: Doctoral
• Awarding institution: University of Oxford
• Language: English
• Keywords: 2d Chiral Conformal Field Theory; conformal nets; fully extended FQFTs; bicommutant category; Functorial Quantum Field Theory
• Subjects: Mathematical Physics