## Periodic Reporting for period 1 - beyondRCFT (Beyond Rationality in Algebraic CFT: mathematical structures and models)

Reporting period: 2019-09-01 to 2021-08-31

What is the problem/issue being addressed?

Conformal quantum field theory (CFT) in low dimensions can be described in a model independent way using the language of operator algebras. This description is mathematically rigorous and it provides natural connections with various areas of pure mathematics, such as the theory of von Neumann algebras, subfactors, tensor categories and vertex operator algebras. The project mainly addresses operator algebraic models of CFT, and the related mathematical objects, beyond the well-studied cases of rational CFTs (those with finitely many “particle excitations” of the vacuum) / finite index subfactors / finite tensor categories.

Why is it important for society?

Despite its great experimental success in the past half century, quantum field theory (our modern understanding of elementary particles and matter at atomic scales) still lacks a commonly accepted and mathematically sound formulation. The project exploits new mathematical tools to study such physical models in the operator algebraic framework, although in the physically simplified situation of low spacetime dimensions and conformal symmetry, and it aims to produce mathematical theorems in support of physical intuitions. Ideas or problems originating in either of these areas across mathematics and physics very often have implications on the others, thus witnessing a constructive interplay between different subjects of fundamental research.

What are the overall objectives?

Algebraic models of CFT in low dimensions have been mainly studied under the so-called rationality assumption. At the same time, and for independent reasons, subfactors and tensor categories have been widely investigated in the finite index / finite spectrum regime. The goal of this research project is to develop and apply new analytical tools to study these structures in more generality, by including non-rational CFTs, infinite index subfactors and infinite tensor categories, and to explore their mutual connections.

Conformal quantum field theory (CFT) in low dimensions can be described in a model independent way using the language of operator algebras. This description is mathematically rigorous and it provides natural connections with various areas of pure mathematics, such as the theory of von Neumann algebras, subfactors, tensor categories and vertex operator algebras. The project mainly addresses operator algebraic models of CFT, and the related mathematical objects, beyond the well-studied cases of rational CFTs (those with finitely many “particle excitations” of the vacuum) / finite index subfactors / finite tensor categories.

Why is it important for society?

Despite its great experimental success in the past half century, quantum field theory (our modern understanding of elementary particles and matter at atomic scales) still lacks a commonly accepted and mathematically sound formulation. The project exploits new mathematical tools to study such physical models in the operator algebraic framework, although in the physically simplified situation of low spacetime dimensions and conformal symmetry, and it aims to produce mathematical theorems in support of physical intuitions. Ideas or problems originating in either of these areas across mathematics and physics very often have implications on the others, thus witnessing a constructive interplay between different subjects of fundamental research.

What are the overall objectives?

Algebraic models of CFT in low dimensions have been mainly studied under the so-called rationality assumption. At the same time, and for independent reasons, subfactors and tensor categories have been widely investigated in the finite index / finite spectrum regime. The goal of this research project is to develop and apply new analytical tools to study these structures in more generality, by including non-rational CFTs, infinite index subfactors and infinite tensor categories, and to explore their mutual connections.

By the work carried out in the project, some possibly infinite index subfactors (the so-called discrete subfactors) are now under mathematical control, their role in CFT has been clarified and some classical results in subfactor theory have been extended to the infinite index case. In particular, Hopf algebras and compact quantum groups have been ruled out as candidates for global gauge symmetries in CFT and the subfactor theoretical Fourier transform (a branch of quantum Fourier analysis) has been extended and partially studied in the infinite index case. Moreover, some general results concerning the realizability of unitary tensor categories and unitary 2-categories (possibly with infinite spectrum) have been established, also using ideas from the theory of planar algebras.

The proposed research is one of the few systematic treatments of infinite index subfactors, non-rational CFTs and infinite tensor categories at the same time. All these infinite structures are natural objects of study and they appear in physical models. Further work and results are expected in the direction of infinite index discrete and semidiscrete subfactors (the most general type of subfactor appearing in the operator algebraic description of CFT extensions), quantum Fourier analysis, representations of 2-categories, index theory for inclusions of von Neumann algebras and quantum information theory.