Periodic Reporting for period 2 - beyondRCFT (Beyond Rationality in Algebraic CFT: mathematical structures and models)
Berichtszeitraum: 2021-09-01 bis 2022-08-31
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 other 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, and some related mathematical objects (such as subfactors and tensor categories), beyond the well-studied cases of “rational” CFTs (those with finitely many “particle excitations” of the vacuum) / finite index subfactors / finite tensor categories. A new type of “gauge symmetry transformation” is also studied in the not necessarily rational / finite index context, using the operator algebraic formulation, where ordinary automorphisms are replaced by unital completely positive maps.
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. It aims to produce mathematical theorems in support of physical intuitions. Ideas or problems originating in either of these areas across mathematics and physics often have implications on the others, stimulating the 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 the project is to develop and apply new analytical tools to study these structures in more general (and natural) situations, 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 other 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, and some related mathematical objects (such as subfactors and tensor categories), beyond the well-studied cases of “rational” CFTs (those with finitely many “particle excitations” of the vacuum) / finite index subfactors / finite tensor categories. A new type of “gauge symmetry transformation” is also studied in the not necessarily rational / finite index context, using the operator algebraic formulation, where ordinary automorphisms are replaced by unital completely positive maps.
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. It aims to produce mathematical theorems in support of physical intuitions. Ideas or problems originating in either of these areas across mathematics and physics often have implications on the others, stimulating the 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 the project is to develop and apply new analytical tools to study these structures in more general (and natural) situations, by including non-rational CFTs, infinite index subfactors, and infinite tensor categories, and to explore their mutual connections.
In the project, we studied a special class of infinite index subfactors (the so-called discrete subfactors, that include all finite index ones). These subfactors are now under good mathematical control, their role in the operator algebraic formulation of 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 uncountable spectrum) have been established, also using ideas from the theory of planar algebras. We investigated the theory of CFT extensions in other mathematical formulations, such as the (categorical) locally covariant AQFT formulation. We studied a more general type of "gauge symmetry transformations" in the operator algebraic formulation, described by unital completely positive maps instead of automorphisms. We applied our ideas also in the quantum information theory context.
Dissemination of the project's results took place via scientific presentations to the specialized audience and experts coming from the various mathematical disciplines involved in the project. The PI presented the results at workshops and international conferences, during the whole duration of the project. The transfer of knowledge also happened through presentations at local seminars and personal interactions. A broad audience presentation of the MSCA program has also been given at the Host Institution, directed to the students of all scientific faculties.
Dissemination of the project's results took place via scientific presentations to the specialized audience and experts coming from the various mathematical disciplines involved in the project. The PI presented the results at workshops and international conferences, during the whole duration of the project. The transfer of knowledge also happened through presentations at local seminars and personal interactions. A broad audience presentation of the MSCA program has also been given at the Host Institution, directed to the students of all scientific faculties.
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 on purely mathematical grounds and they appear in physical models. Future work and impact are expected in the direction of infinite index semidiscrete subfactors, quantum Fourier analysis, representations of 2-categories, index theory for inclusions of arbitrary von Neumann algebras, quantum information theory and in relation to other axiomatic approaches to CFT.