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.
