Tensor Categories and Endomorphisms of von Neumann Algebras

Abstract

Q-systems describe "extensions" of an infinite von Neumann factor N , i.e., finite-index unital inclusions of N into another von Neumann algebra M . They are (special cases of) Frobenius algebras in the C* tensor category of endomorphisms of N . We review the relation between Q-systems, their modules and bimodules as structures in a category on one side, and homomorphisms between von Neumann algebras on the other side. We then elaborate basic operations with Q-systems (various decompositions in the general case, and the centre, the full centre, and the braided product in braided categories), and illuminate their meaning in the von Neumann algebra setting. The main applications are in local quantum field theory, where Q-systems in the subcategory of DHR endomorphisms of a local algebra encode extensions A(O) ⊂ B(O) of local nets. These applications, notably in conformal quantum field theories with boundaries, are briefly exposed, and are discussed in more detail in two separate papers [4,5].

Topological defect lines and renormalization group flows in two dimensions

“…For example, a Y segment connecting either b + and b + , or b − and b − , is even under Z 2 , while one connecting b + and b − is odd under Z 2 . 26 Similarly, a Y Y Y junction ending on either b + , b + , b + , or b + , b − , b − , is even under Z 2 , while the other possibilities are odd. Since these TDL configurations ending on defect operators can be expanded in bulk operators (by locality), the Z 2 -invariance put constraints on the structure constants of the TQFT, which we exploit in the following sections to pin down the TQFT.…”

Generalized orbifold construction for conformal nets

“…There is a direct sum which well-defined on sectors, namely [ρ] ⊕ [σ] is given by the sector of r 1 ρ( · )r * 1 + r 2 σ( · )r * 2 where r * i r j = δ ij 1 and r 1 r * 1 + r 2 r * 2 = 1 is a representation of the generators of the Cuntz algebra O 2 in M . We refer to [BKLR15] for more details.…”

On generalized symmetries and structure of modular categories

“…As a graded algebra, H is identified with C[y 1 ]/ y 4 1 −y 1 where y 1 has degree 1. Hence a basis for H is given by {1, y 3 1 , y 1 , y 2 1 }. To do computations with Mathematica, one can use the fact that ∆(y 1 ) i = ∆(y i 1 ).…”

“…The irreducible modules of H are given by its algebra decomposition. The algebra Span{1 − y 3 1 } gives three modules from the action of counit denoted by {y 3 1 , y 1 , y 2 1 }, and the second algebra Span{y 3 1 , y 1 , y 2 1 } gives itself as a module, called ρ, with the action given by multiplication. The module category is therefore {y 3 1 , y 1 , y 2 1 , ρ}.…”

“…The algebra Span{1 − y 3 1 } gives three modules from the action of counit denoted by {y 3 1 , y 1 , y 2 1 }, and the second algebra Span{y 3 1 , y 1 , y 2 1 } gives itself as a module, called ρ, with the action given by multiplication. The module category is therefore {y 3 1 , y 1 , y 2 1 , ρ}. As {y 3 1 , y 1 , y 2 1 } forms a Vec Z 3 as a fusion category, it remains to understand how ρ fuses with itself and other simple modules.…”