Related papers: Modular Invariants from Subfactors: Type I Couplin…
In these lectures we explain the intimate relationship between modular invariants in conformal field theory and braided subfactors in operator algebras. A subfactor with a braiding determines a matrix $Z$ which is obtained as a coupling…
We consider a type III subfactor $N\subset M$ of finite index with a finite system of braided $N$-$N$ morphisms which includes the irreducible constituents of the dual canonical endomorphism. We apply $\alpha$-induction and, developing…
In this paper we further develop the theory of $\alpha$-induction for nets of subfactors, in particular in view of the system of sectors obtained by mixing the two kinds of induction arising from the two choices of braiding. We construct a…
In this paper we further analyze modular invariants for subfactors, in particular the structure of the chiral induced systems of M-M morphisms. The relative braiding between the chiral systems restricts to a proper braiding on their…
Modular invariants satisfy remarkable fusion rules. Let $Z$ be a modular invariant associated to a braided subfactor $N\subset M$. The decomposition of the non-normalized modular invariants $Z Z^{*}$ and $Z^{*}Z$ into sums of normalized…
In this lecture we explain the intimate relationship between modular invariants in conformal field theory and braided subfactors in operator algebras. Our analysis is based on an approach to modular invariants using braided sector induction…
The semisimple module categories over a braided fusion category $\mathcal{C}$ form a connected fusion 2-category $\text{Mod}(\mathcal{C})$. Its Drinfeld center $\mathcal{Z}(\text{Mod}(\mathcal{C}))$ is a braided fusion 2-category. To any…
We prove that every slightly degenerate braided fusion category admits a minimal nondegenerate extension, and hence that every pseudo-unitary super modular tensor category admits a minimal modular extension. This completes the program of…
We give three applications of general theory about braided endomorphisms from conformal inclusions developed previously by us. The first is an example of subfactors associated with conformal inclusion whose dual fusion ring is…
We study a class of $\Z^{d}$-substitutive subshifts, including a large family of constant-length substitutions, and homomorphisms between them, i.e., factors modulo isomorphisms of $\Z^{d}$. We prove that any measurable factor map and even…
We apply the theory of $\alpha$-induction of sectors which we elaborated in our previous paper to several nets of subfactors arising from conformal field theory. The main application are conformal embeddings and orbifold inclusions of SU(n)…
We call a subfactor trivial if it is isomorphic with the obvious inclusion of N into matrices over N. We prove the existence of type II_1 factors M without non-trivial finite index subfactors. Equivalently, every M-M-bimodule with finite…
To any complex Hadamard matrix H one associates a spin model commuting square, and therefore a hyperfinite subfactor. The standard invariant of this subfactor captures certain "group-like" symmetries of H. To gain some insight, we compute…
It is well known that the existence of a braiding in a monoidal category V allows many structures to be built upon that foundation. These include a monoidal 2-category V-Cat of enriched categories and functors over V, a monoidal bicategory…
Braided subfactors of von Neumann algebras provide a framework for studying two dimensional conformal field theories and their modular invariants. We review this in the context of SU(3) conformal field theories through corresponding SU(3)…
If $N \subset P,Q \subset M$ are type II_1 factors with $N' \cap M = C id$ and $[M:N]$ finite we show that restrictions on the standard invariants of the elementary inclusions $N \subset P$, $N \subset Q$, $P \subset M$ and $Q \subset M$…
Delta finite-type invariants are defined analogously to finite-type invariants, using delta moves instead of crossing changes. We show that they are closely related to the lower central series of the commutator subgroup of the pure braid…
A braided fusion category is said to have Property $\textbf{F}$ if the associated braid group representations factor over a finite group. We verify integral metaplectic modular categories have property $\textbf{F}$ by showing these…
For a word w in the braid group on n-strands, we denote by T_w the corresponding transverse braid in the rotational symmetric tight contact structure on S^3. We exhibit a map on link Floer homology which sends the transverse invariant…
We prove that a type II$_1$ factor $M$ can have at most one Cartan subalgebra $A$ satisfying a combination of rigidity and compact approximation properties. We use this result to show that within the class $\Cal H \Cal T$ of factors $M$…