The n-Category Café

A useful and interesting text is

K.-H. Rehren
On Local Boundary CFT and Non-Local CFT on the Boundary
math-ph/0412049,

in particular its section 4, which goes beyond the discussion in

Longo, Rehren
Local fields in boundary conformal QFT
math-ph/0405067

that this review is based on.

Before talking about that section 4, I should mention this:

Canonical time evolution (“evolution of modular origin”)

The appendix contains, as a review, at least part of the answer to the kind of question we were discussing in the comment section of QFT of Charged $n$-Particle: Canonical 1-Particle, concerning the relation of Alain Connes’ so-called “canonical time evolution” with the standard notion of time evolution in quantum field theory.

Let $A$ be the von Neumann algebra of observables in some local QFT associated to a wedge region, acting on a Hilbert space $H$, and let $\Omega$ be the cyclic and separating vacuum state in $H$.

Connes’ automorphism $\sigma : A \to A$ defined by $$\langle \Omega , a b\Omega \rangle = \langle \Omega , \sigma(b) a \Omega \rangle$$ is induced by conjugation with some $$\Delta \in B(H) \,.$$ This operator also appears in the polar splitting of the closure of the operator $$S : a \Omega \mapsto a^* \Omega$$ as $$\bar S = J \Delta^{1/2} \,.$$

In the given case, where$A = A(W)$ is the algebra of observables associated to a spacelike wedge $A$ of Minkwoski space by a local QFT, $J$ is a CPT operator and $\Delta$ generates the unitary group of Lorentz boosts that preserve the wedge region $W$ $$\Delta^{i t}_{A(W),\Omega} = U(\Lambda_W(-2 \pi t)) \,.$$

I still need to better understand this, and how it comes about. But that’s a start.

Open string algebras

From the AQFT point of view, the Frobenius algebras internal to modular tensor categories, that characterize (rational) 2-dimensional conformal field theory #, arise as Q-systems of DHR endomorphisms of algebras of observables.

I’ll say something about Q-systems below. Here is what K.-H. Rehren says about open strings in section 4 of the above paper.

Given a 2-dimensional worldsheet with boundary, we may associate to every interval $I$ of the boundary an algebra $B(I)$ of observables obtained by restricting the local conformal net of observables in the bulk suitably. This net on the real line will contain a chiral sub-set $$\iota : A(I) \stackrel{\subset}{\to} B(I) \,.$$

Under suitable conditions $B(I)$ is a von Neumann algebra factor and $A(I)$ a subfactor.

Specifying subfactors of vN factors is something of more profound importance than it may seem on first sight. Probably everything there is to know about this, in the AQFT context, is mentioned in

Longo, Rehren,
Nets of subfactors.

Inclusions of vN subfactors are closely related to adjunctions. Think of everything living in the bicategory of von Neumann bimodules, with every morphism of algebras inducing a bimodule structure on the codomain algebra. Think of compositioon of morphisms as composition of bimodules, but , at the same time, as far as morphisms of bimodules are concerned, in the present applications, think of all these bimodules as mere left modules, by right-composing them suitably with left-modules, along the lines indicated in Amplimorphisms.

A morphism $\rho : A \to B$ is said, in the present context, to have a conjugate $\bar \rho : B \to A$ if there are morphisms $\mathrm{Id} \to \rho \circ \bar \rho$ and $\mathrm{Id} \to \bar \rho \circ \rho$ that have a right inverse.

Notice that this condition on inverses would imply, if we were to furthermore impose the obvious zig-zag identities on these morphisms (as we will below), thus turning them into an ambidextrous adjunction, the structure of a special ambidextrous adjunction.

A morphism $\sigma : A \to A$ is called a DHR endomorphism if it is localized and transportable, in the sense described in Amplimorphisms and Quantum Symmetry, I.

Now, consider all morphisms $$a : A(I) \to B(I)$$ which are subobjects of one of the form $\iota \circ \sigma$, with $\iota$ the above inclusion and $\sigma$ a DHR endomorphism.

Then, K.-H. Rehren says, such $a$ model boundary conditions of the theory, and (slightly paraphrased by myself), that strings stretching from an $a$ to a $b$-boundary are described by the bimodule associated with the morphism $$\bar a \circ b : A(I) \to B(I) \to A(I) \,,$$ where $\bar b$ is a (the) conjugate of $b$, in the above sense.

Q-Systems

In the world of AQFT and von Neumann subfactor theory, the name for special symmetric Frobenius algebras of DHR representations is “Q-system”.

Here is how a Q-system is defined in

Y. Kawahigashi, R. Longo
Classification of Local Conformal Nets. Case $c \lt 1$
math-ph/0201015.

Again $A$ is a Frobenius algebra and $\rho : A \to A$ a (DHR) endomorphism. This becomes a Q-system by specifying morphisms $$V : \mathrm{Id}_A \to \rho$$ and $$W : \rho \to \rho \circ \rho$$ that act on the corresponding bimodules by multiplying with corresponding operators $$V,W \in A \subset B(H) \,,$$ which I denote by the same letters, such that $$V^* W = \rho(V^*) W$$ and $$\rho(W) W = W^2 \,.$$

The second condition is nothing but the coassociativity constraint on a coproduct structure on $\rho$, while the first one is the corresponding counit condition. Using the existence of adjoints in $B(H)$, we also get an associtive unital product on $\rho$ this way.

In fact, also the Frobenius property follows automatically by using the existence of adjoint operators, and hence Q-systems are Frobenius algebra structures on DHR representations $\rho : A \to A$ – even special symmetric *-Frobenius algebras, as described in

D.E. Evans and P.R. Pinto,
Subfactor realisation of modular invariants
Commun. Math. Phys. 237 (2003) 309
math.OA/0309174.

A very brief review of this is also given in Ingor Runkel’s Algebra in Braided Tensor Categories and Conformal Field Theory.

It seems to me that K.-H. Rehren’s description of $a$-$b$-string states above would provide the subfactor realization of the special ambidextrous adjunctions that correspond to these special Frobenius algebras. But I can’t remember having seen this discussed in the literature.

So much for now.