The n-Category Café

I reproduce a transcript of the notes I have taken. Comments by myself are set in italics.

Let $H$ be a Hilbert space and $B(H)$ the algebra of bounded operators on that.

For $S \subset B(H)$ write $S' \subset B(H)$ for the commutant of $S$, which is the set of all bounded operators that commute with all those in $S$.

Notice that $B(H)$ is in particular a $*$-algebra. A von Neumann algebra is any $*$-subalgebra $A \subset B(H)$ such that $A'' = A$.

Definition: A net of vN algebras on $\mathbb{R}$ is an inclusion-preserving assignment $$I \mapsto A(I)$$ of vN algebra $A(I) \subset B(H)$, for some fixed Hilbert space $H$, to bounded open intervales $$I \subset \mathbb{R} \,.$$

A simple example is the choice $H = L^2(\mathbb{R})$ with the assignment $$A : I \mapsto A(I) = \{ f \in L^ \infty(\mathbb{R}) | f \mathrm{constant}\; \mathrm{on}\; \mathbb{R}-I \} \,.$$

However, this kind of example is not what one is really interested in. Interesting examples are much harder to describe (and are not described in this talk here).

Given such a net, one can also assign vN algebras to unbounded open subsets $E \subset \mathbb{R}$ by forming the vN algebra generated from all $A(I)$ for all bounded open intervals $I \subset E$.

The notation for the vN algebra obtained this way is $$A(E) \,.$$

Alternatively, one can consider just the $C^*$-algebra generated by all these $A(I)$. The result of that is then called $$A^*(E) \,.$$

(In the following we will mostly be interested in $A^*(E)$. At least one reason for that is, apparently, that the representations of $A(E)$ are rather boring. But I need to better understand this issue of switching from considering vN algebras to $C^*$-algebras.)

Definition Here are a couple of additional properties of nets of vN algebras which we will assume in the following.

A net is called additive if $$A(I \cup J) = A(I)\vee A(J)$$ whenever the intervals $I$ and $J$ have nontrivial intersection, $I \cap J \neq \emptyset$.

(Here $A_1 \vee A_2$ denotes the vN algebra generated from both $A_1$ and $A_2$. This is nothing but the double commutant of the union of both these algebras: $A_1 \vee A_2 = (A_1 \cup A_2)''$.)

A net is called local if $$A(I) \subset A(I')'$$ where $I' = \mathbb{R}-I$ is the complement of $I$ (which is not a bounded interval if $I$ is, so that this means we are making use of the notation introduced above.)

In words: the net is local if the algebras associated to two disjoint intervals mutually commute (as subalgebras of the fixed $B(H)$.)

A net $A$ satisfies Haag duality if the above inclusion is even an equality $$A(I) = A(I')' \,.$$

The main point is to prove

Theorem. Let the net $A$ be additive and satifying Haag duality, then the sectors of $A$ form a braided tensor category.

Here a “sector” is, roughly, a localized representation of $A$. To get to the precise definition, consider the following:

Definition. Let $$\rho,\sigma : A^**(\mathbb{R}) \to A^*(\mathbb{R})$$ be two endomorphisms of the global $C^*$-algebra, then an intertwiner $$\nu : \rho \to \sigma$$ is an operator $u \in B(H)$ such that $$u \rho(a) = \sigma(a) u$$ for all $a \in A^*(\mathbb{R})$.

If $u$ is unitary, then $\rho$ and $\sigma$ are said to be unitarily equivalent.

Remark. When $\rho$ is an endomorphism as above, we get a rep of $A^*(\mathbb{R})$ on $H$ simply by setting $$a_\rho (v) = \rho(a) v$$ for $v \in H$.

The above intertwiners then are nothing but morphisms of these representations.

An endomorphism $\rho$ is called localized in $$I \subset \mathbb{R}$$ if it is the identity outside of some bounded interval $I$, i.e. if $$\rho(a) = a$$ for all $a \in A^*(A')$.

Finally, a sector is a localized endomorphism that is localized, up to unitary equivalence, in any open bounded $I \subset \mathbb{R}$.

Under suitable assumption on $A$, this definition is equivalent to the notion of Doplicher-Haag-Roberts representations (DHR reps).

So were are identifying certain representations of $A$ with certain endomorphisms of $A$. Thinking of these as ordinary representations yields an obvious direct sum structure on all these reps. But thinking of them as endomorphisms yields an obvious tensor product:

Definition. The tensor product on sectors is simply the composition of the corresponding endomorphisms $$\rho \otimes \sigma := \rho \sirc \sigma \,.$$

Lemma.

1) If $\rho$ is localized in $I$ then $\rho(A(I)) \subset A(I)$

2) if $u : \rho \to \sigma$, where $\rho$ and $\sigma$ are localized in $I$, then $u \in A(I)$.

In addition, there is a braiding on this tensor product, induced by moving the localized reps around by unitary operators.

Assume that $\rho$ and $\sigma$ are localized in $I$ and pick an open bounded $J \subset \mathbb{R}$ to the right of $I$ and pick any $u \in U(H)$ such that $$\mathrm{Ad}_u \circ \sigma$$ is localized in $J$.

Lemma: The intertwiner $$c_{\rho,\sigma} : \rho \otimes \sigma \to \sigma \otimes \rho$$ given by the operator $u^* \rho(u)$ is a braiding isomorphism, which does not depend on any of the choices used in its construction.

The talk also covered the proofs of the above statements, and then ended by stating the Kawahigashi-Longo-Müger theorem, which says that under the additional assumption that the net of vN algebras is completely rational and has a modular PCT symmetry the braided abelian monoidal category of DHR reps constructed so far is in fact a modular tensor category.

I’ll say more about this elsewhere.