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April 2, 2007

Oberwolfach CFT, Monday Morning

Posted by Urs Schreiber

Today Arthur Bartels reviewed the standard construction of a modular tensor categories of “DHR representations” from a local net of von Neumann algebras on the real line.

Here is a transcript of my notes.

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


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

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

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

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

A simple example is the choice H=L 2()H = L^2(\mathbb{R}) with the assignment A:IA(I)={fL ()|fconstantonI}. 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 EE \subset \mathbb{R} by forming the vN algebra generated from all A(I)A(I) for all bounded open intervals IEI \subset E.

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

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

(In the following we will mostly be interested in A *(E)A^*(E). At least one reason for that is, apparently, that the representations of A(E)A(E) are rather boring. But I need to better understand this issue of switching from considering vN algebras to C *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(IJ)=A(I)A(J) A(I \cup J) = A(I)\vee A(J) whenever the intervals II and JJ have nontrivial intersection, IJI \cap J \neq \emptyset.

(Here A 1A 2A_1 \vee A_2 denotes the vN algebra generated from both A 1A_1 and A 2A_2. This is nothing but the double commutant of the union of both these algebras: A 1A 2=(A 1A 2)A_1 \vee A_2 = (A_1 \cup A_2)''.)

A net is called local if A(I)A(I) A(I) \subset A(I')' where I=II' = \mathbb{R}-I is the complement of II (which is not a bounded interval if II 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)B(H).)

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

The main point is to prove

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

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

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

If uu is unitary, then ρ\rho and σ\sigma are said to be unitarily equivalent.

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

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

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

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

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

So were are identifying certain representations of AA with certain endomorphisms of AA. 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 ρσ:=ρsircσ. \rho \otimes \sigma := \rho \sirc \sigma \,.

Lemma.

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

2) if u:ρσu : \rho \to \sigma, where ρ\rho and σ\sigma are localized in II, then uA(I)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 ρ\rho and σ\sigma are localized in II and pick an open bounded JJ \subset \mathbb{R} to the right of II and pick any uU(H)u \in U(H) such that Ad uσ \mathrm{Ad}_u \circ \sigma is localized in JJ.

Lemma: The intertwiner c ρ,σ:ρσσρ c_{\rho,\sigma} : \rho \otimes \sigma \to \sigma \otimes \rho given by the operator u *ρ(u)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.

Posted at April 2, 2007 12:35 PM UTC

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