nLab Oberwolfach Workshop, June 2009 -- Thursday, June 11

Here are notes by Urs Schreiber for Thursday, June 11, from Oberwolfach.

Henriques: Invertible conformal nets

André Henriques has notes on this work available on his website;

  • Chris Douglas, André Henriques, Michael Hill, Geometric String structures – notes on 2\mathbb{Z}_2-graded conformal nets (ps)

today: classification of invertible objects in a 3-category

theorem (André Henriques, Arthur Bartels, Chris Douglas): Conformal nets form the objects of a symmetric monoidal 3-category CN3 .

  • there is something about this written up to be found on Henriques’ webpage.

definition A conformal net is a factorization algebra on the category of 1-dimensional balls with values in von Neumann algebras.

  • 1-dim ball is just an interval, of course

  • so a conformal net associates to each interval a vN algebra, and for each inclusion of intervals there is an inclusion of vN algebras

for I,JI, J two intervals equipped with an embedding into interval KK the net assigns

A:(J,KI)(A(J)A(K)A(I)) A : (J,K \subset I) \mapsto (A(J) \otimes A(K) \to A(I))

Freed: why do you call this conformal ?

André: (as far as I understand the reply he gives after my question) the usual covariance condition under Moebius group imposed on conformal nets is taken care of here by havin not intervals in the real line on the line but abstract intervals with maps between them being embeddings. The claim is that this automatically implies the right kind of covariance that is otherwise imposed by hand.


{unitaryfullCFTs}{conformalnets} \{unitary full CFTs\} \simeq \{conformal nets\}

question: do you really mean full instead of chiral CFTs on the right?

answer: yes

symmetric monoidal structure is objectwise the tensor product

theorem aa conformal net AA is invertible in CN3 iff its μ\mu-index μ(A)\mu(A) equals 1.

It is fully dualizable iff μ(A)<\mu(A) \lt \infty

in general μ(A){0}[1,]\mu(A) \in \{0\} \cup [1, \infty]

von Neumann algebras

in the 2-category of vN algebras, a morphism

AB A \to B

is a bimodule AH B{}_A H_B where HH is a Hilbert space

the unit

Id A:AA Id_A : A \to A

is given by the bimodule called L 2(A)L^2(A)

this comes from the following: if XX is a measure space and AA arises as L (X)L^\infty(X) then L 2(A)L^2(A) is L 2(X)L^2(X).

the general L 2(A)L^2(A) is a non-commutative analog of this construction, for AA which are not of the form “measurable functions on a measure space”.

One word about composition, if

AHBKC A \stackrel{H}{\to} B \stackrel{K}{\to} C

given by H ConnesKH \otimes_{Connes} K is the Connes-Fusion operation

associated to AH B{}_A H_B is

dim( AH B){0}[1,] dim({}_A H_B) \in \{0\} \cup [1,\infty]
dim={0 iffH=0 1 iffHisinvertible < iffHhasadual dim = \left\{ \array{ 0 & iff H=0 \\ 1 & iff H is invertible \\ \lt \infty & iff H has a dual } \right.

fact think of an interval II as the upper semicircle. The two actions of A(I)A(I) on H 0=L 2(A(I))H_0 = L^2(A(I)) can be used to define an action of A(J)A(J) on H 0H_0 for every JusbsetS 1J \usbset S^1.

Because if JJ is in the upper semicircle, use the left action, if in the lowwer semicircle use the right action, if it is neither, do something else (not explained here)

so H 0H_0 has an action of A(S 1)A(S^1).

so now everything is invariant projectively with respect to diffeomorphism group


given two circles S a,S bS_a, S_b, let nw(S a)nw(S_a) be north west quarter cicle, etc

consider H 0H_0 as a bimodule over

A(sw(S a))A(ne(S a)) A(sw(S_a)) \otimes A(ne(S_a))


A(nw(S b))A(se(S b)) A(nw(S_b)) \otimes A(se(S_b))

claim: μ(A)=dim(H 0)\mu(A) = dim(H_0)

now some ideas on the proof of the first theorem

define always

A¯(I)=A(I) op=A(I¯) \bar A(I) = A(I)^{op} = A(\bar I)
  • if AA is dualizable, then A¯\bar A is its dual

  • if AA is invertible, then A¯\bar A is its inverse

a 1-morphism in CN3, to be called a defect is

consider the category of bi-colored 1-dimensional manifold is a 1-d manifolds (decomposition into intervals labeled by two things) with morphisms being color-preserving maps

sufficient to look at intervals

  • completely blue

  • half blue, half orange

  • completely orange

now a defect/1-morphism is a conformal net as above on bi-colored 1d manifolds

so suppose three nets and two defects

ADBEC A \stackrel{D}{\to} B \stackrel{E}{\to} C

then define the composition defect net by

DE(blueintervalgluedtoyellowinterval)=insertorangeintervalinthemiddleandaverageoveralllocalevaluations D \otimes E ( blue-interval glued to yellow interval) = insert orange interval in the middle and average over all local evaluations

the diffeomorphism group of the orange interval does act on the space of all possible compositions here, which one shows to all be coherently isomorphic

now construct a defect

D:AA¯1 D: A \otimes \bar A \to 1

by setting

D(interval):=A(bentaroundinterval) D(interval) := A(bent-around interval)

(whatever that means)

Henriques: informal evening session

things we could talk aout:

  • composition of defects

  • examples of conformal nets

  • 2\mathbb{Z}_2-graded version of all that

Mike Hopkins: could we talk about relation full CFT to chiral CFT?

examples for conformal nets

  • from the standard chiral positive energy nets that are usually called “conformal nets” in the literature we get the more general considered here by tensoring two chiral bits (“left” and “right”) and choosing an extension

examples can be obtained as follows

  • loop group nets (S 1LG˜LGS^1 \to \tilde {L G} \to L G)

  • integral lattice

    • moonshine net: take the Leech lattice and do some orbifolding

      • its full automorphism group is isomorphic to the Monster
  • free fermion

  • minimal model = Virasoro

general remarks:

  • recall the Hillber space H 0H_0 obtained from the full cnformal net as the algebra of the upper semi circle

  • but in practice one first constructs H 0H_0 and then from that the net

  • a functor

    {subintervalsofS 1,diffeosbetweenthem}VN \left\{ subintervals of S^1, diffeos between them \right\} \to V N

    is already the same as a functor on all abstract intervals, since that category is equivalent to intervals in the circle

so given an interval II, construc the set of maps

IMaps((I,I),(G,{e})) I \mapsto Maps((I, \partial I), (G, \{e\}))

if II happens to be thought of as embedded in the circle, then this group could be identified with the subgroup of the loop group whose elements send the complement of II to ee.

but the point is there is way to get a conformal net here without assuming embedding into S 1S^1

so the algebra we will assign to II is just the group algebra

I𝒞(Maps((I,I),(G,{e}))] I \mapsto \mathcal{C}(Maps((I, \partial I), (G, \{e\}))]

forgetting all topology, but this still depends on the central extension and it sits densely in the vN algebra obtained by taking the group, letting it act on the vacuum in the Hilbert space H 0H_0 through a rep of LG˜\tilde {L G}

so A(I)A(I) is now defined to be the completion of that algebra in B(H 0)B(H_0)

this here gives a chiral theory

audience: one should think of intervals here as thickened points and of the vN algebras as what is assigned to the point in an extended CFT

algebra assigned to the circle

here is how to get the algebra assigned to the circle:

puff up the circle to an annulus, cut this vertically in two pieces; call the two intervals where the cut goes II and JJ,

let H 0H_0 be the Hilbert space assigned to the boundary circles of the two pieces of the cut annulus. Then the algebra in question is the Connes fusion product of H 0H_0 with itself over A(I)A(J)A(I) \otimes A(J)

A(circle):=H 0 A(I)A(J) ConnesH 0 A(circle) := H_0 \otimes^{Connes}_{A(I) \otimes A(J)} H_0

audience: so the circle doesn’t really appear in this formula, it’s just a thing to help us think about this

answer: yes

now recall the definition of tensor product of nets

(AA op)(I):=A(I)A(I) op \left( A \otimes A^{op} \right)(I) := A(I) \otimes A(I)^{op}

then consider extensions BB of nets of the following form

AA opB A \otimes A^{op} \hookrightarrow B

such is cooked up from every Frobenius algebra object QQ in Rep(AA op)Rep(A \otimes A^{op}) (in vN theorists jargon this is a “QQ-system”)

in the 3-category VN3 this Rep(A)Rep(A) can be written as

Rep(A)=End VN3(Id A)={ A ρ A } Rep(A) = End_{VN3}(Id_A) = \left\{ \array{ & \nearrow \searrow \\ A &\Downarrow^{\rho}& A \\ & \searrow \nearrow } \right\}

to the circle BB assigns QQ


detailed notes with further links and backgeound information are here:

The point of this is to define a notion of twisted differential nonabelian cohomology for every fibration sequence of smooth \infty-groupoids and then establish from that the following list of examples.

We now

  • list fibration sequence of smooth \infty-groupoids

  • and indicate properties of the corresponding differential twisted nonabelian cohomology

Examples / Claim

  • fibration sequence: BU(n)BPU(n)B 2U(1)\mathbf{B}U(n) \to \mathbf{B} PU(n) \to \mathbf{B}^2 U(1)

    • twisting cocycle: lifting gerbe;

    • twisted cocycle: twisted bundles / gerbe modules

    • twisted Bianchi identity: dF =H 3d F_\nabla = H_3

    • occurence: Freed-Witten anomaly cancellation on D-brane

  • fibration sequence: BString(n)BSpin(n)12p 1B 3U(1)\mathbf{B}String(n) \to \mathbf{B} Spin(n) \stackrel{\frac{1}{2}p_1}{\to} \mathbf{B}^3 U(1)

    • twisting cocycle: Chern-Simons 2-gerbe;

    • twisted cocycle: twisted nonabelian String-gerbe with conection

    • twisted Bianchi identity: dH 3F F d H_3 \propto \langle F_\nabla \wedge F_\nabla \rangle

    • occurence: Green-Schwarz anomaly cancellation

    • Proof.

    • for aspects of the twisted case see also

      • AsJu

      • for aspects of the untwisted case see also (Wa)

  • fibration sequence: BFivebrane(n)BString(n)16p 2B 7U(1)\mathbf{B}Fivebrane(n) \to \mathbf{B} String(n) \stackrel{\frac{1}{6}p_2}{\to} \mathbf{B}^7 U(1)

  • fibration sequence: B 2U(1)B(U(1) 2)B 2\mathbf{B}^2 U(1) \to \mathbf{B} (U(1) \to \mathbb{Z}_2) \stackrel{}{\to} \mathbf{B} \mathbb{Z}_2

    • twisting cocycle: 2\mathbb{Z}_2-orbifold;

    • twisted cocycle: orientifold gerbe / Jandl gerbe with connection

    • occurence: unoriented string

    • unwrap the above abstract nonsense and use the above results to find SchrSchwWal and the bosonic part of DiFrMo

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