Here are notes by Urs Schreiber for Thursday, June 11, from Oberwolfach.
André Henriques has notes on this work available on his website;
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 .
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 $I, J$ two intervals equipped with an embedding into interval $K$ the net assigns
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.
claim
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 $A$ is invertible in CN3 iff its $\mu$-index $\mu(A)$ equals 1.
It is fully dualizable iff $\mu(A) \lt \infty$
in general $\mu(A) \in \{0\} \cup [1, \infty]$
in the 2-category of vN algebras, a morphism
is a bimodule ${}_A H_B$ where $H$ is a Hilbert space
the unit
is given by the bimodule called $L^2(A)$
this comes from the following: if $X$ is a measure space and $A$ arises as $L^\infty(X)$ then $L^2(A)$ is $L^2(X)$.
the general $L^2(A)$ is a non-commutative analog of this construction, for $A$ which are not of the form “measurable functions on a measure space”.
One word about composition, if
given by $H \otimes_{Connes} K$ is the Connes-Fusion operation
associated to ${}_A H_B$ is
fact think of an interval $I$ as the upper semicircle. The two actions of $A(I)$ on $H_0 = L^2(A(I))$ can be used to define an action of $A(J)$ on $H_0$ for every $J \usbset S^1$.
Because if $J$ 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 $H_0$ has an action of $A(S^1)$.
so now everything is invariant projectively with respect to diffeomorphism group
definition
given two circles $S_a, S_b$, let $nw(S_a)$ be north west quarter cicle, etc
consider $H_0$ as a bimodule over
and
claim: $\mu(A) = dim(H_0)$
now some ideas on the proof of the first theorem
define always
if $A$ is dualizable, then $\bar A$ is its dual
if $A$ is invertible, then $\bar A$ 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
then define the composition defect net by
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
by setting
(whatever that means)
things we could talk aout:
composition of defects
examples of conformal nets
$\mathbb{Z}_2$-graded version of all that
Mike Hopkins: could we talk about relation full CFT to chiral CFT?
examples can be obtained as follows
loop group nets ($S^1 \to \tilde {L G} \to L G$)
integral lattice
moonshine net: take the Leech lattice and do some orbifolding
free fermion
minimal model = Virasoro
general remarks:
recall the Hillber space $H_0$ obtained from the full cnformal net as the algebra of the upper semi circle
but in practice one first constructs $H_0$ and then from that the net
a functor
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 $I$, construc the set of maps
if $I$ 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 $I$ to $e$.
but the point is there is way to get a conformal net here without assuming embedding into $S^1$
so the algebra we will assign to $I$ is just the group algebra
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 $H_0$ through a rep of $\tilde {L G}$
so $A(I)$ is now defined to be the completion of that algebra in $B(H_0)$
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
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 $I$ and $J$,
let $H_0$ 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 $H_0$ with itself over $A(I) \otimes A(J)$
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
then consider extensions $B$ of nets of the following form
such is cooked up from every Frobenius algebra object $Q$ in $Rep(A \otimes A^{op})$ (in vN theorists jargon this is a “$Q$-system”)
in the 3-category VN3 this $Rep(A)$ can be written as
to the circle $B$ assigns $Q$
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: $\mathbf{B}U(n) \to \mathbf{B} PU(n) \to \mathbf{B}^2 U(1)$
twisting cocycle: lifting gerbe;
twisted cocycle: twisted bundles / gerbe modules
twisted Bianchi identity: $d F_\nabla = H_3$
occurence: Freed-Witten anomaly cancellation on D-brane
fibration sequence: $\mathbf{B}String(n) \to \mathbf{B} Spin(n) \stackrel{\frac{1}{2}p_1}{\to} \mathbf{B}^3 U(1)$
twisting cocycle: Chern-Simons 2-gerbe;
twisted cocycle: twisted nonabelian String-gerbe with conection
twisted Bianchi identity: $d H_3 \propto \langle F_\nabla \wedge F_\nabla \rangle$
occurence: Green-Schwarz anomaly cancellation
Proof.
(with Danny Stevenson and Christoph Wockel: (SSSS)) use BCSS model (BCSS) of $String(n)$ with Brylinski-McLaughlin construction of $\frac{1}{2}p_1$
(using (SaScSt I, SaScSt III):) compute local differential form data after differentiating smooth $\infty$-groupoids to L-infinity algebroids using the formalism of (SaScSt I)
for aspects of the twisted case see also
fibration sequence: $\mathbf{B}Fivebrane(n) \to \mathbf{B} String(n) \stackrel{\frac{1}{6}p_2}{\to} \mathbf{B}^7 U(1)$
twisting cocycle: Chern-Simons 6-gerbe;
twisted cocycle: twisted nonabelian Fivebrane-gerbe with connection
occurence: dual Green-Schwarz anomaly cancellation for NS 5-brane magnetic dual to string
fibration sequence: $\mathbf{B}^2 U(1) \to \mathbf{B} (U(1) \to \mathbb{Z}_2) \stackrel{}{\to} \mathbf{B} \mathbb{Z}_2$
twisting cocycle: $\mathbb{Z}_2$-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