# nLab Oberwolfach Workshop, June 2009 -- Monday, June 8

Here are notes from Urs Schreiber for Monday, June 8, from Oberwolfach.

## Schweigert: CFT and algebra in braided tensor categories, I

### 1. Modular tensor categories and rational CFT

#### 1.1

• consider a rational semisimple conformal vertex algebra $V$

• $\to$ from this we get a representation category $C$, which is a braided monoidal category

• that $V$ is semisimple and rarional makes $C$ in fact a modular tensor category
• $\to$ this gives rise to conformal blocks

Definition: modular tensor category

• abelian category, $\mathbb{C}$-linear (i.e. $Vect_{\mathbb{C}}$ enriched category), semisimple category tensor category

• the tensor unit is a simple object, $I$ a finite set of representatives of isomorphism classes of simple objects

• fusion category

• braided monoidal category

• ribbon category, in particular objects have duals

• modularity a non-degeneracy condition on the braiding given by an isomorphism of algebras

$K(C) \otimes_{\mathbb{Z}} \stackrel{\simeq}{\to} End(Id_C)$

where

$[U] \mapsto \alpha_U$

where the transformation $\alpha_U$ is given on the simple object $V$ by

$\alpha_U(V) = straight V-line encircled by U-loop$

(on the right we use string diagram notation)

#### 1.2

Fact (Reshitikhin-Turaev model?) for any modular tensor category $C$ there is a monoidal functor

$tft_C : cobord_{3,2}^C \to vect_f(\mathcal{C})$
• 1) factorization homomorphism: given a surface $\Sigma_{U,U^*}$ with two marked points labeled by $U$ and $U^*$ consider the 3d cobordism from $\Sigma$ to the result of gluing a handle to $\Sigma$ that connects the two marked points with a $U$-line running inside the handle. the corresponding linear map given by the tft

$fact_U := tft_C(\Sigma_{U,U^*} \to \Sigma)$

we have

• $\oplus_{i \in I} fact_{U_i} : \oplus_{i \in I} tft_C(\Sigma_{U,U^*}) \stackrel{\simeq}{\to} tft_C(X)$
• a representation of the mapping class group $Map(\Sigma)$ on $tft_C(X)$

### 2. CFT correlator

#### 2.1

• $X$ a 2-dimensional conformal manifold, either oriented or unoriented without boundary

• for the purpose of this talk restrict to the oriented case (but unoriented case has been dealt with, too)

• also, from now on $X$ regarded just as a topological surface, no longer a conformal one

#### 2.2

strategy

• decorate $X$

• appropriate spaces of “functions” for correlators

holomorphic factorization

• pass from $X$ to its $\mathbb{Z}_2$ orientation bundle $\hat X$, the orientation double cover (identifying points on the boundary, though) (an orientation twisted version of what is called the double )

• double accounts for what physicists call left- and rightmoving degrees of freedom

goal

• 1) find a decoration for $X$ such that $\hat X \in cobord_{3,2}^C$

• 2) specify the correlator $Cor(X) \in tft_C(\hatt X)$

• a) $Cor(X)$ invariant under $Map(\hat X)^{\mathbb{Z}_2}$ modular invariance

• b) compatibility with factorization homomorphism (technical to state, dropped here)

#### 2.3

Insight. decoration bicategory of special symmetric Frobenius algebras in $C$

• Frobenius algebra in $C$: an object

$(A \in Obj(C), \eta : 1 \to A,\epsilon : A \to 1, m : A \otimes A \to A, \Delta : A \to A \otimes A)$

which is a unital associative algebra and counital coassociative coalgebra in $C$

• being Frobenius means that the coproduct $\Delta : A \to A \otimes A$ is a homomorphism of $A$-bimodules

• being symmetric means that the two obvious nontriviall morphisms $A \to A^*$ that one can build using unit and counit are equal

• being special means that $m \circ \Delta = Id_A$ and $\epsilon \circ \eta = dim(A) Id_1$

#### 2.4

A typical worldsheet $X$: higher genus surface with defect lines and marked points drawn on it

decoration

• to 2-dimensional cells assign special symmetric Frobenius algebras $A$;

• to 1-dimensional cells

• to boundary lines a left or right module over the respective Frobenius algebra (boundary is oriented);

• to a defect line: a bimodule over the respective Frobenius algebras

• to 0-dimensional cells

• to a junction of three defect lines labeled by three bimodules $D_i$, associate an element in $Hom_{A_1| A_3}(D_1 \otimes_{A_2} D_2, D_3)$

• boundary field insertions: for marked point on the boundary $x \in \partial X$ attach a simple object $U \in Obj(C)$ to a marked point on a given defect line (or rather, on a junction of two defect lines) and an element in $Hom_A(D_1 \otimes U, D_2)$

• bulk field insertion: for $x$ not in the boundary but in the inside of $X$, consider the two preimages in $\hat X$, assign simple objects $U, V$ to these, respectively and assign an element in

$Hom_{A_1|A_2}(U \otimes A_1 \otimes V, A_2)$
• here the bimodule structure on these tensor products are given by over- and underbraiding, respectively (depending on orientation)

#### 2.5

correlators from cobordisms

given $X$ consider the cobordism from the empty surface

$\emptyset \stackrel{M_X}{\to} \hat X$

given by

$M_X = (\hat X \times [-1,1])/_{\sigma t \sim -t}$

and set

$Cor(X) = tft_C(M_X) 1 \in tft_C(\hat H)$

Example

$X$ the disk with a defect line running across it from boundary to boundary.

then

• $\hat X$ is the sphere

• $M_X$ is the 3-ball

• $X$ sits inside the equatorial plane of $M_X$: one copy of $[-1,1]$ over each of its interior points connectint its two preimages in $\hat X$, in addtion one copy of the interval connecting each boundary point to its unique preimage in $\hat X$

long discussion with audience ensues: audience wants to better understand why all these constructions are being undertaken. The answer is in the theorem to come: every assignment of correlators as above (compatible with factorization morphism and invariant in the above sense under mapping class group (preserving defect decoration)) is obtained by the recipe presented here

## Runkel: CFT and modular tensor categories, II

• Ingo Runkel, Monday morning 11.15. CFT and algebra in braided tensor categories II; notes by Bruce Bartlett: talk2.pdf?

## Freed

joint work with Jacques Distler and Greg Moore.

(on differential cohomology of background fields in type II string theory, in particular on orientifold backgrounds)

• $\Sigma$ in this talk a compact 2-dimensional manifold: the worldsheet

• (called $X$ in the morning talks)
• $X$ smooth 10-dimensional manifold: spacetime

• fields $\Sigma \stackrel{\phi}{\to} X$ (see sigma-model )

• 2-dimensional theory $\to$ 10-dimensional theory

• more generally: $X$ is an orbifold

• today: a variation of this called orientiold : $X_W \to X$ is a double cover

• physicists think of $X_W$ as the spacetime equipped with an involution, but really spacetime is the $\mathbb{Z}_2$-quotient $X$
• I) $\sigma \in \mathbb{Z}_2$ acts trivially : type I string

• II) section $X_W \stackrel{\leftarrow}{\to} X$ type II string theory

this project started when Jacques Distler showed Dan Freed a certain formula, namely

let

$i : F \hookrightarrow X_w$

be the fixed point

• set
$RR-charge = \pm 2^{something} i_* ( \sqrt{\frac{L'(F)}{L'(\nu)}} )$

where

$L' = \prod \frac{x/4 u}{tanh x/4u}$

where * $u$ is the Bott generator of K-theory * $\nu$ is normal bundle of $F \hookrightarrow X_W$ * with another factor this would be the Hrizebruch L-function, this way it is not

what is it?

this led to thinking about the following

• 1) definition of fields/theory

• 2) derive RR-charge formula over $\mathbb{Z}[1/2]$ from $10 d$

• 3) anomaly cancellation in 2d

• there is paper on arXiv with a summary, but this is still work in progress

diff geoemtric structures that one needs to make sense of this:

• differential cohomology

• twistings

### differential cohomology

suppose $h$ is any cohomology theory

then $h$ with rational coefficients $h(-; \mathbb{Q}) = H(-;h(pt, \mathbb{Q}))^\bullet$

here the coefficient object

$h_Q := h(pt;\mathbb{Q})$

on the right is a graded ring and the bulleted-degree is the corresponding total degree of that and the cohomology degree

consider the homotopy limit which gives differential cohomology

$\array{ h^{v \bullet} &\to& \Omega(M, h_{\mathbb{R}})_{closed} \\ \downarrow && \downarrow \\ h^\bullet(M) &\to& H(M; h_{\mathbb{R}}) }$

question from Mike Hopkins: are there choices in the bottom horizontal map; doesn’t one have to choose a basis of cocycles?

we get from this two exact sequences

$0 \to h(M;h_{\mathbb{R}} \otimes \mathbb{R}/\mathbb{Z})^{q-1} \stackrel{curv}{\to}\Omega(M;h_{\mathbb{R}})^q_{\mathbb{Z}} \to 0$
$0 \to forms \to h^{v \bullet}(M) \to h^q(M) \to 0$
• this was introduced for ordinary cohomology by Cheeger and Simons and generalized by Hopkins and Singer

• we can think of the cohomology group as components of some space

$h^q(M) = \pi_0(Map(M, h_q))$

and

$h^{v q}(M) = \pi_0(--)$
• various people constructed various models for this, such as Bunke and Schick; by Gomi for equivariant cohomology, also Szabo and Alessandro

examples

$H^{v 1}(M) = Map(M, \mathbb{T})$
$H^{1}(M) = \pi_0 Map(M, \mathbb{T})$
$H^{v 2}(M) = \{line bundles with connection\}$
$H^{v 3}(M) = \{line bundle gerbes with connection\}$

remarks

• the application to string theory here will have completely topological flavor

• defining certain terms in an action like Wess-Zumino-Witten term and chern-Simons term, these are nicely understood in terms of differential cohomology: observation goes back to Gawedzki

• in physics the forms are “currents” which say where charges are located, the class in real cohomology is the total charge

• in quantum physics this total charge has to be quantized (sit on a lattice inside the real cohomology)

• so the above pullback diagram says that classical charges are to be combined with quantization condition in order to give physical fields

### twistings of $KR(X_w)$

so consider again $\pi : X_w \to X$ a double cover

• an object in $KR^0(X_W)$ is represented by

• a $\mathbb{Z}_2$-graded complex

vector bundle $E \to X_W$ (in terms of pseudobundles: even part minus odd part)

• $\tilde \sigma : \sigma^* \bar E \to E$

recall that $\sigma$ may have fixedpoints

special cases

• $\sigma$ acts trivially: we get just $KO^0(X_w)$-theory

• $X_w \to X$ has a section: $K^0(X)$

twisting

pass to a locally equivalent groupoid

$Y_W \to Y$
$Y : (Y_0 \stackrel{\leftarrow}{\leftarrow} Y_1 \stackrel{\stackrel{\leftarrow}{\leftarrow}}{\leftarrow} \cdots)$

notation: $V^\phi =$ $V$ if $\phi = 0$ and $\bar V$ otherwise

definition

a twsiting of $KR(X_W)$ is an equivalent thing $Y_w \to Y$ as above

where

$d : Y_0 \to \mathbb{Z}$

continuous

$L \to Y_1$

hermitian line bundle, $\mathbb{Z}_2$-graded

$\theta: L_g^{\phi(f)} \otimes L_f \to L_{g f}$

cocycle condition for $\theta$

recognize these twistings as classified by some cohomology theory

cohomology group

For $K(X)$: \pi_{\{0,1,2,3\}h \simeq \{\mathbb{Z}, \mathbb{Z}_2, 0 , \mathbb{Z}\}

for some $h$ that we are not being told about

For $KO(X_W)$: $\pi_{\{0,1,2\}} k_{0 \lt 0..2\gt} \simeq \{\mathbb{Z}, \mathbb{Z}_2, \mathbb{Z}_2\}$

for $KR(X_W)$ the iso classes are

$H^0(X, \mathbb{Z}) \times H^1(X; \mathbb{Z}) \times H^{w+3}(X, \mathbb{Z})$

as a set

$u \in K^2(pt) \in KR^{\tau_1 +2}(pt)$

speaker is running out of time, coherence is being lost a bit… notetaker misses to take notes on some central statement on these twisted cohomology classes, but see the arXiv article

### string backgrounds

the differential cocycles are background data for the 2-d theory and field data for the 10 d theory (see sigma-model)

def an NS-NS superstring background is

• i) a smooth 10d orbifold with metruc and real function (dilaton field)

• ii) $\pi : X_W \to X$ orientifold double cover

• iii) $\beta^v$ a differential twisting of $KR(X_w)$: the $B$-field

• iv) K : $R(\beta) \to \tau^{KO}(T X -2)$ iso of twistings of $KO(X)$: twisted Spin-structure

Bott shift, leading to equivalent theories $\beta^v \to \beta^v + (\tau^v + 2)$

and something else

Stiefel-Whitney classes

$w_1(X) = t w$
$w_2(X) = t w^2 + a w$

aim: mix iii) and iv)

2d theory: A worldsheet $\Sigma$ with metric, a spin structure on $\hat \Sigma$: the orientation double cover

$\array{ \hat \Sigma &\to& X_w \\ \downarrow && \downarrow \\ \Sigma &\to& X }$
$\phi^* w \simeq \hat w$

and spme spinor fields on $\hat \Sigma$

in the path integral: integrate over all of these pieces of data

$\to$ effective action Pfaffian of Dirac operator

$Pfaff D_{\hat \Sigma}(\phi^*T X - 2) \cdot exp 2 \pi i \int_{\Sigma/S} \phi^* \beta^v$

Pfaffian bit is section of a Pfaffian line bundle over $S$

where?

work over some parameter space $S$

both factors above are sections of a line bundle over $S$

theorem (in preparation) there is a hopefully canonical trivialization of $L_\Psi \otimes L_B \to S$

action being section of bundle instead of function: annomaly:

sources

• 1) integrals over fermions $L_\Psi$: spin structure need not be equivariant under $\mathbb{Z}_2$-action

• 2) simultaneous electric and magnetic current or alternatively self-dual current

interply between 1 and 2 leads to anomaly cancellation

• 3) boundary of topological terms, like WZW, Chern-Simons

what about $L_B$: exotic orientation

## Kevin Costello; part I

### deformation quantization

• classical mechanics: $A^{cl}$ commutative algebra with Poisson brackets $\{-,-\}$

• this is the classical observable algebra

• to quantize this we need to find some associative algebra $A^q$ over the ring $\mathbb{R}[\![\hbar]\!]$

• such that

• 1) $A^q/\hbar A^q = A^cl$

• 2) if $a,b \in A^c{l}$, $\tilde a, \tilde b$ are lifts in $A^q$ then

$\{a,b\} = \frac{1}{\hbar} [\tilde a, \tilde b] mod \hbar$

goal of these lectures: want to give an analog of ths picture for QFT

• 1) need to explain wha plays the role of commutative, Poisson and associative algebras

• 2) explain how classical field theory is encoded in commutative and Poisson

• 3) explain how to quantize

structure that play the role of associative algebras is a factorization algebra

this is a $C^\infty$-analog (i.e. differential geometric analog) of a chiral algebra in the sense of Beilinson and Drinfeld

• let $M$ be a manifold (on which we do QFT)

• Let $B(M) = \{smooth balls in M\}$; this is an $\infty$-dim manifold

audience: would open balls here form a manifold? does it matter? >answer: well, really we don’t think of manifolds but of diffeological spaces, of course (sheaves on manifold)

• Let $B_n(M) = \{n disjoint smooth balls embedded in larger ball in M\}$

there are obvious projection maps

$B(M) \stackrel{q}{\leftarrow} B_n(M) \stackrel{p}{\to} B(M)^n$

everything now joint work with O. Gwilliam

def A factorization algebra is

• a vector bundle $F$ on $B(M)$

• equipped with maps $p^*(F^{\times n}) \to q^*(F)$

• satisfying some evident compatibility condition

• such that everything is invariant under the obvious $S_n$-action on $B_n(M)$ that exchanges the order of the balls

• concretely:

• $F$ assigns a vector space to every ball $B \subset M$

• if we have some configuration of balls, like 2 balls $B_1, B_2$ inside a big one $B_3$ we get a map $F(B_1)\otimes F(B_2) \to F(B_3)$

• these must vary smoothly as the configuration of the balls varies

This is an algebra over the embedded little disk operad, which is a “colored operad” (i.e. a multicategory with more than one object)

• where the colors are $B(M)$

• $n$-ary operations are $B_n(M)$

• with extra conditions such that the vector space we assign to each color forms a smooth vector bundle

• and all operad maps are compatible with this

notice that it happenss that for given in and out colors, there is at least one morphism in the operad.

• there are several different reductions of this structure that are more familiar

• notion of vector bundle comes in three natural flavors

• 1) $C^\infty$

• 2) holomorphic

• 3) locally constant sheaves

• definition of factorization algebra can be modified to the case of 2) and 3)

def a locally constant factorization algebra is like a factorization algebra, except that instead of being a vector bundle $F$ is a locally constant sheaf on $B(M)$ of cochain complexes and the structure maps of locally constant sheaves

question: cohomologically locally constant or really locally constant

> answer: I think cohomologically locally constant

let $F$ be a locally constant factorization algebra in $\mathbb{R}^n$, then since $B(\mathbb{R}^n)$ is contractible $F$ is quasi-isomorphic to a trivial sheaf with fiber $V$ a cochain complex

so then for instance the map V__{B_1} \otimes V_{B_2} \to V_{B_3} depends only on homotopies of the configuration,

so, a locally constant factorization algebra on $V__\mathbb{R}^n$ is an $E_n$-algebra

next specialization: holomorphic factorization algebras

Let $\Sigma$ be a Riemannian surface.

We know what it means for a map from a complex manifold to $B(M)$ to be holomorphic; so we can talk about holomorphic algebras on $B(\Sigma)$

a homomorphic map $U \to B(M)$ is a bundle $M \to U$ all of whose fibers are balls and a map

$\array{ M &\to& \Sigma \\ \downarrow \\ U }$

actually $B(M)$, too, is also not a complex manifold but a sheaf on complex manifold

Andre Henriquez: but with this definition, won’t every map $U \to B(M)$ be holomorphic: > answer: oops, right

??

let’s consider a holomorphic factorization algebra on $\mathcal{C}$, which is translation invariant and dilation invariant

let $V = F$ (any round disk)

if we have a configuratoin of disks with $B_1$ and $B_2$ in $B_3$ with radii $\epsilon_i$ with one disk in the center of the big one and the other at complex parameter $z$, the map

$m_z : V \otimes V \to V$

must vary holomorphically with $z$

so $z \mapsto m_z$ is a holomorphic map $Annulus \to Hom(V \otimes V, V)$, so it has a Laurent expansion

$m_z \sim \sum_{k \in \mathbb{Z}} Z^k a_k$

with $a_k$ in some completion of $Hom(V \otimes V , V)$

question by Ulrich Bunke: algebraic tensor product or not? > answer: no, in examples tensor product will be projective tensor product

reminiscent of vertex operator algebra

notice that Beilinson-Drinfeld make the same def in the algebraic setting, in their case the axioms are equivalent to that of a vertex operator algebra;

they show axioms for chiral algebra on $\mathcal{C}$ are essentially equivalent to those of a vertex operator algebra

claim: structure of factorization algebra: good to encode quantum field theory

notice that factorizations algebras on real line tend to be associativ algebras, so that fits in with the expectation from quantum mechanics.

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