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, $ℂ$-linear (i.e. ${\mathrm{Vect}}_{ℂ}$ 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

  $$K(C){\otimes }_{\mathbb{Z}}\stackrel{\simeq }{\to }\mathrm{End}({\mathrm{Id}}_C)$$K(C) \otimes_{\mathbb{Z}} \stackrel{\simeq}{\to} End(Id_C)

  where

  $$[U]\mapsto {\alpha }_U$$[U] \mapsto \alpha_U

  where the transformation ${\alpha }_U$ is given on the simple object $V$ by

  $${\alpha }_U(V)=\mathrm{straight}V-\mathrm{line}\mathrm{encircled}\mathrm{by}U-\mathrm{loop}$$\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

$${\mathrm{tft}}_C:{\mathrm{cobord}}_{3,2}^C\to {\mathrm{vect}}_f(𝒞)$$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

  $${\mathrm{fact}}_U:={\mathrm{tft}}_C({\Sigma }_{U,U^{*}}\to \Sigma )$$fact_U := tft_C(\Sigma_{U,U^*} \to \Sigma)

  we have

• $${\oplus }_{i\in I}{\mathrm{fact}}_{U_i}:{\oplus }_{i\in I}{\mathrm{tft}}_C({\Sigma }_{U,U^{*}})\stackrel{\simeq }{\to }{\mathrm{tft}}_C(X)$$\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 $\mathrm{Map}(\Sigma )$ on ${\mathrm{tft}}_C(X)$

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 $\hat{X}\in {\mathrm{cobord}}_{3,2}^C$

• 2) specify the correlator $\mathrm{Cor}(X)\in {\mathrm{tft}}_C(hattX)$

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

  $$(A\in \mathrm{Obj}(C),\eta :1\to A,ϵ:A\to 1,m:A\otimes A\to A,\Delta :A\to A\otimes A)$$(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 $m\circ \Delta ={\mathrm{Id}}_A$ and $ϵ\circ \eta =\dim (A){\mathrm{Id}}_1$

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 ${\mathrm{Hom}}_{A_1\mid A_3}(D_1{\otimes }_{A_2}{D}_2,D_3)$

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

  • 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

    $${\mathrm{Hom}}_{A_1\mid A_2}(U\otimes A_1\otimes V,A_2)$$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

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

given by

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

and set

$$\mathrm{Cor}(X)={\mathrm{tft}}_C(M_X)1\in {\mathrm{tft}}_C(\hat{H})$$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 $\Sigma \stackrel{\varphi }{\to }X$ (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 $X_W\overleftarrow{\to }X$ type II string theory

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

let

$$i:F\hookrightarrow X_w$$i : F \hookrightarrow X_w

be the fixed point

• set

  $$\mathrm{RR}-\mathrm{charge}=\pm 2^{\mathrm{something}}{i}_{*}(\sqrt{\frac{L\prime (F)}{L\prime (\nu )}})$$RR-charge = \pm 2^{something} i_* ( \sqrt{\frac{L'(F)}{L'(\nu)}} )

  where

  $$L\prime =\prod \frac{x/4u}{\tanh x/4u}$$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 $10d$

• 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 $h(-;\mathbb{Q})=H(-;h(\mathrm{pt},\mathbb{Q}){)}^{•}$

here the coefficient object

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

$$\begin{array}{ccc}{h}^{v•}& \to & \Omega (M,h_{ℝ}{)}_{\mathrm{closed}}\\ \downarrow & & \downarrow \\ h^{•}(M)& \to & H(M;h_{ℝ})\end{array}$$\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

$$0\to h(M;h_{ℝ}\otimes ℝ/\mathbb{Z}{)}^{q-1}\stackrel{\mathrm{curv}}{\to }\Omega (M;h_{ℝ}{)}_{\mathbb{Z}}^q\to 0$$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

$$0\to \mathrm{forms}\to h^{v•}(M)\to h^q(M)\to 0$$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

  $$h^q(M)={\pi }_0(\mathrm{Map}(M,h_q))$$h^q(M) = \pi_0(Map(M, h_q))

  and

  $$h^{vq}(M)={\pi }_0(--)$$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

$$H^{v1}(M)=\mathrm{Map}(M,𝕋)$$H^{v 1}(M) = Map(M, \mathbb{T})

$$H^1(M)={\pi }_0\mathrm{Map}(M,𝕋)$$H^{1}(M) = \pi_0 Map(M, \mathbb{T})

$$H^{v2}(M)=\{\mathrm{line}\mathrm{bundles}\mathrm{with}\mathrm{connection}\}$$H^{v 2}(M) = \{line bundles with connection\}

$$H^{v3}(M)=\{\mathrm{line}\mathrm{bundle}\mathrm{gerbes}\mathrm{with}\mathrm{connection}\}$$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 $\mathrm{KR}(X_w)$

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

• an object in ${\mathrm{KR}}^0(X_W)$ is represented by

  • a ${\mathbb{Z}}_2$-graded complex

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

* $\tilde{\sigma }:{\sigma }^{*}\overline{E}\to E$

recall that $\sigma$ may have fixedpoints

special cases

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

• $X_w\to X$ has a section: $K^0(X)$

twisting

pass to a locally equivalent groupoid

$$Y_W\to Y$$Y_W \to Y

$$Y:(Y_0\overleftarrow{\leftarrow }{Y}_1\stackrel{\overleftarrow{\leftarrow }}{\leftarrow }\cdots )$$Y : (Y_0 \stackrel{\leftarrow}{\leftarrow} Y_1 \stackrel{\stackrel{\leftarrow}{\leftarrow}}{\leftarrow} \cdots)

notation: $V^{\varphi }=$ $V$ if $\varphi =0$ and $\overline{V}$ otherwise

definition

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

where

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

continuous

$$L\to Y_1$$L \to Y_1

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

$$\theta :L_g^{\varphi (f)}\otimes L_f\to L_{gf}$$\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 $\mathrm{KO}(X_W)$: ${\pi }_{\{0,1,2\}}{k}_{0<0..2>}\simeq \{\mathbb{Z},{\mathbb{Z}}_2,{\mathbb{Z}}_2\}$

for $\mathrm{KR}(X_W)$ the iso classes are

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

as a set

$$u\in K^2(\mathrm{pt})\in {\mathrm{KR}}^{{\tau }_1+2}(\mathrm{pt})$$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 $\mathrm{KR}(X_w)$: the $B$-field

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

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

and something else

Stiefel-Whitney classes

$$w_1(X)=tw$$w_1(X) = t w

$$w_2(X)=t{w}^2+aw$$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

$$\begin{array}{ccc}\hat{\Sigma }& \to & X_w\\ \downarrow & & \downarrow \\ \Sigma & \to & X\end{array}$$\array{ \hat \Sigma &\to& X_w \\ \downarrow && \downarrow \\ \Sigma &\to& X }

$${\varphi }^{*}w\simeq \hat{w}$$\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

$$\mathrm{Pfaff}{D}_{\hat{\Sigma }}({\varphi }^{*}TX-2)\cdot \exp 2\pi i{\int }_{\Sigma /S}{\varphi }^{*}{\beta }^v$$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: $A^{\mathrm{cl}}$ 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 $ℝ[[\hslash ]]$

• such that

  • 1) $A^q/\hslash A^q=A^{\mathrm{cl}}$

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

    $$\{a,b\}=\frac{1}{\hslash }[\tilde{a},\tilde{b}]\mathrm{mod}\hslash$$\{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 $C^{\mathrm{\infty }}$-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 $B(M)=\{\mathrm{smooth}\mathrm{balls}\mathrm{in}M\}$; this is an $\mathrm{\infty }$-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 $B_n(M)=\{n\mathrm{disjoint}\mathrm{smooth}\mathrm{balls}\mathrm{embedded}\mathrm{in}\mathrm{larger}\mathrm{ball}\mathrm{in}M\}$

there are obvious projection maps

$$B(M)\stackrel{q}{\leftarrow }{B}_n(M)\stackrel{p}{\to }B(M{)}^n$$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) $C^{\mathrm{\infty }}$

  • 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 ${ℝ}^n$, then since $B({ℝ}^n)$ 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 ${ℝ}^n$ 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

$$\begin{array}{ccc}M& \to & \Sigma \\ \downarrow \\ U\end{array}$$\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 $𝒞$, 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 ${ϵ}_i$ with one disk in the center of the big one and the other at complex parameter $z$, the map

$$m_z:V\otimes V\to V$$m_z : V \otimes V \to V

must vary holomorphically with $z$

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

$$m_z\sim \sum _{k\in \mathbb{Z}}{Z}^k{a}_k$$m_z \sim \sum_{k \in \mathbb{Z}} Z^k a_k

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

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 $𝒞$ 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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