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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 VV

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

  • \to this gives rise to conformal blocks

Definition: modular tensor category

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

  • the tensor unit is a simple object, II 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) End(Id C) K(C) \otimes_{\mathbb{Z}} \stackrel{\simeq}{\to} End(Id_C)

    where

    [U]α U [U] \mapsto \alpha_U

    where the transformation α U\alpha_U is given on the simple object VV by

    α U(V)=straightVlineencircledbyUloop \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 CC there is a monoidal functor

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

    fact U:=tft C(Σ U,U *Σ) fact_U := tft_C(\Sigma_{U,U^*} \to \Sigma)

    we have

  • iIfact U i: iItft C(Σ U,U *)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 Map(Σ)Map(\Sigma) on tft C(X)tft_C(X)

2. CFT correlator

2.1

  • XX 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 XX regarded just as a topological surface, no longer a conformal one

2.2

strategy

  • decorate XX

  • appropriate spaces of “functions” for correlators

holomorphic factorization

  • pass from XX to its 2\mathbb{Z}_2 orientation bundle X^\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 XX such that X^cobord 3,2 C\hat X \in cobord_{3,2}^C

  • 2) specify the correlator Cor(X)tft C(hattX)Cor(X) \in tft_C(\hatt X)

    • a) Cor(X)Cor(X) invariant under Map(X^) 2Map(\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 CC

  • Frobenius algebra in CC: an object

    (AObj(C),η:1A,ϵ:A1,m:AAA,Δ:AAA) (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 CC

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

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

  • being special means that mΔ=Id Am \circ \Delta = Id_A and ϵη=dim(A)Id 1\epsilon \circ \eta = dim(A) Id_1

2.4

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

decoration

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

  • 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 iD_i, associate an element in Hom A 1|A 3(D 1 A 2D 2,D 3)Hom_{A_1| A_3}(D_1 \otimes_{A_2} D_2, D_3)

    • boundary field insertions: for marked point on the boundary xXx \in \partial X attach a simple object UObj(C)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 1U,D 2)Hom_A(D_1 \otimes U, D_2)

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

      Hom A 1|A 2(UA 1V,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 XX consider the cobordism from the empty surface

M XX^ \emptyset \stackrel{M_X}{\to} \hat X

given by

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

and set

Cor(X)=tft C(M X)1tft C(H^) Cor(X) = tft_C(M_X) 1 \in tft_C(\hat H)

Example

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

then

  • X^\hat X is the sphere

  • M XM_X is the 3-ball

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

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

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

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

  • 2-dimensional theory \to 10-dimensional theory

  • more generally: XX is an orbifold

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

    • physicists think of X WX_W as the spacetime equipped with an involution, but really spacetime is the 2\mathbb{Z}_2-quotient XX
  • I) σ 2\sigma \in \mathbb{Z}_2 acts trivially : type I string

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

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

let

i:FX w i : F \hookrightarrow X_w

be the fixed point

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

    where

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

    where * uu is the Bott generator of K-theory * ν\nu is normal bundle of FX WF \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 [1/2]\mathbb{Z}[1/2] from 10d10 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 hh is any cohomology theory

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

here the coefficient object

h Q:=h(pt;) 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

h v Ω(M,h ) closed h (M) H(M;h ) \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

0h(M;h /) q1curvΩ(M;h ) q0 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
0formsh v(M)h q(M)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)=π 0(Map(M,h q)) h^q(M) = \pi_0(Map(M, h_q))

    and

    h vq(M)=π 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)=Map(M,𝕋) H^{v 1}(M) = Map(M, \mathbb{T})
H 1(M)=π 0Map(M,𝕋) H^{1}(M) = \pi_0 Map(M, \mathbb{T})
H v2(M)={linebundleswithconnection} H^{v 2}(M) = \{line bundles with connection\}
H v3(M)={linebundlegerbeswithconnection} 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)KR(X_w)

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

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

    • a 2\mathbb{Z}_2-graded complex

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

  • σ˜:σ *E¯E\tilde \sigma : \sigma^* \bar E \to E

recall that σ\sigma may have fixedpoints

special cases

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

  • X wXX_w \to X has a section: K 0(X)K^0(X)

twisting

pass to a locally equivalent groupoid

Y WY Y_W \to Y
Y:(Y 0Y 1) Y : (Y_0 \stackrel{\leftarrow}{\leftarrow} Y_1 \stackrel{\stackrel{\leftarrow}{\leftarrow}}{\leftarrow} \cdots)

notation: V ϕ=V^\phi = VV if ϕ=0\phi = 0 and V¯\bar V otherwise

definition

a twsiting of KR(X W)KR(X_W) is an equivalent thing Y wYY_w \to Y as above

where

d:Y 0 d : Y_0 \to \mathbb{Z}

continuous

LY 1 L \to Y_1

hermitian line bundle, 2\mathbb{Z}_2-graded

θ:L g ϕ(f)L fL gf \theta: L_g^{\phi(f)} \otimes L_f \to L_{g f}

cocycle condition for θ\theta

recognize these twistings as classified by some cohomology theory

cohomology group

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

for some hh that we are not being told about

For KO(X W)KO(X_W): π {0,1,2}k 0<0..2>{, 2, 2}\pi_{\{0,1,2\}} k_{0 \lt 0..2\gt} \simeq \{\mathbb{Z}, \mathbb{Z}_2, \mathbb{Z}_2\}

for KR(X W)KR(X_W) the iso classes are

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

as a set

uK 2(pt)KR τ 1+2(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) π:X WX\pi : X_W \to X orientifold double cover

  • iii) β v\beta^v a differential twisting of KR(X w)KR(X_w): the BB-field

  • iv) K : R(β)τ KO(TX2)R(\beta) \to \tau^{KO}(T X -2) iso of twistings of KO(X)KO(X): twisted Spin-structure

Bott shift, leading to equivalent theories β vβ v+(τ v+2)\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)=tw 2+aw w_2(X) = t w^2 + a w

aim: mix iii) and iv)

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

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

and spme spinor fields on Σ^\hat \Sigma

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

\to effective action Pfaffian of Dirac operator

PfaffD Σ^(ϕ *TX2)exp2πi Σ/Sϕ *β 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 SS

where?

work over some parameter space SS

both factors above are sections of a line bundle over SS

theorem (in preparation) there is a hopefully canonical trivialization of L ΨL BSL_\Psi \otimes L_B \to S

action being section of bundle instead of function: annomaly:

sources

  • 1) integrals over fermions L ΨL_\Psi: spin structure need not be equivariant under 2\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 BL_B: exotic orientation

Kevin Costello; part I

deformation quantization

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

  • this is the classical observable algebra

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

  • such that

    • 1) A q/A q=A clA^q/\hbar A^q = A^cl

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

      {a,b}=1[a˜,b˜]mod \{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 C^\infty-analog (i.e. differential geometric analog) of a chiral algebra in the sense of Beilinson and Drinfeld

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

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

there are obvious projection maps

B(M)qB n(M)pB(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 FF on B(M)B(M)

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

      • satisfying some evident compatibility condition

      • such that everything is invariant under the obvious S nS_n-action on B n(M)B_n(M) that exchanges the order of the balls

  • concretely:

    • FF assigns a vector space to every ball BMB \subset M

    • if we have some configuration of balls, like 2 balls B 1,B 2B_1, B_2 inside a big one B 3B_3 we get a map F(B 1)F(B 2)F(B 3)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)B(M)

  • nn-ary operations are B n(M)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 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 FF is a locally constant sheaf on B(M)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 FF be a locally constant factorization algebra in n\mathbb{R}^n, then since B( n)B(\mathbb{R}^n) is contractible FF is quasi-isomorphic to a trivial sheaf with fiber VV 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 nV__\mathbb{R}^n is an E nE_n-algebra

next specialization: holomorphic factorization algebras

Let Σ\Sigma be a Riemannian surface.

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

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

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

actually B(M)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 UB(M)U \to B(M) be holomorphic: > answer: oops, right

??

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

let V=FV = F (any round disk)

if we have a configuratoin of disks with B 1B_1 and B 2B_2 in B 3B_3 with radii ϵ i\epsilon_i with one disk in the center of the big one and the other at complex parameter zz, the map

m z:VVV m_z : V \otimes V \to V

must vary holomorphically with zz

so zm zz \mapsto m_z is a holomorphic map AnnulusHom(VV,V)Annulus \to Hom(V \otimes V, V), so it has a Laurent expansion

m z kZ ka k m_z \sim \sum_{k \in \mathbb{Z}} Z^k a_k

with a ka_k in some completion of Hom(VV,V)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\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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Revised on September 20, 2011 00:31:06 by Urs Schreiber (82.93.78.115)