Skip to the Main Content

Note:These pages make extensive use of the latest XHTML and CSS Standards. They ought to look great in any standards-compliant modern browser. Unfortunately, they will probably look horrible in older browsers, like Netscape 4.x and IE 4.x. Moreover, many posts use MathML, which is, currently only supported in Mozilla. My best suggestion (and you will thank me when surfing an ever-increasing number of sites on the web which have been crafted to use the new standards) is to upgrade to the latest version of your browser. If that's not possible, consider moving to the Standards-compliant and open-source Mozilla browser.

November 28, 2011

Higher Structures 2011

Posted by Urs Schreiber

This week takes place this year’s workshop on

Higher structures in mathematics and physics

in Göttingen.

Myself, I speak about

\infty-Chern-Simons functionals (pdf slides).

I’ll report on some talks in the comment section below.

Posted at November 28, 2011 9:17 AM UTC

TrackBack URL for this Entry:   http://golem.ph.utexas.edu/cgi-bin/MT-3.0/dxy-tb.fcgi/2463

31 Comments & 1 Trackback

Getzler, “Higher analytic stacks”

Ezra Getzler speaks about joint work with Kai Behrend.

Here a brief and incomplete note on what he said.


Let AA be a dg-algebra. Form the simplicial set

N A:=MC(C (Δ[]),A) N_\bullet A := MC(C^\bullet(\Delta[\bullet]), A)

of Maurer-Cartan elements α\alpha

δα+Dα+α 2=0 \delta \alpha + D \alpha + \alpha^2 = 0

on the AA-valued cochains on a simplicial set, where δ\delta is the simplicial differential and DD is the “internal” differential on AA.

Claim: This simplicial set is a quasi-category.

If AA is an ordinary algebra, then this is the nerve of the corresponding multiplicative monoid.

(ed.: Hence it makes sense to write BA:=N(A)B A := N(A).)

Consider now the maximal Kan complex inside this, we may think of it as the delooping of the general linear group with coefficient in AA

BGL(A):=core(N(A))N(A). B GL(A) := core(N(A)) \subset N(A) \,.

Theorem: This also works for differential graded Banach algebras.

Application: for EXE \to X a homomorphic vector bundle, its algebra of forms

A:= iΩ 0,i(X,End(E)) A := \oplus_i \Omega^{0,i}(X, End(E))

is a dg-Banach algebra. Hence we have an application to Kuranishi deformation theory.

Posted by: Urs Schreiber on November 28, 2011 9:58 AM | Permalink | Reply to this

Re: Getzler, “Higher analytic stacks”

bullar = bullet?

Posted by: jim stasheff on November 28, 2011 1:05 PM | Permalink | Reply to this

Re: Getzler, “Higher analytic stacks”

bullar = bullet?

Thanks, I have fixed it now.

Posted by: Urs Schreiber on November 28, 2011 1:14 PM | Permalink | Reply to this

Valette, “The minimal model for the BV-operad”

Bruno Valette speaks on joint work with Gabriel Drummond-Cole on Minimal models for the BV-operad.

Here are brief and incomplete notes from the talk.


Theorem (Baranikov, Kontsevich 98, Manin 99)

Consider a dg-BV-algebra

(A,d,,Δ), (A, d, \cdot, \Delta) \,,

where (A,d)(A,d) is a chain complex, “\cdot” a binary commutative product , Δ\Delta an order 2 operator of degree +1 (or -1 depending on convetion), which satisfies the “dΔd \Delta-condition”

kerdkerΔ(ImdImΔ)=Im(dΔ) ker d \cap ker \Delta \cap (Im d \oplus Im \Delta) = Im(d \Delta)

(Δ=0\Rightarrow \Delta = 0 on H (A,d)H_\bullet(A, d)).

Then there exists on the underlying homology groups H (A,d)H_\bullet(A,d) a natural Frobenius manifold structure.

Let ¯ 0,n+1\bar \mathcal{M}_{0,n+1} be the moduli space of surfaces with nn-inputs

H (¯ 0,n+1)H_\bullet ( \bar \mathcal{M}_{0,n+1})-algebra

dim¯=2(n+2) dim \bar \mathcal{M} = 2(n+2)

Theorem (Kadishvili). Homotopy transfer theorem

Let

(A,d)h(A,d)ip(H,d H) (A,d) \stackrel{h}{\to} (A,d) \stackrel{\overset{p}{\to}}{\underset{i}{\leftarrow}} (H, d_H)

be a homotopy retract i,pi,p with homotopy exhibited by hh.

ipid A=hd A±d Ah. i p - id_A = h d_A \pm d_A h \,.

Now let μ:A×AA\mu : A \times A \to A be an associative product on AA. This Induces a product pμii p \circ \mu \circ i \otimes i on HH, which is in general not associative.

But the given hh can be used to equip it with the structure of an A-infinity algebra.

By means of this one can give a construction that rectifies any A A_\infty-algebra to a dg-algebra.

Task: do something similar for C-infinity algebra in order to have rectification for the BV-operad.

idea: use Koszul duality to pass from the BV-operad to a dg-cooperad.

Theorem (Valette): There is a homotopy transfer for infinity-BV-algebras.

Moreover, there are minimal models for the dg-BV-operad. There is an explicit construction using the homotopy transfer theorem for C C_\infty-algebras. They are unique up to isomorphism.

One finds: where in the Baranikov-Kontsevich theorem above one finds Frobenius manifolds, here one finds, of course, homotopy Frobenius manifolds.

Posted by: Urs Schreiber on November 28, 2011 11:08 AM | Permalink | Reply to this

Re: Valette, “The minimal model for the BV-operad”

Dear Urs, thank you for kindly providing the notes of this interesting talk. Bruno is always a good, very clear, speaker!

To the organizers. This newest workshop was not much visible online and many people did not know that it would be held. The webpage was quite hidden and not enough announced on the web. When I heard of it, relatively recently, google could not find the announcement.

Posted by: Zoran Skoda on November 28, 2011 12:31 PM | Permalink | Reply to this

Re: Valette, “The minimal model for the BV-operad”

Two quibbles:

By means of this one can give a construction that
rectifies any A ∞-algebra to a dg-algebra.

`This’ seems to refer to oo-itization, which hardly rectifies

also A underlies H(A) not vice versa

Posted by: jim stasheff on November 28, 2011 1:03 PM | Permalink | Reply to this

Re: Valette, “The minimal model for the BV-operad”

‘This’ seems to refer to oo-itization

It is supposed to refer to the homotopy transfer theorem.

Posted by: Urs Schreiber on November 28, 2011 1:11 PM | Permalink | Reply to this

Abad, “The A-infinity de Rham theorem and Reidemeister torsion”

Camilo Arias Abad speaks about joint work with Florian Schätz.

Here a quick and incomplete note.


In the first half of the talk he reviews the A-infinity refinement of the de Rham theorem, which over a smooth manifold XX provides an equivalence of A A_\infty-algebras

Ω (X)C (X,) \Omega^\bullet(X) \stackrel{\simeq}{\to} C^\bullet(X, \mathbb{R})

from the de Rham algebra of differential forms under the wedge product and the singular cochains, which becomes an A A_\infty-algebra under the cup product.

In his context, Camilo thinks of ∞-representations of ∞-groupoids XX on (∞,1)-vector spaces, hence ∞-functors

ρ:X(,1)Vect \rho : X \to (\infty,1)Vect

alternatively as A A_\infty-homomorphisms from the linearization of XX (the formal dual to the cochain A A_\infty-algebra on the underlying simplicial set) to the the A A_\infty-category of chain complexes

ρ˜:XdgVect. \tilde \rho: \mathbb{R}X \to dgVect \,.

He says “representation up to homotopy” for this.

(ed.: Maybe “strong homotopy representations” would be more in line with historical convention?)

(Talk almost but not quite over, but my laptop battery is dying right now.)

Posted by: Urs Schreiber on November 28, 2011 1:40 PM | Permalink | Reply to this

evera, “On quantization of Lie quasi-bialgebras”

Pavol Ševera talks about joint work with Stefan Sakáloš.

Here are the notes I took in the first 15 or so minutes, stating the problem. The rest of the talk is detailed algebra giving the solution, which I don’t feel like reproducing here.


The following problem of quantization of Lie bialgebras was already discussed/partly solved by Enriques-Halbent ‘08 and Etingof-Kazhdan 96.

Given a quasi-bialgabra HH consider the monoidal category HModH Mod of its modules.

The tensor product on HModH Mod comes from the coproduct

Δ:HHH \Delta : H \to H \otimes H

on HH and the associator in HModH Mod corresponds to an element

Φ HH 3 \Phi_H \in H^{\otimes^3}

Let now

H=𝒰𝔤[[]] H = \mathcal{U}\mathfrak{g}[ [\hbar ] ]

be a universal enveloping algebra as a k[[]]k[ [\hbar] ]-module

Let the skew-symmetric part of Δ\Delta be

δ:𝔤 2𝔤 \delta : \mathfrak{g} \to \wedge^2 \mathfrak{g}

The associator element corresponds now to some

ϕ 3𝔤. \phi \in \wedge^3 \mathfrak{g} \,.

So we have a Lie algebra 𝔤\mathfrak{g}, and some additional structure δ\delta and ϕ\phi on

Call this a Lie quasi-bialgebra.

Then

d=𝔤𝔤 *, d = \mathfrak{g}\oplus \mathfrak{g}^* ,

is a Lie algebra and the canonical pairing

, \langle -,-\rangle

is an invariant polynomial on it.

Task: given this Lie algebraic data 𝔤d\mathfrak{g} \subset d, like this, (re)construct HH.

Posted by: Urs Schreiber on November 28, 2011 3:09 PM | Permalink | Reply to this

Re: Higher Structures 2011

Thomas Willwacher recalls how Maurer-Cartan elements serve to twist all kinds of higher algebras and asks: can we generally say which algebras over which operads have twistings by MC elements?

I didn’t take notes. Notice that MC elements are secretly homomorphisms, see here.

Posted by: Urs Schreiber on November 28, 2011 4:16 PM | Permalink | Reply to this

Re: Higher Structures 2011

On Tuesday, I speak about

\infty-Chern-Simons Functionals (pdf slides).

I am still fiddling with the slides a bit. All comments are welcome.

Posted by: Urs Schreiber on November 28, 2011 9:08 PM | Permalink | Reply to this

Re: Higher Structures 2011

Slide 37: “Bereziniuan”

Posted by: David Roberts on November 29, 2011 1:15 AM | Permalink | Reply to this

Re: Higher Structures 2011

Thanks! I have fixed it now.

Posted by: Urs Schreiber on November 29, 2011 5:57 AM | Permalink | Reply to this

Rogers, “Higher geometric quantization”

Chris Rogers spoke about higher geometric quantization of 2-plectic geometry.

He provides pdf slides accompanying the talk.

I am fond of the idea that Chris presents, this looks like it is going in the right direction. Eventually one should look into how to lift not only the categorical degree of the prequantum line bundle but also that of the base space, to have a notion of geometric quantization of symplectic infinity-groupoids.

Posted by: Urs Schreiber on November 29, 2011 12:12 PM | Permalink | Reply to this

Waldorf, “Lifting problems and Non-Abelian Gerbes”

Konrad Waldorf speaks about joint work with Thomas Nikolaus.

The first part of the talk is about nonabelian obstruction theory in terms of lifting bundle gerbes, and lifting bundle 2-gerbes, making use of aspects of their article

Four equivalent versions of non-abelian gerbes (arXiv:1103.4815)

on the various equivalent incarnations of 2-groupoids of principal 2-bundles and bundle gerbes.

The second part is about transgression of principal 2-bundles to ordinary principal bundles over the loop space of their base space.

Posted by: Urs Schreiber on November 29, 2011 2:05 PM | Permalink | Reply to this

Re: Higher Structures 2011

Are there any surprising new directions being taken by anyone, or does it feel like the talks fit nicely inside your view of things?

Posted by: David Corfield on November 30, 2011 2:47 PM | Permalink | Reply to this

Re: Higher Structures 2011

Are there any surprising new directions being taken by anyone, or does it feel like the talks fit nicely inside your view of things?

Maybe the choice of correct framework for many of the “higher structures”-questions has stabilized and effort is concentrating more on crayoning the map?

Personally I find the questions that Chris Rogers addressed in his talk are pushing in the direction of an unexplored gold mine of fundamental interest.

I have been chatting about (higher) geometric quantization with him. Thanks to this, I believe I understand now the intrinsic theory of Hamiltonian vector fields and their prequantum operators over general nn-plectic cohesive \infty-groupoids (section 4.8). Only about the right intrinsic notion of polarization I am not sure yet.

The perspective here is this: with the higher Chern-Simons functionals in cohesive \infty-toposes (section 4.6) we have a fairly large class of extended classical TFTs. By the cobordism theorem, for quantizing these to an extended TQFT it should be sufficient to do so in top codimension, hence to geometrically quantize the nn-plectic nn-stack phase spaces that they define.

It seems the only ingredient that I am missing for understanding this generally is the intrinsic formulation of the polarization and understanding how the then resulting quantum state space is fully dualizable in some suitable ambient nn-category.

The 2-plectic geometric quantization that Chris explored in his talk is at the very least a helpful example to play around with for exploring the possibilities here.

Posted by: Urs Schreiber on November 30, 2011 7:29 PM | Permalink | Reply to this

Re: Higher Structures 2011

Knowing nothing about higer quantization, I am curious why you think there should be some notion of a polarization.

Is it by analogy with the symplectic case?

Or is there a physical reason?

Posted by: Eugene Lerman on September 20, 2012 2:35 PM | Permalink | Reply to this

Re: Higher Structures 2011

Knowing nothing about higer quantization, I am curious why you think there should be some notion of a polarization.

This comes from formal reasons as well as from examples:

the formal reasons is this: given a prequantization of an nn-plectic smooth \infty-stack

:XB nU(1) conn \nabla : X \to \mathbf{B}^n U(1)_{conn}

and a prequantum nn-state ψ\psi of it

X ψ V n//B n1U(1) conn B nU(1) conn \array{ X &&\stackrel{\psi}{\to}&& V_n//\mathbf{B}^{n-1}U(1)_{conn} \\ & {}_{\mathllap{\nabla}}\searrow && \swarrow \\ && \mathbf{B}^n U(1)_{conn} }

there for each compact oriented smooth manifold Σ\Sigma of dimension kk of a transgression map that takes this to a prequantum (nk)(n-k)-state on the “configuration space” [Σ,X][\Sigma,X]

[Σ,X [Σ,ψ] [Σ,V n//B n1U(1)] hol Σ k V nk//B nk1U(1) conn [Σ,] [Σ,B nU(1) conn] exp(2πi Σ()) B nkU(1) conn. \array{ [\Sigma,X &&\stackrel{[\Sigma,\psi]}{\to}&& [\Sigma,V_n//\mathbf{B}^{n-1}U(1)] &\stackrel{hol_{\Sigma_k}}{\to}& V_{n-k}//\mathbf{B}^{n-k-1}U(1)_{conn} \\ & {}_{[\Sigma,\mathllap{\nabla}]}\searrow && \swarrow && \swarrow \\ && [\Sigma,\mathbf{B}^n U(1)_{conn}] &\stackrel{\exp(2 \pi i \int_\Sigma(-))}{\to}& \mathbf{B}^{n-k} U(1)_{conn} } \,.

In particular for k=n1k = n-1 the prequantum nn-states transgress to ordinary prequantum states. And these we know need to be polarized. So consistency demands that we impose some condition also on prequantum nn-states that makes them into quantum nn-states such as to ensure that their transgression are quantum states.

Examples that show that some conditions is needed arise already for n=2n = 2. If we consider the extended geometric quantization of the open bosonic or type II string charged under the B-field and consider just the gauge coupling sector of the system, them the prequantum 2-bundle is the B-field itself on spacetime XX. For V//BU(1)V//\mathbf{B}U(1) the canonical nn-dimensional representation of the 2-group/group stack BU(1)\mathbf{B}U(1) on the moduli stack BU(n)\mathbf{B}U(n), a prequantum state now is a rank nn twisted unitary bundle with twist the B-field. These do not exist globally if the B-field has a non-torsion class. But they do exist on “Bohr-Sommerfeld 2-orbits” that impose a polarization condition: namely on D-branes QXQ \hookrightarrow X that satisfy the Freed-Witten-Kapustin anomaly cancellation-condition: in partiular they are isotropic submanifold of the 2-plectic structure.

Now one finds that the above transgression of prequantum 2-states to prequantum states on the string’s configuration space [S 1,X][S^1, X] is the Freed-Witten Kapustin anomaly cancellation mechanism: it takes the B-field to its holonomy line bundle on loop space, it takes a twisted bundle with connection to its twisted holonomy line bundle on loop space, where their classes cancel and give the genuine anomaly-free action functional of the string.

There is a similar kind of story known for the membrane charged under the supergravity C-field regarded as a prequantization of a 3-plectic structure. Again the “Bohr-Sommerfeld 3-ordbits” at least need to be isotropic before we can take 3-sections to produce quantum 3-states.

So there are lots of hins and indications for what has to happen. But I am still lacking a fully general abstract notion of higher polarization in higher prequantum physics.

By the way, Chris Rogers tells me he will soon have out more details on the n=2n = 2-case for XX a group manifold (I think).

Posted by: Urs Schreiber on September 20, 2012 5:42 PM | Permalink | Reply to this

Re: Higher Structures 2011

Thanks. I am looking forward to Chris Rogers’s paper.

Posted by: Eugene Lerman on September 20, 2012 8:00 PM | Permalink | Reply to this

Henriques, “Three-tier CFTs”

André Henriques speaks about a spin-off idea of his joint work with Chris Douglas and Arthur Bartels.

The idea of the talk is to start with 3d Chern-Simons theory regarded as an extended TQFT

Z CS:Bord 33Vect Z_{CS} : Bord_3 \to 3 Vect

and then “compactify” it on the interval II by making sense of the expression

( IZ CS):ΣZ CS(Σ×I). (\int_I Z_CS) : \Sigma \mapsto Z_{CS}(\Sigma \times I) \,.

If one imagines that one can choose “conformal boundary conditions” on the boundaries (Σ×I)\partial (\Sigma \times I), then something like this should give the 2d CFT called the WZW model, regarded as an “extended CFT”.

One formalization of this instance of the holographic principle is what underlies the FRS formalism. André is pushing here towards a more systematic understanding of this process from the point of view of extended QFT, reminiscent (to me) of what I once talked about at 2-functorial CFT.

Unfortunately the talk ended too early.

Posted by: Urs Schreiber on November 30, 2011 4:15 PM | Permalink | Reply to this

Runkel, “Centres of algebras and two-dimensional field theory”

Ingo Runkel talks about his joint work with Alexei Davydov and Liang Kong.

Since this, too, appeared in our book, I won’t produce notes here but just point you there.

Posted by: Urs Schreiber on November 30, 2011 4:24 PM | Permalink | Reply to this

Meyer, “Desingularisation of non-Hausdorff groupoids”

R. Meyer indicated in his talk a construction of a resolution of any topological groupoid that may fail to be degreewise Hausdorff by a topological 2-hypergroupoid that is.

I would think that there is a construction that does this very generally – for all \infty-groupoids modeled on any site that has small coproducts and fiber products (such as Hausdorff spaces).

This follows from a slight variant of Dugger’s resolution in the projective model structure on simplicial presheaves. The argument is written out as prop. 2.1.49 here.

Posted by: Urs Schreiber on November 30, 2011 6:53 PM | Permalink | Reply to this

Rehren, “Tensor categories and 2-categories in local quantum field theory”

Karl-Henning Rehren reviews the construction and properties of the DHR category associated with a local net of observables.

Posted by: Urs Schreiber on December 1, 2011 9:29 AM | Permalink | Reply to this

Prat-Waldron, “Differential K-Theory via Euclidean Field Theories”

Arturo Prat-Waldron reviews first differential cohomology and differential K-theory, and then reviews the relation between Supersymmetric field theories and generalized cohomology.

(ed.: It was long expected that if one takes the relation between (1,1)-dimensional Euclidean field theories and K-theory and instead of dividing out concordance of QFTs over some space XX, just divides out equivalences, one gets differential K-theory. The claim of Arturo and Alexander Kahle is maybe that they can now show this.)

Posted by: Urs Schreiber on December 1, 2011 10:28 AM | Permalink | Reply to this

Han, “Equivariant Cohomology via Gauged Field Theories”

Fei Han reviews the setup of Supersymmetric field theories and generalized cohomology and then passes for the latter to equivariant cohomology. The statement is that for the former this means to pass to gauged field theory of sorts.

Posted by: Urs Schreiber on December 1, 2011 11:07 AM | Permalink | Reply to this

Teichner, “Field Theories as Generalized Cocycles”

Peter Teichner reviews ideas of Supersymmetric field theories and generalized cohomology.

Posted by: Urs Schreiber on December 1, 2011 2:30 PM | Permalink | Reply to this

Bunke, “Differential algebraic K-theory”

Ulrich Bunke talks about joint work with David Gepner.

The claim here is: a deep theorem by Beilinson about the “Borel regulator” for the cyclotomic field which may look a bit mysterious, apparently has a natural interpretation in the differential cohomology refinement of the corresponding algebraic K-theory.


Let RR be any number ring. The running example to keep in mind is the the cyclotomic field

R=/(1+ξ++ξ p1). R = \mathbb{Z}/ (1 + \xi + \cdots + \xi^{p-1}) \,.

Write then

BGL(R) B GL(R)

for the classifying space for RR-module bundles and

BGL(R)Quillenconstr(BGL(R)) + B GL(R) \stackrel{Quillen constr}{\hookrightarrow} (B GL(R))^+

for the Quillen construction, such that for

KRK R the algebraic K-theory spectrum, the one whose homotopy groups are the algebraic K-theory groups K i(R)K_i(R)

K i(R)π i(BGL(R)) * K_i(R) \simeq \pi_i (B GL(R))^*

we have

Ω KRBGL(R) +×K 0(R). \Omega^\infty K R \simeq B GL(R)^+ \times K_0(R) \,.

Now for any choice σ:R\sigma : R \hookrightarrow \mathbb{C}, which for the running example: would be given by ξμU(1)\xi \mapsto \mu \in U(1) a ppth root of unity, and a bundle V σBGL(R)V \otimes_\sigma \mathbb{C} \to B GL(R), an observation by S. Götte gives a corresponding bundle 𝒱\mathcal{V} over the +-construction fitting into

V σ 𝒱 BGL(R) (BGL(R)) + \array{ V \otimes_\sigma \mathbb{C} &\to& \mathcal{V} \\ \downarrow && \downarrow \\ B GL(R) &\hookrightarrow& (B GL(R))^+ }

and such that 𝒱\mathcal{V} is locally filtered.

The Borel regulator construction starts with a K-class

xK 2n1(R) x \in K_{2n-1}(R)

classified by a map

x:S 2n1BGL(R) + x : S^{2n-1} \to B GL(R)^+

and considers on the pullback bundle

x *𝒱 S 2n1 \array{ x^* \mathcal{V} \\ \downarrow \\ S^{2n-1} }

a choice of connection \nabla that preserves the locally filtered structure.

Choose also a hermitean structure and let *\nabla^* be the corresponding adjoint connection

Then the Borel regulator of xx is the expression

r σ(x)= S 2n1ch 2n1( *,), r_\sigma(x) = \int_{S^{2n-1}} ch_{2n-1}(\nabla^* , \nabla) \in \mathbb{R} \,,

where

ch 2n1( *,)Ω 2n1(S 2n1) ch_{2n-1}(\nabla^*, \nabla) \in \Omega^{2n-1}(S^{2n-1})

is the relative Chern-Simons form between the two connections.

This defines a map

r:K 2n1(R) r: K_{2n-1}(R) \to \mathbb{R}

and the big question is: what is the image of this map? This is complicated, and the answer in general is not known.

But a theorem of Beilinson says that for RR the cyclotomic field as above, there exist qq \in \mathbb{Q} and xK 2n1(R)x \in K_{2n-1}(R) such that

r σ(x)=qRe[1(2πi) nLi 2n1(μ)] r_\sigma(x) = q Re\left[ \frac{1}{(2 \pi i)^n} Li_{2 n -1}(\mu) \right]

where

Li k(z):= n1 z nn k Li_k(z) := \sum_{n \geq 1}^\infty \frac{z^n}{n^k}

is a “higher logarithm”.

The proof that Beilinson gives is complicated. The claim in the following is: this is also a consequence of a statement that naturally ought to be true in index theory for differental algebraic K-theory.

Index theorem is about comparison of two pushforwards, one analytic, one homotopy theoretic.

Fix a bundle π:WB\pi : W \to B with compact fibers and being a proper submersion (no need for any kind of orientation in the following).

In the running example we look at lens spaces

S 2n1/ pL p 2n+1P n. S^{2n-1}/{\mathbb{Z}_p} \simeq L_p^{2n+1} \to \mathbb{C}P^n \,.

Consider local systems of RR-modules on WW and BB. The claim is then that there is a diagram as follows, to be explained now:

Loc(W) cycle K^R 0(W) analytictransfer higheranalytictorsion BecherGottliebtransfer Loc(B) cycle K^R 0(B) \array{ Loc(W) &\stackrel{cycle}{\to}& \hat K R^0(W) \\ {}^{\mathllap{analytic transfer}}\downarrow &\neArrow_{\simeq}^{higher analytic torsion}& \downarrow^{\mathrlap{Becher-Gottlieb transfer}} \\ Loc(B) &\stackrel{cycle}{\to}& \hat K R^0(B) }

Here K^R\hat K R is the differential K-theory spectrum

Loc(B)Loc(B) is isomorphism classes of geometrically constant sheaves of finitely generated projective RR-modules on BB,

where the “geometric” is choices of hermitean metrics on the complexification of the sheaves (which are sheaves of sections of complexes of flat vector bundles)

In the example we have

π 1(L p 2n+1)= p \pi_1(L_p^{2n+1}) = \mathbb{Z}_p

take VV te rank 1, free, with holonomy ξ\xi

Now definition of K^R\hat K R following Hopkins-Singer.


[to be polished from here on, no time now]

We have the spectrum KRK R and can form smash product with the Moore spectrum, the result being equivalent to an Eilenberg-MacLane spectrum

KRKRMHA. K R \to K R \wedge M \mathbb{R} \stackrel{\simeq}{\to} H A \,.

where A =K(R)A_\bullet = K\bullet(R) \otimes \mathbb{R}

give a de Rham construction for this

H(Ω(B,A))deRhamtheoremH(A) B H(\Omega(B,A)) \stackrel{de Rham theorem}{\to} H(A)^B

form the homotopy pullback

K^R 0(B) H(σ 0Ω(B,A)) I KR B H(Ω(B,A)) \array{ \hat K R^0(B) &\to& H(\sigma^{\geq 0} \Omega(B,A)) \\ {}^{\mathllap{I}}\downarrow && \downarrow \\ K R^B &\to& H(\Omega(B,A)) }

KR 1rΩ 1(B,A)/imDaK^R 0(B) K R^{-1} \stackrel{r}{\to} \Omega^{-1}(B,A)/im D \stackrel{a}{\to} \hat K R^0(B)

the kernel of aa is the Borel regulator from before

task: give geometric models for elements in this hopullback

Theorem: there exists a cycle map

cycl:LocK^R cycl : Loc \to \hat K R

g Vg^V is the geometry

I(cycle(V,g V))=[V] I(cycle(V, g^V)) = [V]

inKR 0(B)[MVBGL(R)Ω KR] in K R^0(B) [M \stackrel{V}{\to} B GL(R) \to \Omega^\infty K R]

R(cycle(V,g V))=containsch( *,) R(cycle(V, g^V)) = contains ch(\nabla^* , \nabla)

now

B^G \hat B G

transfer choose Riem structure on Π\Pi

pushforward induced from the following morphism of diagrams

KR W H(Ω(W,A)) H(Σ 0Ω(B,A)) canonical W/Re ()... KR B H(Ω(W,A)) H(Σ 0Ω(B,A)) \array{ K R^W &\to& H(\Omega(W,A)) &\leftarrow& H(\Sigma^{\geq 0} \Omega(B,A)) \\ \downarrow &\neArrow_{canonical}& \downarrow^{\int_{W/R} \wedge e^{\nabla}} && \downarrow^{\int (-) \wedge ...} \\ K R^B &\to& H(\Omega(W,A)) &\leftarrow& H(\Sigma^{\geq 0} \Omega(B,A)) }

analytic transfer

(V,g V)(R iπ *V,g L R V) (V, g^V) \mapsto (R^i \pi_* V, g_{L^R}^V)

Conjecture:

cycle(transfer analytic(V,g V))tr^(cycle(V,G V))=a(τ) cycle(transfer^{analytic}(V, g^V)) - \hat tr(cycle(V, G^V)) = a(\tau)

τ \tau

is higher analytic torsion form.

Posted by: Urs Schreiber on December 2, 2011 10:25 AM | Permalink | Reply to this

Huerta, “A categorified supergroup for string theory”

John Huerta spoke about his thesis work.

This has been mentioned here many times before, but let me suggest to look at it from the following perspective.

The Green-Schwarz action for sigma-models with supermanifold target spaces is strictly of (coset) higher WZW-model type. See here.

What John calles the superstring 2-group is the total space of the corresponding WZW-gerbe / WZW-2-bundle over the super-translation group. Maybe it would eventually be better to call it the GS-superstring 2-group to distinguish it from the role of the string 2-group for the heterotic string, which is analogously the WZW gerbe on SpinSpin and the joint generalization, which will be a WZW gerbe on all of the super-Poincaré-Lie algebra.

By the way, one may wonder why the M5-brane does not appear in the brane scan (the NS5-brane does). I guess it’s clear: the necessary 7-cocycle that is missing on the (10,1)(10,1)d super Poincaré-Lie algebra does reappear on its “𝔪2𝔟𝔯𝔞𝔫𝔢 gs\mathfrak{m}2\mathfrak{brane}_{gs}” super Lie 3-algebra extension only. It is the 7-cocycle that defines the supergravity Lie 6-algebra.

Posted by: Urs Schreiber on December 5, 2011 9:49 AM | Permalink | Reply to this

Cattaneo, “Classical and quantum Lagrangian field theories with boundaries”

Alberto Cattaneo spoke about joint work with Pavel Mnev and Nicolai Reshitikhin.

They are aiming to get a kind of extended quantization from the BV-formalism by taking the boundary term in the variation of the action into account.

Parts of the discussion reminded me a bit of what I was talking about with Igor Khavkine here.

Pdf slides for the talk are available, but I am not sure if I may link to them.

Posted by: Urs Schreiber on December 5, 2011 1:54 PM | Permalink | Reply to this

Re: Cattaneo, “Classical and quantum Lagrangian field theories with boundaries”

I wrote:

Pdf slides for the talk are available, but I am not sure if I may link to them.

No need to worry about that anymore, the article corresponding to Cattaneo’s talk has now appeared on the arXiv:

  • A. Cattaneo, P. Mnev, N. Reshetikhin, Classical BV theories on manifolds with boundaries (arXiv:1201.0290)
Posted by: Urs Schreiber on January 5, 2012 11:20 AM | Permalink | Reply to this
Read the post Mathematical Aspects of String and M-Theory in Oxford
Weblog: The n-Category Café
Excerpt: A workshop in Oxford on mathematical aspects of string theory.
Tracked: January 11, 2012 7:29 PM

Post a New Comment