February 5, 2006

Seminar on 2-Vector Bundles and Elliptic Cohomology, IV

Posted by urs Transcript of part 3 of our first session.

3) $K\left(V\right)$ and Elliptic Cohomology

Warning. This third part is extremely sketchy, due to three reasons. 1) It is the technically most demaning part, but 2) was dealt with only near the end of our seesion when 3) I was not paying due attention to some details. This part is planned to be dealt with in more detail in future sessions. So if the following is about as mysterious to you as it is to me, just read it like you would read a horoscope. ;-)

Question. What is elliptic cohomology, anyway??

1) Homotopy Theorist’s answer

Theorem [Hopkins, Miller, Goerss, and others]

There is a spectrum (a generalized cohomology theory) called TMF = “topological modular forms” that is universal with respect to all elliptic cohomology theories.

So if $E$ are homology theories that send elliptic curves to formal group laws, then

(1)$\mathrm{TMF}=\mathrm{lim}E\phantom{\rule{thinmathspace}{0ex}}.$

(Or something along these lines. As I have said, this is sketchy. More than sketchy, actually.)

Here is another vague statement of this sort.

“TMF gives a topological version of the Witten genus.”

Whatever TMF is, BDR, have proven the following:

Theorem. [BDR]

$K\left(V\right)$ is not quite the same as TMF, but it “sees as much” as the stable homotopy groups of spheres.

“Elliptic cohomology should have something to do with CFTs”.

This is where 2-vector bundles enter the game. We can regard parallel transport in an ordinary vector bundle with connection as a form of a 1-dimensional field theory, one which assigns “Hilbert spaces of states” (fibers of the bundle) to points and whose “propagator” is the parallel transport.

Hence, in as far as K-theory is about classes of vector bundles, it is about classes of 1-dimensional field theories. (This is explained in much more detail in section 3 of What is an elliptic object?).

So one might expect that there is something like parallel surface transport in a 2-vector bundle such that this is similarly related to elliptic cohomology.

Reminder.

For ordinary vector bundles we have

(2)$\left[X,K\left(U\left(n\right)\right)\right]=\mathrm{Gr}\left(\mathrm{Vect}\left(X\right)\right)\phantom{\rule{thinmathspace}{0ex}},$

which says that the homotopy classes of maps from a space $X$ to the K-theory space of unitary vector bundles form the Grothedieck group completion of classes of vector bundles over $X$.

Question. What do maps $X\to K\left(V\right)$ see?

Theorem. [BDR] For “nice” $X$ (for instance CW complexes) the homotopy classes of maps $\left[X\to K\left(V\right)\right]$ are given by the colimit over cyclic Serre fibrations $Y\to X$ over the Grothedieck group completion of classes of 2-vector bundles over $Y$.

But what is a 2-vector bundle?

That’s easy to describe once some general 2-bundle language is established. A BDR 2-vector bundle is simply a 2-bundle with structure “2-group” ${\mathrm{GL}}_{n}\left(B\right)$. Only that this is not a 2-group, but just a 2-monoid.

So a 2-vector bundle in the sense of BDR is defined in terms of local trivializations as a 2-bundle whose transition functions take values in objects of ${\mathrm{GL}}_{n}\left(ℬ\right)$ and whose transition modifications on triple overlaps take values in morphisms of that category. Note that ${\mathrm{GL}}_{\left(}ℬ\right)$ is not a 2-group, but just a 2-monoid. This means that transitions in BDR 2-vector bundles have a direction. If we can make a transition in a BDR 2-bundle from ${U}_{i}$ to ${U}_{j}$ we cannot in general make a transition from ${U}_{j}$ to ${U}_{i}$.

Our first session endet with the following statement

Hope. [BDR] 2-vector bundles with 2-connection might realize parallel surface transport that realizes elliptic objects.

Posted at February 5, 2006 12:47 PM UTC

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Re: Elliptic Cohomology IV

It would be great if you could tell us more about TMF! I came across it recently while browsing the M-theory literature, that is, in

Type II string theory and modularity I. Kriz and H. Sati hep-th/0501060

which goes into F-theory. Is there a better short guide to TMF than

Algebraic Topology and Modular Forms M. J. Hopkins ICM 2002 Vol III 1-3

?

Posted by: Kea on February 5, 2006 8:19 PM | Permalink | Reply to this

Re: Elliptic Cohomology IV

Is there a better short guide to TMF […]

Unfortunately, I am not really the right person to answer this question.

All I can say at the moment is that for somebody without any true background in this area TWF 197 works wonders.

Depending on how much background you have, you might find the lecture notes by Matthew Ando a helpful source of keywords and list of results.

Posted by: urs on February 6, 2006 2:16 PM | Permalink | Reply to this

Re: Seminar on 2-Vector Bundles and Elliptic Cohomology, IV

Hi Urs,

You might be amused to know that Jacob Lurie has given a geometric characterization of the topologists’ spectrum TMF. The basic idea (so far as I remember his lecture) is as follows: There’s a bijection between one-dimensional formal group laws over the complex numbers and periodic oriented cohomology theories. The additive group corresponds to the usual integral cohomology (with a generator inverted in degree 2). The multiplicative line corresponds to complex K-theory. And to every complex torus (thought of as an abelian group), there is an elliptic cohomology theory. TMF, introduced by Hopkins and Miller, is the universal such theory.

Lurie’s contribution (again, as I understand it) is to observe that we can reinterpret this result as saying that there exists a very funny geometric space, which is like the moduli space of elliptic curves in that its points are elliptic curves, but which differs greatly in that its structure sheaf assigns to each open set a commutative ring spectrum and does so in such a way that the “local ring spectrum” at a point is exactly the elliptic cohomology spectrum associated to the elliptic curve represented by the point. Once you construct this space – and I’m not sure anything has been published yet – you recover the TMF spectrum by taking the global sections of the structure sheaf.

Posted by: A.J. on February 6, 2006 4:15 AM | Permalink | Reply to this

Re: Seminar on 2-Vector Bundles and Elliptic Cohomology, IV

Lurie’s contribution […]

Thanks for this hint.

Is this discussed anywhere in his survey of elliptic cohomology? (I haven’t read that yet.)

Posted by: urs on February 6, 2006 2:37 PM | Permalink | Reply to this

Re: Seminar on 2-Vector Bundles and Elliptic Cohomology, IV

Oh cool, I didn’t know Jacob had written anything up yet! Yes, the ideas are explained in that survey, although you might want to look at the first two sections of his thesis to get some idea what this “derived algebraic geometry” stuff is about.

Posted by: A.J. on February 6, 2006 7:17 PM | Permalink | Reply to this

Re: Seminar on 2-Vector Bundles and Elliptic Cohomology, IV

I did notice that the time stamp on that text is Jan 30, 2006 and wondered if that was just the date of the latest modification. :-)

I had a look at the first few pages of this text. To me it is intriguing that in the later sections he is talking about 2-equivariance. (I have some ideas about 2-equivariance of 2-bundles.) But I haven’t yet made it far enough to understand what he means by that. Do you?

Posted by: urs on February 6, 2006 8:22 PM | Permalink | Reply to this

Re: Seminar on 2-Vector Bundles and Elliptic Cohomology, IV

Urs, I haven’t gotten there yet. Soon though. What exactly do you mean by 2-bundle? I’m imagining that this is an object classified by H^2(X,GL_n).

Posted by: A.J. on February 9, 2006 6:14 AM | Permalink | Reply to this

Re: Seminar on 2-Vector Bundles and Elliptic Cohomology, IV

What exactly do you mean by 2-bundle?

A 2-bundle is a bundle internal to $\mathrm{Cat}$.

For the case of principal bundles this term was introduced by Toby Bartels, but the basic idea was considered before, in particular in form of the 2-vector (2-)bundles of BDR which I mentioned. Toby is writing a thesis which should contain much more details on principal 2-bundles.

If one restricts attention to a certain special case of 2-bundles, these have been shown to have the same classification as (nonabelian) bundle gerbes. Apart from that not much is known.

I like to regard a 2-bundle with 2-connection as a transport 2-functor from surfaces to a transport 2-groupoid. (The image of all points under this 2-functor is the bundle in $\mathrm{Cat}$.)

For instance, a bundle gerbe with connection can neatly be described as a “pre-trivialization” of a 2-functor with values in $\mathrm{Vect}$. A similar description holds for nonabelian bundle gerbes.

This really is a special case of KV-2-vector bundles (with 2-connection). I am in the process of preparing some notes on this.

There is a more or less straightforward categorification of the notion of equivariance of a bundle to that of (2-)equivariance of a 2-bundle. This is unpublished work which I hesitate to reveal, but some comments along these lines can be found here and here.

Posted by: urs on February 9, 2006 1:01 PM | Permalink | Reply to this

Re: Seminar on 2-Vector Bundles and Elliptic Cohomology, IV

I have a question on TMF and the Witten genus. Originally Witten was interested in
finding a genus for “the Dirac operators on the free loop space of M”, and assumed that this would be found in circle equivariant K-theory of the
free loopspace LM on M.

Now it seems that TMF cohomology of the manifold itself is the favoured candidate for where the index of this operator should live.

Here is the question:
Is there any relation between TMF(M) and K_{S^1}(LM) ?

Posted by: Marcel Bokstedt on February 6, 2006 2:18 PM | Permalink | Reply to this

Re: Seminar on 2-Vector Bundles and Elliptic Cohomology, IV

Is there any relation between $\mathrm{TMF}\left(M\right)$ and ${K}_{{S}^{1}}\left(\mathrm{LM}\right)$?

Posted by: urs on February 6, 2006 4:12 PM | Permalink | Reply to this

Re: Seminar on 2-Vector Bundles and Elliptic Cohomology, IV

Is there any relation between $\mathrm{TMF}\left(M\right)$ and ${K}_{{S}^{1}}\left(\mathrm{LM}\right)$?

I have now been told that this question has been addressed by I. Grojnowski and Ioanid Rosu. Possibly there is more known about this than has been written up, I was told, but one should have a look at this paper:

Posted by: urs on February 7, 2006 5:24 PM | Permalink | Reply to this

Re: Seminar on 2-Vector Bundles and Elliptic Cohomology, IV

Thanks.. this is kind of fun. The way I understand the Grojnowski stuff is that he describes a rational (or rather complex) elliptic cohomology as the global sections in a sheaf of chain complexes over an elliptic curve. One point (I think) he makes, is that his definition gives uncompleted groups, while the usual topological definition using MU and a genus gives the completion of his version. BTW, the 2-equivariant theories of Lurie should be more fancy versions of these “uncompleted” theories. Then Rosu tells you how to translate in characteristic 0 between this and K-theory. This is excellent if you are only interested in a genus, but if you like torsion information it’s not optimal.

On the other hand, in his survey article Lurie claims (remark 5.10) that elliptic cohomology “heuristically” is the same as circle equivariant K-theory on the loop space, and adds that this is strictly true if you restrict to elliptic cohomology close to infinity, and to the circle fixed points of LM (which is just a sad copy of M). I don’t understand this, but it has a nice sound.

Posted by: Marcel on February 7, 2006 10:44 PM | Permalink | Reply to this

Re: Seminar on 2-Vector Bundles and Elliptic Cohomology, IV

that he describes a rational (or rather complex) elliptic cohomology as the global sections in a sheaf of chain complexes over an elliptic curve

Interesting. I haven’t looked at this yet. What are these chain complexes like? Is there a heuristic way to understand how this is related to the other definition(s)?

elliptic cohomology ‘heuristically’ is the same as circle equivariant K-theory on the loop space

From the 2-vector bundle perspective on elliptic cohomology this sounds very natural. On the other hand, the fact that there are indications that we need to consider parallel transport of closed as well as open strings in these 2-vector bundles raises the question if we might want to consider not K-theory of loop space, but of some sort of path space. Have you ever seen anything like this discussed?

Posted by: urs on February 8, 2006 1:50 PM | Permalink | Reply to this

Re: Seminar on 2-Vector Bundles and Elliptic Cohomology, IV

I belive that it is unclear what the exact relation is between elliptic cohomology in the sense leading to tmf, and the 2-vector bundle approach of BDR.

What seems to be true is that these theories “see the same spaces”, that is, a space has the same generalized homology as a point (ie, it is trivial for the theory) with respect to one of the theories, if and only if it is for the other. This is a very weak form of equivalence of spectra. In algebra, this corresponds to saying that two modules M,N over a ring R are equivalent if and only if for every chain complex C of free R-modules the chaincomplex we get by tensoring C with M (over R) is acyclic if and only if we get an acyclic complex when we tensor N with C. Lots of very different R-modules will become equivalent under this equivalence.

Precisely, BDR consider an equivalence class of p-local spectra (p a prime) which they call “of chromatic type 2”. This goes back to work by Morava, Ravenel, Hopkins and many other good people. Most elliptic cohomology theories have this complexity, but for example periodic (not conneted) K-theory has chromatic type 1.

They conjecture (plausibly) that the algebraic K-theory of the category of (their version of) 2-vector spaces has the same homotopy type as algebraic K-theory of the connected spectrum derived from topological K-theory. This algebraic K-theory has been studied by Ausoni and Rognes, and it is known that it is as close to having cromatic type 2 as one can expect from a connected spectrum.

Even if one can relate some group to 2-vector space cohomology, it is quite unclear to me if and how one can derive from this that it is related to tmf, because the equivalence relation we consider between spectra is so weak. But of course one could hope that eventually one would be able to give more precise information about the relation between say the spectrum defining 2-vector space cohomology and tmf. Maybe relating both to equivariant K-theory of the loop space could be a start.

Posted by: Marcel on February 8, 2006 7:23 PM | Permalink | Reply to this

Re: Seminar on 2-Vector Bundles and Elliptic Cohomology, IV

Ever alert to bring Connes and Grothendieck together as Cartier suggests (p. 402) we should, I notice:

“Heuristically, the elliptic cohomology of a space M can be thought of as the S^1-equivariant K-theory of the loop space LM.”

from Lurie’s survey, and

“In the special case of A = k[G], G a discrete group, HC_*(A) may be identified with the Borel S^1-equivariant homology of the natural S^1 action on the free loop space of BG.”

from p.2 of Fiedorowicz’s The Symmetric Bar Construction. The latter is cyclic homology, something explained by Connes on p. 27 of this. Although I see Lurie pp. 10-11 draws a distinction between Borel equivariance and the one he’s after, he also says how they’re related.

Posted by: David Corfield on February 8, 2006 8:49 PM | Permalink | Reply to this
Read the post Philosophy of Real Mathematics
Weblog: The String Coffee Table
Excerpt: David Corfield on Jacob Lurie's survey of elliptic cohomology.
Tracked: February 7, 2006 5:10 PM
Read the post Seminar on 2-Vector Bundles and Elliptic Cohomology, I
Weblog: The String Coffee Table
Excerpt: Review of the 2-vector approach towards elliptic cohomology. Part I.
Tracked: August 7, 2006 1:10 PM

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