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February 9, 2012

The Moduli 3-Stack of the C-Field

Posted by Urs Schreiber

We are in the process of finalizing a little article

Domenico Fiorenza, Hisham Sati, U.S., The E 8E_8 moduli 3-stack of the C-field in M-theory

Abstract The higher gauge field in 11-dimensional supergravity – the C-field – is constrained by quantum effects to be a cocycle in some twisted version of ordinary differential cohomology. We argue that it should indeed be a cocycle in a certain twisted nonabelian differential cohomology. We give a simple and natural characterization of the full smooth moduli 3-stack of configurations of the C-field, the field of gravity and the (auxiliary) E8-Yang-Mills field. We show that the truncation of this moduli 3-stack to a bare 1-groupoid of field configurations reproduces the differential integral Wu structures that Hopkins-Singer had shown (HS02) to formalize Witten’s argument (Wi96) on the nature of the C-field. Finally we give a similarly simple and natural characterization of the moduli 2-stack of boundary C-field configurations and show that it is equivalent to the smooth moduli 2-stack of anomaly free heterotic supergravity field configurations (SSS12).

This may be read as a companion to the article that I mentioned last time, at Multiple M5-branes, String 2-connections, and 7d nonabelian Chern-Simons theory

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A pdf of the article is behind the above link. Any comment you might have would be most welcome.

Posted at February 9, 2012 9:11 AM UTC

TrackBack URL for this Entry:   http://golem.ph.utexas.edu/cgi-bin/MT-3.0/dxy-tb.fcgi/2495

10 Comments & 1 Trackback

Re: The Moduli 3-Stack of the C-Field

Some typos from section 1:

“such that the B-field and the RR-fields” such AS

“an integral class divisble by 2.” divisible

“using this machninery”

“Only truncations and reductions … sits” sit

Posted by: David Corfield on February 9, 2012 1:25 PM | Permalink | Reply to this

Re: The Moduli 3-Stack of the C-Field

Thanks! Fixed.

Posted by: Urs Schreiber on February 9, 2012 2:13 PM | Permalink | Reply to this

Re: The Moduli 3-Stack of the C-Field

Since that was a rather boring comment, when you write

This is the correct starting point for any actual quantization of the system (as an effective low-energy gravitational higher gauge theory, as it were, but conceivably of relevance also to the full M-theory),

is this the point at which the hard-nosed string theorist takes note?

Posted by: David Corfield on February 9, 2012 5:08 PM | Permalink | Reply to this

Re: The Moduli 3-Stack of the C-Field

Since that was a rather boring comment,

Much of the process of finalizing an article is boring, unfortunately. But since it has to be done, I am grateful to you for bothering to look for typos! We fixed a bunch more now. Many probably still remain for the moment.

is this the point at which the hard-nosed string theorist takes note?

It’s a step in the right direction, I would think. Your question made me add a sentence to that “essay”.

More could be said here. Let me tell you what I am currently intrigued by:

there have been, roughly three major attempts for formalizing, mathematically, what higher supergravity is all about. I am thinking of

1) exceptional generalized geometry,

2) twisted generalized differential cohomology,

3) D’Auria-Fré theory.

We know now that all three find their natural home in nonabelian differential cohomology (section 4.3, section 4.4).

We currently understand some overlap between the three, but it seems clear that there is a natural unification of all three in a single natural cohesive structure.

Posted by: Urs Schreiber on February 9, 2012 6:50 PM | Permalink | Reply to this

Re: The Moduli 3-Stack of the C-Field

There are also the Wu characteristic classes mod p > 2.
Do they show up detecting higher torsion in your setting?

Posted by: jim stasheff on February 10, 2012 1:08 PM | Permalink | Reply to this

Re: The Moduli 3-Stack of the C-Field

There are also the Wu characteristic classes mod p>2p \gt 2. Do they show up detecting higher torsion in your setting?

Let’s see:

The appearance of ordinary Wu classes originates here in the quadratic refinement of the intersection pairing on integral/differential cohomology, notably in higher dimensional Chern-Simons theory action functionals of the form

exp(iS(G^))=exp(2πi ΣG^^G^) \exp(i S(\hat G)) = \exp(2 \pi i \int_\Sigma \hat G \hat \cup \hat G)

as they appear for instance in 7d supergravity dual to the M5-brane.

However, the full M-theory Chern-Simons term involves, in fact, the diagonal of a trilinear intersection pairing

exp(iS(G^))=exp(2πi ΣG^^G^^G^) \exp(i S(\hat G)) = \exp(2 \pi i \int_\Sigma \hat G \hat \cup \hat G \hat \cup \hat G)

(or rather “secondary” pairing, since here dimΣ=11dim \Sigma = 11 with degG^=4deg \hat G = 4).

and hence is asking for a cubic refinement. Generally one can consider higher dimensional cup product Chern-Simons functionals

exp(iS(G^))=exp(2πi Σ(G^) ^n) \exp(i S(\hat G)) = \exp(2 \pi i \int_\Sigma (\hat G)^{\hat \cup n})

for all nn \in \mathbb{N}. This will be asking for nn-fold refinements.

Will that involve Wu classes with coefficients in p\mathbb{Z}_p for p>2p \gt 2? I haven’t thought about this question yet. Will need to look up something on these Wu classes with other coefficients…

Posted by: Urs Schreiber on February 10, 2012 2:36 PM | Permalink | Reply to this

Re: The Moduli 3-Stack of the C-Field

Note: The Wu classes $q_n$ for p>2 are the analogs of Stiefel-Whitney classes, NOT the analog of the mod 2 Wu classes you were referring to. See my book with Milnor p. 228

Posted by: jim stasheff on February 11, 2012 1:25 PM | Permalink | Reply to this

Re: The Moduli 3-Stack of the C-Field

Hi Jim,

you write:

The Wu classes q nq_n for p>2p \gt 2 are the analogs of Stiefel-Whitney classes, NOT the analog of the mod 2 Wu classes you were referring to. See my book with Milnor p. 228

The q nq_n are not called Wu classes in your book with Milnor. It’s the v iv_i that are, page 229. And these are the ones of relevance here, too.

But maybe I see now what’s going on: a little earlier in

Milnor, On Characteristic Classes for Spherical Fibre Spaces, Comm. Math. Helv. (1968),

the terminology is still different, there the Stiefel-Whitney classes are called Wu classes. (A pdf-witness of this is page 3 of Byun: On vanishing of characteristic numbers in Poincaré complexes (pdf))

Anyway, as I tried to recall above, what matters for the discussion of quadratic refinement of the intersection pairing are mod2mod \, 2 Wu classes as on page 229 of your book with Milnor.

I understood – and still understand – your question as asking whther I can see a role also for other modpmod \,p-Wu classes in the present discussion.

To which my reply still is: an evident naive guess is that they might be relevant in refinements of higher powered intersection pairings. Unfortunately, I still don’t know, but I’ll try to think about it.

Posted by: Urs Schreiber on February 11, 2012 11:31 PM | Permalink | Reply to this

Re: The Moduli 3-Stack of the C-Field

Thanks for setting the record straight - almost.
In our index, Wu class does list pages for both interpretations.

Posted by: jim stasheff on February 12, 2012 1:45 PM | Permalink | Reply to this

Re: The Moduli 3-Stack of the C-Field

We have added one more subsection,

4.4 Horava-Witten boundaries and higher orientifolds (p. 25).

This discusses how to lift the Horava-Witten boundary condition from 3-forms to differential cocycles and further to the moduli 3-stack of configurations that satisfy the flux-quantization condition.

Posted by: Urs Schreiber on February 11, 2012 1:21 PM | Permalink | Reply to this
Read the post What is homotopy type theory good for?
Weblog: The n-Category Café
Excerpt: An example in mathematical string theory where appeal to homotopy type theory helps solve a subtle problem.
Tracked: May 10, 2012 1:25 AM

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