## April 28, 2008

### Dual Formulation of String Theory and Fivebrane Structures

#### Posted by Urs Schreiber

We would like to share the following:

Hisham Sati, U.S. and Jim Stasheff
Dual Formulation of String Theory and Fivebrane Structures
(pdf)
Update: now available as arXiv:0805.0564

Abstract. We study the cohomological physics of fivebranes in type II and heterotic string theory. We give an interpretation of the one-loop term in type IIA, which involves the first and second Pontrjagin classes of spacetime, in terms of obstructions to having bundles with certain structure groups. Using a generalization of the Green-Schwarz anomaly cancelation in heterotic string theory which demands the target space to have a String structure, we observe that the “magnetic dual” version of the anomaly cancellation condition can be read as a higher analog of String structure, which we call Fivebrane structure. This involves lifts of orthogonal and unitary structures through higher connected covers which are not just 3- but even 7-connected. We discuss the topological obstructions to the existence of Fivebrane structures. The dual version of the anomaly cancelation points to a relation of String and Fivebrane structures under electric-magnetic duality.

This expands on some of the material announced in section 3 of

H. S., U.S., J. S.
$L_\infty$-connections and application to String- and Chern-Simons transport
(arXiv, blog pdf)

but so far concentrates on the topological aspects of Fivebrane structures. A discussion of the differential geometry of Fivebrane 6-bundles with connection – which are nonabelian differential cocycles that are to super 5-branes as String 2-bundles with connection are to superstrings and as ordinary Spin bundles with connection are to spinning particles – as well as of the Chern-Simons 7-bundles with connection obstructing their existence, will be given elsewhere, following the general approach described in

On nonabelian differential cohomology
(pdf).

I’d be grateful for comments, but should add that I’ll be travelling in Ireland until 3rd of May, which will reduce my responsiveness here for that period.

Posted at April 28, 2008 3:12 AM UTC

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### Re: Dual Formulation of String Theory and Fivebrane Structures

Is this fivebrane really exactly like a string, but with 5 dimensions? What I mean is, with branes I heard that we get duals, but only for the classical version of the string or as soliton solutions formed by the endings of open branes.

So, is this really a higher dimensional dual with, with quantization, renomalizable, etc?

Posted by: Daniel de França MTd2 on April 28, 2008 1:28 PM | Permalink | Reply to this

### Re: Dual Formulation of String Theory and Fivebrane Structures

Thanks for the question. I just have half a second, no time to really reply right now – but you can essentially find the answer here:

More later.

Posted by: Urs Schreiber on April 30, 2008 3:08 PM | Permalink | Reply to this

### Re: Dual Formulation of String Theory and Fivebrane Structures

For completeness I should add the following, in case it wasn’t clear:

a truly satisfactory $\sigma$-model description of the super 5-brane is to date lacking, due to, as far as I can see, the usual problems with quantizing the Green-Schwarz-like formulation of susy $\sigma$-models and absence of another formulation here.

But even without this full quantization the respective anomalies can be computed. See also

Dixon, Duff, Plefka, Putting the string/fivebrane duality to the test.

On p. 3 and 4 they review the worldsheet description of the relevant terms for the string and then look at the corresponding situation for the 5-brane on p. 6.

Posted by: Urs Schreiber on May 3, 2008 1:17 PM | Permalink | Reply to this

### Re: Dual Formulation of String Theory and Fivebrane Structures

5 am in a hotel in Dublin. While she is in the bathroom, I have another 1.5 seconds to reply to the above question:

yes, the (super-)5-brane is a gadget pretty much like the (super-)string, just 5+1 instead of 1+1 dimensional.

In 10 dimensions, the critical number of dimensions for the string, the 5-brane is the magnetic dual to the ordinary (“electric”) string and there are various reasons to expect that heterotic string theory, which arises as the effective target space theory (the second quantization) of the heterotic superstring has a dual formulation with the superstring replaced by the super 5-brane.

On the target space one such reason is the fact that there is a well known dual formulation of the target space theory: the ordinary target space theory of the heterotic string is a higher gauge theory involving a 2-bundle/1-gerbe: the (twisted) Kalb-Ramond field, which is the “electric” background field for the string.

The dual formulation is a higher gauge theory involving a 6-bundle/5-gerbe: the magnetic dual of the Kalb-Ramond field, which is the “magnetic” background field for the fivebrane.

This dual formulation of the heterotic background theory is originally due to Chamseddine, as far as I am aware.

To this dual target space theory one expects a corresponding $\sigma$-model for a super-5-brane from which the former arises as the effective background theory.

This super 5-brane worldsheet theory is the one discussed in the article I mentioned. As discussed there, it is in principle clear how and that this works, even though one technical subtlety concerning the interpretation of a prefactor of 1/2 remains.

But the upshot is that from the target space theory alone one can deduce, as I discussed here that heterotic string theory, in order to cancel the anomalies induced by the fermions it contains (gravitino, dilatino and gaugino) needs to contain a specific electric string and magnetic fivebrane current.

The electric string current, measured by a 4-form, had famously also been derived from the point of view of the worldsheet theory of the String. Originally by Killingback (review here). From this point of view it arises as a higher generalization of the condition that the target space for the super particle (the spinning particle) has to have Spin structure: the target of the heterotic superstring even has to have what is called a “String structure”.

Similarly one expects that the $\sigma$-model for the super 5-brane is consistent only if a certain structure on the target trivializes which carries an 8-form curvature. Such a thing is a 7-bundle or 6-gerbe with connection, or degree 8 differential cocycle. And indeed, as the article I linked to demonstrates, that’s what one finds.

In our article we discuss this “Fivebrane” structure as a generalization of the formerly known String structure.

Posted by: Urs Schreiber on May 3, 2008 5:14 AM | Permalink | Reply to this

### Re: Dual Formulation of String Theory and Fivebrane Structures

If we disregard the gauge bundle for a moment (regarding the case that it is trivial, for instance), we have:

$X$ is Spin if $X$ is orientable and $w_2(X)$ vanishes.

$X$ is String if X is $Spin$ and $\frac{1}{2}p_1(X)$ vanishes.

$X$ is Fivebrane if X is $String$ and $\frac{1}{6}p_2(X)$ vanishes.

(These fractional Pontrjagin classes are reviewed/explained in section 4.4.)

If the gauge bundle is nontrivial the situation becomes a little more intricate. In that case the structure that obstructs the existence of the Fivebrane structure has a curvature 8-form which is a linear combination of the second Pontrjagin characteristic class of the Spin bundle and the fourth Chern class of the gauge bundle and of a couple of characteristic forms which are wedge products of nontrivial characteristic forms.

We discuss various special cases in which these decomposable characteristic forms vanish anyway. But more generally the question is:

what is the integral singular cohomology class of a 7-bundle/6-gerbe (whatever you call it) which has a curvature 8-form being the sum of given characteristic forms.

I think that, using the construction by Brylinski-McLaughlin, one can show that the integral class depends only on the indecomposable characteristic forms appearing in the curvature $n+1$-form. The other contributions suspend to 0 and hence do not contribute in that construction to the underlying topology, but just to the connection put on it.

This will be discussed in more detail elsewhere.

Posted by: Urs Schreiber on May 3, 2008 6:45 AM | Permalink | Reply to this

### Re: Dual Formulation of String Theory and Fivebrane Structures

Just one minor correction, so the youngun’s don’t get misled:

A section will trivialize a principle bundle
but to trivialize a vector bundle you need a nowhere 0 section

Posted by: jim stasheff on May 1, 2008 3:14 AM | Permalink | Reply to this

### Re: Dual Formulation of String Theory and Fivebrane Structures

Ah, so even the old experts misspell ‘principal bundle’!

As Allen Knutson realized, a ‘principle bundle’ is one with moral fiber.

Later, James Dolan started using ‘moral fiber’ to mean ‘homotopy fiber’ — since in homotopy theory, the ‘morally correct’ version of the fiber is the homotopy fiber.

Putting this all together, I think we should use ‘principle bundle’ to denote the homotopy-theoretic analogue of a principal bundle. Its moral fibers will be $A_\infty$ spaces.

Posted by: John Baez on May 2, 2008 11:06 PM | Permalink | Reply to this

### Re: Dual Formulation of String Theory and Fivebrane Structures

LOL ;-)
but don’t
a generation ago we already had principal bundles with A\infty fibres acting in an A\infty fashion on the total space; e.g. the based loop space as fibre

Posted by: jim stasheff on May 4, 2008 1:17 PM | Permalink | Reply to this
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