## March 4, 2010

### 2-Branes and Supergravity Theories

#### Posted by John Baez

A while back I mentioned a very old review article by Duff. If you look at his brane scan you’ll see he lists superstring theories in 3, 4, 6, and 10 dimensions. He also lists 2-brane theories in 4, 5, 7 and 11 dimensions, which give the superstring theories upon dimensional reduction. Now John Huerta and I are wondering: are all of these 2-brane theories associated to theories of supergravity? The 2-brane theory in 11 dimensions is rather famously associated to 11d supergravity. But what about the other cases?

In particular, John Huerta has been perusing Supergravity and Superstrings: A Geometric Perspective by Leonardo Castellani, Riccardo D’Auria and Pietro Fré, and this book seems to say there’s no 5d supergravity theory of the sort one might hope for. Something about spinors being complex. What’s up with that?

But I’m also dying to know the stories in dimensions 4 and 7. Is the 4d theory of 2-branes associated to one of the famous 4d supergravity theories? And what about dimension 7?

Posted at March 4, 2010 6:13 PM UTC

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### Re: 2-Branes and Supergravity Theories

[ this is a message from Hisham Sati that I am forwarding with kind permission ]

– begin forwarded message–

In general, brane solutions in lower dimensions can usually be obtained from higher dimensions by dimensional reduction and/or dualization. One can view this as:

1. either creating a new supergravity theory by dimensional reduction of the parent supergravity (typically 11d sugra) and then finding a solution to the resulting theory;

2. or, by dimensional reduction of a solution to the original theory, i.e. a brane, a black hole etc.

Not all solutions in lower dimensions can be ‘lifted’ (aka “oxidized”) to solutions in higher dimensions. Such solutions are called “stainless”. But this is not the case here.

Now 2-branes in 4, 5, 7 dimensions are obtained by vertical dimensional reduction from the 2-brane in 11 dimensions, viewed as ‘fundamental object’. The case of 5 and 7 dimensions is straightforward, while the case of 4 dimensions requires some care due to appearance of some divergences, because this is a result of reducing a ($D=5$, $d=D-3=2$) solution to a ($D-1=4$, $d=D-3=2$) solution, producing a domain wall. Such classes of reductions are special, which can be seen when one writes down the explicit solution. In the end, however, it is essentially the same process and one can say that all 2-branes above are the result of dimensional reduction of the 2-brane from eleven dimensions and are solutions to supergravity theories.

As for supersymmetry, there is a distinction between Minkowski and curved space and between $N=1$ supersymmetry and $N \gt 1$ supersymmetry. Indeed in 5 and 7 dimensional Minkowski space there are no W, M, pM, MW, pMW spinors. However, this is evaded when one passes to “extended” supersymmetry (in fact the number $N$ of supersymmetries should be even to define a needed symplectic form on the the space of supercharges). In dim 5 the spinors are spM (symplectic pseudo Majorana) and in dim 7 they are sM (symplectic Majorana). So such supergravity theories in 5 and 7 dimensions do exist.

To see how a membrane might arise: for instance the supermultiplet in 7 dimensions contains a 2-form potential $B_2$. This can be dualized to a 3-form potential $B_3$, which is pairs with a membrane worldvolume (think: $H_3=dB_2$ and $H_4=dB_3$ are Hodge dual in 7 dimensions).

– end forwarded message–

Posted by: Urs Schreiber on March 4, 2010 8:32 PM | Permalink | Reply to this

### Re: 2-Branes and Supergravity Theories

Thanks a lot! I’m a bit confused though… I’m just learning this stuff:

You say that the 2-branes in 4, 5, and 7 dimensions are all solutions to supergravity theories. But you say there are no $N = 1$ supergravity theories in 5 and 7 dimensions. Am I to conclude that the 2-branes in 5 and 7 dimensions are solutions of $N \gt 1$ supergravity theories?

The only reason this seems funny to me is that John Huerta and I have studied some of the math surrounding 2-branes in 4, 5, 7 and 11 dimensions and it seems to be very similar in each case. So, I would naively imagine that just as there’s a theory of $N = 1$ 2-branes in 11 dimensions, there’s also one in 4, 5 and 7 dimensions — and that in each case these 2-branes are associated to an $N = 1$ supergravity theory.

Clearly that naive guess is wrong. So I’m wondering at precisely what step it’s wrong.

Posted by: John Baez on March 4, 2010 9:14 PM | Permalink | Reply to this

### Re: 2-Branes and Supergravity Theories

So, I would naively imagine that just as there’s a theory of $N=1$ 2-branes in 11 dimensions, there’s also one in 4, 5 and 7 dimensions — and that in each case these 2-branes are associated to an $N=1$ supergravity theory.

[…] Clearly that naive guess is wrong. So I’m wondering at precisely what step it’s wrong.

But what are your steps? Have you told us?

It is clear that if the theory in $d \lt 11$ dimensions arises from dimensional reduction of $N=1$ $d=11$-SUGRA, then it will have more supersymmetry: the spinors in lower dimensions have less components and so what used be a single spinor for $Spin(d+(11-d))$ will appear as a tuple of spinors for $Spin(d)$ in $d$ dimensions. That’s why the famous $d=4$ supergravity theory has $N=8$, because 8 spinors in $d=4$ are the components of one spinor in $d = 11$.

[John Baez: Sorry, now that I’m back home I can only reply this way.

Urs wrote:

But what are your steps? Have you told us?

I described ‘steps’ that were not steps of logical reasoning, but steps where I read some stuff and made a bunch of wild guesses. I can be a bit more explicit:

Step 1: There are $N = 1$ super-2-brane models in Minkowski spacetime of dimensions 4, 5, 7 and 11. This is discussed by Duff in that old paper I mentioned. Hisham seems to confirm that this ‘step’ is correct:

The 2-branes in the question are obtained with the constraint on the worldvolume to be supersymmetric. This requires that the number of bosonic degrees of freedom $N_B=(D−3)$ equal to the number of fermionic degrees of freedom $N_F$ on the worldvolume. This gives $N=1$ spacetime supersymmetry in spacetime dimensions $D=4,5,7$, and $11$.

Step 2: It seems that sometimes supermembrane theories are associated (in some way) to supergravity theories. So, by Step 1, maybe there should be supergravity theories in spacetime dimensions $D = 4,5,7,11$. And maybe these supergravity theories should have $N = 1$ supersymmetry, since the supermembrane theories did.

Maybe step 2 is wrong. Hiram seems to be saying it’s wrong:

So it seems to me, from the constraint, that it is a trade-off: either require worldvolume supersymmetry but not be able to embed in supergravity, or lose worldvolume supersymmetry and be able to embed in supergravity.

I don’t understand this very well, but it sounds like the super-2-brane theories I’m talking about (except for $D = 11$ and maybe $D = 4$) are ones where you ‘require worldvolume supersymmetry but not be able to embed in supergravity’.

I’d love to know if I’m understanding Hiram’s remarks correctly, and if I am, I’d also love a few elementary hints as to why things work this way.]

Posted by: Urs Schreiber on March 4, 2010 10:05 PM | Permalink | Reply to this

### Re: 2-Branes and Supergravity Theories

[ forwarded message from Hisham Sati ]

– begin forwarded message–

The 2-branes in the question are obtained with the constraint on the worldvolume to be supersymmetric. This requires that the number of bosonic degrees of freedom $N_B=(D-3)$ equal to the number of fermionic degrees of freedom $N_F$ on the worldvolume. This gives $N=1$ spacetime supersymmetry in spacetime dimensions $D= 4, 5, 7$, and $11$.

I realized I was doing something else. I was looking at membranes from the spacetime point of view. So the ones I am talking about, while can be viewed as solutions of supergravity (see e.g. page 33 of Lectures on supergravity $p$-branes ), would not be worldvolume supersymmetric (if one checks the formula $N_F=N_B$).

So it seems to me, from the constraint, that it is a trade-off: either require worldvolume supersymmetry but not be able to embed in supergravity, or lose worldvolume supersymmetry and be able to embed in supergravity.

Examples of supermultiplets in various dimensions can be found in Table 3 of

– end forwarded message–

Posted by: Urs Schreiber on March 4, 2010 10:48 PM | Permalink | Reply to this

### Re: 2-Branes and Supergravity Theories

I do not have a full answer, but here are some comments: There are a number of ways to understand your question about the relation between a brane theory and the corresponding sugra:

The strongest version would be that at low energies, the brane theory induces the sugra equations of motion. This is the case for example in the various 10d string theories: There, one has at least three possibilities to derive the sugra equations: First, one can consider the brane theory (the sigma model) in a general curved background and ask for those backgrounds in which there is no conformal anomaly on the world-sheet. At 1-loop level, the resulting equations are the sugra equations of motion, at higher loop level corrected by higher curvature terms. Second, one can compute string scattering amplitudes and check that (now at space-time tree level) these are equivalent to the e.o.m.’s. Third, one can work out which multiplet the zero modes (fancy word for center of mass coordinate plus superpartners) are in. One finds the supergravity multiplet and then in 10d the action is pretty much unique due to the high amount of supersymmetry.

That was the stringy situation. Already for membranes in 11d, it’s not so easy: The world-sheet theory is not conformal, thus it does not make sense to worry about the conformal anomaly. Second, there is no reliable way to compute scattering amplitudes. However, Dasgupta, Nicolai and Plefka worked out vertex operators for the supermembrane which are consistent with what one would assume from a sugra perspective. Last, the zero modes live in the sugra multiplet in 11d and the theory is unique thus, mission accomplished.

None of these methods is directly available in lower dimensions. But at least the last one should be able to tell you the field content of the sugra you are looking for. The membrane theory in a flat target is pretty much always the same: It’s just the range of the indices and the (space-time) type (Majorana and or Weyl) that differs.

Let me point out another thing: For good BPS membranes, you want the sugra algebra to have central charges that are 2-forms. I don’t know when that is the case. Furthermore, I would expect there to be a 3-form gauge potential which would couple to the membrane world volume. But maybe a 1-form can do as well with a Chern-Simons-type coupling. Note that the 3-form can also look like a d-5 form upon Hodge dualization (NB the field strengths should be dual).

Lastly, one might expect that there are membrane like solutions to the sugra equations of motion as mentioned above. Again, those would need some form field to be BPS and preserve some susy.

I don’t know if that can be related to membranes, but there is a 5d sugra theory that for many purposes serves as a toy version of 11d sugra (e.g. regarding dualities and hidden symmetries), see:

• E. Cremmer, Supergravities in five dimensions, in: S. Hawking, M. Rocek (Eds.), Superspace and Supergravity. Proceedings of the Nuffield Gravity Workshop, Cambridge, Jun 22 - Jul 12, Cambridge Univ. Press, Cambridge, 1980.

That one has a 1 form A and like the 11d theory has a AFF coupling.

Posted by: Robert on March 5, 2010 4:23 PM | Permalink | Reply to this

### Re: 2-Branes and Supergravity Theories

I emailed Jacques Distler asking about the 5 dimensional supergravity theory. He wrote:

Spinors in D=5 are either Dirac or (equivalently) symplectic-Majorana.

So the minimal supersymmetry has 8 real supercharges.

I gather Cremmer calls that “N=2” (which is what it would be called in 4d), but I think that nomenclature is perverse.

So, the “N = 2” is not exactly counting the number of spinor irreps, which is what I had thought it was for. Then I looked more closely at Castellani, D’Auria and Fre. They only use one spinor irrep to build the theory. They decompose their spinor into two components which are related by what they call a “pseudo-Majorana” condition. Undoubtedly, this is the same as the symplectic-Majorana condition Jacques refers to. I think the “N = 2” nomenclature may somehow be related to these two components.

I would call this theory “N = 1”, and (confusingly!), so do the authors, just a few pages after they claimed no such theory existed!

Posted by: John Huerta on March 6, 2010 2:57 AM | Permalink | Reply to this

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