### 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?

## Re: 2-Branes and Supergravity Theories

[

this is a message from]Hisham Satithat 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:

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;

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).

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