The n-Category Café

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

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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).

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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.

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!