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Self-dual strings and M5-brane anomalies

Last night, David Berman gave a very nice talk about his recent work with Jeff Harvey,

Incidentally, most of the people in the room were British - they apparently want to take over Cambridge once again.

Imagine several parallel M5-branes and M2-branes stretched between them. The boundaries of these M2-branes inside the M5-branes look like strings; they are self-dual strings because they are the sources of the self-dual three-form field strength in the six-dimensional (2,0) multiplet.

Moreover, because we separated the M5-branes, we study the theory on the Coulomb branch. That's the setup that we were considering with Ori Ganor, and that was later but independently looked at, using the more powerful tools based on anomalies, by Ken Intriligator.

If you think about the geometry of the configuration above, you will realize that the direction in which the M5-branes are separated is singled out, and the remaining 10 dimensions preserve the symmetry

spin(1,1) x spin(4) x spin(4)

The Lorentzian group lives inside the string. The first spin(4) group rotates the remaining 4 dimensions inside the 5-brane, while the final spin(4) belongs to the R-symmetry of the fivebrane - it rotates 4 of its 5 transverse dimensions which are also transverse to the string, of course.

Note that the first spin(4) looks much like the other spin(4), and they can be correlated. In fact, the fermionic zero modes (the Goldstinos from 8 broken supercharges) living on the self-dual string can either be left-moving (charge "+1/2" under spin(1,1)), or right-moving (charge "-1/2") - and these two types of fermions transform differently under the groups spin(4) x spin(4). The total representation is

(+1/2; 1,2; 1,2) (+) (+1/2; 2,1; 2;1) (+)
(+) (-1/2; 1,2; 2,1) (+) (-1/2; 2,1; 1,2)

The first line are the left-movers, the second line are right-movers. Because of the different representations of the left-moving and the right-moving fermions, they can give you an anomaly for the spin(4) R-symmetries of your two-dimensional theory. It's partly a matter of convention whether you call it a R-symmetry anomaly or a gravitational anomaly - one can move the anomalies around a little bit.

The correlation of the two spin(4) groups was also known to me and Ori when we wrote down the topological (skyrmionic, as Kenneth later called it even though the details are different from the usual skyrmions) terms giving you the string solution. It's sort of entertaining how such a membrane, connecting two parallel fivebranes, can be described purely in terms of geometry of the fivebranes, and let me say a couple of sentences about it.

The fivebranes have codimension five. There are five dimensions (x6,x7,x8,x9,x10) transverse to their worldvolume. The asymptotic separation is a vector V from R^5 - imagine that it is in the x10 direction. The separation near the center of the self-dual string is a different vector V(x2,x3,x4,x5) from R^5 - a vector that depends on the four coordinates inside the M5-brane, which are however transverse to the string; the string is stretched in (x0,x1).

OK, now normalize the vector V(x2,x3,x4,x5) so that it becomes a point on S^4, a unit sphere in the space (x6,x7,x8,x9,x10). It's obvious that you can now identify this S^4 with the S^4 arising as the compactification of the R^4 generated by (x2,x3,x4,x5) - if you add a point at infinity. Therefore, you can find a topologically nontrivial solution localized in (x2,x3,x4,x5). We argued, using several independent arguments, that the "density of the S^4 solid angle" is really sourcing the self-dual B-field inside the fivebrane - and therefore the nontrivial wrapping of one S^4 onto another S^4 really carries the same charges as the self-dual string.

Ken Intriligator used a better argument based on anomalies to derive the existence of related couplings on the fivebrane worldvolume. There have been a few anomaly-based papers afterwards that were not quite correct. But Berman and Harvey, hopefully, fixed all these errors, and they re-analyzed which anomalies can be canceled by which terms.

Their main new question was how the anomalies of the self-dual string scale with Q2 and Q5, which are the numbers of membranes and fivebranes, respectively. The final result for the "number of the degrees of freedom" happens to be "Q2" times "Q5 squared". The non-linearity in Q5 is certainly interesting, and it is definitely related to the usual "Q5 cubed" scaling of the degrees of freedom of the fivebrane, even though a gravitational counting of the "Q2" times "Q5 squared" does not exist yet - but it may be analogous to the five-brane counting, although the exact SUGRA solution describing the self-dual string is unlikely to be found analytically.

More generally, the scaling "Q5 cubed" is something truly interesting, and many paradigms to explain this scaling have been proposed. Note that this replaces the number "N squared" or "Q3 squared" for the number of fields in a U(Q3) gauge theory - the usual counting of the dimension of adjoint representations defined on D-branes. This "Q5 cubed" therefore indicates, in a sense, that the "fundamental fields" defining the theory on Q5 M5-branes carry three indices.

It is one of the example how M-theory in 11 dimensions replaces the number two by the number three. The dimension of membrane worldvolume is 3 instead of 2, which is the dimension of worldsheets. All important formulae in M-theory depend on l_{planck}^3, as opposed to l_{string}^2 in string theory. This number 3 is also found in the lattices of E_k. Moreover, the group E_8 - an important group in 11 dimensions - has cubic (3) invariants, much like the U(N) groups relevant for D-branes only have quadratic (2) invariants.

This is powerful stuff, but it's still numerology. Is there some specific, constructive way to see why the number of degrees of freedom on the fivebranes scales like "Q5 cubed"? Well, there are BPS pants configurations of membranes stretched between three fivebranes, aren't there? By the way, anyone knows where can one read about them? Nima Arkani-Hamed likes to explain the cubic scaling by some "1/Q5" fractional strings in the (2,0) theory, and by a choice of cutoffs in the deconstruction of that theory.

It would be, of course, much more satisfactory to have a Lagrangian-like formulation of that six-dimensional theory that would illuminate all these questions.

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reader Urs said...

Hi Luboš!

Thanks for letting us know about Berman&Harvey's talk.

I have written a followup on your comment at the String Coffee Table, see Since I am not sure if your blog supports trackback pings I thought I'd drop a note here.