It's known that gauge group for type-I superstring theory should be SO(32) for chiral anomalies to cancel. This is realized by presense of 16 space-time filling D9-branes, together with 16 space-time filling O9-planes with opposite charge: D9-branes bring U(16) gauge symmetry and due to orientifold planes the symmetry gets enhanced to SO(32). But what about other Dp-branes which can exist in type-I theory: wouldn't they bring anomaly - because open strings can end on those branes and create gauginos from gauge group different from required SO(32) - and will that anomaly cancel eventually (people make physical constrctions with other Dp-branes e.g. while studying BH in superstring theory)? Or in any picture we are always supposed to be able to perform enough T-duality transformations and get our 16 D9-branes and 16 O9-planes, but there will be additional branes. Probably the answer to my question is that all other possible branes in that picture will be part of 16 space-time filling branes, and therefore gauge YM group will be then SO(32) anyway and other gauge groups will be just subgroups of it - hence in the whole system anomalies will cancel, is it true?


2 Answers 2


Dear Mikhail, good interest of yours. Some points that will hopefully store all the answers:

  1. Type I string theory is defined by having its orientifold O9-plane. It only has one such a plane; there is no sense in which there are 16 of them (you only look into one mirror; there are no open strings ending on orientifold planes, so no degeneracy may be introduced, either). Type I may be defined as type IIB with the spacetime-filling O9-plane - which makes all strings unorientable. And the 16 spacetime-filling D9-branes and their images have to be added to cancel the anomalies - either in spacetime, or in worldsheet.

  2. In spacetime, the relevant anomaly for 9+1 dimensions is given by one-loop "hexagon" graph with gauginos and gravitinos and other chiral fermions running in the loop. That wouldn't cancel but in 1984, Green and Schwarz began the first superstring revolution when they showed that there is an extra tree-level graph, with a group of 2 and another group of 4 external lines attached to an internal propagator, and all the anomalies cancel for $SO(32)$ or $E_8 \times E_8$.

  3. The supersymmetric D-branes in type I string theory include D9-branes, D5-branes, and D1-branes. In type IIB string theory where we began, there were also D7-branes, D3-branes, and D(-1)-branes (D-instantons), but those three types were removed from the supersymmetric spectrum by the orientifold projection (or at least became non-supersymmetric and unstable). D1-branes become heterotic strings in the S-dual heterotic $SO(32)$ string theory; D5-branes become the heterotic fivebranes in the same strong-coupling limit.

  4. The D1-branes of type I string theory - the heterotic strings - also have potential world volume anomalies and they do cancel because of the 1-9 strings (whose massless states are chiral on the D1-brane), open strings connecting the D1-branes with the 16 spacetime-filling D9-branes. This cancellation also works but isn't equivalent to the cancellation of the spacetime anomalies in type I - or the world sheet anomalies in type I, for that matter.

  5. Similarly, the D5-branes become the heterotic fivebranes in the limit, so that's enough to see that they carry a $(1,0)$ supersymmetry in six dimensions. Because this asymmetric counting guarantees that the theory is left-right asymmetric, one must also check the anomalies and indeed, they still cancel. But this cancellation is again non-equivalent to the cancellations for D9-branes and D1-branes described above.

  6. Type I string theory may be T-dualized but what we get is not type I. In particular, type I is type IIB with an extra orientifold. So just like in any type IIB, the T-duality with respect to an odd number of circles gives you type IIA, and only if the number of circles is even, you get type IIB back. The T-duality with respect to $N$ circles transforms the O9-plane and D9-branes to O$(9-N)$-plane(s) and D$(9-N)$-branes.

  7. At the beginning, I said that there is "one" spacetime-filling O9-plane in type I. However, if you T-dualize $N$ dimensions as in the previous point, you will see that there are $2^N$ different loci with O$(9-N)$-planes. It's because if you have a periodic coordinate $\phi$ and the orientifold acts by flipping the sign of $\phi$, there are 2 fixed points on the circle of length $2\pi$, namely $\phi=0$ and $\phi=\pi$.

  8. If one T-dualizes D-branes, he has to deal with Wilson lines. Each $U(1)$ wrapped on a circle has a Wilson line around the circle which gets translated to the position of the dual (lower-dimensional) D-brane on the T-dual circle.

  9. The T-dual theories to type I are called type IA (also type I', read "one prime") and perhaps type IAA etc., depending on the number of T-dualities.

  10. The lower-dimensional D-branes arising by T-dualizing the original D9-branes may have different positions. The dilaton sources are only eliminated locally - and the dilaton stays constant across the spacetime - if we uniformly distribute the 16 D-branes and their images to the $2^N$ different loci of the orientifold O$(9-N)$-planes. For type IA theory, we deal with 2 loci of O8-planes and there are 8 (plus 8 images) D8-branes at each locus. So the gauge group is SO(16) at each of them. Similarly, for the doubly T-dualized type I, or IAA, one has 4 places with O7-planes and each of them carries an SO(8) group from the D7-branes. That division corresponds to the Wilson lines of the original type I theory whose $U(1)$ parts are maximally equally divided between trivial Wilson lines and $(-1)$ from the $U(1)$ groups.

  11. There are no anomalies localized on the D$(9-N)$ planes in the T-duals of type I. In particular, the D8-branes and other even-dimensional branes - in an orientifold of type IIA - are non-chiral. In particular, D-branes that are moved away from the orientifold planes become ordinary type II D-branes which are always non-chiral but carry no anomalies. Note that the bulk physics of type I differs from type IIB because it only has one gravitino, among other things; however, type IA, IAA, IAAA, IAAAA etc. are equivalent to type IIA, IIB, IIA, IIB... almost everywhere in the bulk and the gravitinos only miss some states exactly at the orientifold planes

  12. The T-duals for other choices of the Wilson lines - e.g. the $SO(32)$ point of type I on a circle itself - distribute the D9-branes asymmetrically and the dilaton is inevitably nonconstant in spacetime, increasing (or decreasing?) from the orientifold planes with a deficit of D-branes to those with an excess of D-branes (or to the D-branes themselves); this dilaton profile has to be taken into account when you talk about the anomalies; of course, localized D-branes in type II string theory have no anomalies, and neither have the O-planes, but the properly variable dilaton around all these objects - depending on the charges - plays the role in cancelling the anomaly; in some sense, the original anomaly in type I gets divided between the different places where the T-dual O-planes and D-branes are localized

Cheers LM


LM beat me to this question this morning. There is not much I can add by writing. Polchinski’s vol II of “String Theory” has from p 138 to 146, section titled T-duality of type I string, which covers much of this.


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