The strings must satisfy some boundary conditions at their boundaries in order to cancel the boundary terms in the variation of the worldsheet action. The Neumann boundary conditions,

- partial/partial sigma (X) = 0,

were thought to be the only meaningful boundary conditions for the open strings until several smart people, most notably Joe Polchinski, pointed out that the Dirichlet boundary conditions are equally important, consistent, and related to Neumann boundary conditions by T-duality. Moreover, they lead to all D-branes and all this wonderful stuff.

One of the trivial consequences of this reasoning is that the heterotic strings can't be open: the boundary conditions relate the left-moving and right-moving degrees of freedom but the heterotic strings have a different number of degrees of freedom on their two sides. Setting the "extra" ones equal to zero at the boundary is far too constraining; it would allow no nontrivial solutions.

Let me summarize. There seem to be no open heterotic strings and consequently no heterotic D-branes which was the main paradox discussed in

Now another physicist whose name is again Joe Polchinski, and you may speculate that it is not a coincidence :-), writes a note that there exist

after all. More precisely, he argues that they only exist in the case of the SO(32) heterotic string theory (HO) and not E_{8} x E_{8} heterotic string theory (HE). The previous sentence may sound particularly bizarre because these two heterotic string theories are related by T-duality, as you can learn in a textbook written by another Gentleman whose name is also Joe Polchinski.

So it is not clear what happens with the open heterotic SO(32) string if you try to switch to a T-dual description. But Joe argues that this contradiction does not exist, the SO(32) string is global in the given compactification, and the broken HE string therefore can't be constructed.

This latter Joe is really speaking about macroscopic open strings - those that may be used for cosmic strings which is a topic that yet another Joe Polchinski was recently interested in - and there is some extra room for slightly "non-local" boundary conditions that may evade the speedy no-go conclusions at the beginning of this text. A potential paradox that the B-field charge is not conserved is avoided by an anomaly inflow and by mixing of the field strength H with the Chern-Simons three-form.

The higher instanton numbers

As we clarified in a debate with Davide Gaiotto, Joe's configuration looks as follows. Take a straight string in the HO theory. It is surrounded by an seven-sphere. If it has an endpoint, this endpoint is surrounded by an eight-sphere. Now take the SO(32) gauge field to have a nonzero higher instanton charge, namely one whose

- integral (S
^{8}) F /\ F /\ F /\ F

That's very cool because you can then count the number of gaugino zero modes localized near the endpoint. You will find that they transform as the fundamental 24 representation of SO(24). You may now imagine that the boundary conditions for the 10 bosonic coordinates are just like in type II; the boundary conditions for 8 fermions on the supersymmetric side and 8 gauge fermions on the bosonic side are defined much like in type II again; and the remaining 24 gauge fermions on the bosonic side (ones that are responsible for the extra "c=12" on this side) are "swallowed" by the gaugino zero modes.

I find this picture very cool. It is, among other things, the most physical way to connect bosonic string theory and superstring theory and reconcile their apparently different sets of degrees of freedom. In this picture, the endpoints themselves are probably very massive; recall that the endpoints carry no extra mass term in type II theories. It remains to be seen whether there is also a microscopic description of "small" open heterotic strings.

Recall that the Green-Schwarz mechanism is based on the term

- int B /\ Tr(F /\ F /\ F /\ F)

The open HO string is S-dual to an open D1-brane in type I theory. In this case we may think that the gauge fields are identical like in the HO theory, but moreover we may identify a nice configuration of tachyons. In this context it is useful to take D9-branes and anti-D9-branes with the gauge group "SO(16) x SO(8)" and define the tachyon in terms of SO(9) gamma matrices.

Note that the endpoints are probably very massive and the mass is stringy times a negative power of "g". It's because the higher instantons want to shrink as much as possible since their mass is proportional to radius to the fourth power. Only the 4D instantons have size moduli because the "F /\ F" topological charge has the same dimension as the energy density "F /\ *F". In lower dimensions, the flux want to dissolve while in higher dimensions than four, they want to shrink.

This means that you should include a mass term for every boundary of your open heterotic strings - a big mass times the proper length or time of the boundary. I also claim that this will have the effect of shrinking the boundary on the worldsheet. A cylinder will be infinitely long. The heterotic stringy cylinder diagrams will only have a contribution from the degenerate, infinitely long cylinder, I think.

If my comments are correct, and many people disagree, the right way to quantize a fundamental open heterotic string is to imagine that the sigma coordinate goes from "-infinity" to "+infinity". I should look at it although I must also prepare the lecture and notes for the Wigner-Eckart theorem tomorrow.

Doubts

Let me now try to extract the negative opinions about the open heterotic strings. It now seems that many people say - probably including Joe - that the energy of the endpoint is infinite by a simple scaling argument (integrate the "F /\ *F" energy density in polar coordinates to get a divergence); it becomes finite if you compactify some of the transverse dimensions. The question about the existence of the endpoints is not just a topological question; the actual dynamics and energetics matters, too, as SM emphasizes. There are some reasons to think that the flux tubes prefer to shrink into a string even in Joe's configuration.

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