The String Coffee Table

Recently I had talked about how I think Pohlmeyer invariants are related to boundary states of D-branes with gauge field and fluctuations turned on. I concluded by saying:

It just remains to note that by using the relation to DDF invariants it is easy to generalize the Pohlmeyer invariants to the superstring (section 2.3.3 of hep-th/0403260), which should carry all of the above disucssion over to its supersymmetric extension.

Indeed the super Pohlmeyer invariants, as I will call them, which are obtained this way seem to be useful and interesting. Here I will look at them in more detail.

The idea is as follows: Similar to the considerations presented in hep-th/0312260 we start with the supersymmetric vertex operator for the gauge field and construct a suitable path ordered multi-integral of this guy around the closed string such that the Ishibashi conditions are satisfied, which say that the result must be super-reparameterization invariant along the string (which in loop space differential geometric language means that the result must be $d_K$-closed).

The authors of hep-th/0312260 discuss one way to do this. Currently I fail to see if and how the boundary state (3.9) which they give is gauge covariant. Maybe it need not be, if only the correct states decouple in the amplitudes calculated from it.

The issue arises due to the fact that the supersymmetric vertex operator $V$ of the gauge field looks something like

where the fermionic term on the right is the exterior derivative of $A$ for the mode $k$ - but not the gauge covariant exterior derivative, but the ordinary one. This makes it very non-manifest that exponentiating this object through a Wilson line gives something gauge covariant (and does it?).

On the other hand, my point is that if we use the DDF$\leftrightarrow$Pohlmeyer relation to construct a Wilson-line-like boudnary state for non-abelian $A$ something manifestly gauge covariant is obtained.

This works as follows:

Start with the supersymmetric DDF oscillator in equation (2.58) of hep-th/0403260 and take its Fourier transform as done in equation (2.51) for the bosonic part of that operator. The resulting quasi local invariant observable now contains also a fermionic contribution, but this is a total $\sigma$-derivative, as one can easily see. That’s crucial.

Namely when one now inserts this integrated and Fourier transformed vertex into an ordinary Wilson line, the derivative term leads to boundary terms in the multi-integrals, in just such a way that the covariant field strength appears in the fermionic terms. More precisely, the result is

This is the explicit form of the super Pohlmeyer invariant. Written this way it is very unobvious that this is really an invariant, so that it commutes with the fermionic Virasoro generators. But, being constructed purely from DDF invariants by the same trick that the bosonic Pohlmeyer invariants are, it is still true, and so in particular this does respect the Ishibashi conditions.

What is manifest, however, is the gauge covariance of this object under $A\mapsto {\mathrm{UAU}}^{†}+U({\mathrm{dU}}^{†})$.

Another nice consequence of the fact that this object comes from exponentiated DDF operators is that it is easy to check for divergences when this is applied to a bare boundary state. The calculation turns out to be completely analogous to that in section 3.5. of hep-th/0407122, which means that to lowest order the condition for vanishing of the divergences in the super Pohlmeyer invariant are just the Yang-Mills equations for $A$.

The authors of hep-th/0312260 obtain the same result, but with a rather different method. They regularize their Wilson line so that divergences are canceled by construction. But then invariance is potentially broken and they confirm that it is preserved by the regularizartion precisely if the equations of motion hold.

So maybe the super Pohlmeyer invariant discussed above is just a complementary view on the same result obtained by these authors. It certainly describes some gauge field background, it has the form of a Wilson line, is well defined as an operator and satisfies the correct physical conditions. Plus - it is manifestly gauge covariant.