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August 19, 2004

Everybody agrees on the non-abelian boundary state!

Posted by Urs Schreiber

[Update next day: What I am saying here is the content of pp. 236-237 of

O. D. Andreev & A. A. Tseytlin: Partition-Function representation for the open superstring effective action: Cancellation of Möbius infinities and derivative corections to Born-Infeld lagrangians (1988) ]

As I have probably mentioned before, I am currently writing up some notes on how super-Pohlmeyer invariants give boundary states for non-abelian gauge fields when applied to the boundary state of a bare D9 brane.

However, up until about two minutes ago I thought there was one big unsolved problem. I was just about to write in the conclusion section a paragraph about how this apparent problem remains unsolved, when suddenly the gods of reserch showed mercy and enlighted me. It’s abasingly simple. There is no problem at all, in fact everything is in much better shape than I thought.

This is what I was puzzled about:

As I menioned before and as far as I am aware, there are two versions of what the boundary state for a non-abelian gauge field should look like. One is that given in equation (3.7) of hep-th/0312260, the other the non-abelian generalization of the state considered in JHEP2000/04/023 which I generalized in hep-th/0407122 to include a nonabelian BB field.

Now in the introduction of hep-th/0312260 the ideas in JHEP2000/04/023 are addressed as ‘another approach’ and somehow this seriously confused me. I was under the impression that there are two different proposals for how the boundary state for a nonabelian gauge field looks like and worried about how that could be. I still don’t know what precisely the authors of that paper meant by ‘another approach’ - but the simple truth is that the boundary states considered in all these papers are all one and the same, identical!

Even though it’s very simple, let me spell it out:

The boundary state given in hep-th/0312260 is the Wilson line

(1)Pexp(∫dσdθV(X(σ,θ))) \mathrm{P} \exp\left( \int d\sigma\, d\theta\, \mathbf{V}\left( \mathbf{X}\left( \sigma,\theta \right) \right) \right)

over the string, with the integrals being super-integrals over the bosonic parameter σ\sigma and the Grassmann parameter θ\theta over the gluon super-vertex

(2)V(X(σ,θ))=A μ(D μX μ)e ik⋅X μ(σ,θ)=(−aA μℰ †μ−iθ(X ′μ+iℰ †μk⋅ℰ †))e ik⋅X(σ), \mathbf{V}\left( \mathbf{X} \left( \sigma,\theta \right) \right) = A_\mu (D_\mu \mathbf{X}^\mu) e^{i k\cdot \mathbf{X}^\mu} \left( \sigma,\theta \right) = \left( -a A_\mu \mathcal{E}^{\dagger \mu} -i \theta \left( X^{\prime \mu} + i \mathcal{E}^{\dagger \mu} \, k \cdot \mathcal{E}^\dagger \right) \right) e^{i k\cdot X} \left( \sigma \right) \,,

where in my wacky notation ℰ †∝ψ+iψ¯\mathcal{E}^\dagger \propto \psi + i\bar \psi is the polar combination of the worldsheet fermions which constitutes a 1-form on loop space.

P in the integral above denotes path ordering, but because the integral is in superspace this is super-path ordering which is enforced by the super-step function Θ(σ n+1−σ n+iθ n+1θ n)\Theta(\sigma_{n+1} - \sigma_n +i \theta_{n+1}\theta_n) which expands to

(3)Θ(σ n+1−σ n+iθ n+1θ n)=Θ(σ n+1−σ n)+iθ n+1θ nδ(σ n+1−σ n). \Theta(\sigma_{n+1} - \sigma_n +i \theta_{n+1}\theta_n) = \Theta(\sigma_{n+1} - \sigma_n) + i \theta_{n+1}\theta_n \delta(\sigma_{n+1} - \sigma_n) \,.

And there you go. I didn’t think about this carefully when reading their paper, but it is very easy to see that this second term quadratic in the Grassmann variables glues gluon-super vertices together in such a fashion that the gauge-covariant derivative appears in the Wilson line, so that after performing the Grassmann integrals one is left with

(4)Pexp(∫dσdθV(X(σ,θ)))=Pexp(∫dσ(iA μX ′μ+c(F A) μνℰ †μℰ †ν)), \mathrm{P} \exp \left( \int d\sigma\, d\theta\, \mathbf{V}\left( \mathbf{X} \left( \sigma,\theta \right) \right) \right) = \mathrm{P} \exp\left( \int d\sigma\, \left( iA_\mu X^{\prime\mu} + c (F_A)_{\mu\nu} \mathcal{E}^{\dagger \mu} \mathcal{E}^{\dagger \nu} \right) \right) \,,

which is precisely the boundary state that I consider in hep-th/0407122 and hep-th/upcoming in various contexts.

Phew.

Posted at August 19, 2004 5:40 PM UTC

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