The String Coffee Table

Open string backgrounds from boundary state deformations

Posted by Urs Schreiber

Today Eric Forgy has asked me to summarize some aspects of results by Koji Hashimoto concerning the description of open string backgrounds by deformations of boundary states.

With the time and energy that I have I will certainly not succedd in giving a complete description starting from first principles, but I’ll try to convey the basic ideas and add some comments concerning my perspective on these matters. (The details are left to the comment section ;-)

So here’s the story: While thinking about how the deformations of closed string worldsheet theories of the form which we discussed a while ago in this entry generalize to open strings I began of course to think about boundary state formalism and some literature search then turned up the papers

Koji Hashimoto: Generalized supersymmetric boundary state (2000)

T. Maeda & T. Nakatsu & T. Oonishi: Non-linear Field Equations from Boundary State Formalism (2004)

Koji Hashimoto: The shape of nonabelian D-branes (2004)

Here is my perspective on what’s going on:

We had seen that all (massless NS, at least) backgrounds of the closed string come from deformations of the super-Virasoro constraints of the form $d_K \to e^{-W}d_K e^W$, $d^\dagger_K \to e^{W^\dagger }d^\dagger_K e^{-W^\dagger}$, where $d^{(\dagger)}_K \sim iT_F \pm \bar T_F$ are polar combinations of the left- and rightmoving supercurrents.

Some of these transformations are pure gauge. Namely if $W^\dagger = -W$ then (and only then!) is the above transformation a global unitary tranformation of the entire super-Virasoro algebra. It has been shown how such unitary transformations encode gauge shifts and dualities of the background fields, precisely as expected (in retrospect, in my case) from string field theory considerations.

In particular, it can be shown that the gauge transformation $W = i\oint d\sigma\,X^{\prime\mu}(\sigma)A_\mu(X(\sigma))$ corresponds to turning on a gauge field background, which couples trivially to closed strings, manifesting itself as a gauge-trivial shift $B \mapsto B + d A$ of the 2-form NS-NS field. This can be understood by regarding a closed string as consisting of two glued open strings. The gauge field couples to the endpoints of these open strings and since they stick together (in this picture) one coupling cancels the other.

But this alrready suggests how the deformation generalizes to the open string: We have to rewrite the opens string theory as that of a closed string split in half.

This is precisely the moral content of the boundary state formalism. I have a brief description of this technique in what is currently appendix C.2 ‘Boundary states’ of my OSFT notes:

Imagine an open string propagating with both ends attached to some D-brane. The worldsheet is topologically the disk (with appropriate operator insertions at the boundary). This disk can equivalently be regarded as the half sphere glued to the brane. But from this point of view it represents the worldsheet of a closed string with a certain source at the brane. Therefore the open string disk correlator on the brane is physically the same as a closed string emission from the brane with a certain source term corresponding to the open string boundary condition. The source term at the boundary of the half sphere can be represented by an operator insertion in the full sphere. The state corresponding to this vertex insertion is the boundary state.

This is very similar in sprit to the method of mirror charges in elementary electrostatics. Simple but profound.

So what we have to do in fact consider just closed strings, and the deformations known of these, and then consider the modified inner product

where $|\alpha\rangle$ is the closed string state which encodes the desired boundary condition for the open string.

One curious observation is that the boundary state $|\alpha_0\rangle$ which describes a flat space-filling D9 brane is nothing but the constant 0-form on loop space! For anyone knowing the respective formulas this is trivial, but to me this seems to be important, conceptually. For details of what I am talking about see what is currently section 3.6 ‘Boundary states and loop space formalism’ in my OSFT notes.

So let $U(A) = \mathrm{Tr}\mathcal{P} \exp(\oint d\sigma\, X^{\prime\mu}A_\mu)$ be the deformation inducing a gauge field background for the closed string, as in equation (3.51) of my hep-th/0401175.

This suggests we study something like $|\alpha\rangle = U(A)|\alpha_0\rangle$ if we want to describe open superstrings on $N$ flat D9 branes with a $U(N)$ gauge field turned on.

And, indeed, this is essentially what Hashimoto showed in JHEP 04 (2000) 023 to be the correct choice, at least with regard to the bosonic degrees of freedom. In addition one want the boundary state to be BRST-closed, i.e. annihilated by the BRST operator, because that sort of makes is an honest physical state of the closed string (a coherent state, in fact, describing the macroscopic excitation of a brane with that gauge field). So one has to add a fermionic component. This can be guessed and checked as in the above paper or derived with superfield formalism as in hep-th/0312260, but in any case the result is that we also need to multipliy by the unitary operator

where $\mathcal{E}^{(\dagger)}$ are the form creators/annihilators on loop space and $F = d_A A$ is the field strength. This is natural, since according to the closed string formalism of section 3.3.2 these operators are asscociated with deformations turning on 2-form backgrounds of the form $dA$.

So this is already the basic idea: A unitary and hence gauge-trivial deformation of the closed string becomes a non-trivial deformation of the open string formalism by ‘cutting it in half’, i.e. by inserting it only on one side of the inner product, instead of on two sides, so to say.

It should also not be a miracle how such deformations arise in the context of

A. Recknagel & V. Schomerus: Boundary deformation theory and Moduli spaces of D-branes,

because that was already clarified by J. Klusoň as reviewed (and referenced) in section 2.4 ‘CFTs from string field backgrounds’ in my notes (I’ll have to expand on that summary, though…).

The above method generalizes strightforwardly to tachyon backgrounds, finite number of gluon excitations (instead of coherent states) and what not. This and the derivation of the correct background equations of motion from the BRST-closedness condition is shown in great detail by Koji Hashimoto in hep-th/0312260 (thereby solving for the case of open striongs the little excercise that a kind referee suggested to me in the context of closed strings ;-)

It is nice to see how much physics this method captures. For instance on p. 5 of hep-th/0401043 Koji Hashimoto mentions that a set of D0 branes whose distribution is encoded in the matrices $M^i$ can be obtained from the simple boundary state $|\alpha_{D0}\rangle$ of a single D0-brane simply by applying the unitary ‘translation’ operator with respect to the non-commutative coordinates $M^i$:

Note that if we tranlated these branes in this way and at the same time added a gauge field in the sense of the boundary state

this is nothing but applying a Pohlmeyer operator to the original boundary state. Maybe I was not that far off with my outlook in hep-th/0403260 after all…

Posted at June 11, 2004 9:12 PM UTC