Super-Pohlmeyer invariants and boundary states for non-abelian gauge fields

Posted by Urs Schreiber

$MathML-enabled post (click for more details).$

I have just submitted the following to the preprint server:

Urs Schreiber: Super-Pohlmeyer invariants and boundary states for non-abelian gauge fields (2004)

Abstract:

Aspects of the supersymmetric extension of the Pohlmeyer invariants are studied, and their relation to superstring boundary states for non-abelian gauge fields is discussed. We show that acting with a super-Pohlmeyer invariant with respect to some non-abelian gauge field $A$ on the boundary state of a bare D9 brane produces the boundary state describing that non-abelian background gauge field on the brane. Known consistency conditions on that boundary state equivalent to the background equations of motion for $A$ hence also apply to the quantized Pohlmeyer invariants.

Introduction:

This paper demonstrates a relation between two apparently unrelated aspects of superstrings: boundary states for nonabelian gauge fields and (super-)Pohlmeyer invariants.

On the one hand side superstring boundary states describing excitations of non-abelian gauge fields on D-branes are still the subject of investigations [1,2,3] and are of general interest for superstring theory, as they directly mediate between string theory and gauge theory.

On the other hand, studies of string quantization focusing on non-standard worldsheet invariants, the so-called Pohlmeyer invariants, done in [4,5,6,7] and recalled in [8], were shown in [9,10] to be related to the standard quantization of the string by way of the well-known DDF invariants. This raised the question whether the Pohlmeyer invariants are of any genuine interest in (super-)string theory as commonly understood.

Here it shall be shown that the (super-)Pohlmeyer invariants do indeed play an interesting role as boundary state deformation operators for non-abelian gauge fields, thus connecting the above two topics and illuminating aspects of both them.

A boundary state is a state in the closed string’s Hilbert space constructed in such a way that inserting the vertex operator of that state in the path integral over the sphere reproduces the disk amplitudes for certain boundary conditions (D-branes) of the open string. In accord with the general fact that the worldsheet path integral insertions which describe background field excitations are exponentiations of the corresponding vertex operators, it turns out that the boundary states which describe gauge field excitations on the D-brane have the form of (generalized) Wilson lines of the gauge field along the closed string [11,12,1,2,3].

Long before these investigations, it was noted by Pohlmeyer [7], in the context of the classical string, that generalized Wilson lines along the closed string with respect to an auxiliary gauge connection on spacetime provide a ‘complete’ set of invariants of the theory, i.e. a complete set of observables which (Poisson-)commute with all the Virasoro constraints.

Given these two developments it is natural to suspect that there is a relation between Pohlmeyer invariants and boundary states. Just like the DDF invariants (introduced in [13] and recently reviewed in [9]), which are the more commonly considered complete set of invariants of the string, commute with all the constraints and hence generate physical states when acting on the worldsheet vacuum, a consistently quantized version of the Pohlmeyer invariants should send boundary states of bare D-branes to those involving the excitation of a gauge field.

Indeed, up to a certain condition on the gauge field, this turns out to be true and works as follows:

$MathML-enabled post (click for more details).$

If $X^\mu(\sigma)$ and $P_\mu(\sigma)$ are the canonical coordinates and momenta of the bosonic string, then $\mathcal{P}_\pm^\mu(\sigma) := \frac{1}{\sqrt{2}T}(P_\mu(\sigma) \pm T \eta_{\mu\nu}X^{\prime\nu}(\sigma))$ , (where $T$ is the string’s tension and a prime denotes the derivative with respect to $\sigma$ ) are the left- and right-moving bosonic worldsheet fields for flat Minkowski background (in CFT context denoted by $\partial X$ and $\bar \partial X$ ) and for any given constant gauge field $A$ on target space the objects

(1)

W_\pm^{\mathcal{P}}[A] := \mathrm{Tr}\, \mathrm{P} \exp \left( \int_0^{2\pi}d\sigma\, A \cdot \mathcal{P}_\pm\left(\sigma\right) \right)

(where $\mathrm{Tr}$ is the trace in the given representation of the gauge group’s Lie algebra and $\mathrm{P}$ denotes path-ordering along $\sigma$ ) Poisson-commute with all Virasoro constraints. In fact the coefficients of $\mathrm{Tr}(A^n)$ in these generalized Wilson lines do so seperately, and these are usually addressed as the Pohlmeyer invariants, even though we shall use this term for the full object above.

Fundamentally, the reason for this invariance is just the reparameterization invariance of the Wilson line, which can be seen to imply that it remains unchanged under a substitution of $\mathcal{P}$ with a reparameterized version of this field. In [9] it was observed that an interesting example for such a substitution is obtained by taking the ordinary DDF oscillators

(2)

A_m^\mu \propto \int_0^{2\pi} d\sigma\, \mathcal{P}_-^\mu(\sigma)e^{im\frac{4 \pi T}{k\cdot p}\, k\cdot X_-(\sigma)}

(where $k$ is a lightlike vector on target space, $X_-$ is the left-moving component of $X$ , $p$ is the center-of-mass momentum, and an analogous expression exists for $\mathcal{P}_+$ ) and forming ‘quasi-local’ invariants

(3)

\mathcal{P}_-^{R\mu}\of{\sigma} := \frac{1}{\sqrt{2\pi}} \sum_{m=0}^\infty A_m^\mu e^{im\sigma}

from them. (We dare to use the same symbol $A$ for the gauge field and for the DDF oscillators in order to comply with established conventions. The DDF oscillators will always carry a mode index $m$ , however, and it should always be clear which object is meant.)

One finds

(4)

W^{\mathcal{P}}[A] = W^{\mathcal{P}^R}[A]

and since the quantization of the $\mathcal{P}^R$ in terms of DDF oscillators is well known, this gives a consistent quantization of the Pohlmeyer invariants. This is the quantization that we shall use here to study boundary states.

The above construction has a straightforward generalization to the superstring and this is the context in which the relation between the Pohlmeyer invariants and boundary states turns out to have interesting aspects, (while the bosonic case follows as a simple restriction, when all fermions are set to 0).

So we consider the supersymmetric extension of the above, which, by convenient abuse of notation, we shall also denote by $A_m^\mu$ :

(5)

A_m^\mu \propto \int_0^{2\pi} d\sigma\, \left( \mathcal{P}_-\left(\sigma\right) +im \frac{\pi \sqrt{2T}}{k\cdot p} k\cdot\Gamma_-\left(\sigma\right)\, \Gamma_-^\mu\left(\sigma\right), e^{im\frac{4 \pi T}{k\cdot p}\, k\cdot X_-\left(\sigma\right)} \right) \,,

where $\Gamma_\pm(\sigma)$ denote the fermionic superpartners of $\mathcal{P}_\pm$ . From these we build again the objects $\mathcal{P}^R$ and finally $W^{\mathcal{P}^R}[A]$ , which we address as the super-Pohlmeyer invariants.

Being constructed from the supersymmetric invariants $\mathcal{P}^R$ , which again are built from the $A_m^\mu$ , these manifestly commute with all of the super-Virasoro constraints. But in order to relate them to boundary states they need to be re-expressed in terms of the plain objects $\mathcal{P}$ and $\Gamma$ . This turns out to be non-trivial and has some interesting aspects to it.

After these peliminaries we can state the first result to be reported here, which is

that on that subspace $\mathbf{P}_k$ of phase space where $k\cdot X_-$ is invertible as a function of $\sigma$ (a condition that plays also a crucial role for the considerations of the bosonic DDF/Pohlmeyer relationship as discussed in [9]) the super-Pohlmeyer invariants built from the supersymmetric $A_m^\mu$ are equal to

(6)

\left. W^{\mathcal{P}^R}[A] \right|_{\mathbf{P}_k} = \mathrm{Tr}\, \mathrm{P} \exp \left( \int_0^{2\pi} d\sigma\, \left( i A_\mu + \left[A_\mu,A_\nu\right] \frac{ k \cdot \Gamma\, \Gamma^\nu }{2k\cdot \mathcal{P}} \right) \mathcal{P}^\mu \right) \,,

that this expression extends to an invariant on all of phase space precisely if the transversal components of $A$ mutually commute,

and that in this case the above is equal to

(7)

Y[A] := \mathrm{Tr}\,\mathrm{P} \exp \left( \int_0^{2\pi}d\sigma\, \left( i A_\mu \mathcal{P}^\mu + \frac{1}{4}\left[A_\mu,A_\nu\right] \Gamma^\mu \Gamma^\nu \right) \right) \,.

The second result concerns the application of the quantum version of these observables to the bare boundary state $|\text{D9}\rangle$ of a space-filling D9-brane (see for instance appendix A of [3] for a brief review of boundary state formalism and further literature). Denoting by $\mathcal{E}^\dagger(\sigma) = \frac{1}{2}\left(\Gamma_+\left(\sigma\right) + \Gamma_-\left(\sigma\right)\right)$ the differential forms on loop space (cf. section 2.3.1. of [3] and section 2.2 of [14] for the notation and nomenclature used here, and see [15] for a more general discussion of the loop space perspective) we find that for the above case of commuting transversal $A$ the application of $W^{\mathcal{P}^R}$ to $|\text{D9}\rangle$ yields

(8)

\mathrm{Tr}\, \mathrm{P} \exp \left( \int_0^{2\pi} d\sigma\, \left( i A_\mu \mathcal{P}^\mu + \frac{1}{4}(F_A)_{\mu\nu}\Gamma^\mu\Gamma^\nu \right) \right) |\text{D9}\rangle = \mathrm{Tr}\, \mathrm{P} \exp \left( \int_0^{2\pi} d\sigma\, \left( i A_\mu X^{\prime \mu} + \frac{1}{4}(F_A)_{\mu\nu} \mathcal{E}^{\dagger \mu} \mathcal{E}^{\dagger \nu} \right) \right) |\text{D9}\rangle \,.

which is, on the right hand side, precisely the boundary state describing a non-abelian gauge field on the D9 brane [1,3].

In summary this shows that and under which conditions the application of a quantized super-Pohlmeyer invariant to the boundary state of a bare D9 brane produces the boundary state describing a non-abelian gauge field excitation.

Conclusion:

The super-Pohlmeyer invariants, whose construction principle was only briefly mentioned at the end of [9], have been studied here in more detail. In particular their expression in terms of local fields has been worked out on that part of phase space where it exists. The extension of this expression to the full phase space has been found to be invariant on all of phase space if the transversal components of the gauge field mutually commute.

Quantizing the result and applying it to the boundary state of a bare D9 brane was shown to yield boundary states of the form considered before in [1,3], which are straightforward non-abelian generalizations of those studied in [11].

This result shows that the Pohlmeyer invariants, which before have only appeared in the connection with vague hopes to circumvent well-known quantum effects like the critical dimension, do have a role to play in (standard) string theory.

While this might not appear as much of a surprise in light of the result [9] which showed that the Pohlmeyer invariants are a subset of all DDF invariants that have the crucial property of mapping physical states to physical states by commuting with all the (super-)Virasoro constraints, it seems noteworthy that it is a priori not obvious that this ability to generate the physical spectrum translates also to a generation of boundary states that satisfy the Ishibashi condition.

And indeed, it was shown above that a precise match between super-Pohlmeyer invariants and boundary states is manifest only in the special case where the transversal components of the gauge field mutually commute. On the other hand, we did not show that in other cases the result of acting with the super-Pohlmeyer invariant on the bare D9-brane boundary state does not produce a new some boundary, but it is at least not obvious.

Of course, naively it is obvious that the result of acting with a super-Pohlmeyer invariant on a state which satisfies the Ishibashi conditions still satisfies these conditions, simply due to the very invariance property of the Pohlmeyer invariants. But the result will in general need regularization and/or a condition on the gauge field that ensures vanishing of contact term divergences, and for the case where the transversal components of $A$ do not mutually commute we did not show that the result has this property.

This is in contrast to the case where the transversal components do mutually commute, and where the lightlike component of $A$ which is non-parallel to $k$ vanishes. In this case we did show that the result of acting with the super-Pohlmeyer invariant associated with that $A$ -field on $|\text{D9}\rangle$ does produce the boundary state of the non-abelian gauge field $A$ . And this result is known to be well defined if (and only if) $A$ satisfies its background equations of motion.

This can either be seen by introducing a regularization and checking that this regularization preserves the invariance properties iff the background equations of motion hold, as was done in [1]. Or, alternatively, no regularization is used and the resulting divergences are shown to vanish when $A$ satisfies its equations of motion. This was done for the abelian case to second order in [11,12] and for the non-abelian case to first order in [3].

While this relation between quantum divergences and background equations of motion is perfectly natural in the context of boundary states, it is a new aspect in the study of Pohlmeyer invariants. It shows that even though these have a consistent quantization in terms of DDF invariants in the sense of constituting a closed algebra of quantum operators that commute with the constraints, not all of them have a well defined application on all states of the string’s Hilbert space. Namely, even though they consist of well-behaved DDF operators, these appear in infinite sums and divergences may occur when acting with these on some states.

Posted at August 20, 2004 7:29 PM UTC

TrackBack URL for this Entry: https://golem.ph.utexas.edu/cgi-bin/MT-3.0/dxy-tb.fcgi/418

The String Coffee Table

Skip to the Main Content

August 20, 2004