[Review] Type II on AdS_3, Part I: Lightcone spectrum
Posted by Urs Schreiber
Strings on are a toy model for the more interesting (and more difficult) scenario. Here I’ll review some aspects of the analysis of type II strings in this background. The goal is to discuss a calculation of the superstring’s spectrum by first going to the (pp-wave) Penrose limit and then making a perturbative calculation in curvature corrections. Since strings on are exactly solvable this is nothing but a warmup for more interesting cases where such a perturbative calculation is inevitable.
In this first part I define the setup by writing down the metric, giving a set of invariant vector fields and defining the Penrose limit in terms of these vector fields, which involves magnifying the vicinity of a lightlike geodesic moving around the equator of .
Then I discuss the exact light-cone spectrum for superstrings with respect to this lightlike direction extending a result given in
A. Parnachev, D. Sahakyan Penrose limit and string quantization in .
The target space comes with the metric (we can ignore the factor)
together with a -field that provides the parallelizing torsion such that superstrings in this background are described by the super Wess-Zumino-(Novikov)-Witten (SWZW) model.
A possible choice of left/right invariant vector fields on the two group manifolds is
for and
for
These vectors are normalized so as to yield the standard non-vanishing Lie brackets
The non-vanishing inner products are
from which one gets the usual quadratic Casimir
Now the Penrose limit is obtained by concentrating on the vicinity of a lightlike geodesic which runs around the equator of the factor. The following vector fields are adapted to the nature of this limit:
where is a real number. The limit will correspond to taking the Penrose limit. In terms of these new vector fields the Casimir simplifies somewhat:
and are the two lightlike directions with respect to which the superstring spectrum can now be analyzed conveniently:
Let be the level of a bosonic current algebra (the straightforward supersymmetric extension is discussed at the end) and let be the eigenvalue of the Casimir of that current algebra. The Virasoro constraint on a state of level number reads
where is a given normal ordering constant that we leave unspecified for the moment.
The eigenvalues and of the zero modes of and (the momenta along and ) can be written as
where grows by one for every and every excitation and is reduced by one for every and excitation (due to and ) and analogously for .
The eigenvalues of the lightcone Hamiltonian and of the longitudinal momentum are now defined by
and the task is to express these quantities as functions of each other and of the transverse excitations of the string:
using the above physical state condition. After a bit of algebra one finds the following
or equivalently
This gives the exact lightcone spectrum of strings in the SWZW model. The Penrose limit is again obtained by taking . Interestingly, the longitudinal momentum has only a first order correction in . This means that doing a first order perturbative calculation of starting in the Penrose limit and taking curvature corrections of the full background into account will already yield the exact result. This will be discussed in a future entry to this blog.
In closing I remark that the above calculation directly generalizes to the superstring by realizing that one has to use the so-called total currents (sum of bosonic plus fermionic SWZW currents) for the lightcone momenta so that one simply has to substitute
in all the above expressions.
Re: [Review] Type II on AdS_3, Part I: Lightcone spectrum
The rest of what I wanted to say here can now be found in the second part of
hep-th/0311064.
I have had a little discussion about this with Luboš here.