### [Review] Type II on AdS_3, Part I: Lightcone spectrum

#### Posted by Urs Schreiber

Strings on ${\mathrm{AdS}}_{3}\times {\mathrm{S}}^{3}\times {\mathrm{T}}^{4}$ are a toy model for the more interesting (and more difficult) ${\mathrm{AdS}}_{5}\times {\mathrm{S}}^{5}$ 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 ${\mathrm{AdS}}_{3}\times {\mathrm{S}}^{3}\times {\mathrm{T}}^{4}$ 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 ${\mathrm{S}}^{3}$.

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 ${\mathrm{AdS}}_{3}\times {\mathrm{S}}^{3}$.

The target space ${\mathrm{AdS}}_{3}\times {\mathrm{S}}^{3}$ comes with the metric (we can ignore the ${\mathrm{T}}^{4}$ factor)

together with a $B$-field that provides the parallelizing torsion such that superstrings in this background are described by the $\mathrm{SL}(2,\mathrm{R})\times \mathrm{SU}(2)$ super Wess-Zumino-(Novikov)-Witten (SWZW) model.

A possible choice of left/right invariant vector fields on the two group manifolds is

for ${\mathrm{AdS}}_{3}$ and

for ${\mathrm{S}}^{3}$

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 ${\mathrm{S}}^{3}$ factor. The following vector fields are adapted to the nature of this limit:

where $1/k$ is a real number. The limit $k\to \mathrm{\infty}$ will correspond to taking the Penrose limit. In terms of these new vector fields the Casimir $C$ simplifies somewhat:

$F$ and $J$ are the two lightlike directions with respect to which the superstring spectrum can now be analyzed conveniently:

Let $k$ be the level of a bosonic $\mathrm{SL}(2,\mathrm{R})\times \mathrm{SU}(2)$ current algebra (the straightforward supersymmetric extension is discussed at the end) and let $-h(h+1)+j(j+1)$ be the eigenvalue of the Casimir ${\eta}_{\mathrm{ab}}{J}_{0}^{a}{J}_{0}^{b}$ of that current algebra. The ${L}_{0}$ Virasoro constraint on a state of level number $N$ reads

where $a$ is a given normal ordering constant that we leave unspecified for the moment.

The eigenvalues ${h}^{3}$ and ${j}^{3}$ of the zero modes of ${K}_{0}^{3}$ and ${J}_{0}^{3}$ (the momenta along $t$ and $\psi $) can be written as

where ${N}_{\mathrm{SU}}^{{\textstyle \prime}}$ grows by one for every ${J}_{-n}^{+}$ and every ${\psi}_{-n}^{+}$ excitation and is reduced by one for every ${J}_{-n}^{-}$ and ${\psi}_{-n}^{-}$ excitation (due to $[{J}_{0}^{3},{J}_{n}^{\pm}]=\pm {J}_{n}^{\pm}$ and $[{J}_{0}^{3},{\psi}_{n}^{\pm}]=\pm {\psi}_{n}^{\pm}$) and analogously for ${N}_{\mathrm{SL}}^{{\textstyle \prime}}$.

The eigenvalues of the lightcone Hamiltonian $H\sim J$ and of the longitudinal momentum ${p}_{-}\sim F$ 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 ${\mathrm{AdS}}_{3}\times {\mathrm{S}}^{3}$ SWZW model. The Penrose limit is again obtained by taking $1/k\to 0$. Interestingly, the longitudinal momentum ${p}_{-}$ has only a *first* order correction in $k$. This means that doing a first order perturbative calculation of ${p}_{-}$ starting in the Penrose limit and taking curvature corrections of the full ${\mathrm{AdS}}_{3}\times {\mathrm{S}}^{3}$ 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.