## January 26, 2004

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

#### Posted by Urs Schreiber

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

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

(1)${\mathrm{ds}}^{2}={R}_{\mathrm{SL}}^{2}\left(-{\mathrm{cosh}}^{2}\left(\rho \right)\phantom{\rule{thinmathspace}{0ex}}{\mathrm{dt}}^{2}+d{\rho }^{2}+\mathrm{sinh}\left(\rho \right)d{\varphi }^{2}\right)+{R}_{\mathrm{SU}}^{2}\left({\mathrm{cos}}^{2}\left(\theta \right)\phantom{\rule{thinmathspace}{0ex}}d{\psi }^{2}+d{\theta }^{2}+\mathrm{sin}\left(\theta \right)d{\chi }^{2}\right)\phantom{\rule{thinmathspace}{0ex}}.$

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

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

(2)${K}_{3}:=-\frac{i}{2}{\partial }_{t}+\frac{i}{2}{\partial }_{\varphi }$
(3)${K}_{+}:=\frac{1}{2}\left({e}^{+i\left(\varphi +t\right)}\mathrm{tanh}of\rho {\partial }_{t}-{\mathrm{ie}}^{+i\left(\varphi +t\right)}{\partial }_{\rho }+{e}^{+i\left(\varphi +t\right)}\mathrm{coth}of\rho {\partial }_{\varphi }\right)$
(4)${K}_{-}:=\frac{1}{2}\left(-{e}^{-i\left(\varphi +t\right)}\mathrm{tanh}of\rho {\partial }_{t}-{\mathrm{ie}}^{-i\left(\varphi +t\right)}{\partial }_{\rho }-{e}^{-i\left(\varphi +t\right)}\mathrm{coth}of\rho {\partial }_{\varphi }\right)$

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

(5)${J}_{3}:=-\frac{i}{2}{\partial }_{\psi }-\frac{i}{2}{\partial }_{\chi }$
(6)${J}_{+}:=\frac{1}{2}\left(-{e}^{+i\left(\chi +\psi \right)}\mathrm{tan}of\rho {\partial }_{\psi }-{\mathrm{ie}}^{+i\left(\chi +\psi \right)}{\partial }_{\theta }+{e}^{+i\left(\chi +\psi \right)}\mathrm{cot}of\rho {\partial }_{\chi }\right)$
(7)${J}_{-}:=\frac{1}{2}\left({e}^{-i\left(\chi +\psi \right)}\mathrm{tan}of\rho {\partial }_{\psi }-{\mathrm{ie}}^{-i\left(\chi +\psi \right)}{\partial }_{\theta }-{e}^{-i\left(\chi +\psi \right)}\mathrm{cot}of\rho {\partial }_{\chi }\right)$

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

These vectors are normalized so as to yield the standard non-vanishing Lie brackets

(8)$\left[{K}_{3},{K}_{±}\right]=±{K}_{±}$
(9)$\left[{K}_{+},{K}_{-}\right]=-2{K}_{3}$
(10)$\left[{J}_{3},{J}_{±}\right]=±{J}_{±}$
(11)$\left[{J}_{+},{J}_{-}\right]=+2{J}_{3}\phantom{\rule{thinmathspace}{0ex}}.$

The non-vanishing inner products are

(12)${K}_{3}\cdot {K}_{3}=\frac{{R}_{\mathrm{SL}}^{2}}{4}$
(13)${K}_{+}\cdot {K}_{-}=-2\frac{{R}_{\mathrm{SL}}^{2}}{4}$
(14)${J}_{3}\cdot {J}_{3}=-\frac{{R}_{\mathrm{SU}}^{2}}{4}$
(15)${J}_{+}\cdot {J}_{-}=-2\frac{{R}_{\mathrm{SU}}^{2}}{4}$

from which one gets the usual quadratic Casimir

(16)$C=-{K}_{3}\left({K}_{3}+1\right)+{K}_{-}{K}_{+}+{J}_{3}\left({J}_{3}+1\right)+{J}_{-}{J}_{+}\phantom{\rule{thinmathspace}{0ex}}.$

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:

(17)$F=\frac{1}{k}\left({K}_{3}-{J}_{3}\right)$
(18)$J:={K}_{3}+{J}_{3}$
(19)${P}_{1}=\frac{1}{\sqrt{k}}{K}_{+}$
(20)${P}_{1}^{ast}:=\frac{1}{\sqrt{k}}{K}_{-}$

(21)${P}_{2}:=\frac{1}{\sqrt{k}}{J}_{+}$
(22)${P}_{2}^{ast}:=\frac{1}{\sqrt{k}}{J}_{-}\phantom{\rule{thinmathspace}{0ex}},$

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

(23)$\frac{1}{k}C=-F\left(J+1\right)+{P}_{1}^{ast}{P}_{1}+{P}_{1}^{ast}{P}_{1}\phantom{\rule{thinmathspace}{0ex}}.$

$F$ 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 $\mathrm{SL}\left(2,\mathrm{R}\right)×\mathrm{SU}\left(2\right)$ current algebra (the straightforward supersymmetric extension is discussed at the end) and let $-h\left(h+1\right)+j\left(j+1\right)$ 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

(24)$-\frac{h\left(h+1\right)}{k}+\frac{j\left(j+1\right)}{k}+N=a\phantom{\rule{thinmathspace}{0ex}},$

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

(25)${h}^{3}=h+{N}_{\mathrm{SL}}^{\prime }$
(26)${j}^{3}=j+{N}_{\mathrm{SU}}^{\prime }$

where ${N}_{\mathrm{SU}}^{\prime }$ 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 $\left[{J}_{0}^{3},{J}_{n}^{±}\right]=±{J}_{n}^{±}$ and $\left[{J}_{0}^{3},{\psi }_{n}^{±}\right]=±{\psi }_{n}^{±}$) and analogously for ${N}_{\mathrm{SL}}^{\prime }$.

The eigenvalues of the lightcone Hamiltonian $H\sim J$ and of the longitudinal momentum ${p}_{-}\sim F$ are now defined by

(27)$H={h}^{3}+{j}^{3}$
(28)${p}_{-}=\frac{1}{k}\left({h}^{3}-{j}^{3}\right)$

and the task is to express these quantities as functions of each other and of the transverse excitations of the string:

(29)$H=H\left({p}_{-},N,{N}^{\prime }\right)$
(30)${p}_{-}={p}_{-}\left(H,N,{N}^{\prime }\right)$

using the above physical state condition. After a bit of algebra one finds the following

(31)$H=-1+{N}_{\mathrm{SL}}^{\prime }+{N}_{\mathrm{SU}}^{\prime }+\frac{N-a}{{p}_{-}-\left({N}_{\mathrm{SL}}^{\prime }-{N}_{\mathrm{SU}}^{\prime }\right)/k}$

or equivalently

(32)${p}_{-}=\frac{N-a}{1+H-{N}_{\mathrm{SL}}^{\prime }-{N}_{\mathrm{SU}}^{\prime }}+\frac{{N}_{\mathrm{SL}}^{\prime }-{N}_{\mathrm{SU}}^{\prime }}{k}\phantom{\rule{thinmathspace}{0ex}}.$

This gives the exact lightcone spectrum of strings in the ${\mathrm{AdS}}_{3}×{\mathrm{S}}^{3}$ 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 ${\mathrm{AdS}}_{3}×{\mathrm{S}}^{3}$ 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

(33)${N}_{}^{\prime }\to {h}_{\mathrm{fer}}+{N}_{\mathrm{bos}}^{\prime }+{N}_{\mathrm{fer}}^{\prime }$

in all the above expressions.

Posted at January 26, 2004 9:41 PM UTC

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## 1 Comment & 0 Trackbacks

### 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.