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

Posted by Urs Schreiber

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

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 $\mathrm{AdS}_3 \times \mathrm{S}^3$ .

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

The target space $\mathrm{AdS}_3 \times \mathrm{S}^3$ comes with the metric (we can ignore the $\mathrm{T}^4$ factor)

(1)

ds^2 = R_{\mathrm{SL}}^2 \left( -\cosh^2(\rho)\, dt^2 + d\rho^2 + \sinh(\rho) d\phi^2 \right) + R_{\mathrm{SU}}^2 \left( \cos^2(\theta)\, d\psi^2 + d\theta^2 + \sin(\theta) d\chi^2 \right) \,.

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

(2)

K_3 := -\frac{i}{2}\partial_t + \frac{i}{2}\partial_\phi

(3)

K_+ := \frac{1}{2} \left( e^{+i(\phi+t)}\tanh\of{\rho}\partial_t - ie^{+i(\phi+t)}\partial_\rho + e^{+i(\phi+t)}\coth\of{\rho} \partial_\phi \right)

(4)

K_- := \frac{1}{2} \left( - e^{-i(\phi+t)}\tanh\of{\rho}\partial_t - ie^{-i(\phi+t)}\partial_\rho - e^{-i(\phi+t)}\coth\of{\rho} \partial_\phi \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(\chi+\psi)}\tan\of{\rho}\partial_\psi - ie^{+i(\chi+\psi)}\partial_\theta + e^{+i(\chi+\psi)}\cot\of{\rho} \partial_\chi \right)

(7)

J_- := \frac{1}{2} \left( e^{-i(\chi+\psi)}\tan\of{\rho}\partial_\psi - ie^{-i(\chi+\psi)}\partial_\theta - e^{-i(\chi+\psi)}\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)

[K_3,K_\pm] = \pm K_\pm

(9)

[K_+,K_-] = -2 K_3

(10)

[J_3,J_\pm] = \pm J_\pm

(11)

[J_+,J_-] = +2 J_3 \,.

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 (K_3 + 1) + K_- K_+ + J_3(J_3 + 1) + J_- J_+ \,.

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^\ast_1 := \frac{1}{\sqrt{k}} K_-

(21)

P_2 := \frac{1}{\sqrt{k}} J_+

(22)

P^\ast_2 := \frac{1}{\sqrt{k}} J_- \,,

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(J+1) + P^\ast_1 P_1 + P^\ast_1 P_1 \,.

$F$ 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 $\eta_{ab}J_0^a J_0^b$ of that current algebra. The $L_0$ Virasoro constraint on a state of level number $N$ reads

(24)

-\frac{h(h+1)}{k} +\frac{j(j+1)}{k} + N = a \,,

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 $K^3_0$ and $J^3_0$ (the momenta along $t$ and $\psi$ ) can be written as

(25)

h^3 = h + N^\prime_{\mathrm{SL}}

(26)

j^3 = j + N^\prime_{\mathrm{SU}}

where $N^\prime_{\mathrm{SU}}$ 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 $[J^3_0,J^\pm_{n}] = \pm J^{\pm}_n$ and $[J^3_0,\psi^\pm_{n}] = \pm \psi^{\pm}_n$ ) and analogously for $N^\prime_{\mathrm{SL}}$ .

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(p_-,N,N^\prime)

(30)

p_- = p_-(H,N,N^\prime)

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

(31)

H = -1 + N^\prime_{\mathrm{SL}} + N^\prime_{\mathrm{SU}} + \frac{N-a}{p_- - (N^\prime_{\mathrm{SL}} - N^\prime_{\mathrm{SU}})/k}

or equivalently

(32)

p_- = \frac{N - a} { 1+ H - N^\prime_{\mathrm{SL}} - N^\prime_{\mathrm{SU}}} + \frac{N^\prime_{\mathrm{SL}} - N^\prime_{\mathrm{SU}}}{k} \,.

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

(33)

N^\prime_{} \to h_\mathrm{fer} + N^\prime_{\mathrm{bos}} + N^\prime_{\mathrm{fer}}

in all the above expressions.

Posted at January 26, 2004 9:41 PM UTC

TrackBack URL for this Entry: https://golem.ph.utexas.edu/cgi-bin/MT-3.0/dxy-tb.fcgi/297

The String Coffee Table

Skip to the Main Content

January 26, 2004