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

Posted by Urs Schreiber

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

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

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

(1)

{ds}^{2} = R_{SL}^{2} (- \cosh^{2} (ρ) {dt}^{2} + d ρ^{2} + \sinh (ρ) d ϕ^{2}) + R_{SU}^{2} (\cos^{2} (θ) d ψ^{2} + d θ^{2} + \sin (θ) d χ^{2}) .

together with a $B$ -field that provides the parallelizing torsion such that superstrings in this background are described by the $SL (2, R) \times SU (2)$ 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_{ϕ}

(3)

K_{+} : = \frac{1}{2} (e^{+ i (ϕ + t)} \tanh of ρ \partial_{t} - {ie}^{+ i (ϕ + t)} \partial_{ρ} + e^{+ i (ϕ + t)} \coth of ρ \partial_{ϕ})

(4)

K_{-} : = \frac{1}{2} (- e^{- i (ϕ + t)} \tanh of ρ \partial_{t} - {ie}^{- i (ϕ + t)} \partial_{ρ} - e^{- i (ϕ + t)} \coth of ρ \partial_{ϕ})

for ${AdS}_{3}$ and

(5)

J_{3} : = - \frac{i}{2} \partial_{ψ} - \frac{i}{2} \partial_{χ}

(6)

J_{+} : = \frac{1}{2} (- e^{+ i (χ + ψ)} \tan of ρ \partial_{ψ} - {ie}^{+ i (χ + ψ)} \partial_{θ} + e^{+ i (χ + ψ)} \cot of ρ \partial_{χ})

(7)

J_{-} : = \frac{1}{2} (e^{- i (χ + ψ)} \tan of ρ \partial_{ψ} - {ie}^{- i (χ + ψ)} \partial_{θ} - e^{- i (χ + ψ)} \cot of ρ \partial_{χ})

for $S^{3}$

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_{SL}^{2}}{4}

(13)

K_{+} \cdot K_{-} = - 2 \frac{R_{SL}^{2}}{4}

(14)

J_{3} \cdot J_{3} = - \frac{R_{SU}^{2}}{4}

(15)

J_{+} \cdot J_{-} = - 2 \frac{R_{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 $S^{3}$ factor. The following vector fields are adapted to the nature of this limit:

(17)

F = \frac{1}{k} (K_{3} - J_{3})

(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_{-},

where $1 / k$ is a real number. The limit $k \to ∞$ 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_{1}^{ast} P_{1} + P_{1}^{ast} 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 $SL (2, R) \times SU (2)$ 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 $η_{ab} J_{0}^{a} J_{0}^{b}$ 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 $K_{0}^{3}$ and $J_{0}^{3}$ (the momenta along $t$ and $ψ$ ) can be written as

(25)

h^{3} = h + N_{SL}^{'}

(26)

j^{3} = j + N_{SU}^{'}

where $N_{SU}^{'}$ grows by one for every $J_{- n}^{+}$ and every $ψ_{- n}^{+}$ excitation and is reduced by one for every $J_{- n}^{-}$ and $ψ_{- n}^{-}$ excitation (due to $[J_{0}^{3}, J_{n}^{\pm}] = \pm J_{n}^{\pm}$ and $[J_{0}^{3}, ψ_{n}^{\pm}] = \pm ψ_{n}^{\pm}$ ) and analogously for $N_{SL}^{'}$ .

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} (h^{3} - j^{3})

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^{'})

(30)

p_{-} = p_{-} (H, N, N^{'})

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

(31)

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

or equivalently

(32)

p_{-} = \frac{N - a}{1 + H - N_{SL}^{'} - N_{SU}^{'}} + \frac{N_{SL}^{'} - N_{SU}^{'}}{k} .

This gives the exact lightcone spectrum of strings in the ${AdS}_{3} \times S^{3}$ 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 ${AdS}_{3} \times S^{3}$ 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_{}^{'} \to h_{fer} + N_{bos}^{'} + N_{fer}^{'}

in all the above expressions.

Posted at January 26, 2004 9:41 PM UTC

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January 26, 2004