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January 26, 2004

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

Posted by Urs Schreiber

Strings on AdS 3 ×S 3 ×T 4 are a toy model for the more interesting (and more difficult) AdS 5 ×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 ×S 3 ×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 ×S 3 .

The target space AdS 3 ×S 3 comes with the metric (we can ignore the T 4 factor)

(1)ds 2 =R SL 2 (cosh 2 (ρ)dt 2 +dρ 2 +sinh(ρ)dϕ 2 )+R SU 2 (cos 2 (θ)dψ 2 +dθ 2 +sin(θ)dχ 2 ).

together with a B-field that provides the parallelizing torsion such that superstrings in this background are described by the SL(2 ,R)×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 :=i2 t+i2 ϕ
(3)K +:=1 2 (e +i(ϕ+t)tanhofρ tie +i(ϕ+t) ρ+e +i(ϕ+t)cothofρ ϕ)
(4)K :=1 2 (e i(ϕ+t)tanhofρ tie i(ϕ+t) ρe i(ϕ+t)cothofρ ϕ)

for AdS 3 and

(5)J 3 :=i2 ψi2 χ
(6)J +:=1 2 (e +i(χ+ψ)tanofρ ψie +i(χ+ψ) θ+e +i(χ+ψ)cotofρ χ)
(7)J :=1 2 (e i(χ+ψ)tanofρ ψie i(χ+ψ) θe i(χ+ψ)cotofρ χ)

for S 3

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

(8)[K 3 ,K ±]=±K ±
(9)[K +,K ]=2 K 3
(10)[J 3 ,J ±]=±J ±
(11)[J +,J ]=+2 J 3 .

The non-vanishing inner products are

(12)K 3 K 3 =R SL 2 4
(13)K +K =2 R SL 2 4
(14)J 3 J 3 =R SU 2 4
(15)J +J =2 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=1 k(K 3 J 3 )
(18)J:=K 3 +J 3
(19)P 1 =1 kK +
(20)P 1 ast:=1 kK

(21)P 2 :=1 kJ +
(22)P 2 ast:=1 kJ ,

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

(23)1 kC=F(J+1 )+P 1 astP 1 +P 1 astP 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)×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 abJ 0 aJ 0 b of that current algebra. The L 0 Virasoro constraint on a state of level number N reads

(24)h(h+1 )k+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 ±]=±J n ± and [J 0 3 ,ψ n ±]=±ψ n ±) and analogously for N SL .

The eigenvalues of the lightcone Hamiltonian HJ and of the longitudinal momentum p F are now defined by

(27)H=h 3 +j 3
(28)p =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 +Nap (N SL N SU )/k

or equivalently

(32)p =Na1 +HN SL N SU +N SL N SU k.

This gives the exact lightcone spectrum of strings in the AdS 3 ×S 3 SWZW model. The Penrose limit is again obtained by taking 1 /k0 . 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 ×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 h fer+N bos +N fer

in all the above expressions.

Posted at January 26, 2004 9:41 PM UTC

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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

I have had a little discussion about this with Luboš here.

Posted by: Urs Schreiber on March 9, 2004 2:07 PM | Permalink | Reply to this

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