CFT2

11 May 2000

Physics Letters B 480 Ž2000. 348–354

Logarithmic operators in AdS 3rCFT2 Alex Lewis 1 Department of Mathematical Physics, National UniÕersity of Ireland, Maynooth, Ireland Received 21 February 2000; accepted 27 March 2000 Editor: P.V. Landshoff

Abstract We discuss the relation between singletons in AdS 3 and logarithmic operators in the CFT on the boundary. In 2 dimensions there can be more logarithmic operators apart from those which correspond to singletons in AdS, because logarithmic operators can occur when the dimensions of primary fields differ by an integer instead of being equal. These operators may be needed to account for the greybody factor for gauge bosons in the bulk. q 2000 Published by Elsevier Science B.V. All rights reserved.

One particularly interesting example of the AdSrCFT correspondence w1–3x is the AdS 3rCFT2 correspondence, which relates supergravity on AdS 3 = S 3 to a 2-dimensional CFT. One advantage of this is that 2-dimensional conformal field theories are very well understood, and that makes AdS 3 especially suitable for studying the relation between singletons on AdS and logarithmic conformal field theories ŽLCFT., since almost all previous work on LCFT has concentrated on the 2-dimensional case. According to w2,3x, at the boundary of AdS Dq 1 we have a coupling between bulk fields F i Ž x, z . and boundary fields Oi Ž x ., Hd D xF i Oi , where the boundary fields are subject to the boundary condition

F i Ž x, z . s l i Ž x, R ., with z s R the boundary of AdS dq 1. The relation between correlation functions in CFTD and the bulk supergravity action is

Ý Hd ²e

i

D

x liO i

: s eyS wF i 4x

Ž 1.

This relation was used in Refs. w4,5x to show that, if there are singletons in AdS dq 1 , the theory on the boundary is in fact an LCFT. A theory of free singletons is formulated in terms of a dipole-ghost pair of fields A and B which satisfy w6x

Ž E mEm q m2 . A q B s 0, Ž E mEm q m2 . B s 0

Ž 2.

these fields have the bulk AdS action S s d Dq 1 x g Ž g mnEm A En B y m2AB y 12 B 2 .

H

1

E-mail: [email protected]

'

Ž 3.

The fields A and B couple to boundary fields C and D and using Eq. Ž1. the two-point functions of C

0370-2693r00r$ - see front matter q 2000 Published by Elsevier Science B.V. All rights reserved. PII: S 0 3 7 0 - 2 6 9 3 Ž 0 0 . 0 0 3 9 0 - 7

A. Lewisr Physics Letters B 480 (2000) 348–354

and similarly for L0 , where for singletons we have h s h and so D s 2 h. The theories with this type of operators are called Logarithmic CFTs and their properties have been studied extensively w8x since they were introduced in Ref. w7x. Applications of LCFT to strings and D-brane scattering were developed in Refs. w9,10x. A recent paper relevant to AdS 3 is w11x One way to see if fields of this type exist in a theory is to look at the four-point functions of ordinary fields. If there are no logarithmic operators, the operator product expansion for primary fields has the form Oi Ž x 1 . Oj Ž x 2 . ; Ý i

f ikj < x 12 < D iq D jy D k

O k Ž x 1 . q PPP

Ž 6. and ² Oi Ž x 1 . Oj Ž x 2 .: s < x 12
f ikj f ilj < x 12 < Diq D j < x 34 < D iq D j

FŽ x.

Ž 7.

where x s x 12 x 34rx 13 x 24 and F Ž x . has an expansion in powers of x, without any logarithmic singularity. If there are logarithmic operators however, the OPE has to be modified and we have instead 1 Oi Ž x 1 . Oj Ž x 2 . ; D q C ln < x 12 < 2 . D iq D jy D Ž < x 12 < q PPP

Ž 8.

which together with the two-point functions for C and D leads to four point functions of the form Ž7., x 0 but with F Ž x . x 2 D ln x. Indeed, logarithmic singularities have been found in four point functions calculated in supergravity on AdS 5 w12,13x, and it is possible that these could be an indication that there is an LCFT on the boundary of AdS 5 . However, these logarithms could also be accounted for as the perturbative expansion of anomalous dimensions in CFT4 , with no need for logarithmic operators w14x. The clearest evidence for the existence of logarithmic operators in AdSrCFT comes from calculations of grey-body factors in AdS 3 . Since grey-body factors are related to two-point functions in CFT, logarithms here are a clear indication that we have logarithmic operators on the boundary. The grey-body factor Žor absorption cross section. for a field in AdS 3 which couples to a field O Ž x . in the CFT on the boundary is related to the two point function in the CFT by w15,16x

™

6

and D are found to be Žsee Ref. w5x for details of the calculation. ² C Ž x . C Ž y . : s 0, c ²C Ž x . D Ž y . : s ² D Ž x . C Ž y . : s , < x y y < 2D 1 ² DŽ x . DŽ y. : s Ž d y 2 cln < x y y < . Ž 4 . < x y y < 2D with the dimension D given by DŽ D y D . s m2 , c s DŽ2 D y D . and d s 2 D y D. these are the usual two-point functions for logarithmic operators in CFT w7,8x. These correlation functions occur if the Hamiltonian Žin two dimensions, the Virasoro generator L 0 . is non-diagonalizable, and has the Jordan form L0 < C : s h < C : , L0 < D : s h < D : q < C : Ž 5.

349

p sabs s

v

Hd

2

x G Ž tyie , x . yG Ž tqie , x .

Ž 9.

where G Ž t, x . s ² O Ž x,t . O Ž0.: is the thermal Green’s function in imaginary time. This can be determined from the periodicity in imaginary time and the singularities of the Green’s function w15x, which if O is a primary field with weights h,h, are given by ² O Ž t , x . O Ž 0. : ;

CO

Ž 10 .

2h 2h xq xy

with x "s t " x. G Ž t, x . has the form

GŽ t, x. sC

ž ž

=

p Tq R sinh Ž p Tq xq .

p Ty R sinh Ž p Ty xy .

2h

/ /

2h

Ž 11 .

2 2 for a BTZ black hole with mass M s rq y ry , angular momentum J s 2 rq ry, left and right temperatures T "s Ž rq" ry .r2p , and Hawking temper-

A. Lewisr Physics Letters B 480 (2000) 348–354

350

ature given by 2rTH s 1rTqq 1rTy w17x. The absorption cross section is then w15,16x

sabs Ž h,h . s

2 hy1

v Ž 2p Ty R . 2 hy1 sinh 2TH G Ž 2 h. G Ž 2 h.

p C Ž 2p Tq R . v

ž

= G hqi

v 4p Tq

ž /

/ž

G hqi

v 4p Ty

gb sabs s p 2v R 2 1 q v Rsln Ž 2 v Rs .

S sabs s

/

2 hy1

Ž 2p Ty R . 2 hy1 G Ž 2 h. G Ž 2 h.

p C Ž 2p Tq R . v

2

v

Ž 12 .

This expression can be obtained either using the effective string method for supergravity w15x or using the AdsrCFT correspondence w16,18x. A large number of classical calculations of absorption cross sections have given results which are consistent with Ž12. Žor a similar expression for fermions. w15x, including calculations for several fields for the BTZ black hole w19,16x. However, in Ref. w19x the cross section for gauge bosons with spin 2, which couple to fields with h,h s Ž2,0. or Ž0,2. on the boundary was found to have logarithmic corrections which cannot be accounted for by Eq. Ž12.. In Ref. w20x, the grey-body factor for singletons was calculated, and while this does have a logarithmic correction to the cross section Ž12., it was found that it still does not give the correct cross section for the gauge bosons. The question we would like to address in this letter is, are there other kinds of logarithmic operators in AdS 3rCFT2 , and can they correctly account for the greybody factor for the gauge bosons? The greybody factor for the gauge bosons with spin s s 2 in AdS 3 , in the low temperature limit v 4 T ", was found to be w19x

Ž 13 .

In the low temperature limit, Eq. Ž12. becomes, up to a normalization whch is proportional to C Ž D s h q h.,

sabs Ž h,h . ; v 2 D y3

function for an ordinary primary field. The greybody S factor for a singleton is therefore given by sabs s EsabsŽ h,h.rED w20x, and so

Ž 14 .

So that the second term in Eq. Ž13. is an indication that the gauge bosons cannot just couple to ordinary primary fields on the boundary. The greybody factor for a singleton can also be calculated from Eq. Ž9., using the relation ² DŽ t, x . DŽ0.: s EDE ² C Ž t, x . DŽ0.:, since ² C Ž t, x . DŽ0.: is the same as the two point

ž / ž ž /

=sinh

2TH

v

=G h q i

v

G hqi 2

4p Tq

1 EC C ED

4p Ty

/

q ln Ž 2p Tq R .

qln Ž 2p Ty R . y c Ž h . y c Ž 2 h .

½ž

q 12 c h q i

ž

qc h q i

v

/ ž / ž

4p Tq

v 4p Tq

v

qc hyi

qc hyi

4p Tq

v 4p Tq

/5

/ ,

Ž 15 . which in the low temperature limit reduces to S sabs ; v 2 D y3 Ž 2ln Ž v R . q cX .

™

Ž 16 .

In an LCFT we always have the freedom to shift D by D D q lC, which leaves Eq. Ž5. invariant, and this can be used to choose any value for the constant cX . However, comparing Eqs. Ž16. and Ž13., we can see that the logarithmic term in Ž13. is multiplied by an extra factor of v R and is thus of a sub-leading order compared to Ž16.. The gauge boson cannot therefore be represented by a singleton in AdS 3 w20x. However, we cannot immediately conclude, as was said in Ref. w20x, that the gauge boson has nothing to do with the AdSrLCFT correspondence, because there is potentially a much richer spectrum of logarithmic operators in a two dimensional LCFT than has been considered so far. The logarithmic operators we have looked at so far arise when the dimensions of two of the primary fields O k in the OPE Ž6. become degenerate, which leads to logarithms in the four-point functions and the OPE has to be modified to include the logarithmic pair C and D, as in Eq. Ž8.. In fact, logarithms will also arise in the four

A. Lewisr Physics Letters B 480 (2000) 348–354

point function if two of the primary fields have dimensions which are not equal, but differ by an integer, so that it is a descendant of one primary field which becomes degenerate with the other primary field. This is because the function F Ž x . in the four-point function Ž7. usually satisfies a Fuchsian differential equation, such as a hypergeometric equation, and when there are no degenerate dimensions the solutions have the form

351

CFT, and we do not expect them in AdS Dq 1 for D ) 2. The logarithmic pair C and D still have the same correlation functions Ž4., and CX is just an ordinary primary field with the usual two point function ² CX Ž xq , xy . CX Ž y0 . : ;

1 2Ž hyN . 2 h xq xy

Ž 21 .

`

F Ž x . ; x Di

Ý an x n

Ž 17 .

ns0

but when two of the dimensions differ by an integer, say D2 s D1 q N, the second solution instead has the form `

F Ž x . ; x Di

Ý Ž a n x n q bn x n log x .

Ž 18 .

ns0

in this case we again have a logarithmic pair with the higher of the two dimensions which as before make the Hamiltonian non-diagonalizable, as in Eq. Ž5.. In addition the C field satisfies in both cases the usual condition for a primary field L n < C : s 0,

nG1

Ž 19 .

However, in the earlier situation where two primary fields became degenerate, D also satisfied this condition, while in the case where we have two fields whose dimension differs by an integer N we have instead N Ž L1 . < D : s b < CX : , L n < D : s 0, n G 2

Ž 20 .

where CX is another primary field, with conformal weights Ž h y N,h., and b is some constant. C is now not really a primary field, but rather a descendant of CX : < C : s syN < CX :, where syN is some combination of Virasoro generators and, in general, the other generators of the chiral algebra of the CFT, of dimension N. Eq. Ž19. then implies that C must be a null vector of the CFT, that is w L n , syN x s 0 for n G 1 w21x Žwhich is why the two-point function ² CC : s 0.. This type of logarithmic operator therefore cannot exist with any dimension, but only with those dimensions for which there are null vectors of the algebra. Because of this, we can only have these generalized logarithmic operators in 2-dimensional

so it seems that these new fields cannot give us anything new when we compute greybody factors. However, it is easy to see that we can reproduce the greybody factor for the gauge bosons Ž13. if we assume that they correspond not to one of the fields C, D or CX in the LCFT, but to a linear combination of all three. This might happen, for example, if the bosons can be thought of as arising from the fusion of two primary fields, since C, D and CX must always appear together in any OPE. Then if CX has dimension D s 2, as is expected for the gauge bosons w20x, and CX ,C, D form a representation of the type discussed above with N s 1, the greybody factor will have exactly the right form, with the logarithmic term being of sub-leading order. Of course, this would imply that the representation which includes the primary field CX must have a null vector at level 1. This would be true if CX has Ž h,h. s Ž0,2. Žor Ž2,0.., as then Ly1 < CX : Žor Ly1 < CX :. is a null vector. If CX has Ž h,h. s Ž1,1., there could still be a null; vector if, for example, the CFT on the boundary has a conserved current for which Jy1 < CX : s 0. Now that we know there could be fields in an LCFT on the boundary that give the correct greybody factor for the gauge bosons, the next question we address is, what sort of fields in the bulk can couple to these fields on the boundary? To answer this question, we start be reviewing how the conformal weights of fields on the boundary determine the mass and spin of fields in the bulk when there are no logarithmic operators. We write the metric for AdS 3 in the form ds 2 s l 2 Ž ycosh2r dt 2 q sinh2r d f 2 q d r 2 .

Ž 22 .

In these coordinates the Virasoro generators L 0 , L " 1 , with commutators w L0 , L " 1 x s .L " 1 and w L1 , Ly1 x

A. Lewisr Physics Letters B 480 (2000) 348–354

352

s 2 L0 , for spin s fields are Ž u s t q f , Õ s t y f . w22,23x: L0 s i Eu ,

ž

Ly1 s ieyi u coth2 rEu y

1 sinh2 r

EÕ

Ds uqÕqf Ž r . C

i i q Er y scoth r , 2 2

ž

i

1 sinh2 r

EÕ

D s u q Õ y 2 iln Ž cosh r . q d C

i

y Er q scoth r 2 2

/

l

™

eyi Ž h uqh Õ.

Ž 24 .

Ž cosh r . hq h

The second Casimir of sl Ž2, R . is 2

1 2

L s Ž L1 Ly1 q Ly1 L1 .

y L20

Ž 25 .

and similarly for L2 , so that, using Eqs. Ž22. and Ž23., the sum of the two Casimirs is L2 q L2 s yl 2E mEm q s 2 coth2r

Ž 26 .

For a primary field, Ž L2 q L2 .F s Ž2 hŽ h y 1. q 2 hŽ h y 1..F , so Eq. Ž26. can be written as

ž

m

yE Em q

s2 l 2 sinh2r

/

F s m2F

Ž 27 .

which is the equation of motion for a field with spin s and mass m in AdS 3 , with the mass l 2 m2 s 2 h Ž h y 1. q 2 H Ž h y 1. y s 2 s D Ž D y 2.

Ž 30 .

Ž 23 .

and similarly for L0 , L " 1 with u Õ and s ys. For a primary fields F , the conditions L0F s hF and L0F s hF , and L1F s L1F s 0 can then be solved to give s s h y h and

F;

Ž 29 .

where the function f Ž r . will depend on what type of logarithmic operator we have. In the simplest case, which we expect to give us singletons, we have L1 D s L1 D s 0, which has the solution

/

L1 s ie i u coth2 rEu y

on the boundary. C satisfies the same conditions as F above, so we find C ; eyi Ž h uqh Õ.rŽcosh r . hqh. The conditions L0 D s hD q C and L 0 D s hD q C then imply that

Ž 28 .

So we can see that the conformal weights h and h on the boundary completely determine the mass and spin of the fields in AdS 3 Žand vice versa.. We can repeat this analysis for a logarithmic pair C and D

where d is an arbitrary constant, which we can set to any value using the freedom to shift D by an amount proportional to C. Evaluating the second Casimirs gives the equations of motion for the fields in AdS 3 which will couple to C and D:

ž ž

yE mEm q

yE mEm q

s2 l 2 sinh2r s2 l 2 sinh2r

/ /

C s m2 C,

D s m2 D q 4 Ž D y 1. C

Ž 31 .

with m2 again given by Eq. Ž28.. When s s 0, these are just the expected equations of motion for singleton dipole-pair Ž2. Žapart from a different normalization of C ., with the expected relation between the singleton mass m and the dimension of the logarithmic operator D. Thus we can see that the mass and spin in AdS 3 are still completely determined by the data of the LCFT on the boundary when we have singletons and logarithmic operators. This also give us another way of seeing, as was found in Ref. w5x that there can be no logarithmic operators with D s 1, corresponding to m2 s y1. Now we can use this map between AdS 3 and CFT2 to see what kind of operators will couple to the other kinds of logarithmic operators. In this case we have three fields C, D and CX , but since C and CX are both primary fields they will both have the same form as before, but with weights Ž h,h. for C and Ž h y N,h. for CX . D is then given by Eq. Ž29. with f Ž r . a solution of the Ž N q 1.’th order differential

A. Lewisr Physics Letters B 480 (2000) 348–354

equation Ž L1 . Nq 1 D s 0. The second casimirs then give the equations of motion in AdS 3 as

ž ž

s2

m

yE Em q

yE mEm q

l 2 sinh2r s2 l 2 sinh2r

/ /

C s m2 C,

D s m2 D q 4 Ž D y 1. C q C

Ž 32 . Which is the same as the equations of motion for the singleton except that we have the new field is C s Ly1 L1 D. C will therefore be a descendant of CX , or in AdS 3 it will correspond to some derivative of the field which couples to the primary field CX , and the action for the singleton Ž3. should be modified by adding a term which couples the singleton to the new field. We will therefore have an interacting theory instead of a free singleton, with an action of the form S s d3 x g Ž g mnEm A En B y m2AB y 12 B 2

H '

ql AC .

353

body factor for the spin-2 gauge bosons. It is an interesting question why the same interactions cannot be introduced for singletons which do not have special values for the mass, which would lead to a contradiction in the CFT, but is not obviously forbidden from the three-dimensional point of view. Possibly related is the question of why these type of fields can exist in AdS 3 but not in AdS Dq1 for D ) 2 – since the full Virasoro algebra applies only to CFT in 2 dimensions, there are no null vectors in D ) 2 and so these type of logarithmic do not exist, although there can be singletons and the ordinary logarithmic pair of C and D in any dimension.

Acknowledgements This work was supported by Enterprise Ireland grant no. SCr98r739.

References

Ž 33 .

where C is a derivative of a field with spin N, for a spinless singleton. In addition, it is important that C is also a descendant in this case, and so the field A in the above action is also a derivative of the field which couples to CX and is not a fundamental field itself. This is especially significant for the case when N s 1, since then CX has no descendant at level 1 except C itself, and so the action in AdS 3 in this case is the same as for the ordinary singleton, except that B is now a derivative of a field BX with spin 1. Of course, we also need to add to the action the kinetic and mass terms for the field BX to treat this field properly. Although we have seen that new kinds of logarithmic operators can exist in AdS 3rLCFT2 , they cannot exist for just any values of m2 and s - we have to have a null vector in the CFT on the boundary. This means that to determine if such fields really exist we need to know more about the structure of the CFT, or to calculate four-point functions, from which the OPE could be deduced. However, it seems to be clear that at least one example of this type of operator is needed to give the correct grey-

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