Type IIB string theory on AdS5×Tnn′

12 August 1999

Physics Letters B 460 Ž1999. 281–287

X

Type IIB string theory on AdS5 = T nn Dileep P. Jatkar

a,b,1

, S. Randjbar-Daemi

a,2

a

b

Abdus Salam International Centre for Theoretical Physics, Strada Costiera 11, Miramare, Trieste 34100, Italy Mehta Research Institute of Mathematics and Mathematical Physics 3 , Chhatnag Road, Jhusi, Allahabad 211 019, India Received 5 May 1999 Editor: L. Alvarez-Gaume´

Abstract X

We study Kaluza-Klein spectrum of type IIB string theory compactified on AdS5 = T nn in the context of AdSrCFT correspondence. We examine some of the modes of the complexified 2 form potential as an example and show that for the states at the bottom of the Kaluza-Klein tower the corresponding d s 4 boundary field operators have rational conformal dimensions. The masses of some of the fermionic modes in the bottom of each tower as functions of the R charge in the boundary conformal theory are also rational. Furthermore the modes in the bottom of the towers originating from q forms on T 11 can be put in correspondence with the BRS cohomology classes of the c s 1 non critical string theory with ghost number q. q 1999 Published by Elsevier Science B.V. All rights reserved.

1. Introduction It has recently been suggested by Klebanov and Witten w1x that the world volume super Yang Mills theory of parallel D 3-branes near a conical singularity of a Calabi-Yau manifold are related to string theory on AdS5 = T 11 . They argue that string theory on AdS5 = T 11 , with T 11 s Ž SUŽ2. = SUŽ2..rUŽ1. can be described in terms of an N s 1 superconformal SUŽ N . = SUŽ N . gauge theory. In the absence of a rigorous proof of the AdSrCFT correspondence w2,3x one needs to have the Kaluza-Klein spectrum of the supergravity in the above geometry. The Kaluza-Klein masses are used to identify the anomalous dimensions of the operators on the Yang Mills side w4,5x. 1

E-mail: [email protected] E-mail: [email protected] 3 Permanent address. 2

The manifold T 11 also enters in the geometrical description of c s 1 matter compactified on a circle at the self dual radius coupled to the d s 2 gravity, the so called non critical d s 2 string theory. It has been shown in w6x that the symmetry algebra of the volume preserving diffeomorphisms of the cone based on T 11 is in fact in a 1 y 1 correspondence with the symmetry algebra of Ž1,0. and Ž0,1. conformal primary fields of the c s 1 theory at the self dual radius. It is therefore tempting to ask if there is any relationship between the c s 1 theory and the type IIB string theory compactified on AdS5 = T 11 . We will also touch upon the relation of Kaluza-Klein spectrum on T n n and c s 1 string theory at n-times the self dual radius in the last section. The Kaluza-Klein modes are characterised by a set of quantum numbers  l,lX , s4 , where l and lX characterise the SUŽ2. = SUŽ2. multiplets and s, related to the R charge in the Yang Mills side, is a UŽ1. charge taking integer or 1r2 integer values.

0370-2693r99r$ - see front matter q 1999 Published by Elsevier Science B.V. All rights reserved. PII: S 0 3 7 0 - 2 6 9 3 Ž 9 9 . 0 0 8 1 0 - 2

D.P. Jatkar, S. Randjbar-Daemir Physics Letters B 460 (1999) 281–287

282

The lowest values of l and lX will depend on s. For any given s there is an infinite tower of Kaluza Klein modes. We shall show that for any given s and for some Žbut not all. of the modes with the smallest possible values of l and lX there corresponds well defined cohomology classes in the ghost numbers 0, 1 and 2 sectors of the c s 1 theory. More precisely we shall show that the Kaluza Klein modes in AdS5 originating from q forms in T 11 are in correspondence with the cohomology classes with ghost number q on the c s 1 side. They have identical SUŽ2. = SUŽ2. quantum numbers. However the modes originating from the components of the metric in the internal space seem to be unmatched. Along the way we shall show that the derivation of the Kaluza-Klein spectrum can be reduced to the problem of the diagonalization of differential operators acting on tensors and spinors in the background of monopole fields on S 2 = S 2 . As an example we give the Kaluza-Klein masses of some of the bosonic and fermionic fields. Possible correspondence between c s 1 theory at self dual radius and the type IIB strings on conifold singularities have been noted in the past by Ghoshal and Vafa w7x. We shall comment on this paper at the end the present note. X

2. The manifolds T nn and the Kaluza-Klein modes Although manifolds T n n are related to the c s 1 string theory,X Kaluza-Klein spectrum can be studied on any T n n for n / nX . Below we will study the Kaluza-Klein spectrum of Type IIB string theory X compactified on T n n . In the end when we will relate our results to c s 1 string theory we will take n s nX . X The geometry of the manifolds T n n is defined in terms of the metric w8x ds 4 s c 2 Ž dy5 y ncos y 1 dy 2 y nX cos y 3 dy4 .

2

q a 2 Ž dy12 q sin2 y 1 dy 22 . q aX 2 Ž dy 32 q sin2 y 3 dy42 . .

be taken to beX integers. We shall denote these manifolds by T n n . Locally they look like S 2 = S 2 = S 1. Their isometry group is SUŽ2. = SUŽ2. = UŽ1., where the UŽ1. factor is due to the translational invariance of the coordinate y 5. It has been argued by Klebanov and Witten that the UŽ1. factor should be identified with the R symmetry of the world volume N s 1, d s 4 superconformal field theory which arises as a consequence of the AdSrCFT correspondence. We shall see that it can also be put in correspondence with the UŽ1. group generated by the Liouville mode of the c s 1 string theory. In this way the R charges of the boundary d s 4 superconformal theory will be set in correspondence with the Liouville momenta of the cX s 1 theory. In order for AdS5 = T n n to be a supersymmetric solution of the type IIB field equations it is necessary that a s a X s 1 , n s "nX and ec s y 3n1 ,

'6 N e N

where, e 2 is related to the AdS5 cosmological constant through Rmn s y4 e 2 gmn . For the present discussion we can keep the background parameters arbitrary. Our strategy for the Kaluza Klein expansion is to dispose of the y 5 coordinate by Fourier expansion. As a result of this we obtain infinite number of fields living on AdS5 = S 2 = S 2 . Writing every object in an orthonormal basis of the S 2 = S 2 manifold we obtain infinite number of fields coupled to a magnetic monopole field of charge ns on the first S 2 and nX s on the second S 2 . The harmonic expansion on the magnetic monopole background has been studied in detail in w9x. We will use the formalism of this reference to write down the eigenvalues of the S 2 = S 2 Laplacian acting on any field. In this way we will obtain the Kaluza Klein modes which depend on the coordinates of AdS5 only. To implement this idea we need the components X of the 5-bein and the spin connections on T n n . They are given by e11 s

Ž 1.

All the coordinates y a are angles, such that y s Ž y 1, y 2 . parametrise a S 2 of radius a while yX s Ž y 3, y 4 . parametrise a S 2 of radius aX . The angle y 5 ranges from 0 to 4p . The constant c is the radius of the circle defined by y 5. The constants n and nX will

e33 s e45 s

1 a 1 a

,

e22 s

,

e44 s nX

X

a cot y 3

,

1 asin y 1 1 X

a sin y 3 e55 s

1 c

e25 s

,

n acot y 1

,

, .

Ž 2.

D.P. Jatkar, S. Randjbar-Daemir Physics Letters B 460 (1999) 281–287

From these one can calculate the components of the spin connections. They are given by vawbcx s va´bc , v5wabx s e ´ab ,

vawb5x s ye ´abvmwnlx s vm ´nl , v5wmnx s e ´mn ,

vmwn5x s ye ´mn ,

Ž 3.

where v a s Ž v 1 s 0, v 2 s y 1a cot y 1 . and v m s Ž v 3 s 0, v 3 s y a1X cot y 3 .. These are just the components of a magnetic monopole potential on each S 2 . For what follows it is important to note that e a5 s yn v a and e m 5 s ynXv m . To clarify the scheme let us look at the derivatives of a d s 10 scalar F Ž x, y 5, y, yX .. First we write 5

X

F Ž x , y , y, y . s

is y5 s

e F Ž x , y, y . .

Ý sg

1 2

X

Ž 4.

Z

5

Since the angle y ranges over a period of 4p , s takes integer as well as X1r2 integer values. The derivative operator on T n n then acts on the Fourier modes F s according to

In these formulae the spheres are regarded as the coset manifolds SUŽ2.rUŽ1.. The SUŽ2. group ele. ment L y represents the point y g S 2 and Dnl s, lŽ Ly1 y are the matrix elements of the unitary irreducible representations of SUŽ2. carrying spin l. Their most important property for us is that

= " Dql , l Ž Ly1 y . i

s

l y1 '2 a1 (Ž l . q . Ž l " q q 1. Dq , l Ž L y . .

D5F Ž x ,, y, y . s

is c

'2

the UŽ1. basis by a prime, viz, = "X s 1 Ž= 3 .i= 4..

'2

From these relations we can easily read the eigenvalues of the Laplacian action on the charge q objects on S 2 , viz, l y1 =a=aDql , l Ž Ly1 y . s Ž =q =yq =y =q . Dq , l Ž L y .

1

X

F Ž x , y, y . ,

s =aF s Ž x , y, y . , Dm F s Ž x , y, yX . s Ž Em y inX s vm . F s Ž x , y, yX . s =m F s Ž x , y, yX . .

Ž 5.

We thus see that = aF s Ž x, y, yX . and = m F s Ž x, y, yX . are precisely the orthonormal frame components of the covariant derivatives of a charged scalar field coupled to UŽ1. monopole fields on each sphere. Furthermore the UŽ1. charge of F s on the first S 2 is ns and on the second S 2 it is nX s. The above covariant derivatives dictate the form of harmonic expansions for each F s. Following the notation of w9x we can write

Ž 2 l q 1.

X

l y1 X X =Dnl s, l Ž Ly1 y . Dn s, l Ž L y . .

Ž 6.

Ž 2 lX q 1 .

l GNn sN,N l NFl X

s

2

ž / c

q =a =a q =m =m ,

Ž 10 .

we can simply read the AdS5 mass of the mode X X Ž x . as Fls,l,l ,l 2

m s

s2 c

2

1 q a X2

a ,l X Fls,l ,l Ž x .

Ý

Da D a s y

1

X

Ž 9.

Noting that

q 1r2

X

Ž Dm D m q Da D a . F Ž x , y 5 , y, yX . s 0.

1r2

lGNn 1 sN,N l NFl

=

Ž l Ž l q 1. y q 2 . Dql , l Ž Ly1 y ..

The eigenvalues of the 5 dimensional Laplacian on F s can now be read immediately. As an example let us consider the complex dilaton. The linearised equation for this field is given by

X

Ý

a2

Ž 8.

DaF s Ž x , y, yX . s Ž Ea y ins va . F s Ž x , y, yX .

F s Ž x , y, yX . s

Ž 7.

The UŽ1. basis " on the first S 2 are defined by = "s 1 Ž= 1 .i= 2.. On the second S 2 we shall denote

sy X

283

2

Ž l Ž l q 1. y Ž ns . 2 .

Ž lX Ž lX q 1. y Ž nX s . 2 . ,

Ž 11 .

where l GN ns N and lX GN nX s N . For the next example we consider the complex, 2-form potential BM N . Firstly, The components Bmn X are T n n scalars. Therefore they must be treated

D.P. Jatkar, S. Randjbar-Daemir Physics Letters B 460 (1999) 281–287

284

object Bp q of charge p on the first S 2 and charge q in the second S 2 we have,

exactly in the same way as F . The remaining components can be decomposed into various tensors on S 2 = S 2 . Here we list the result, Da Da Bm 5 s y

s

ž / c

=aBp q s Ž Ea y ipva . Bp q

2

q =a =a q =m =m y 4 e 2

and =m Bp q s Ž Em y iq vm . Bp q , where E a and E m denote the partial derivatives in the orthonormal basis. The Ž p,q . charges of Bm 5, Bm ", BmX ", and B " Ž " .X are, respectively, Ž ns,nX s ., Ž ns " 1,nX s ., Ž ns,nX s " 1. and Ž ns " 1,nX s " 1.. These charges determine the lower bounds of l and lX in the harmonic expansion of Bp q on S 2 = S 2 . The expansion of Bp q is identical to the one of F given above except that we should put the lower bounds l GN p N and lX GN q N . Other bosonic fields can be treated in a similar way. We tabulate the Ž p,q . charges of all of the bosonic fields of the type IIB supergravity in Table 1. The technique outlined above can be used to obtain the Kaluza Klein masses of small perturba-

=Bm5 y 2 e ´ab=aBm b y 2 e ´mn=m Bm n ,

Ž 12 . Da Da Bm "s y

ž

s c

2

.e

q =a =a q =m =m y e 2

/

=Bm ". 2 ie= " Bm 5 , Da Da BmX "s y

ž

s c

Ž 13 .

2

.e

/

q =a =a q =m =m y e 2

=BmX ". 2 ie= "X Bm 5 .

Ž 14 .

In these relations the covariant derivatives are defined by the UŽ1. charge of each object. For an

Table 1 X The quantum numbers of the KK spectrum of type IIB string theory on AdS5 = T n n . Correspondence with the c s 1 model at the self dual X radius holds for n s n s 1 and for each KK tower only for the modes at the bottom of the tower. The rank of the differential form on T 11 is mapped to the ghost number on the c s 1 side. The c s 1 spectrum given in the table is at self dual radius and for s G 0 only. Similar analysis can be done for negative s KK excitations of IIB

X

c s 1 analog

X

q XŽ . Ž . Ysq1, p z Os, p z Ys,qp Ž z . Osy1, pX Ž z .

Constraints on l and l X

X

hmn , Amnrs , Bmn , F

l GN sn N , l GN sn N

hmq, Bm q, Amnr q,

l GN sn q 1 N , l GN sn N

Amnr yX , hm yX , Bm yX

l GN sn N , l GN sn y 1 N

hmy, Bm y, Amnr y

X

X

Os, p Ž z . Os, pX Ž z .

X

X

X

l GN sn y 1 N , l GN sn N X

X

X

X

Amnr qX , hm qX , Bm qX

l GN sn N , l GN sn q 1 N

hm 5 , Amnr 5 , Bm 5 h "" h " Ž " .X hŽ " .X Ž " .X h" 5 hŽ " .X 5 h 55

l GN sn N , l GN sn N X X l GN sn " 2,0 N , l GN sn N X X l GN sn " 1 N , l GN sn " 1 N X X l GN sn N , l GN sn " 2,0 N X X l GN sn " 1 N , l GN sn N X X l GN sn N , l GN sn " 1 N X X l GN sn N , l GN sn N

Amnqy, AmnqXyX , Bqy, BqXyX

l GN sn N ,l GN sn N

X

X

X

Amn " Ž " .X , B " Ž " .X

l GN sn " 1 N , l GN sn " 1 N

Amnq5 , Bq5

l GN sn q 1 N , l GN sn N

X

X

X

X

AmnyX 5 , ByX 5

l GN sn N , l GN sn y 1 N

Amny5 , By5

l GN sn y 1 N , l GN sn N

AmnqX 5 , BqX 5

X

X

X

X

l GN sn N , l GN sn q 1 N

Osy1, p Ž z .Ys,qpX Ž z . q XŽ . Os, p Ž z .Ysq1, p z aŽ z, z . Os, p Ž z . Os, pX Ž z .

Ys,qp Ž z .Ys,qpX Ž z . Ysq" 1, p Ž z .Ysq" 1, pX Ž z . q XŽ . Ž . aŽ z, z .Ysq1, p z Os, p z q aŽ z, z .Ys, p Ž z . Osy1, pX Ž z .

aŽ z, z . Osy1, p Ž z .Ys,qpX Ž z . q XŽ . aŽ z, z . Os, p Ž z .Ysq1, p z

D.P. Jatkar, S. Randjbar-Daemir Physics Letters B 460 (1999) 281–287

tions around the background solution AdS5 = T n n. The full analysis, although straightforward, is quite lengthy and will not be given here. Here we shall give the result of calculation of the masses of those modes which decouple without too much labour from the rest of the spectrum. The easiest one is the complex scalar for which we gave the spectrum of the masses in the previous section. We shall next consider the complex 2-form potential BM N . Note that for each s there is an infinite tower of modes in each field. We shall consider only some of these towers whose mass can easily be deduced. For each s this happens for the modes with the smallest values of l and lX . Consider first the modes with ns G 1. In this case the modes Bms,yn sy1, n s decouple from the rest of the system. Using the background values of 1ra2 s 6 e 2 ,1rec s y3n we obtain n sy1, n s 2 D m Dw m Bns,xy 2

y e 2 Ž 3ns q 1 . y 1 Bns,yn sy1, n s s 0.

Ž 15 .

s, n s, n sy1 Identical equation will result for BnX y . For the range of ns F y1 it is the leading modes s,y n s,y n sy1 of Bns,qy n sy1,y n s and BnX q which decouple. Their equation of motion becomes y n sy1,y n s 2 D m Dw m Bns,xq 2

y e 2 Ž 3ns y 1 . y 1 Bns,qy n sy1,y n s s 0

285

The same phenomenon also happens for the complex dilaton modes. This was observed by Gubser w11x. From our Eq. Ž11. it follows that for any given s the smallest masses are obtained for l s lX sN ns N , which is m2 s e 2 wŽ3 N ns N q2. 2 y 4x. Plugging this in the formula for the dimension with p s 0 produces the positive root D s 4 q 3 N ns N . Gubser has conjectured that for a given s this will happen to every mode in the bottom of the tower with that value of s. We can also obtain the fermion masses. The easiest one is the dilatino. It satisfies the equation

Ž Du x q iDu y q e . x s 0.

Ž 17 .

We categorise the spectrum of iDu y using the quantum number s. For the sake of simplicity let us set n s nX s 1. Let us take N s NG 1r2. In this case we have four subsectors. We will use the notation e Ž s . to denote sign of s and define e Ž0. s 1 in the fourth sector below. We shall also introduce the 2 function n Ž l, s . through n Ž l, s . s 1a Ž l q 12 . y s 2 X X X and define n Ž l , s . s n Ž l , s .. In the first sector we have only one eigenvalue

(

lŽ l,lX . s l Ž y 12 q N s N ,y 12 q N s N . s ye Ž s . e Ž 1 q 3 N s N . .

Ž 18 .

Ž 16 .

s,y n s,y n sy1 and an identical equation for BnX q . Following w10x we shall identify e 2 wŽ3ns q 1. 2 y s, n s, n sy1 1x as the AdS mass 2 of Bms,yn sy1, n s and BnX y . 2 2 wŽ Likewise e 3ns y 1. y 1x will be identified with s,y n s,y n sy1 the AdS mass of Bns,qy n sy1,y n s and BnX q . According to the AdSrCFT correspondence every bulk field in AdS5 should correspond to a well defined gauge invariant operator in the boundary d s 4 theory. For an AdS mode of mass m which originates from a p form field in the internal space the conformal dimension D of the d s 4 theory is given by Ž D y p .Ž D q p y 4. s m2 . Clearly only very small subset of modes will produce a rational solution for D. This is what happens to the modes singled out above, namely, the conformal dimension s, n s, n sy1 of the fields dual to Bms,yn sy1, n s and BnX y turn Ž . out to be 3 1 q ns and those of the fields corres, n s, n sy1 sponding to Bms,qn sy1, n s and BnX q are equal to 3ns q 1.

The second sector is defined by l sN s N y1r2, lX GN s N q1r2. Here we have two towers parametrised by lX

l " Ž y 12 q N s N ,lX . s ye Ž s .

e 2

(

"

n X2 q

e2 4

1 q 12 N s N Ž 1 q 3 N s N .

Ž 19 .

and in particular for lX sN s N q1r2 the eigenvalue is rational

l " Ž y 12 q N s N , 12 q N s N . s ye Ž s .

e 2

"

NeN 2

Ž 6 N e N q5. .

Ž 20 .

D.P. Jatkar, S. Randjbar-Daemir Physics Letters B 460 (1999) 281–287

286

In the third sector, which is defined by l GN s N q1r2, lX sN s N y1r2, for each s again we have two towers parametrised by l l " Ž l,y 12 q N s N . e s ye Ž s . 2

(

"

n 2q

where l j indicates the eigenvalues of iDu y and j runs over various sectors. The fermionic modes in the direction of the Killing spinor in T n n satisfy equations similar to those in Ref. w10x.

e2

1 q 12 N s N Ž 1 q 3 N s N . Ž 21 . 4 and as in the case of sector two, for l sN s N q1r2 the eigenvalue is rational. i.e., l " Ž 12 q N s N ,y 12 q N s N . e NeN s ye Ž s . " Ž 6 N e N q5. . Ž 22 . 2 2 In the fourth sector, we will relax the condition to N s NG 0. Thus the s s 0 modes are included in this sector. This sector is defined by l GN s N q1r2, lX GN s N q1r2. Here we have four towers and they are given by e l1" Ž l,lX , s . s ye Ž s . 2

(

2

" n 2 q n X 2 q e 2 Ž 3 N s N q 12 . , X l" 2 Ž l,l , s . s e Ž s .

(

e 2 2

" n 2 q n X 2 q e 2 Ž 3 N s N y 12 . .

Ž 23 . X

In this case, however, we find that for l s l sN s N q1r2, only two eigenvalues, l2 become rational, except for the case s s 0 when l1 also becomes rational. l1" Ž 12 q N s N , 12 q N s N . e NeN '36 s 2 q 108 N s N q49 , s ye Ž s . " 2 2 1 1 l" 2 Ž 2 q N s N , 2 q N s N. e NeN se Ž s. " Ž 6 N s N q7. . Ž 24 . 2 2 From the above list of the eigenvalues of iDu y one can also obtain the masses of the AdS gravitinos as well as the gamma trace of the gravitino, g m cm . This latter field satisfies an equation similar to Ž17. in which e is replaced by 3e, whereas the Dm traceless and gamma traceless gravitino fm satisfies Du x fŽjm . y Ž l j y e . fŽjm . s 0.

3. Relation to c s 1 string theory The fundamental fields of the c s 1 theory are two scalars X and f and the b,c ghost fields. X is targeted on a S 1 while f is coupled to a background charge. At the self dual radius R s 1r '2 of the circle there is a SUŽ2. = SUŽ2. symmetry. The BRS cohomology classes are organised according to the representations of this group. These classes are also labeled by their ghost numbers. Of interest to us are the ghost number zero, one and two operators given q Ž . XŽ . respectively by Os, p Ž z . Os, pX Ž z ., Ysq1, p z Os, p z as q well as aŽ z, z . Os, p Ž z . Os, pX Ž z . and Ys, p Ž z .Ys,qpX Ž z .. The complex conjugates of these operators should also be added to the list. In each case the subscript s characterises the SUŽ2. = SUŽ2. content of each object. For a given integer or 1r2 integer s the indices p and pX range from ys to qs w6x. Now consider a Kaluza-Klein tower originating from a q-form field in T 11 . For a given UŽ1. charge s we consider the modes at the bottom of each tower Žthose which presumably have rational conformal dimensions in the boundary gauge theory.. Our observation is that these Kaluza-Klein modes are in correspondence with the ghost number q cohomology classes in the c s 1 theory. In Table 1 we have listed all the Kaluza-Klein modes and the corresponding objects in the c s 1 model. Note that the modes originating from the components of the metric in T 11 Žwhich are not q-forms in the internal space!. do not seem to have a counterpart in the c s 1 side. Furthermore the modes corresponding to an operator containing aŽ z, z . in the c s 1 side can actually be gauged away in the Kaluza-Klein side. The c s 1 theory has an infinite dimensional algebra given in terms of the volume preserving diffeomorphisms of the quadric cone a1 a 2 y a 3 a4 s 0. It is known that the base of this cone is isomorphic to T 11 . In the context of present discussion the 3 com-

D.P. Jatkar, S. Randjbar-Daemir Physics Letters B 460 (1999) 281–287

plex dimensional Ricci flat cone is in fact identical to the subspace transverse to the D 3-brane solution of the type IIB supergravity. Our Kaluza-Klein background is a near horizon approximation to this D 3 brane geometry. Thus the cone seems to be the common geometrical entity in the two very different looking theories 4 . Let us consider the quadric cone a1 a 2 y a 3 a4 s 0. This cone is singular at its apex a1 s a 2 s a 3 s a 4 s 0. One can resolve the singularity by deforming the defining equation into a1 a2 y a3 a 4 s m. From the point of c s 1 theory m corresponds to the 2-dimensional cosmological constant. One can also consider a topological s model targeted on a CY three fold near a conical singularity, for which the local equation is the same as our quadratic expression. For both of these theories the free energies can be evaluated as a function of m and can be expressed as a genus expansion. In w7x Ghoshal and Vafa argued that in fact the two theories must be the same. They observed that, at the self dual radius, the g s 0, 1 and 2 contributions to the free energy of the c s 1 theory agree with the corresponding terms of the free energy of the topological sigma model near the conifold singularity. Subsequently, assuming the type II-Heterotic duality, the results of w13x gave further support to the Ghoshal Vafa conjecture. These authors calculated the coefficient of the term R 2 F 2 gy2 in the effective action of the Heterotic theory compactified on K 3 = T 2 and realised that for any g the coefficient is also given by the genus g term of the partition function of the c s 1theory at the self dual radius. More recently Gopakumar and Vafa w14x have calculated the s model partition function near a conifold singularity and have proven the conjecture made in w7x. A better understanding of the correspondences noted above may require the unraveling of the relevance of the volume preserving diffeomorphisms of the cone in the D 3 brane context. On the basis of the observations made in this note we would like to

4

Similar remarks can be made about the manifolds T n n . In this case we should consider the cs1 string theory at n times the self-dual radius w12x.

287

think that the c s 1 theory at the self dual radius has a role to play in organising the chiral primaries of the boundary SU Ž N . = SU Ž N . superconformal gauge theory.

Acknowledgements We would like to thank Mathias Blau, Edi Gava, D. Ghoshal, S.F. Hassan, M. Moriconi, K.S. Narain, Ashoke Sen, George Thompson and specially Cumrun Vafa for useful conversations. One of us ŽD.P.J.. would like to thank Abdus Salam ICTP for warm hospitality.

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