Duality symmetries in string theory

ELS I-iVl ER

UCLEAR PHYSICS

Nuclear Physics B (Proc. Suppl.) 45B,C (1996) 46-58

PROCEEDINGS SUPPLEMENTS

Duality Symmetries in String Theory Ashoke Sen a*

aMehta Research Inst i tute for Mathematics and Mathetical Sciences 10 Kas turba Gandhi Marg, Allahabad 211002, INDIA

We review recent progress in our understanding of duality symmetries in string theory.

1. I N T R O D U C T I O N massless scalar fields. These consist of

In these lectures I shall review some of the recent progress in our understanding of strong-weak coupling duality in four, three, two and six dimensional string theories. Throughout these lectures a D dimensional string theory will refer to heterotic string theory compactified on a ( 1 0 - D ) dimensional torus unless specified otherwise. These lectures will be based mostly on refs.[1]-[15].

Although these theories are unphysical (even the four dimensional theory has N = 4 supersymmetry, and hence is not capable of describing tile real world) there are several reasons why the s tudy of duality symmetries in these theories can be useful. First of all, s tudy of duality symmetries around diverse backgrounds may give us a clue to the underlying symmetries of a general background independent formulation of string theory. On a more pragmatic level, s tudy of duality symmetries in these unphysical theories may throw some light on duality symmetries in more real- istic string theories (with less number of supersymmetries). Some progress in these directions has already been made[16].

2. F O U R D I M E N S I O N A L T H E O R Y

We star t our analysis with the four dimensional theory[I], represented by heterotic string theory compactified on a six dimensional torus. We shall describe the various duality transformations by specifying their action on various massless neutral fields in the theory. First let us concentrate on the

*On leave of absence from Tata Institute of Fundamental Research, Homi Bhabha Road, Bombay 400005, INDIA

1. A 28 x 28 matr ix valued scalar field M satisfying

M L M T = L, M T = M , (1)

where L is the 28 x 28 matr ix

L = , (2) o- 1

-116

where

L has six eigenvalues +1 and twenty two eigenvalues - 1 . Since M preserves L, M is an element of the group O(6,22). Furthermore the symmet ry of M implies that it actually parametrizes the coset 0(6, 22) / ( (0(6) x 0(22)) . To see this, note that since M = M T, we can express M as VF v where V is an arbi t rary O(6,22) matrix. The expression for M remains un- changed if we multiply V by an 0(6) x 0(22) matr ix from the right. The total number of parameters labelling M is thus

d im(O(6 , 22)) - d i m ( O ( 6 ) ) - d im (O (22 ) )

= 132.

0920-5632/96/$15.00 o 1996 Elsevier Science B.V. All rights reserved. SSDI: 0920-5632(95)00628-1

(4)

A. Sen~Nuclear Physics B (Proc. Suppl.) 45B, C (1996) 46-58 47

Thus M corresponds to 132 scalar fields. Physically these 132 scalars correspond to

c2_(m) R(~o) the internal components ~ m n , ~ r n n and A(10)I (4 < m < 9, 1 < I < 16) of the ten 7 / 2 , . . . .

dimensional fields.

2. The dilaton (I), and the axion ko obtained by dualizing the anti-symmetric tensor field B~. in four dimensions. It is convenient to combine these two fields into a complex scalar fiels

A = g2 + ie - ¢ --= kl + iA2. (a)

Both the set of fields M and A have flat poten- tial; as a result (k) and (M) are arbitrary. These label different classical vacua of the theory. A and M are known as the moduli fields. Since {e - ¢ ) is proportional to the inverse of the string loop expansion parameter 92, we see that

1 (A2} ~ g-V (6)

We are now in a position to specify the action of various duality transformations on the scalar fields. This theory has two different kinds of duality symmetries:

1. T-duality: Under this t ransformation

M ~ f ~ M f l T , A - , ,~, (7)

where f~ is an 0(6,22; Z) transformation, 1.e. it satisfies the conditions:

f ~ L f l T -- L , (8)

and tha t it preserves an even, self-dual, Lorentzian lattice A2s with metric L. The choice of the lattice A2s is arbi trary (and can be traded in with the choice of (M)), but once we have chosen a A2s we must stick to it. We shall choose A2s to be spanned by the vectors of the form:

Z, e Aus×E , (9)

where AE8 x Es denotes the Es x / i s root lattice. One can easily verify tha t these vectors span an even, self-dual lattice if we use the metric L to compute inner products.

The important point one should note is tha t the T-duali ty t ransformation does not act on A. Thus it does not act on the string loop expansion parameter g 2 and hence does not mix different orders in the string per turba- tion theory. Hence this duality symmet ry is easy to test, and has been shown to be a symmet ry of string theory to all orders in 92 .

2. S-duality: This duality t ransformation acts on the scalar fields as

M ~ M , A ~ P A + q (10) r A + s '

where p, q, r, s are integers satisfying p s - qr = 1. This describes an SL(2, Z) group of transformations. Note tha t this duality t ransformation acts on A and hence on the string loop expansion parameter 9- As a result this t ransformation mixes different orders in the string per turbat ion theory and does not hold order by order in g 2. This makes it more difficult to test this symmetry, but at the same time this symmet ry is more interesting since this can relate a theory at strong coupling to a theory at weak coupling.

In order to gain further insight into the nature of these duality t ransformations we shall s tudy their action on the gauge fields. At a generic point in the moduli space this theory has a U(1) 2s gauge symmetry and hence contains 28

U(1) gauge fields ,a(~).~ for 1 < a < 28, 0 <_ # _< 3. The origin of these gauge fields in the ten dimen-

~(10) B},{o) sional theory is from the components ~ m u ,

and A (l°)/~ . If F (~)u~ denote the field strengths associated with these U(1) gauge fields then the T duality t ransformation acts on these fields as

F(~) ~(b) (11) -~ f~ab~'~v ,

and S-duality t ransformation acts as

F(~ ) --~ (rA1 4- s)F(~ ) 4- r A 2 ( M L ) a b F ( ~ ) • (12)

48 A. Sen~Nuclear Physics B (Proc. Suppl.) 45B, C (1996) 46-58

From the above transformation laws we see that T-dual i ty transforms electric fields to electric fields, whereas S-duality transformation mixes electric fields with magnetic fields. This in turn shows tha t when we t ry to extend these transformation laws on states in the theory tha t carry charge, then T-dual i ty t ransformation transforms elementary string states (which carry electric charge but no magnetic charge) into states carrying electric charge only, and hence the trans- formed states can also be part of the elementary st:ring excitation spectrum. On the other hand, S-duality transforms electrically charged elementary string states into magnetically charged states which are not present in the elementary string spectrum and must arise as solitons in the theory. Thus S-duali ty t ransformation mixes elementary excitations with solitons. This again demonstrates the non-trivial nature of this symmetry.

Let us now turn to the problem of testing S- duality. Since S-duality t ransformation takes the theory from weak coupling (large A) to strong coupling (small Im(A)), it is clear tha t it cannot be tested order by order in string pertur- b:~tion theory, and can be tested only on those quantities which are known exactly. In general there are no such quantities known in a generic string theory, but the particular string theory under consideration has N = 4 local supersymmetry which gives rise to several non-renormalization theorems. These theorems gurantee tha t for cer- tain quantities in the theory the tree level result is in fact exact. Thus we can test S-duality on these quantities. We now make a list of such quantities.

1. Low energy effective action involving massless neutral fields: The low energy dynamics of this string theory is described by an ef-

fective action S ( M , A A (~) ~ f e r m i o n s ) . /x ~ Y t t u ~

In general this would get corrected by string loop effects, but in this case local N = 4 supersymmetry fixes the form of the action completely. As a result the tree level answer is exact, and one can ask if this low energy effective theory is S-duality invariant. It can be shown that the equations of motion derived from this effective action

2.

.

are indeed invariant under both, S- and T- duality transformations.

Allowed spectrum of electric and magnetic charges: We can define the electric and magnetic charges associated with a given state in the theory by looking at the asymp- totic form of the field strength around tha t state:

~ ( a ) r ) ( a ) r'(a) ,.o "~et ~(a) ..¢mag (13)

Or - - 7,2 , ~ 0 r --~ r 2 ,

for large r. Normally spect rum ~'f r)(~) and v ~ ,.t5 e l

Q(~) of a state will be renormalized by m a g

quantum corrections due to the renormalization of the quantum of electric charge, but this does not happen in an N = 4 supersymmetric theory. Thus the tree level result is exact. The result is

Q(a) = Lab~b m a g

Q(a) _ 1----~-(Mab)(olb -~- (/~l}~b), (14) d (A2)

where 4,/3 c A2s. Using the t ransformation

laws of F (a) A and M under the various duality transformations we can determine the transformation laws of (~ and/3. They a r e

under T-duality, and

- r ~ + sf~ } '

(15)

(16)

under S-duality transformation. Thus we see that c~ ~, ~ , d" and ~/, all belong to the lattice A2s. This shows that the allowed spectrum of electric and magnetic charges are invariant under both, T- and S-duali ty transformations.

Bogomol'nyi bound: Let us now turn to the mass spectrum of the theory. In general the masses of various elementary string


states are renormalized by quantum corrections and we do not have any information about the mass spectrum of the theory that is valid non-perturbatively, However, in the present case, because of N = 4 supersym- merry algebra, there is a lower bound to the mass of a state carrying a given electric and magnetic charge. This bound is known as the Bogomol'nyi bound and is given by,

r \ ,,--~ (b) m2 > (A2>{Q~)(L(M}L+./~bw¢I - 16

+Q~)g(L<M}L + L~ja b r)(b)~m~j ~ riM(a) ()(a) ------- s, '~et ,"~m~g, (A}, <M>). (17)

If S-duality is a symmetry of this theory then the function f defined in the above equation must be invariant under the duality transformation. It can be easily checked that this is indeed the case. (This function is also invariant under T-duality transfor- marion).

4. Spectrum of Bogomol'nyi saturated states: Let us consider a state carrying charge quantum numbers (5, ~) and saturating the Bogomol'nyi bound:

= , O (a) m 2 f (Q~) .~ma9 , ()~}, <M>). ( 1 8 )

These are known as BPS states and form a 16 component representation of the N = 4 supersymmetry algebra. Let N(5,/3") denote the degeneracy of such BPS supermul- tiplets. It has been argued by Olive and Witten[17] tha t this number is independent of (A) (and (M}) and hence the degegener- acy of BPS multiplets, calculated at weak coupling, is the exact answer for all values of the coupling. Hence if S-duality is a symmetry of the theory, then N(5 , ~) must satisfy:

N(5 , f l ) = N(p~ - q ~ , - r ~ + s~). (19)

This gives a set of testable predictions of S-duality. In particular, this gives

N(5, O) = N(pS, (20)

The left hand side of this equation is calcu- lable from the known spectrum of elementary string states. On the other hand, the right hand side corresponds to the degeneracy of dyonic solitons, - quantum states of solitons carrying both electric and magnetic charges. Thus S-duality predicts the degeneracy of dyonic solitonic states. Note that since p, q, r and s are integers satisfying ps - qr = 1, p and r must be relatively prime (i.e. they cannot have any common factor).

The value of N(5 , 0) in four dimensional heterotic string theory depends on the combination ctaLabCt b. In particular N(d , 0) vanishes for OlaLabOt b < - - 2 , is equal to 1 for o~anabo~ b = - - 2 , i s equal to 24 for olaLabC~ b -~

0, is equal to 324 for OlaLabOl b -= 2 and so on. This way S-duality makes an infinite number of predictions for the degeneracy of BPS soliton states. Of these the case olaLabO~ b = --2 is best understood, so we shall spend some time discussing how we compute N ( p S , - r S ) in this case and verify the S-duality conjecture.

As has already been pointed out, due to the Witten-Olive argument, N(5 , j ) is independent of <A> and (M). Thus we shall compute this number by i) taking Im((A)) to be sufficiently large so that semi-classical approximation is valid, and ii) choosing (M) appropriately so that we can use a low energy effective field theory as a good approximation. In particular we shall choose (M) in such a way that the function f ( p S , - r j , (A>, (M)) is small (compared to the string scale) for the particular 5 we are considering. In this case there are nearly massless particles in the spectrum of elementary string states carrying electric charge vector + & T h e low energy effective field theory describing the dynamics of neutral massless fields together with these nearly massless charged particles is N = 4 supersymmetric Yang-Mills theory with gauge group SU(2) x U(1) 27, with the extra nearly massless particles providing


the charged vector bosons required to com- plete the SU(2) gauge multiplet. SU(2) is broken to U(1) by the vacuum expectation value of the adjoint Higgs (which is also part of the gauge supermultiplet) at a scale small compared to the string scale. The states with quantum numbers (pc~, -rc~) represent BPS monopole / dyon states in this theory carrying - r units of magnetic charge and p units of electric charge. Thus the problem of verifying the S-duality predictions for N ( p & - r ~ ) reduces to the problem of finding degeneacy of BPS dyon states in N = 4 super-Yang-Mills theory.

This is done as follows. Classical r- monopole solution in this field theory is parametr ized by 4r parameters, which, for well separated monopoles, can be taken to be the three spatial coordinates and one U(1) phase for each monopole. This gives a 4r dimensional moduli space of solutions. This moduli space is known to have the structure

= ( n 3 × s × 347)/z ), (21)

where R 3 is labelled by the center of mass coordinate i of the r monopoles, S 1 is la-

belled by the overall U(1) phase O and AJ (°) is labelled by the 4r - 4 relative coordinates and phases ~. The action of Zr leaves 2 invariant, t ransforms ~ to O + 27c/r, and acts

as a diffeomorphism on AA(°): # --* fit(F ).

The space 34 (o) is equipped with a natural hyper-Kahler metric. It can be shown that the problem of finding the low lying states of the soliton reduces to the problem of supersymmetr ic quantum mechanics of a particle moving on 34r. Using this correspondence one finds that a BPS supermultiplet with r units of magnetic chrge and p units of electric charge is in one to one correspondence with a harmonic differential

form on 34(0) which picks up a phase of exp(-27cip/r) under the action of Zr. Thus the prediction of S-duality can be stated as follows:

For every pair of integers (p,r) which are

relatively prime, the space 2t4(r °) must contain a unique harmonic differential form which picks up a phase exp(-27cip/r) un- tier #

Furthermore the fact that the form is unique implies that it must be a self-dual or anti-self-dual (2r - 2) form, since otherwise using Poincare duality one can produce a new harmonic form from every harmonic differential form.

The case r = 1 and p any integer is trivial,

since here ~4~ °) is just a point and the phase factor exp(-27rip/r) is unity. The required harmonic form is just the constant function with support at tha t point.

The case r -- 2, p odd is more interest-

ing. In this case AA~ °) is a non-trivial four dimensional space and the phase fac-

tor exp(-27rip/r) is - 1 . Metric on M~ °) has been calculated by Atiyah and Hitchin[18], and using this metric the required self-dual harmonic form of rank 2 has been con- structed explicitly[2].

The case r ~ 3 is more complicated. In this

case the metric on the space JM(~ °) is not known explicitly, and more general topo- logical arguments are necessary to establish the existence of the required harmonic forms. Some progress in this direction has been made by G. Segal.

Let us turn to the c a s e o~anabO! b > O. The case c~aLabC~ b = 0 has been analyzed to some extent by Gaunt le t t and Harvey[19]. In this case the dual solitons are known as H-monopoles. In order to verify the prediction of duality we need to establish tha t N ( p G , - r d ) for H-monopoles is 24. Unfor- tunately so far no definite calculation of the degeneracy of H-monopoles and dyons have been performed, mainly because the calculation is plagued by both infra-red and ul- traviolet problems.

Another viewpoint that one can adopt for the oPLab(~ b >_ 0 states is the following.

A. Sen/Nuclear Physics B (Proc. Suppl.) 45B, C (1996) 46-58 51

It can be shown that in this case the BPS states have the same mass-charge relations as extremal black holes. This suggests that perhaps we should interprete the extremal black holes in string theory as the (peT, -roT) states for OLaLabO~ b ~ O. In tha t case, veri- fication of the S-duality predictions will re- quire us to compute the degeneracy of ex- t remal black hole states with quantum numbers (pal,-roT), and comparing it with the degeneracy of elementary string states with quantum numbers (c7,0). At present this seems to be a formidable problem.

5. Finally one can show tha t the invariance of the Bogomol'nyi formula under S-duality also shows tha t the Yukawa coupling between massless neutral scalar fields M and I , and the charged BPS states, are invariant under the S-duality transformation. This is a further piece of eveidence for S-duality in this theory.

3. T H R E E D I M E N S I O N A L T H E O R Y

We shall now leave the subject of four dimensional string theory and begin our study of three[4] dimesional string theory obtained by compactifying heterotic string theory on a seven dimensional torus. In this case the massless bosonic fields in the theory consist of:

1. The dilaton ~.

2. 30 x 30 matr ix valued scalar field ~I satisfying

~ I L ~ / T = L, ~-I T = i/~/, (22)

where L is the 30 x 30 matrix

O" 1

L = (23)

(T 1

- - 1 1 6

/~I parametrizes the coset 0(7, 23) / (0(7) x 0(23)) and corresponds to 161 independent

scalar fields, which originate from the com- • , - , (10) R ( 1 0 ) A (10)I (3 < r~,5 < ponen~s ~ f i , ~ , ~ and --m - -

9) of the ten dimensional fields.

3. 30 U(1) gauge fields $(a) (1 < g < 30, --~tL - - - -

0 <_ /2 _< 2) originating from the compo- ~(10) R(10) and A ( 1 0 ) I nents _ ~ , _ ~ . .~ of the ten di-

mensional fields. In three dimensions gauge fields are dual to scalar fields; using this duality we can trade in the 30 U(1) gauge fields for 30 scalar fields ~pa.

4. The canonical Einstein metric g , - .

The total number of scalar fields in this theory is 192. It can be shown tha t these 192 scalar fields can be arranged into a symmetric 0(8 , 24) matr ix M satisfying

Jld£214 T = £ , M T = 3, t , (24)

where £ is the 32 x 32 matr ix

c = . ( 2 5 )

(71

- - I 1 6

M parainetrizes the coset 0 ( 8 , 2 4 ) / ( 0 ( 8 ) x o(24)) .

The T-duali ty t ransformation in this three dimensional theory is generated by an 0(7, 23; Z) matrix ~ tha t leaves gp~ invariant and acts on Ad a s

M - - - , ( 12 ~ ) . h d ( I2 ~tT) , (26)

where ~) satisfies

~)L~) T = L , (27)

and that it preserves an even, self-dual, Lorentzian lattice A30 with metric L. The lattice A30 is spanned by the vectors of the form:

mi E Z, fl~ C AEsxE s . (28)


In order to see what other kind of duality symmetry might be present in this theory, let us consider the three dimensional theory as a four dimensional theory compactified on a circle. In this case it is natural to assume that the original S- duality symmetry of the four dimensional theory is still present. It is a straightforward exercise to work out the action of this S-duality transformation on the field A4, and this transformation takes the form:

. ~ --~ 5AAfi, (29)

w h e r e

(0~r 0 0 - q s r 0

i ~= q p 0 (30) 0 0 s

I2s

Note that this transformation does not in general commute with the T-duality transformation given in eq.(26). Together these transformations generate a much bigger group, which turns out to be 0(8, 24; Z). This transformation acts on Jc[ ~ks

M -* f i M f i T , (31)

where ~ satisfies

t}/2a T = £ , (32)

and that it preserves an even, self-duai, Lorentzian lattice A32 with metric £. The lattice A32 is spanned by the vectors of the form:

rr~i E Z, ]~ E AEsxEs - (33)

This shows that upon compactification S- and T- duality transformations can combine into a much bigger group. This has been called unified duality or U-duality by Hull and Townsend[7].

4. T W O D I M E N S I O N A L T H E O R Y

We now turn to the duality symmetry group of two dimensional string theory[6] obtained by

compactifying heterotic string theory on an eight dimensional torus. The bosonic fields in this theory consist of a scalar dilaton field ~ and a 32 x 32 symmetric O(8, 24) matrix M parametrizing the coset O(8,24)/(O(8) × 0(24)). T-duality transformation in this theory leaves ~ invariant and transforms M to ~/~r~T as usual, where ~ is an O(8,24; Z) matrix. Unfortunately S-duality

transformation acts on M in a non-local fashion. In order to understand the origin of this problem, note that T-duality, for example, acts as a local

~(10) whereas transformation on the components "-'23 , S duality acts as a local transformation on the

~(10) i~(10) /~(10) dual ~23 of ~23 in two dimensions, k~23 is the dimensional reduction of the axion field of the four dimensional theory.) This shows that in order to have a formalism in which both, S- and T-duality transformations act naturally, we

/~(10) and ~(lo) must include both ~23 "--'23 on equivalent footing in our formalism. It turns out that such a formalism is provided by introducing an auxil- iary set of variables that makes the equations of motion first order. In this formalism, we trade in ~r(x) for a field V(x,v) which is a 0(8, 24) valued function (not necessarily symmetric) of the two dimensional coordinate x and a real variable v known as the spectral parameter. V(x, v) con-

tains the same set of informations as M(x) up to gauge transformations, except for extra zero modes. (Thus for example V(x, v) contains infor-

m(10) mation about the zero modes of -23 as well as

/~(10) whereas M does not contain information 23 , about the zero mode of/~2(~°).) The advantage

of using this new variable ~2 is that both, S- and T-duality transformations act naturally on this variable. In fact the general form of both transformations is:

V(x, v) --~ g(v)9(x, v)g(x, v), (34)

where g(v) is an 0(8,24) valued function of v and H(x, v) is determined completely in terms of g(v). Thus the duality transformations are labelled by

9(v). The T-duality transformation 0(8 , 24; Z) men-

tioned above corresponds to the choice

g(v) = ~, 5 c 0(8, 24; Z) . (35)


On the other hand the S-duality transformation corresponds to the choice:

p 0 0 - q v \ 0 s r v -1 0

g(v) = 0 qv p 0 . (36) - - rv -1 0 0 s

I28

The full duality group G of the theory is at least the group generated by S- and T-duality transformations. In order to gain some insight into this

group let us define the group 0(8, 24; Z) to be the group of 0(8, 24) valued functions g(v) with the property that g(v) has an expansion of the form

M

g(v) = E gnv n, (37) 2"~=-- N

such that each 32 × 32 matrix gn, acting on any vector in the lattice A32, gives us another vector of the lattice Aa~.

It is easy to see that both the S- and T-duality transformations satisfy this criteria. Thus the group G generated by S- and T-duality trans-

formations is certainly a subgroup of 0(8, 24; Z). The question that we would like to ask now is:

'Are there elements of 0(8, 24; Z) that are not in G?' The answer to this question turns out to be yes. The example of such an element is:

g0(v) = v -~ (38)

/30

In order to see that such an element cannot be generated by a combination of S and T duality transformations we note that for all g(v) corre- sponding to S- and T-duality transformations, g(v = 1)g- l (v = - 1 ) is connected to the iden- t i ty element of 0(8, 24). Since this property of g(v) is preserved under group multiplication, all elements of G satisfy this property. On the other hand for the element go(v) given in eq.(38),

g0(1)gol ( -1 ) = - 1 , (39) I30

which is not connected to identity in 0(8,24).

Thus go(v) represents an element of 0(8, 24; Z) which is not in G.

It is still tempting to conjecture that the full duality symmetry group in two dimensions is

0(8,24; Z), and not just the group G. But this requires us to make new duality conjectures in two dimensional string theory, like invariance of

the theory under the 0(8, 24; Z) transformation go(v).

5. S I X D I M E N S I O N A L T H E O R Y

Finally I shall discuss some aspects of the conjectured equivalence between heterotic string theory compactified on T 4 and type IIA string theory compactified on K317]-[12]. I shall begin by list- ing the set of evidence which led to this conjecture in the first place.

1. N u m b e r of S u p e r s y m m e t r i e s : In ten dimensions type IIA theory has double the number of supersymmetries compared to heterotic string theory. However, compactification on K3 surface kills half of the supersymmetries, whereas compactification on T 4 preserves all the supersymmetries. As a result the two theories under consideration have the same number of supersymmetries in six dimensions. In fact both of them have non-chiral N=2 supersymmetry in six dimensions.

2. S p e c t r u m o f mass less s tates[13]: At a generic point in the moduli space, heterotic string theory compactified on T 4 has the following spectrum of massless bosonic states:

• Dilaton 6p.

• Graviton Gu..

• Antisymmetric tensor field Bu. .

• 80 scalars M which belong to the coset 0(4, 20) / (0(4) x 0(20)) .

• 24 U(1) gauge fields A(~ a).

On the other hand, type IIA string theory compaetified on K3 has the following set of bosonic fields:

• Dilaton ~ .


3.

4.

! • Graviton Gu~.

• Anti-symmetric tensor field B ~ .

• 80 scalar fields M' E 0(4, 20) / (0(4) x 0(20)).

Ll/(a) • 24 U(1) gauge fields ~.~ .

Thus we see that the two theories have the same spectrum of massless bosonic fields. The same result holds for massless fermionic fields as well. Note tha t the counting of the number of scalars and vectors in the second theory depends crucially on the cohomol- ogy of K3; hence the agreement between the two spectra is a non-trivial result.

L o w e n e r g y e f fec t ive ac t ion : The dynamics of massless bosonic fields in the heterotic string theory compactified on T 4 is described by an effective ac-

tion S(G~,, Bu~, A(~)u , ff~, M). Similarly, the dynamics of massless bosonic fields in type I IA string theory compactified on K3 is described by an effective action

' n ' A '(~) m' M') . These two effec- S ( G u . , ~ . , - ~ u , - , tive actions give equivalent equations of motion if we make the identification:

M ' = M, ~ ' = -~5,

At} a) = A (a) , G#v = e-~'G#v,

x/Z--Oe-e HU~O = l_ei*,'m,~ H' 6 o.Tt~

(40)

where,

Hl,~ p = O[,B~p I + Chern-Simons terms,

' ' . ( 4 1 ) H~p = O[t~Bvp ]

T - d u a l i t y G r o u p s : In both the theories the T-duali ty group is O(4,20;Z). In the heterotic string theory this arises through a combination of global gauge transformation involving the gauge fields and anti- symmetric tensor fields, global diffeomorphism of T 4, and R ---+ 1/R duality transformation. In the type IIA theory on K3 this

arises through a combination of global diffeomorphism on K3, global gauge transformation involving the anti-symmetric tensor field, and mirror symmetry[14].

Let us now discuss the relationship between this six dimensional string-string duality, and S- duality in four dimensions. For this we com- pactify both the theories further on a two dimensional torus. In this case the heterotic string theory on T a has a T-dual i ty group 0(6, 22; Z), and the conjectured S-duality group SL(2, Z)s. On the other hand, the type IIA theory on K 3 has a T-duali ty group 0(4, 20; Z ) ' x SL(2, Z)' T x SL(2, Z)'~, and the conjectured S-duality group SL(2, Z)' s. The O(4 ,20 ;Z) ' is the T-dual i ty group associated with K3, SL(2, Z)b is the mod- ular group acting on the complex structure of the two torus, and SL(2, Z)' T acts on the p parameter of T 2 whose real part consists of the anti- symmetric tensor field, and the imaginary par t the size of T 2. On the heterotic side, it is convenient to work with a subgroup O(4,20; Z) x SL(2, Z)u x SL(2, Z)T of the 0(6, 22; Z) group. 0(4, 20; Z) acts on the parameters of T 4, whereas SL(2, Z)u and SL(2, Z)T again acts on the complex structure and the p parameter of T 2 respec- tively.

Since the heterotic string theory on T 6 and type IIA string theory on K3 x T 2 are expected to be equivalent by the string-string duality conjecture in six dimensions, it is a natural to ask how the duality groups in the two theories are related. The answer is obtained by studying the relationship between the fields in the two theories given in eq.(40). The result is,

0(4 , 20;Z) --~ O(4 ,20 ;Z) '

SL(2, Z)u --, SL(2, Z)b

SL(2, Z)s --+ SL(2, Z)'r SL(2, Z)T -~ SL(2, Z)'s. (42)

Since T duality of a theory is testable order by order in per turbat ion theory, the SL(2, Z)T and SL(2, Z)' T symmetries are on a much firmer footing than any of the S-duality symmetries. How- ever, since string-string duality maps SL(2, Z)s to SL( 2, Z)'T, and SL( 2, Z)' s to SL(2, Z)T, w e see

A. Sen~Nuclear Physics B (Proc. SuppL) 45B, C (1996) 46-58 55

that the string-string duality conjecture makes the case for S-duality much stronger both in the type I IA theory on K3 x T 2, and the heterotie theory on T 6. In other words, if T-duali ty is a symmet ry of both the theories, and if string- string duality holds, then S-duality must also be a symmetry of both the theories.

Let us now pose a puzzle and a question. In type IIA string theory compactified on K3, all the gauge fields arise from the Ramond-Ramond sector of the theory. It is well-known that none of the elementary string excitations carry any charge under Ramond-Ramond gauge fields. On the other hand, in the heterotic string theory the gauge fields arise in the Neveu-Schwarz sector, and there are certainly elementary string states carrying these gauge charges. Thus if the two theories are equivalent, then the type IIA theory must contain charged states which arise as solitons in the theory. The puzzle is, how can there be solitons in the theory carrying electric charges 'if there are no electrically charged fields in the theory to start with?

The question that we pose is the following. Heterotic string theory certainly contains macroscopic fundamental heterotic string states. The question is: where are these states in type IIA string theory compactified on K 3 ? Type IIA theory of course contains macroscopic fundamental type I IA string states, but one can easily see that these are distinct from the fundamental heterotic string states. The reason is that the fundamental type I IA string will carry electric type B~.

/ charge since the B , , field directly couples to the type I IA string world sheet. On the other hand, heterotic string carries electric type Bu~ charge. By eq.(40) this corresponds to magnetic type B~. charge. Thus the fundamental heterotic string must be represented by a soliton solution in the r~ype I IA string theory compactified on K3.

Fortunately such a solution exists[15]. In the variables of the type IIA theory, the solution is given by,

{ 7 ds ,2 = - d r 2 + (dx5) 2 + (1 + ~ ) d r 2

+(c + r )da

e - ~ ' = ( l + C ) -1, A ~ a ) = 0 , r 2

M' = I24 , H~j k = - 2 C c i j k , ( 4 3 )

where df~3 is the line element on an S 3, c is the volume form on the same S 3, and C is a nmnerical constant related to the tension of the solitonic string. The last line in the above equation shows that the solution does carry magnetic type B~. charge, as required. The solution looks singular near r = 0, but it can be easily seen to be due to a wrong choice of coordinates. In particular, if we choose coordinate p - in r, then near the point r = 0 (p = - o e ) the metric and the dilaton looks like,

d J 2 ~_ - d t 2 + (dxS) 2 + C d p 2 + C d f ~ ,

-2p. (44)

Thus the space-time looks like the product of S a x M (2) and a semi-infinite line labelled by p. The dilaton is linear in p, showing tha t the con- formal field theory defined in this background is well defined. However the coupling of the type IIA string theory (e ~) blows up as p approaches - e c .

Once we have obtained a candidate for the soliton in the type IIA theory tha t might represent the fundamental heterotic string, we can ask: does this solution have the properties of the heterotic string? In particular, the fundamental heterotic string carries 20 left moving and 4 right moving currents on the world sheet associated with the internal coordinates conjugate to the Narain lattice. Does this soliton have the same property? In order to show this we need to show tha t the soliton solution allows 24 parameter deformations representing charge / current carrying strings. I t turns out tha t such deformations of the solution can indeed be generated by solution generating techniques[ll,12]. The final form of the deformed solution is given by,

ds '2 = A - 1 / 2 [ - ( r 2 + C)dt 2

+C(cosh a - cosh/3)dtdx 5

+@2 + C cosh c~ cosh/3) (dx5) 2]

dr2 { + da ) (45) + A1/2 \ r 2


_~, r 2

A1/2 ,

H~ k = - C ( c o s h a + cosh/3)si jk,

(46)

(47)

Air(a) rt(a)

- 2 v ~ A C sinh ct

1 x { r 2 cosh/3 + ~C(cosh c~ + cosh/3)}

for 1 < a < 20, p(a-20)

- - - C s i n h / 3

2v A 1

x { r z cosh c~ + ~C(eosh a + cosh/3)}

for 21 < a < 24,

(48)

n(a) - 2 v ~ A C s i n h a

x { r 2 + 1Ccosh /3 (cosh a + cosh/3)} 2

for 1 < a < 20, p(a-20)

- - - C sinh/3 2v A

1 C cosh (~(cosh a + eosh ~)} x {r 2 + 2

for 21 < a < 24,

(49)

Pnnr QnPT' (50) .~I'=I28+ Qpn T p p p r J ,

where,

A = r 4 + Cr2(1 + c o s h a c o s h / 3 )

C 2 + - T ( c o s h c~ + cosh/3) 2 ,

C 2 P = ~ sinh 2 a sinh 2/3,

(51)

(52)

~ : - C A - 1 s inh a s inh /3

x { r 2 + 2 C ( 1 + c o s h a cosh/3)}, (53)

Here a and t3 are two boost parameters , g is a 20 dimensionM unit vector, and/~ is a 4 dimensional unit vector.

Near r = 0, we again use the coordinate p = ln r. In this coordinate sys tem the metric near r = 0 takes the form:

2 ds '2 ~

cosh a + cosh fi

x ( - dt 2 + cosh ct cosh/3(dx5) 2

+(cosh a - cosh/3)dtdx 5)

C +~- (cosh a + cosh/3)(dp 2 + d a 2 ) ,

(54)

and describes a completely non-singular geometry. We also see from eqs.(48), (49) t ha t the gauge fields are non-singular as r --4 0.

We define the electric charge per unit length (q(~)) and electric current (j(~)) carried by the solution in terms of the asympto t ic values of the gauge fields (r --* oo):

F,.'(a) q(a) ~"(~) "~ J(~) (55) t ~ r3 , ~r5 -- r3 •

Compar ing eq.(55) with eqs.(48), (49) we get

n(a) q(~) - C s i n h a cosh/3,

for 1 < a < 20, p(a-20)

- C sinh 13 cosh a ,

for 21 < a < 24, (56)

and,

j(a) n(a) - - - C s i n h c ~ ,

for 1 < a < 20, p(a-20)

- - - C sinh fl ,

for 21 < a < 24.

Thus, for small c~,/3,

n(a) q(a) = j(a)

q(a) = --j(a) ~_

C a

for p(a-20) - - C / 3

for

1 < a < 2 0 ,

21 < a < 24.

(57)

(58)


This shows that the deformed solution does represent a flmdamental heterotic string carrying 20 left-moving and 4 right moving currents on its world-sheet. Treating q(~) / j(~) as collective coordinates, and taking into account other collective coordinates associated with the broken transla- tion and supersymmetry generators, we get an effective lagrangian for these collective coordinates which coincide with the world-sheet lagrangian for the heterotic string theory. Thus we see that heterotic string emerges as a soliton of the type IIA string theory compactified on K3.

Notice that we have also resolved the puzzle that was posed earlier; the deformed soliton solutions that we have found definitely carries electric charge! It turns out that the resolution of the puzzle is hidden in the non-standard from the field equations satisfied by the gauge fields in the type IIA theory. This takes the form:

1 D ; [ ( M ' L ) ~ b F '(b)t'~] + - ~ ( ~ ) - - l E r e # c ~ P V

H ' F '(~) = 0 (59) O" p t,,' - T ~ *

The second term in the above equation can act as a source for the electric charge. To see that this is the source of the electric charge in the charged string solution written down earlier, let us briefly examine this solution. From eq.(48), (49) we can easily verify that the total electric flux per unit length of the string through a surface of constant r (or p) vanishes as r -+ 0 (p --+ - ~ ) . This shows that there is no source of electric field hidden at the far end of the semi-infinite geometry (p ~ -oo) . In fact, a more detailed examina- tion of the solution shows that the total electric charge carried by the solution is indeed given by the integral of the second term in eq. (59) over the whole space, therby proving that this is the only source responsible for the electric charge of the soliton.

We can also ask the reverse question: Is it possible to get a soliton solution of the heterotic string theory that will represent the fundamental type IIA string? Such a soliton must be charac-

f terized by an electric type B , . charge, and hence a magnetic type B~. charge. It turns out that such a solution does exist in the heterotic string

theory[15] and is given by,

ds 2 = - d t 2 + (dx5) 2 + (1 + )dr 2

+(c' +

e-+ = ( 1 + c ' ) -1 , A(;)=o, T2

M = I24, Hi3k = - 2 C ' g i j k , (60)

C' is a constant related to the string tension of the type IIA string. As in the previous case, this solution is non-singular at r = 0, but the coupling constant of the heterotic string theory (e ~) grows as r - ~ 0.

Let us now ask the following question: do we get charged type IIA string soliton by starting with the neutral type IIA string soliton, and applying the solution generating transformations on this solution? If the answer is yes, it would imply that the type IIA string soliton does not satisfy the properties of a fundamental type IIA string, since it is known that the fundamental type IIA string does not carry any world-sheet current that couples to the gauge fields originating from the Ramond-Ramond sector. It turns out tha t this particular solution is invariant under the solution generating transformations, and hence we do not generate any new solutions by applying solution generating transformations to (60)! This is pre- cisely what we want, since if we had gotten a new solution using the solution generating transformation, it would have almost certainly corresponded to a charge carrying solution, thereby showing that the soliton does not have the right properties for being identified as the fundamental type IIA string. This shows that the fundamental type IIA string also emerges as a soliton solution in the heterotic string theory.

R E F E R E N C E S

1. A. Sen, Int. J. Mod. Phys. A9 (1994) 3707 (hep-th/9402002) and references therein.

2. A. Sen, Phys. Lett. B329 (1994) 217 (hep- th/9402032).

3. N. Marcus and J. Schwarz, Nucl. Phys. B2 2 8 (1983) 145.

4. A. Sen, (hep-th/9408083).


5. P. Breitenlohner and D. Maison, Ann. Inst. Poincare 46 (1987) 215; H. Nicolai, preprint DESY-91-038.

6. A. Sen, (hep-th/9408083). 7. C. Hull and P. Townsend, preprint QMW-94-

30 (hep-th/9410167). 8. C. Vafa, Unpublished. 9. M. Duff, preprint CTP-TAMU-49/94 (hep-

th/9501030); M. Duff and R. Khuri, Nucl. Phys. B411 (1994) 473 (hep-th/9305142).

10. E. Witten, preprint IASSNS-HEP-95-18 (hep-th/9503124).

11. A. Sen, preprint TIFR/TH/95-19 [hep- th/9504027].

12. J. Harvey and A. Strominger, preprint EFI- 95-16 (hep-th/9504047).

13. N. Seiberg, Nucl. Phys. B303 (1988) 286. 14. P. Aspinwall and D. Morrison, preprint DUK-

TH-94-68 (hep-th/9404151). 15. M. Duff, R. Khuri and J. Lu, preprint

CTP/TAMU-67/92 (hep-th/9412184), and references therein.

16. S. Kachru and C. Vafa, preprint hep- th/9505105.

17. E. Witten and D. Olive, Phys. Lett. B78 (1978) 97.

18. M. Atiyah and N. Hitchin, 'Geometry and Dynamics of Magnetic Monopoles', Princeton Univ. Press (1988).

19. J. Gauntlett and J. Harvey, preprint hep- th/9407111.

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Duality symmetries in string theory