Center for Theoretical Physics. Physics Department. College Station, Texas ABSTRACT

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1 CTP-TAMU-6/94 hep-th/ PUTTING STRING/STRING DUALITY TO THE TEST y M. J. Du and R. Minasian Center for Theoretical Physics Physics Department Texas A & M University College Station, Texas ABSTRACT HEP-TH After simultaneous compactication of spacetime and worldvolume on K3, the D = 0 heterotic vebrane with gauge group SO(3) behaves like ad= 6 heterotic string with gauge group SO(8) SU(), but with Kac{Moody levels dierent from those of the fundamental string. Thus the string/vebrane duality conjecture in D = 0 gets replaced by a string/string duality conjecture in D = 6. Since D = 6 strings are better understood than D =0vebranes, this provides a more reliable laboratory in which to test the conjecture. According to string/string duality, the Green{Schwarz factorization of the D = 6 spacetime anomaly polynomial I 8 into X 4 X4 ~ means that just as X 4 is the -model anomaly polynomial of the fundamental string worldsheet so X ~ 4 should be the corresponding polynomial of the dual string worldsheet. To test this idea we perform a classical dual string calculation of X4 ~ and nd agreement with the quantum fundamental string result. This also provides an aposteriori justication for assumptions made in a previous paper on string/vebrane duality. Finally we speculate on the relevance of string/string duality to the vacuum degeneracy problem. June 994 y Research supported in part by NSF Grant PHY

2 Introduction In D spacetime dimensions an extended object of worldvolume dimension d is dual, in the sense of Poincare duality, to another extended object of dimension ~ d given by ~d = D d (.) the most familiar example being D = 4, where an electric monopole (d = ) is dual to magnetic monopole ( d ~ = ). More recently, attention has focussed on D = 0, where a superstring (d = ) is dual to a supervebrane ( d ~ = 6) []{[48]. In this paper, however, we shall focus on D = 6 where a string (d = ) is dual to another string ( d ~ = ) [8, 4]. There is now a good deal of circumstantial evidence to suggest that the string and the vebrane theories may actually be dierent mathematical formulations of the same physics with the strongly coupled string corresponding to a weakly coupled vebrane, and vice-versa. We shall refer to this idea as the string/vebrane duality conjecture. In a certain sense, it is a generalization of the electric/magnetic duality conjecture of [49]. In previous papers [7, 8, 9, 9], it was pointed out that the Green{Schwarz [50] factorization of the spacetime anomaly polynomial I into X 4 X8 ~ provides a non-trivial check on this string/vebrane duality [,]. Just as X 4 is the -model anomaly polynomial of the d = string worldsheet, so X ~ 8 should be the corresponding polynomial of the d ~ =6vebrane worldvolume. To test this idea, a classical vebrane calculation of X8 ~ was performed [] and found to be in agreement with the string one-loop result, thus supporting the conjecture. Although the vebrane calculation of the Yang{Mills contribution to X 8 is relatively straightforward, however, an educated guess had to be made for the gravitational contributions; a guess which has recently been questioned in [39, 40]. In this paper we repeat the test for string/string duality ind= 6. Now the factorization of the spacetime anomaly polynomial I 8 into X 4 X4 ~ means that X ~ 4 should be the -model anomaly polynomial of the d ~ = dual string worldsheet. Since D = 6 strings are better understood than D = 0vebranes, this provides a more reliable theoretical laboratory in which to test the conjecture. In particular, the gravitational contributions to the -model anomaly for a string are well-known, thus avoiding the guesswork [] that had to be made for those of the vebrane. We perform a classical dual string calculation of X4 ~ and once again nd agreement with the fundamental string one-loop result. The chiral, anomaly-free, fundamental string we shall consider will be the heterotic string obtained by spacetime compactication on K3 oftheso(3) heterotic string in D = 0. The resulting D = 6 string has gauge group SO(8) SU() with Kac{Moody levels k SO(8) = and k SU() =. The dual string is obtained by simultaneous compactication of spacetime and vebrane worldvolume on K3. As discussed in section 3, this string is also chiral and has the same gauge group but with dierent Kac{Moody levels: k ~ SO(8) = 4 and k ~ SU() = 0. Moreover, the SU() worldsheet fermions have the opposite chirality. The calculation of X4 ~ can be done either directly with the D = 6 dual string or else by starting with the D =0vebrane and In fact a string can be dual to another string even in D < 6 if the vebrane wraps around 4 of the compactied directions [, 38].

3 using ~X 4 = Z K3 ~X 8 (.) Both ways yield the same result, and this provides an a posteriori justication for the gravitational contributions to ~ X8 assumed in []. Review of string/string duality We begin by recalling some facts about duality ind= 6 [4]. There are two formulations of D = 6 supergravity, both with a 3-form eld strength. The bosonic part of the usual action takes the form I 6 = Z d 6 x p ge [R +4(@) :3! H 3 + :::] (.) To within Chern{Simons corrections to be discussed below, H 3 is the curl of a -form B H 3 = db + ::: (.) The metric g ( =0; :::; 5) is related to the canonical Einstein metric g c by where the D = 6 dilaton. ~I 6 = Z g = e g c (.3) Similarly, the dual supergravity action is given by d 6 x q ~ge [ R ~ +4(@) H :3! ~ 3 + :::] (.4) To within Chern{Simons corrections to be discussed below, ~ H3 is also the curl of a -form ~ B The metric ~g MN is related to the canonical Einstein metric by The two supergravities are related by Poincare duality: ~H 3 = d ~ B + ::: (.5) ~g = e g c (.6) ~H 3 = e H 3 (.7) where denoted the Hodge dual. (Since this equation is conformally invariant, it is not necessary to specify which metric is chosen in forming the dual.) This ensures that the roles of eld equations and Bianchi identities in the one version of supergravity are interchanged in the other. As eld theories, each supergravity seems equally as good. In particular, provided we couple them to SO(8)SU() super Yang-Mills, then both are anomaly-free. 3 Since the usual Our use of the symbol conforms with the notation of section 6 of [4] where 8 = (D ) and where = p for D =6. 3 There are other many other anomaly-free groups in D = 6 [5, 5] but this is the one we shall focus on in this paper.

4 version corresponds to the eld theory limit of the heterotic string, it was natural to conjecture [8, 4] that other version corresponds to the eld theory limit of a \dual string". The bosonic action of the fundamental D = 6 heterotic string is given by S = 0 Z p d i j X g M i j X B + ::: where i (i =;) are the worldsheet coordinates, ij is the worldsheet metric and ( 0 ) is the string tension T. The metric and -form appearing in S are the same as those in I 6. This means that under the rescalings with constant parameter : both actions scale in the same way g! g B! B e! e (.8) ij! ij (.9) I 6! I 6 S! S (.0) We therefore assume that the bosonic action of the dual D = 6 heterotic string given by ~S = Z d ~ q ~~ i j X ~g ~ 0 ~M i j X B ~ + ::: (.) where ~ i (i =;) are the dual worldsheet coordinates, ~ ij is the dual worldsheet metric and ( ~ 0 ) is the dual string tension T. ~ Furthermore we assume that the metric and -form appearing in S ~ are the same as those appearing in I6 ~. The virtue of these identications is that under the recalings with constant parameter : ~ both actions again scale in the same way: ~g! ~ ~g ~B! ~ B ~ e! ~ e ~ ij! ~ ~ ij (.) ~I 6! ~ I6 ~ ~S! ~ S ~ (.3) The duality relation (.7) is invariant under both rescalings. Since the dilaton enters the dual string equations with the opposite sign to the fundamental string, it was argued in [8, 4] that the strong coupling regime of the string should correspond to the weak coupling regime of the dual string: g dual string =<e >=g string (.4) 3

5 where g string and g dual string are the string and dual string loop expansion parameters. Further evidence for this duality is provided by the complementary discovery that the dual string emerges as a soliton solution of the fundamental string, and vice-versa [4]. The combined supergravity-source action I 6 + S admits the singular elementary string solution [53] where ds = ( a =r )( d + d )+( a =r ) dr +( a =r )r d 3 (.5) e = a =r (.6) e H 3 = a 3 (.7) a = T= 3 (.8) and 3 is the volume of S 3. The source-free action I 6 also admits the non-singular solitonic string solution [4] whose tension ~ T is given by ds = d + d +( ~a =r ) dr + r d 3 (.9) e = ~a =r (.0) H 3 = ~a 3 (.) The Dirac quantization rule relating the Noether \electric" charge to the topological \magnetic" charge ~a = ~ T=3 (.) e = p ZS 3 e H 3 (.3) g = p ZS 3 H 3 (.4) translates into a quantization conditon on the two tensions [5]: = n() 3 0 ~ 0 ; n = integer (.5) Similarly, the dual supergravity-source action I6 ~ + S ~ admits the dual string as the fundamental solution and the fundamental string as the dual solution. When expressed in terms of the dual metric (.6), however, the former is singular and the latter non-singular. Both the string and dual string soliton solutions break half the supersymmetries, both saturate a Bogomol'nyi bound between the mass and the charge. These solutions are the extreme mass equals charge limit of more general two-parameter black string solutions [, 4]. Duality mixes up string and dual string loops in the sense that the roles of worldsheet loop expansions and spacetime loop expansions are interchanged [8, 4]. Consequently what is a quantum eect for the string might be a classical eect for the dual string, and vice versa. At higher loop orders, this leads to an innite number of non-renormalization theorems 4

6 (including the vanishing of the cosmological term) all of which are consistent with known string calculations to higher orders both in 0 (worldsheet loops) and g string (spacetime loops). It is this loop mixing which allows us to test string/string duality, in spite of our ignorance of how to quantize the dual string. If duality is correct, we should be able to reproduce string loop eects from tree-level dual strings! In this paper we shall show in particular how to reproduce the Green{Schwarz spacetime anomaly corrections to the H 3 eld equations (a fundamental string one-loop eect) from the Chern{Simons worldsheet anomaly corrections to the H3 ~ Bianchi identities (a dual string tree-level eect). 3 The fundamental string The fundamental string we shall consider will be the D = 6 heterotic string obtained by compactication of the D = 0 heterotic string on K3. We choose this because it is essential for our purposes that both the string and the dual string be chiral. K3 is a four-dimensional compact closed simply-connected manifold. It is equipped with a self-dual metric and hence its holonomy group is SU(). It entered the physics literature as a gravitational instanton [54, 55], its Pontryagin number being p = 8 ZK3 trr = 48 (3.) However, it was then invoked in a Kaluza-Klein context in [56, 57] where it was used, in particular, as a way of compactifying D = 0 supergravity tod= 6. Half the spacetime supersymmetry remains unbroken as a consequence of the SU() holonomy, and hence it gives rise to an N = supergravity ind=6. There are four N = ;D = 6 supermultiplets to consider: Supergravity A g ; L ;B L T ensor B R ; A R; Hypermatter a R; Y ang Mills A ; A L The scalars parametrize a quaternionic Kahler manifold of the form G=H Sp(). The index A =; labels Sp() and the index a lables one of the representations of H. All spinors are symplectic Majorana{Weyl. The -forms B L and B R have 3-form eld strengths that are self-dual and anti-self-dual, respectively. Only with the combination of one supergravity multiplet and one tensor multiplet do wehave a conventional covariant Lagrangian formulation. In the case of K3, the massless sector of the D = 6 theory coming from the supergravity multiplet in D = 0 consists of this combination plus 0 hypermatter multiplets. The 80 scalars belong to the coset SO(0; 4)=SO(0) SO(3) Sp() [58, 59, 60], this being the moduli space of K3. There are no vector multiplets since K3 has no isometries and is simply connected. This argumentwas generalized to supergravity-yang-mills with gauge groups G 0 = SO(3) or G 0 = E 8 E 8 in [6], where it was emphasized that the resulting N =;D = 6 supergravity 5

7 would be chiral and anomaly free 4. Counting the number of D =6multiplets coming from the Yang-Mills sector is more subtle. As discussed in [67], the values of the background elds, R 0 and F 0, are not independent. The requirement that H 3 be globally dened leads, on integrating (A.0), to the constraint Z M 4 (tr SO(3) F 0 tr SO(;9) R 0 )=0 (3.) for any closed sub-manifold M 4 of the ten-dimensional spacetime. In this background the eective lower-dimensional theory has a reduced gauge symmetry. If the non-zero elds F 0 span a subgroup H G 0 then the gauge group in the lower dimension will be given by G such that G 0 G H (3.3) The adjoint representation of G 0 can be decomposed into a sum of representations adj G 0 = X i (L i ;C i ) (3.4) where L i and C i are irreducible representations of G and H respectively. In particular, for G 0 = SO(3), X dim L i dim C i = dim G 0 = 496 (3.5) i In the present context we choose M 4 to be K 3 and the backgrounds R 0 and F 0 to lie only in the four compactied directions. Specically we \embed the the holonomy group in the gauge group" [68, 6] by taking F 0 =R 0 to lie in the SU() subgroup 5 of SO(3). Then the Yang-Mills supermultiplets are those of G = SO(8) SU(). Thus L = (378; ) C = L =(;3) C 3 = L 3 = (8; ) C = L 4 =(;) C 4 =3 The number of left-handed spinor mutiplets in the representation L i of G is given by an index theorem: = N i = Z [ 8 M 4 tr C i F dim C itrr 0 ] (3.6) In the case of K3, on using (3.) and (3.3), this reduces to N i = =(dim C i R i ) (3.7) where R i is the ratio R i = tr C i F 0 trf 0 (3.8) 4 K3was thus the forerunner of Calabi-Yau compactication [6] from D =0toD= 4. This manifold has a curious way of cropping up in the superstring literature in a variety of apparently unrelated contexts [56, 57, 59, 60, 6,63,64, 65, 66]. 5 Dierent embeddings were chosen in [6]. 6

8 So coming from the Yang-Mills sector in D =0wehave N (378;) = = 0= N (;3) = = 0= N (8;) = = = 0 N (;) = = 3 48 = 45; (3.9) the rst two being the adjoint representation left-handed gaugino superpartners of the G = SO(8) SU() gauge elds belonging to the Yang-Mills supermultiplets, and the last two being the right-handed superpartners of a in the hypermatter multiplets. 4 Worldsheet and spacetime anomalies Let us rst consider the G Yang{Mills and Lorentz Chern{Simons corrections to H 3. Let us dene F = da + A where the gauge elds A = A M dx M are matrices in the adjoint representation of the gauge groups and R = d!+! where the Lorentz connections! =! M dx M are in the vector representation. Let us further dene [69] I 4 = d! 3 = I 4 h () X l k l i TrF l +trr g l! 3 = d! (4.) where the sum is taken over the gauge groups appearing in G = G G G l, k l are the corresponding levels of the Kac{Moody algebra and g l are the dual Coxeter numbers. Then the action S can be modied so as to be both gauge invariant and Lorentz invariant provided [70] and hence the gauge invariant eld strength is given by B = 0 ()! (4.) H 3 = db 0 ()! 3 dh 3 = 0 () I 4 (4.3) In the case of K3 compactication of the SO(3) string, we have and the fundamental representation decomposes as SO(3) SO(8) SU() (4.4) 3! (8; ) + (; )+(;) (4.5) 7

9 so the worldsheet fermions carrying the SO(8) SU() symmetry correspond to Kac{Moody levels k SO(8) = and k SU() =. Similarly, the gravitational anomaly is as in D = 0 but where R now belongs to SO(; 5) instead of SO(; 9). Thus the Yang-Mills and Lorentz Chern{Simons corrections are given by h i dh 3 = 0 trf SO(8) trf SU() + trr (4.6) 4 where we have used TrF SO(8) = 6trF SO(8), TrF SU() = 4trF SU() and g SO(8) = 6, g SU() =. This modication to the Bianchi identity isthus a classical string eect (i.e. tree level in the D = 6 string loop expansion). This is conrmed by the observation that there is no dilaton dependence and that dh 3 is independent of and that the factors cancel out. By supersymmetry, the same combination of Yang-Mills eld strengths and curvatures appearing in (4.6) also appears in the D = 6 tree level Lagrangian: where S (0) YM = Z d 6 x p ge 0 8 th trf SO(8) F SO(8) trf SU() F SU() + trr R i (4.7) t = (g g g g ) (4.8) Next we turn to the Green{Schwarz anomaly cancellation mechanism [50]. The spacetime anomaly polynomial I 8 of this D = 6 string has been calculated by Erler [65] who nds, as expected, that it factorizes in the form I 8 = X 4 ~ X4 (4.9) where and X 4 = I 4 = 4() h trf SO(8) ~X 4 = h trfso(8) +44trF 4() SU() trf SU() + trr i trr i (4.0) (4.) As a consistency check, one notes that factorization requires the absence of a trr 4 term in I 8 and that this is guaranteed if [6] 4X i= N i = dim L i = 4 (4.) as may beveried from (3.9). Dening ~! 3 by X ~ 4 = d~! 3, the consistent anomaly is then cancelled by adding to the eective action Z = M 6 0 () B ~ X 4 + 3! 3 ~! 3 (4.3) 8

10 and recalling the transformation rule for B given in (4.). Now with the normalization of the kinetic term for B given in (.), the addition of (4.3) modies the eld equation to d(e H 3 ) = = 0 () ~ X 4 h trfso(8) +44trF () 3 0 SU() trr i (4.4) This modication to the eld equations is thus a string one-loop eect. This is conrmed by the dilaton dependence on the left hand side and by noting that the right hand side is linear in and involves a factor =() 3 appropriate to a one-loop Feynman integral in D =6. By supersymmetry, the same combination of Yang-Mills eld strengths and gravitational curvatures appearing in (4.4) also appears in the D = 6 one loop Lagrangian S () YM = Z d 6 x p g 0 8 th trf SO(8) F SO(8) i +44trF SU() F SU() trr R (4.5) After this summary of the D = 6 string, we are now in a position to test string/string duality by reproducing (4.4) and (4.5) from the classical dual string. 5 String/string test in D =6 After compactication to D = 6, ad = 0vebrane will appear as a vebrane, a fourbrane, a threebrane, a membrane or a string according as it wraps around 0; ; ; 3 or 4 of the compactied directions. The vebrane is trivial having no degrees of freedom in D = 6. The dual supergravity theory in D = 6 obtained by compactifying the dual D = 0 supergravity onk3, will consist of the same combination of supergravitymultiplet and one tensor multiplet as in the fundamental theory but instead of 0 hypermultiplets, there will be 9 linear multiplets of type and one of type. Inatype linear multiplet [7] one of the four scalars (an SU() singlet) is swapped for a 4-form b ;inatype linear multiplet three of the four scalars (an SU() triplet) are swapped for three 4-forms. The appearance of these 4-forms means that the dual string alone cannot be responsible for this low-energy limit; the threebrane is also contributing. Only the odd-branes will display sigma-model anomalies [8, ] and we might therefore expect contributions to the D = 6 spacetime anomaly polynomial I 8 of the form X 0 X8 ~, X X6 ~, and X 4 X4 ~. However, X 0 is trivially zero and X will involve trr which is zero and trf which is also zero since our gauge group has no abelian factors. For the purposes of the sigma-model anomaly, therefore, we may focus just on the the massless states of the dual string worldsheet. These follow from the compactication of the vebrane worldvolume M ~ 6 to M ~ K3. The -symmetric, spacetime supersymmetricvebrane action (in the absence of internal symmetry) has been constructed in [7] using the Green{Schwarz variables x ; with =0; :::; 9 and =; :::; 6. Since the scalar d'alembertian just splits as 6 = + K3, the number of massless scalar elds is unchanged under compactication. So the Green{Schwarz variables x 9

11 will remain the same, except that only six of them will be regarded as spacetime coordinates. Similarly,by spacetime supersymmetry, the which transformed as a 6 of SO(; 9) are now interpreted as a (4; 4) of SO(; 5) SO(4). This counting is exactly the same as that of the fundamental Green{Schwarz string. Hence, the contribution to the gravitational sigma-model anomaly will be the same as that of the fundamental string. A complete covariant -symmetric Green{Schwarz action for the heterotic vebrane (i.e. with the internal symmetry included) is still lacking. If the eld theory limit of the heterotic vebrane is indeed to coincide with the SO(3) anomaly-free supergravity, however, we must include this internal symmetry in some way. In [], we suggested twoways that would yield the same -model anomaly and that mimic the heterotic string: Weyl fermions in the fundamental of SO(3) or a level WZW model. In this paper, we focus on the fermionic formulation. The calculation of the spin = fermions on the dual string worldsheet parallels that of section 3. The fundamantal representation of G 0 can be decomposed into a sum of representations X fund G 0 = (l i ;c i ) (5.) i where l i and c i are irreducible representations of G = SO(8) SU() and H = SU() respectively. The number of left-hand spinor multiplets in the representation l i of G is given by an index theorem: = n i = Z [ 4 M 4 tr c i F dim c i trr 0 ] = 4(dim c i r i ) (5.) where r i is the ratio r i = tr c i F 0 trf 0 This diers from (3.6) by a factor of 4 since we are going from Weyl fermions in ~ d =6to Majorana{Weyl in ~ d =, as opposed to Majorana{Weyl in D = 0 to symplectic Majorana{ Weyl in D = 6. Since, under SO(3)! SO(8) SU() SU(), we have (5.3) 3! (8; ; ) + (; ; ) (5.4) n (8;) = n (;) = = 4( 0)=4= ~ k SO(8) = 4( ) = 40 = ~ k SU() (5.5) where ~ k SO(8) = 4 and ~ k SU() = 0 are the dual Kac{Moody levels 6. So in analogy with (4.3) the complete sigma-model anomaly may be written 7 d ~ H 3 = ~0 () ~ I4 (5.6) 6 As far as we are aware, the numbers (4,0) appearing here have no direct connection to the numbers (4,0) describing the moduli space of K3 7 The change of sign relative to (4.3) is chosen so as to accord with the vebrane conventions of [] and Appendices. 0

12 where from (4.) ~I 4 = h i 4trFSO(8) +40trF () SU() + trr The negative sign in the SU() case corresponds to the opposite chirality of fermions on the worldsheet. This is not yet in agreement with X ~ 4 of (4.) but previous experience with the vebrane in D = 0 suggests that we should not expect it to be []. First we note that I ~ 4 X4 ~ = h i trf () SO(8) trf SU() + trr =X4 (5.8) so that the discrepancy is twice the sigma-model anomaly of the fundamental string. Up until now wehave been assuming that the supergravity ~ B appearing in (.5) should be identied with the dual string ~ B appearing in (.). However, there is an ambiguity in this denition. In general, we must allow (5.7) ~ 0 ~ B (supergravity) = ~ 0 ~ B (dual string)+ m 0B (string) (5.9) where, in order to preserve the quantization condition on the WZW term, m must be some integer. We nd from (5.8) that the correct result is obtained by the choice 8 m =. In this case, d ~ H3 (supergravity) = ~ 0 () ~ X4 = ~0 4 h trfso(8) +44trF SU() trr i (5.0) This modication to the dual string Bianchi identity is thus a classical dual string eect (the tree level in the D = 6 dual string expansion). This is conrmed by noting that there is no dilaton dependence and that d ~ H 3 is independent of and that the factors cancel out. Once again, by supersymmetry, the same combination of Yang-Mills eld strengths and gravitational curvatures appearing in (5.0) also appears in the dual D = 6 Lagrangian ~S (0) YM = Z q d 6 x ~0 ~ge 8 th trf SO(8) F SO(8) i 44trF SU() F SU() + tr R ~ R ~ (5.) Now here is the crucial step: using the duality condition (.7), and the Dirac quantization rule (.5) with n =, this classical modication to dual string Bianchi identity (5.0) and tree level action (5.) are seen to identical to the string one-loop correction to the fundamental string eld equations (4.4) and action (4.5), respectively. This is the main result of the paper. 8 As we shall see in Appendix B this admits the D =0interpretation of p =4 where p is the Pontryagin number of K3 given in (3.).

13 6 Conclusion According to string/string duality, the Green-Schwarz factorization of the D = 6 spacetime anomaly polynomial I 8 into X 4 X4 ~ means that just as X 4 is the -model anomaly polynomial of the fundamental string worldsheet so X4 ~ should be the corresponding polynomial of the dual string worldsheet. To test this idea we have performed a classical dual string calculation of X4 ~ and found agreement with the quantum fundamental string result. Moreover, as we discuss in the Appendices, the same result for X ~ 4 can be obtained by starting with the D =0vebrane and using Z ~X 4 = ~X 8 (6.) where X ~ 8 is the anomaly polynomial of the d =6vebrane worldvolume calculated in a previous paper [] on string/vebrane duality. This therefore provides an aposteriori justication for assumptions made in [] on the gravitational contribution to X8 ~, although we agree with [39, 40] that an a priori justication is still lacking. However, we disagree with these authors concerning their criticism of [] that the gauge fermions of the covariant heterotic vebrane cannot belong to the 3 of SO(3). They claim that this is inconsistent with Strominger's result [] that the fermions of the gauge-xed solitonic vebrane belong to the (; 8) of SU() SO(8). The vebrane soliton, with its SU() instanton in the four transverse directions, breaks the spacetime plus internal symmetry SO(; 9) SO(3) to SO(; 5) SU() SO(8). (Note that the embedding SO(; 9)SO(3) SO(; 5)SO(9) [39] is not the minimal embedding and corresponds to two minimal instantons [40]). Assuming that the fermions in the unbroken phase transform as a (6; 3) is entirely consistent with a (4; ; 8) in the broken phase. Despite the success of our consistency check, however, some questions remain unresolved. Firstly, wehave concentrated on the D = 6 string obtained from the SO(3) heterotic string in D = 0. It would be interesting to repeat the exercise for the E 8 E 8 heterotic string and for the Type IIB string but we have not yet done so. Secondly, the dual string seems very dierent from the fundamental string. In particular, whereas for the fundamental string the number of left and rightmoving gauge fermions (n L ;n R )is(0;4) and the Kac-Moody levels are k SO(8) = and k SU() =, for the dual string we have (60; 44) and k SO(8) = 4 and k SU() = 0. Although this dual heterotic string still has n L n R = 6, there seems to be a problem with conformal invariance since we get the wrong central charge. There are some caveats to be made in this connection, however. For the purpose of calculating -model anomalies, we have focussed only on the massless states of the dual string worldsheet. There will also be massive Kaluza- Klein modes on the worldsheet coming from compactifying the vebrane worldvolume on K3 and these could contribute to the conformal anomaly. Nor should we forget that the fourbrane, threebrane and membrane are also present, even though they make no contribution to the - model anomaly. Thirdly, the SU() Yang-Mills kinetic energy terms for the dual supergravity action (5.) also appears to have the wrong sign. However, it should be borne on mind that the fundamental action and the dual action are equivalent up to duality transformations and that S (0) YM and S ~ (0) YM contibute to both. The dierence lies only in the loop expansion: in particular what is tree-level for the fundamental string Yang-Mills kinetic energy is one-loop for the dual string ( S ~ (0) YM = S () YM), and vice-versa. This may beveried by converting to K3

14 the appropriate -model metric and counting powers of e. On the subject of loop expansions, it was conjectured in [5] that the number of vebrane loops L6 ~ was related to the Euler number of the vebrane worldvolume ~ 6 by the formula ~ 6 = ( L6 ~ ) in just the same way that the number of string loops L is related to the Euler number of the string worldsheet by the formula = ( L ). In the present context M ~ 6 = M ~ K3 and the Euler number is given by ~ 6 = 48( L ~ ) and hence ~L 6 =4~ L 3 (6.) In summary, the success in achieving the right anomaly has still left unanswered the question of whether the dual theory (string plus membrane, threebrane and fourbrane) is itself quantum mechanically consistent. Perhaps the most important lesson to be learned is that the nature of the dual string depends crucially on the compactication. The inconsistencies (if there are any) may thus be a blessing in disguise: perhaps the requirement that both the fundamental and the dual string be consistent will thus provide a non-perturbative way of narrowing down the range of allowed superstring vacua. One mighteven entertain the idea that in the perfect vacuum the dual string is identical to the fundamental string. For example, requiring that the dual string be chiral with non-vanishing ~ X 4 means that I 8 must be non-vanishing and hence that the spacetime theory must be chiral. This would rule out toroidal compactication, for example, which is not ruled out perturbatively 9. It is obviously of interest in this context to see whether four spacetime dimensions is superior to six. 7 Acknowledgements We are grateful to John Dixon and Ergin Sezgin for many useful discussions. Conversations with Jerey Harvey, Jim Liu and Paul Townsend are also gratefully acknowledged. A Review of string/vebrane test in D =0 D= 0 supergravity-yang-mills admits two anomaly-free [50, 5] formulations: one with a three-form eld strength H 3 [73, 74] and the other with a seven-form eld strength ~ H 0 7 [75]. They are related by Poincare duality: ~H 0 7 = e ^ H 3 (A.) where ^ isthed= 0 dilaton and denotes the Hodge dual using the canonical metric g c MN (M =0;; :::; 9). Taking the exterior derivative of both sides of (A.) reveals that the roles of eld equations and Bianchi identities of the three-form version of supergravity are interchanged in going to the seven-form version. The former corresponds to the eld theory limit of the heterotic string while the latter is conjectured to be the eld theory limit of an extended object 9 There is something rather asymmetrical about toriodal compactication to four spacetime dimensions: the target space duality of the fundamental string is O(6; ;Z) and that of the dual string SL(6;Z) SL(;Z) [3, 4,7, 9, 38]. 3

15 dual to the string: the \heterotic vebrane" [, ]. Just as the -form potential B MN couples to the d = string worldsheet via the term S = 0 Z M d i x j x N B MN = 0 Z M B (A.) where i (i =;) are the worldsheet coordinates and ( 0 ) is the string tension, so the 6-form potential BMNPQRS ~ (M =0;;:::;9) couples to the d ~ =6vebrane worldvolume via the term ~S 6 = = Z () 3 ~ 0 Z () 3 ~ 0 ~M 6 d 6 ~ 6! i x j x k x l x m x n x S ~ BMNPQRS ~M 6 ~ B6 (A.3) where ~ i (i =;:::;6) are the worldvolume coordinates and [() 3 ~ 0 ] is the vebrane tension. The two tensions obey the Dirac quantization rule [5] 0 = n() 5 0 ~ 0 ; n = integer (A.4) where 0 is the D = 0 gravitational constant. To within O( 0 ) Chern{Simons corrections to be discussed below, H 3 is the curl of B H 3 = db + O( 0 ) (A.5) Similarly, uptoo( ~ 0 ) Chern{Simons corrections to be discussed below, ~ H 7 is the curl of ~ B6. ~H 7 = d ~ B6 + O( ~ 0 ) (A.6) We are tempted to identify ~ H 0 7 appearing in (A.) with ~ H 7 appearing in (A.6). However, as pointed out in [], this identication needs to be modied when we include the gravitational Chern{Simons corrections which are of higher order in the low-energy expansion than those of Yang{Mills. Accordingly, we wrote [] ~H 0 7 = ~ H 7 48 ~ 0 0 trr H 3 (A.7) Here R = d! +! and the Lorentz connections! =! M dx M are in the vector representation. The choice of coecient =48 is signicant and we shall return to this later on. In [], the O( 0 ) corrections to H 3 and the O( ~ 0 ) corrections to ~ H 0 7 were examined as a test of the string/vebrane duality conjecture. One begins with the observation of that duality mixes up string and vebrane loops: what is a one loop eect for the string might be a tree level eect for the vebrane, and vice versa. It is this loop mixing which allows us to test string/vebrane duality, in spite of our ignorance of how to quantize the vebrane. If duality is correct, we should be able to reproduce string loop eects from tree-level vebranes! 4

16 To see this, let us rst consider the well-known SO(3) Yang{Mills and Lorentz Chern{ Simons corrections to H 3. Let us dene F = da + A where the gauge elds A = A M dx M are matrices in the fundamental representation of SO(3). Let us further dene I 4 = d! 3 = I 4 () h trf + trr i! 3 = d! (A.8) then the action S can be modied so as to be both gauge invariant and Lorentz provided [70] B = 0 ()! (A.9) and hence the gauge invariant eld strength is given by H 3 = db 0 ()! 3 dh 3 = 0 () I 4 (A.0) This modication to the Bianchi identity is thus a classical string eect (i.e. tree level in the D = 0 string loop expansion). This is conrmed by the observation that there is no dilaton dependence and that dh 3 is independent of 0 and that the factors cancel out. Next we turn to the Green{Schwarz anomaly cancellation mechanism [50]. The anomaly receives contributions from the gravitino and the dimension 496 adjoint representation gauginos of the D = 0 theory, both of which are Majorana{Weyl. As emphasized by Green and Schwarz [50], the miracle of SO(3) is that I factorizes: where I = X 4 ~ X8 (A.) X 4 = I 4 (A.) and ~X 8 = h () 4 4 trf 4 9 trf trr (trr ) + 9 trr4i (A.3) The factors of = in front ofi and I 4 arise because both D = 0 spacetime fermions and the d =worldsheet fermions are Majorana. Dening ~! 0 7 by ~ X 8 = d~! 0 7, the consistent anomaly is then cancelled by adding to the eective action = Z M 0 0 () B ~ X 8 + 3! 3 ~! (A.4)

17 and recalling the transformation rule for B given in (A.0). Now for B normalized as in (.), its kinetic term is Z = M 0 e H 3 ^H 3 (A.5) 0 and hence the addition of (A.4) modies the eld equation to d(e H 3 )= 0 0 () ~ X 8 (A.6) This modication to the eld equations is thus a string one-loop eect. This is conrmed by the dilaton dependence on the left hand side and by noting that the right hand side is linear in 0 and involves a factor =() 5 appropriate to a one-loop Feynman integral in D = 0. So far, all our considerations started with the string worldsheet. The acid test for string/vebrane duality is to reproduce (A.6) starting from the vebrane worldvolume. Let us dene ~I 8 = () 4 h 4 trf 4 96 trf trr (trr ) + 9 trr4 i d~! 7 = ~ I8 ~! 7 = d~! 6 (A.7) Then, as shown in [] the action S ~ 6 of section can be modied so as to be both gauge invariant and Lorentz invariant, provided B6 ~ = ~ 0 () 4 ~! 6 (A.8) and hence the gauge invariant eld strength is given by ~H 7 = d B6 ~ + ~ 0 () 4 ~! 7 (A.9) d H ~ 7 = ~ 0 () 4 I8 ~ (A.0) Note that I ~ 8 is not quite identical to X ~ 8 but ~I 8 X8 ~ = 48 () trr X 4 (A.) and so invoking (A.7), (A.0), (A.) and (A.), the gauge-invariant eld strength ~ H 0 7 satises d ~ H 0 7 = ~ 0 () 4 ~ X8 (A.) This modication to the Bianchi identity is thus a classical vebrane eect (tree-level in the D = 0 vebrane loop expansion). This is conrmed by noting that there is no dilaton dependence and that d ~ H 0 7 it is independent of 0 and that the factors of cancel. Now here is the crucial step: using the duality condition (A.) and the Dirac quantization rule (A.4), this classical modication to the vebrane Bianchi identity (A.) is seen to be identical to the string one-loop correction to the string eld equations (A.6). 6

18 B D =6results from D =0 In this Appendix, we reproduce the D = 6 dual string results of section 5 starting from the D = 0 dual vebrane results of [] reviewed in Appendix A. First we note that the dual string worldsheet ~ M is obtained by compactication on K3 of the dual vebrane worldvolume ~ M6 = ~ M K3. In particular, the -form of ~ B of (.) is related to the 6-form ~ B 6 of (A.3) by where and V is the volume of K3. Similarly, ~ 0 ~ B = ~ 0 = 0 () 3 ~ 0 ZK3 () 3 ~ 0 V ~B 6 (B.) (B.) = V (B.3) So the D = 6 Dirac quantization rule (.5) follows from the D = 0 rule (A.4). As a consequence of (B.), we have Z d H ~ ~ 0 3 = () ZK3 ~ d H ~ 7 =~ 0 () ~I 8 (B.4) 0 K3 on using (A.). This provides us with the consistency check. Under the compactication of section 3, trf SO(3) 4 trf SO(3) trr SO(;9) 4 trr SO(;9) = trf SO(8) +trf SU() +trf 0 = trf SO(8)4 +trf SU()4 +6trF 4 SU() trf 0 +trf 0 = trr SO(;5) + trr 0 4 = trr SO(;5)4 + trr 0 (B.5) Using the fact that trr 0 =trf 0, together with (3.) we can now perform the integration bearing in mind that only F 0 and R 0 lie in the K3 subspace. Thus a th order polynomial with arbitrary coecients a; b; c; d; e yields Z i hatrf 4 +b(trf ) + ctrf trr + d (trr ) + etrr 4 () 4 4 K3 = h 4(b + c) i trfso(8) + 4(3a + b +c)trf () SU() +4(c+d)trR SO(;5) (B.6) In particular, from (A.3) and (A.7) Z Z K3 K3 ~I 8 = ~ I 4 ~X 8 = ~ X 4 (B.7) 7

19 in complete agreement with (5.6) and (5.0). It is now ofinterest to see how the D = 6 shift (5.9) follows from the D = 0 shift (A.7). In particular, the factor =48 was rather mysterious from D = 0 point of view []. Now we see the signicance of 48 in the D = 6 context as the Pontryagin number of K3, and the integer m is given by m = 48() ZK3 trr 0 = p 4 = (B.8) The above results (B.6), (B.7) and (B.8) thus provide an aposteriori justication for for the choice in [] of polynomial I ~ 8 in (A.7) and shift in (A.). The point being that the coecients c; d; e chosen in [] on the basis of an educated guess of the mixed and gravitational vebrane sigma-model anomalies, yield the well-known gravitational sigma-model anomaly polynomial for the string as well as the correct Yang{Mills terms. In a similar way wemay derive the terms in the D = 6 dual Lagrangian (5.) quadratic in the Yang-Mills eld strengths and gravitational curvature. We begin with the tree-level heterotic vebrane action which, as discussed in [7], must be quartic in the eld strengths: ~S (0) YM = Z q d 0 x ~ge ^=3 4 tijklmnpqh trf IJ F KL F MN F PQ 8 trf IJF KL trr MN R PQ + 3 trr IJR KL trr MN R PQ + i 8 trr IJR KL trr MN R PQ where ~g MN = e =6 g c MN is the vebrane -model metric [5] and where ~ 0 (B.9) t IJKLMNPQ = (gik g JL g IL g JK )(g MP g NQ g MQ g NP ) (gkm g LN g KN g LM )(g PI g QJ g PJ g QI ) (gim g JN g IN g JM )(g KP g LQ g KQ g LP ) + (gjk g LM g PN g QI + g JM g NK g LP g QI + g JM g NP g KQ g LI + permutations) (B.0) Integrating over K3 as in (B.6) above yields the D = 6 dual string action (5.). References [] M. J. Du, Class. Quantum Grav. 5 (988) 89. [] A. Strominger, Nucl. Phys. B343 (990) 67. [3] M. J. Du and J. X. Lu, Nucl. Phys. B347 (990) 394. [4] A. Font, L. Ibanez, D. Lust and F. Quevedo, Phys. Lett. B49 (990) 35. [5] M. J. Du and J. X. Lu, Nucl. Phys. B354 (99) 9. 8

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