EQUIVALENCE OF DUAL FIELD THEORETICAL LIMITS OF SUPERSTRING THEORIES. J. M. F. Labastida *

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1 October, 1985 EQUIVALENCE OF DUAL FIELD THEORETICAL LIMITS OF SUPERSTRING THEORIES J. M. F. Labastida * The Institute for Advanced Study Princeton, New Jersey 08540, USA ABSTRACT The equivalence of type IA and type IB field theoretical limits of superstring theories is shown by using a new type of duality transformation. Compactifications of the type IB theory to four dimensions are studied and it is shown that they induce the same vacuum configurations as the type IA theory provided the field equations are satisfied. * Research supported by U. S. Department of Energy Contract No. DE-AC02-76ER02220.

2 1. Introduction. Superstring theories [1] are presently considered as promising candidates for the unification of all fundamental interactions. They provide a framework which is free of perturbative ultraviolet divergences and gauge, gravitational and mixed anomalies [2,3]. Furthermore, recent work has shown that their low energy spectrum consists of chiral matter fields interacting through gauge and gravitational forces in a suitable form to describe the physical world [4,5]. The study of the field theoretical limit of type I (with gauge group SO(32)) [6] and heterotic [7] string theories has played a very important role in the analysis of the low energy behavior of these theories [4,5]. The massless sector of this limit consists of the d=10, N=1 supergravity multiplet coupled to a non-abelian vector multiplet with gauge groups SO(32) (type I or heterotic) or E 8 E 8 (heterotic). Part of the field theoretical action corresponding to this limit is known. The form of this action which has been utilized more often in the analysis of the low energy behavior consists of the locally supersymmetric invariant action of Chapline and Manton [8,9] plus some counterterms that are responsible for the cancelation of gauge, gravitational and mixed anomalies [3]. This form of the field theoretical limit has been denominated theory IA [11]. Recently, a new candidate for the field theoretical limit of type I or heterotic strings has been proposed by Gates and Nishino [11]. This theory, which has been denominated theory IB, consists of the dual version of the locally supersymmetric invariant action of Chapline and Manton [10,11] plus some counterterms which make the theory free of all gravitational, gauge and mixed anomalies. As pointed out in [11], the method used to construct theory IB suggests that it is a new quantum theory. In particular, the structure 1

3 of the anomaly cancelation looks different than in theory IA. Furthermore, a superficial analysis of four-dimensional vacuum configurations seems to indicate that the difference between both theories would arise in the topology of compactifications. However, a more detailed study of theory IB reveals that the arguments forward its inequivalence to theory IA do not hold and that, on the contrary, everything seems to indicate that both theories are in fact equivalent. In this note it is shown that there is not any reason to believe that both theories are quantum mechanically inequivalent and it is conjectured that theories IA and IB are different realizations of the same theory. This is achieved by using a new type of duality transformation. Our analysis places the equivalence of both theories at the same level of strictness as the equivalence between ordinary dual theories. Dual theories are expected to be quantum mechanically equivalent [12,13]. The only known source of inequivalence appears when considering dual theories in spaces with non-trivial topology [14]. The analysis of this question in the theories at hand lies outside the scope of this note. 2. Ordinary duality transformations. Ordinary duality transformations applied to field theories in d dimensions transform p-form potential fields into q-form potential fields. The ranks of these forms satisfy p + q + 2 = d, i.e. the sum of the ranks of their corresponding field strengths equals the space-time dimension. Ordinary duality transformations can be performed when the actions depend only on the field strengths of these potential fields. Explicit appearance in the action of the potential field invalidates the transformation if locality of both dual actions is required. The new type of duality transformation described in this note provides a procedure 2

4 to relate certain class of local actions with explicit dependence on potential fields. Necessary conditions for their applicability are described in the next section. In this section we review ordinary duality transformations [15]. Consider a field theory that involves a p-form potential field A with field strength F, F = da + X, (1) and Bianchi identity, df = dx, (2) where X isa(p + 1)-form which depends on some of the rest of the fields of the theory. Assume that the theory is described by an action S that depends on A only trhough its field strength, S[F,...], and that, besides the gauge symmetry of F (F is invariant under the transformation δa = dλ), this action is invariant under a certain set of symmetry transformations parametrized by s, δ s S = 0. The duality transformation is performed by introducing an auxiliary q-form field P and constructing the following first order action*: S[F,...] S i [G, P,...] =S[G,...]+ P(dG dx ), (3) where the integration corresponds to the integral of a d-form on a d-dimensional manifold and G is an arbitrary (p + 1)-form. The s-transformations of G and P can be arranged in such a way that S i is s-invariant. First, assign to G the same s-transformation as the one of F. Second, perform an s-symmetry transformation on S i. From the variations of S in (3) one obtains a term proportional to (dg dx) since this variation must vanish in the case that (2) is satisfied. From the second * Wedges in exterior products are omited whenever it does not lead to confusion. 3

5 term in (3) one obtains two contributions, one from the variation of P and the other one from the variation of G and X. The second contribution vanishes because of (1). The first contribution has the right form to be combined with the variation coming from the first term of S i to define the variation of P in such a way that S i is invariant under s-transformations. Integrating by parts the last term of S i in (3) we observe that S i depends on P only through its field strength (q + 1)-form Q, Q = dp, (4) that satisfies the Bianchi identity dq =0. (5) The last step of the duality transformation consists of the elimination of G in the action by using its field equation. We assume that the action is suitable to carry out this elimination algebraicaly so the locality of the dual action is ensured. Solving for G in terms of Q and the rest of the fields of the theory and substituting its value back into S i we obtain the dual form of the theory, S[F,...] S[Q,...]. (6) By construction, S is invariant under s-symmetry transformations. Dual theories related in the way described above are expected to be equivalent at the quantum level [12,13]. 3. The new type of duality transformation. The new type of duality transformation is defined as follows. Consider a field theory that involves a p-form potential field B with field strength H, H = db + Y, (7) 4

6 and Bianchi identity, dh = dy, (8) where Y is a (p + 1)-form which depends on some of the rest of the fields of the theory and perhaps on B but only through H. Assume that the theory is described by an action A that besides containing a B-dependence through its field strength H it contains a linear term in B, A[H, B,...] =A 0 [H,...]+ ZB, (9) where Z is a closed (q +2)-form which depends on some of the rest of the fields of the theory. Suppose as before that this action possesses some symmetries parametrized by s, δ s A =0. If we proceeded (as in the ordinary case) demanding the Bianchi identity (8) for a p-form J by using an auxiliary field we would have obtained a differential field equation for J because of the explicit appearance of the potential field B in the action. The key point of the new type of duality transformations is to demand eq. (7) instead of eq. (8) by using an auxiliary field. Consider the auxiliary (q + 1)-form field N and the following first order action: A[H, B,...] A i [J, B, N,...] =A[J, B,...]+ N (J db Y), (10) where J is an arbitrary (p + 1)-form. Following the same reasoning as in the ordinary case it is easy to see that the variation of N under s-transformations can be defined in such a way that A i is invariant. First, assign to J the same s-transformation as the one of H. Then perform the variation of A in (10). This variation is proportional to (J db Y) since it must vanish when (7) is satisfied. From the variation of the second term of A i in (10) one obtains two contributions. The one that contains the variation of N is combined 5

7 with the term obtained from the variation of A to define the s-transformation of N. The other one vanishes in virtue of (7). Thus, the intermediate action A i is invariant under s-transformations. The next step consists of the elimination of B by using its field equation after integration by parts in (10), Z +( 1) q+1 dn =0. (11) This equation plays the role of the Bianchi identity satified by the field strength N. Notice that, contrary to the ordinary case, the auxiliary field is not the potential field but the field strength itself. Equation (11) is consistent with the fact that Z is a closed form. It is worth to remark that at this point the intermediate action A i does not contain any dependence on B. The last step consists of the elimination of J in A i by using its field equation. We assume this can be done algebraicaly to ensure locality of the dual action. Solving for J in terms of N and the rest of the fields of the theory and substituting its value back into A i we obtain the dual form of the theory, A[H, B,...] Ã[N,...] (12) where N is a field strength with Bianchi identity (11). Since Z is a closed form, there exists at least locally a (q+1)-form Z 0 such that Z = dz 0. Then, eq. (11) can be solved for N in terms of an arbitrary q-form potential M such that N = dm +( 1) q Z 0. (13) By construction, Ã is invariant under s-transformations. We expect that theories related 6

8 by this new type of duality transformation are quantum mechanically equivalent. Similar analysis as the ones discussed in [13] could be used also in this case. It is important to emphasize that this new type of duality transformation can be applied only to theories whose action has a linear dependence on the potential field B as shown in (9) with Z closed. This is, of course, very restrictive. It is so restrictive that in most of the cases if a theory is suitable for a duality transformation of the new type, it is also suitable for an ordinary one. If Z is exact, i.e. Z 0 can be defined globally, after integration by parts in (9), the action A does not have any explicit dependence on the potential field B. Therefore, an ordinary duality transformation can be performed. If Z is closed but not exact, the only way to proceed is the new one. However, in this case (13) is only valid locally. In the case that the two types of duality transformation can be performed, both lead to the same action up to field redefinitions. By performing an ordinary duality transformation starting with the action A, the dual action contains a q-form M such that the Bianchi identity satisfied by its field strength, N = dm, is trivial, i.e. dn = 0. However it is possible to rewrite the resulting action in terms of another q-form M whose field strength satisfies the non-trivial Bianchi identity (13) by doing the field redefinition, N = N ( 1) q Z 0. (14) The importance of the new type of duality transformation in this case resides in the fact that it provides a direct way to write the dual form of the theory with field strengths that do not have trivial Bianchi identities. As we will see when considering dual field theoretical limits of superstrings, it seems that a representation of the theory with non-trivial Bianchi identities is preferred because it facilitates the analysis of symmetry and anomaly properties of the theory. The equivalence of theories IA and IB could be shown by using an ordinary 7

9 duality transformation and performing the appropriate field redefinition. However, this procedure is not very elegant. The new type of duality transformation provides a new way of verifying directly this equivalence. 4. Equivalence of theories IA and IB. The field content of theory IA consists [8,9] of the d=10, N=1 supergravity multiplet {e A M,ψ M,B MN,λ,φ}, respectively, zehnbein, Rarita-Schwinger field, 2-form potential, spinor and scalar; and the d=10, N=1 vector multiplet {A M,χ}, respectively, vector and spinor. Fields ψ M and λ have opposite chirality and A M and χ are Lie-algebra valued fields in the adjoint representation of the gauge groups SO(32) or E 8 E 8. The action of this theory is an extension of the Chapline and Manton action [9] A CM. This extension, which makes theory IA anomaly free, has two aspects [3,16,17]: i) The field strength of the 2-form potential B has an additional term with a Lorentz Chern-Simons form, H = db + ω 3L 1 30 ω 3Y, (15) where ω 3L = tr(ωr 1 3 ω3 ) and ω 3Y = Tr(AF 1 3 A3 ). We adhere to the following notation. Traces over forms involving gauge fields are denoted by Tr as a reminder that they are taken over the adjoint representation of the gauge group; traces over forms involving spin connections are denoted by tr as a reminder that they are taken over the fundamental representation of SO(9, 1). The purpose of this modification of H is to have extra terms in the Lorentz transformation of B such that by adding counterterms to A CM (see ii) the Wess-Zumino consistency conditions [16] are satisfied. 8

10 ii) Counterterms are added to A CM in the following form [3,17]: A IA = A CM [H,...]+ξ ( 3BX 8 2(ω 3L 1 30 ω 3Y )X 7 ), (16) where ξ is a constant (see [18] for an explicit computation) and X 8 = 1 24 TrF (TrF 2 ) TrF 2 trr (trr2 ) trr4, (17) with X 7 being a generalized Chern-Simons form, dx 7 = X 8. (18) Notice that in virtue of (18) this action has the proper form to be dual transformed by using the new procedure (compare (16) to (9)). Following the same steps that led to eq. (10) we introduce the auxiliary 7-form field N and construct the following first order action, A IAi [J, B, N,...] =A CM [J,...]+ [ + ξ (3BX 8 2(ω 3L 1 ) 30 ω 3Y )X 7 + N ( db J + ω 3L 1 30 ω ) ] 3Y. (19) Integrating out B we obtain the Bianchi identity satisfied by N, 3ξX 8 + dn =0. (20) Taking into account (18) one can write the solution to (20) in terms of a 6-form potential field M in the following way: N = 3ξ(dM + X 7 ), (21) where the factor 3ξ in front of dm has been introduced for later convenience in comparing the resultant dual action to the action of theory IB. This equation defines the transformation properties of M by requiring N to be invariant under gauge and Lorentz transformations. 9

11 This is essential in probing cancellation of gauge, gravitational and mixed anomalies. The next step of the transformation consists of the elimination of the field J in the intermediate action (19) by using its algebraic field equation. First, we need to know the explicit dependence of A CM on J. The form of this action [8,9] is (κ = 1)*: A CM [J,...] = ( d 10 xe φ 2 JMNP J MNP + 1 ) 3 8 φ 4 JMNP F MNP + rest, (22) where, F MNP = ψ Q Γ QMNPR ψ R +6ψ M Γ N ψ P 2ψ Q Γ MNP Γ Q λ +Tr(χΓ MNP χ) (23) and rest stands for terms in the action which do not contain any dependence on J. Using the algebraic field equation for J in (19) we find, J MNP = 1 24 φ 3 4 F MNP !7! e 1 φ 3 2 ɛ Q 1...Q 7 MNP N Q1...Q 7. (24) Introducing this expression back into the intermediate action A IAi, we obtain the dual action of theory IA, Ã IA [N,M,...] = + ξ ( 1 d 10 xe 384 F MNPF MNP 1 3!7! 1 3!7! φ 3 2 N M 1...M 7 N M 1...M 7 + rest 1 6 ( (ω 3L 1 30 ω 3Y )X 7 +3M(trR TrF 2 ) 1 24 φ 3 4 e 1 ɛ Q 1...Q 7 MNP N Q1...Q 7 F MNP ) where we have used (21) to write out the last term with an explicit dependence on M. In this form it is easy to check that ÃIA is in fact the action of theory IB found in [11] up to field redefinitions. This can also be concluded by observing that (24) and the last integral in (25) are the same as their corresponding counterparts in [11]. Notice also that, contrary to * Our conventions are the same as in ref. [9] except for the replacement F (YM) MNP 2J MNP. This replacement ), (25) avoids the appearance of factors 2 in (15). 10

12 the procedure done in [11], we do not need to remove any term linear in both M and TrF 2. The only difference with theory IB is the factor 3ξ in the Bianchi identity (21). However, this is not a discrepancy. The role of this Bianchi identity is to define the transformation properties of M assuming that N is gauge and Lorentz invariant. Clearly, we can multiply N by any constant without modifying the transformation of M. This means that in fact Gates and Nishino could have constructed a one-parameter family of anomaly free theories. Instead, they chose the simplest case in which the factor between N and X 7 is 1. Acording to (21), the theory dual to IA selects one member of such a family in which the factor is 3ξ. 5. Vacuum configurations. Our next goal is to study compactifications of this theory based on the dual action obtained above. As pointed out in [11], due to the fact that in this case the Bianchi identity (21) involves an 8-form, it seems that the structure of the vacuum configurations could be very different. The purpose of this section is to analyze this question. We proceed deriving conditions for a background field configuration M 4 K (K is a six-dimensional compact manifold) to be perturbatively supersymmetric. Similar arguments can be used for other kinds of compactifications. We follow the strategy utilized in [4]. Our notation is the following: upper case latin indices run over (0,1,...,9), greek indices will run over (0,1,2,3) (M 4 ) and lower case latin indices over (4,5,...,9) (K). Conventions regarding the Dirac matrices will be the same as in ref. [4]. First, let us assume that the background field configuration is maximally symmetric. For Fermi fields this requires that they vanish. For fields with space-time indices this implies 11

13 A µ = 0, and N M1...M 7 = 0whenever the number of M-indices which are greek is different from four. We do not have necessarily vanishing vacuum configurations for φ, A m and N µνρσmnp = e 4 ɛ µνρσ Tmnp, (26) where e 4 is the four-dimensional vierbein determinant, ɛ µνρσ is the four-dimensional epsilon symbol and T mnp is a rank-3 antisymmetric tensor in K. To require invariance of the vacuum under supersymmetry implies that the variations of the background fields under supersymmetry transformations vanish. If we demand this for the Fermi fields, by supersymmetry it also holds for Bose fields. In order to study the implications of these conditions in the case of the dual form of theory IA we define the following 3-rank antisymmetric tensor in K, T qrs φ 3 2 e 1 6 ɛ qrsmnp Tmnp, (27) where e 6 is the six-dimensional vierbein determinant and ɛ mnpqrs is the six-dimensional epsilon symbol. Introducing (27) into the variations of the Fermi fields under supersymmetry transformations, which can be easily obtained by using (24) in the transformations given in ref. [9], we find, δψ µ = µ ε φ 3 4 (γµ γ 5 T )ε, (28.1) δψ m = m ε φ 3 4 (γm T 12T m )ε, (28.2) δλ = 3 8 2φ 1 (γ m m φ)ε φ 4 Tε, (28.3) δχ = 1 4 φ 3 8 Fmn γ mn ε, (28.4) where ε is the parameter of the supersymmetry transformation, M is the supergravity- Yang-Mills covariant derivative and, T T pqr γ pqr, T m T mqr γ qr. (29) 12

14 Equations (28) are in fact the starting point of the analysis carried out in ref. [4]. We recall here their consecuences. Requiring the vanishing of this equations it is found that the geometries of M 4 and K are restricted in the following way: i) M 4 must be Mikowski space-time (e 4 = 1). ii) K must be a semi-khäler manifold of vanishing first Chern class with complex structure J mn satifying, T mnp = 1 12 φ 3 4 e6 ɛ mnprst r J st. (30) satifies, Furthermore, it is also obtained that φ is constant and that the gauge field strength F mn γ mn η =0, (31) where η is a gauge singlet spinor in K verifying, ( m 3 4 φ 3 4 Tm )η =0. (32) Comparing (27) and (30) we can identify T mnp as proportional to the exterior derivative of the complex structure: T mnp =9φ 3 4 [m J np]. (33) On the other hand, from the Bianchi identity (20) we obtain that d T =0, (34) after using (21) and the fact that X 8 vanishes when F and R take their background values. Equations (33) and (34) are consistent because the background configurations have constant φ. 13

15 The geometrical restrictions for the dual theory are the same as for the ordinary theory. However, contrary to what occurs in the ordinary case, in the case at hand we do not obtain any relation between the exterior derivative of T and the 4-forms trr 2 and TrF 2. After this analysis, we observe that the dual theory produces almost the same vacuum configurations as theory IA. This is in fact what one would expect to occur. Dual theories are equivalent on-shell [13]. This means that compactifications of dual thoeries will give the same kind of vacuum configurations if one takes into account the background field equations. In our analysis above we have proceeded without using the background field equations because we wanted to extract the maximum information of the nature of the compactification without making use of them. The reason behind this is that we are working only with a truncation of the field theoretical limit of superstring theories and such equations could be very different if the whole limit were considered. For the only purpose of comparing the dual theories at hand we will take into account the background equations of motion. Our next task is to compute the resultant field equation from ÃIA in (25) when making a variation respect to M. After using (26) and (27) and performing some algebra we find, dt =trr TrF 2. (35) This is just the equation we were seeking to see how eq. (15) comes out in the contest of the dual theory. To check the complete equivalence of the vacuum configurations of both theories we need to work out which one is the condition implied by the field equation of H in the ordinary theory. Using (15), (16) and (22) it is easy to find out that this condition is that the exterior derivative of the dual form of H must vanish. This is just eq. (34) in the case of the dual theory. These results are just an example of the fact that dual theories interchange Bianchi identities by field equations. This ends our analysis on the equivalence 14

16 of the vacuum configurations of this theories. 6. Concluding remarks. The analysis presented in this note shows that theories IA and IB are dual and so equivalent except for possible differences originated by global aspects of space-time [14]. As shown explicitly in [3] and [11] both dual theories are free of gauge, gravitaional and mixed anomalies. The study of vacuum configurations in the dual version has shown the interplay between Bianchi identities and field equations in dual theories. If the full field theoretical limit of superstrings admits dual versions, this interplay implies that our lack of knowledge of the full field equations carry a lack of knowledge of Bianchi identities. From this point of view, eq. (17) could be incomplete. We now address the question of the existence of dual versions of the full field theoretical limit of superstring theories. Based on the work by Fradkin and Tsytlin [19] it seems that the answer to this question differs depending on which kind of superstring is considered. The effective action approach proposed in [19] suggest that the heterotic string prefers the 2-form potential field. The reason for this is that in their effective action the coupling of the 2-form potential field B to the bosonic modes of the string occurs via a Wess-Zuminolike term which can not be generalized to the case of a 6-form potential field. This force us to conjecture that the full field theoretical limit of the heterotic string does not admit dual versions. Our arguments dealing to a possible incompletness of the Bianchi identity generated by eq. (17) do not hold for the heterotic string and it may be possible that we already know its complete form. 15

17 The situation is very different in the case of type I superstrings. Contrary to the case of the heterotic string, for type I superstrings the 2-form potential field B comes from the fermionic part of the string and so it couples to fermionic modes. The couplings of B in the effective action presented in [19] admit a generalization to the case of the 6-form potential field M. Thus, it seems that the field theoretical limit of type I superstrings may admit a dual version at the full level. Finally, we would like to do some remarks about anomalies in theories resulting from the compactifications of the dual field theories at hand. For the ordinary theory, the analysis presented in [17,20] show that compactifications to four and six dimensions are anomaly free without requiring the field equations to be satisfied. The essential tool of their analysis is eq. (17). In general, one is supposed to use the background field equations in this kind of analysis. However, it just happens that the anomaly cancellations can be proven without making use of them in this case. Similar analysis as the ones in [17,20] have been carried out recently [21] for the dual theory regarding compactifications to six dimensions. In this case, as well as in the case of compactifications of the dual theory to four dimensions, it turns out that the field equations are needed to prove cancellation of anomalies. This is what one should expect since the powerfull tool (eq. (17)) used in the ordinary case appears now as a field equation. I would like to thank J. Bagger, C. Burgess, T. Morris and M. Mueller for discussions. References 16

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