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The low-energy effective action for perturbative heterotic strings on $K

时间:2011-09-03


ULB-TH-97/17 hep-th/9709129

The low-energy e?ective action

arXiv:hep-th/9709129v2 19 Jun 1998

for perturbative heterotic strings on K3 × T 2 and the d=4 N=2 vector-tensor multiplet

R. Siebelink
Service de Physique Th? eorique, Universit? e Libre de Bruxelles, Campus Plaine, CP 225, Bd du Triomphe, B–1050 Bruxelles, Belgium

Abstract

We consider d = 4 N = 2 supergravity theories which serve as low-energy e?ective actions for heterotic strings on K3 × T 2 . At the perturbative level we construct a new version of the heterotic e?ective action in which the axion has been traded for an antisymmetric tensor ?eld. In the string frame the antisymmetric tensor doesn’t transform under Poincar? e supersymmetry into the dilaton-dilatini system. This indicates that in this frame the antisymmetric tensor ?eld and the dilaton are not contained in an N = 2 vectortensor multiplet. Instead, we ?nd that the heterotic dilaton is part of a compensating hypermultiplet, whereas the antisymmetric tensor is part of the gravitational multiplet. In order to obtain our results we use superconformal techniques. This enables us to comment on the range of applicability of this particular framework.

1

Introduction

During the last few years much progress has been made in the study of supersymmetric ?eld theories and string theories in various dimensions thanks to the discovery of a whole set of duality symmetries. Four dimensional models with N = 2 supersymmetry have proven to be particularly interesting because of the rich physical phenomena that appear in this context. Moreover these phenomena can be studied in great detail thanks to the speci?c structure imposed by the N = 2 supersymmetry. A by-now famous example concerns the duality that relates heterotic strings on K3 × T 2 to type IIA strings on K3 -?bered Calabi-Yau manifolds [1, 2, 3, 4]. This duality has very powerful consequences. Suppose for instance that one is describing the low-energy dynamics of these strings in terms of an e?ective N = 2 supergravity action with vector and hypermultiplet couplings. The vector multiplet sector of such an e?ective action can in principle be determined exactly by performing a tree level computation in the type II picture. Thanks to the heterotic-type II duality one can reinterprete this exact result in a heterotic context. This automatically solves a strong coupling problem because in terms of the heterotic variables the vector multiplet action receives tree level, one-loop and non-perturbative contributions. The interpretation of the type II result in a heterotic language is not so obvious however, because the variables that are naturally inherited from the type II side turn out to be complicated functions of the “natural” heterotic variables. The latter transform in a simple way under the SO(2, n) T-duality group, whereas the type II variables in general don’t. Amongst other things this implies that one needs to perform a whole set of ?eld rede?nitions before the e?ective action can really be compared to direct perturbative heterotic string computations. Consider for instance the dilaton-axion-like scalar ?eld S = φ ? ia which on the type II side is de?ned as the complexi?ed K¨ ahler modulus of the IP1 base of the K3 -?bration. This ?eld S is an N = 2 “special coordinate” and it is shifted by a purely imaginary constant under a (quantised) PecceiQuinn symmetry. It is well-known [5, 6] that S is not invariant under SO(2, n) transformations once loop and non-perturbative contributions are taken into account. This indicates that φ, which coincides with the true heterotic dilaton at the string tree level, starts to di?er from it at the one-loop level or non-perturbatively. Therefore it is necessary to express the ?eld φ as a function of the true dilaton φinv and the other moduli, before one can properly separate the non-perturbative e?ects from the perturbative ones. It is also known [7] that one has to perform a change of variables at the level of the vector ?elds. The reason is that the SO(2, n) transformations mix the ?eld strengths for the type II inherited vectors1 with their duals, which implies that these vectors themselves transform in a non-local way. In order to avoid this, one can perform an electro-magnetic duality transformation on one of the vectors, such that one ends up with a new set of “stringy” vectors which transform just linearly into eachother under SO(2, n). In addition to the above-mentioned ?eld rede?nitions there is a last change of variables which so far has been less well under control, and which we intend to study in the course of the present work. What we have in mind here is the duality transformation which trades the axion ˇ?ν . Of course this transformation can only be implemented a for an antisymmetric tensor ?eld B after going to the perturbative region of the vector multiplet moduli space, where the PecceiQuinn symmetry is continuous instead of being quantised. The axion can then be identi?ed with the zero form gauge potential associated to this continuous symmetry and as such it can be dualised into an antisymmetric tensor ?eld. In [5] it was conjectured that in the N = 2 context this duality transformation would replace the vector multiplet which originally contained the ˇ?ν would ?nd their scalar S by a so-called vector-tensor multiplet [8, 5] in which φinv and B
Incidently we will refer to these vector ?elds as the ST U vectors, because they lead to a vector multiplet action which is characterised by a prepotential F (X ) of the form ST U + more. Here T and U stand for the moduli of the heterotic T 2 .
1

1

natural place. In order to test this conjecture a systematic study of vector-tensor supergravities was undertaken in [9]2 . An interesting class of interacting vector-tensor theories came out of this analysis, but quite surprisingly the sought-after heterotic vector-tensor theory was not found. In this article we further investigate the issue of a possible vector-tensor structure in the antisymmetric tensor e?ective action for heterotic strings. We do this by explicitly constructing this antisymmetric tensor e?ective action, together with its associated N = 2 supersymmetry transformation rules. The outcome is ?rst of all that we have to be careful before drawing rigorous conclusions about the possible (non)existence of the heterotic vector-tensor multiplet because the N = 2 supersymmetry of our ?nal model is only realised on-shell. This means that one ?rst has to decide on which variables one uses as the fundamental ones, before one can identify the kind of multiplets these variables belong to. When the string metric and the ˇ?ν corresponding string gravitinos are used as fundamental variables, the antisymmetric tensor B is not linked to dilaton-dilatini system by supersymmetry. As a result there is no vector-tensor multiplet in the string frame theory. In this article we also wish to comment on other issues which are of interest from a purely supergravity point of view. Since it is not always possible (nor desirable) to keep the heterotic string and supergravity ideas completely separated throughout the main text, we summarise the di?erent lines of thought already here, for the sake of clarity.

1.1

Construction of the antisymmetric tensor e?ective action for perturbative heterotic strings with N=2 supersymmetry

Our strategy for obtaining the antisymmetric tensor e?ective action for heterotic strings consists of performing a sequence of duality transformations. As a starting point we take a conventional N = 2 supergravity theory coupled to a set of vector and hypermultiplets, and we use the wellknown vector multiplet prepotential for type II strings on K3 -?bered Calabi-Yau manifolds. This yields what we call the ST U version of the heterotic e?ective action. We review the basic properties of this model including its Peccei-Quinn and SO(2, n) symmetries. We brie?y discuss how the SO(2, n) symmetry can be made manifest at the lagrangian level by implementing the higher-mentioned duality transformation on the vector gauge ?elds [7]. Before we dualise the axion we broaden our point of view and study generic vector multiplet theories containing a Peccei-Quinn symmetry. We show that the class of Peccei-Quinn invariant models is in fact quite restricted and precisely comprises the cases discussed in [9] plus the case which is relevant for perturbative heterotic strings. The various Peccei-Quinn invariant models can be dualised in a uni?ed way and this explains why the resulting antisymmetric tensor theories share a similar gauge structure. Most notably there exists a particular U (1) gauge symmetry which acts as a shift symmetry on the antisymmetric tensor. For those cases that allow for an o?-shell treatment along the lines of [9] this U (1) transformation is nothing but the central charge transformation. We show that in the cases of [9] the central charge-like structure is really indispensable (even on-shell!) whereas this is not so for the heterotic antisymmetric tensor theory. In the latter case the central charge-like structure can be completely removed, and one ends up with a set of vector ?elds which all appear on an equal footing. This of course re?ects the underlying SO(2, n) invariance of the heterotic model. It is worth mentioning that very little string information is used in our construction of the “heterotic” antisymmetric tensor theory. In fact we just start from the most general PecceiQuinn invariant vector multiplet theory and then select the case which contains an SO(2, n) symmetry. At the end of the whole dualisation procedure we ?nd that several known string theoretical properties are manifestly realised in our model, which shows that in fact they can
2

More recently superspace descriptions of the vector-tensor multiplet have appeared in [10].

2

be thought o? as being a consequence of N = 2 supersymmetry alone, rather then being of an intrinsic stringy nature. First of all we nicely reproduce the speci?c couplings of the dilaton and the antisymmetric tensor to the other moduli. Furthermore we see that the supersymmetry transformation rules for all the ?elds are (almost) completely independent of the one-loop part of the theory. In this respect the antisymmetric tensor formulation is simpler then the other possible formulations of the heterotic e?ective action in terms of vector and hypermultiplets only.

1.2

N=2 superconformal supergravity and the perturbative heterotic string

It so happens that the low-energy e?ective action for perturbative heterotic strings is a perfect laboratory to address some interesting supergravity issues. These supergravity issues form a second main ingredient of this paper. In particular we focus on the superconformal framework for d = 4 N = 2 supergravity [11, 12, 13] which is very well suited for our present purposes. The general philosophy of the superconformal approach to N = 2 supergravity is the following. Although the ultimate goal of the superconformal techniques is to construct super Poincar? e theories describing the on-shell interactions of a certain set of physical ?elds, one starts o? with various multiplets that form a representation of the o?-shell superconformal algebra (which is considerably larger then the super Poincar? e algebra). Due to this high degree of symmetry several expressions —like the supersymmetry transformation laws for the ?elds— have a relatively simple form. In a second step one reduces the symmetry algebra to the super Poincar? e algebra by implementing a partial gauge choice. Moreover one eliminates several auxiliary ?elds. In doing so the supersymmetry transformations in general acquire a more complicated form, but in any case they can be obtained by a number of well-de?ned algorithmical steps. What makes the perturbative heterotic string so interesting from an N = 2 supergravity point of view is that it gives us two di?erent examples of theories where the direct application of the superconformal ideas is problematic. The ?rst example arises when one goes to the stringy vector formulation for which a prepotential — which is a crucial ingredient in any o?-shell superconformal vector multiplet theory— doesn’t exist [7].3 The second example concerns the heterotic antisymmetric tensor theory, which, as we already said, escaped any direct superconformal treatment along the lines of [9]. Although we know in advance that somehow the o?-shell superconformal framework must break down when we dualise towards these “problematic” theories, we perform our computations in a superconformal setting. First of all this enables us to verify which ingredient of the standard superconformal framework is incompatible with the dualisation procedure. We ?nd that the duality transformations that are used to obtain the two problematic theories both affect the Weyl multiplet, such that the latter is no longer realised o?-shell4 . In this sense the nonexistence of a conventional superconformal description for both the stringy vector multiplet version and the antisymmetric tensor version of the heterotic e?ective action has a common origin. There is no problem in preserving the various superconformal symmetries during the duality transformations. This is important from a technical point of view, because it implies that we can determine the supersymmetry transformation laws for the stringy vectors and antisymmetric tensor ?eld before having to impose the conventional superconformal gauge choices and even before eliminating the auxiliary ?elds. In this way the necessary computations can be kept relatively simple.
This case can be treated in an elegant way if one goes on-shell, where it is possible to write down an action [14] in terms of symplectic sections rather then the prepotential itself. In the stringy basis these symplectic sections remain well-de?ned even though they are no longer based on an underlying prepotential. 4 Note that this departure from o?-shellness does not occur in the cases decribed in [9] because there the Weyl multiplet is never touched.
3

3

One of the beautiful aspects of the superconformal setup is that it allows for some ?exibility in the Poincar? e reduction. We can bene?t from this ?exibility by choosing a dilatation gauge which immediately leads to a string frame Poincar? e action. It is satisfactory to see that from a purely supergravity point of view the string frame is selected as a very natural one. It can be de?ned as the only dilatation gauge which makes the supersymmetry transformations of the stringy vector ?elds dilaton independent. Once this gauge has been chosen one may verify ˇ?ν does not transform into the dilaton or dilatini, which means that the antisymmetric tensor B that in the string frame these ?elds don’t combine into a vector-tensor supermultiplet. What happens instead is that the dilaton and the dilatini e?ectively become part of the compensating hypermultiplet, whereas the antisymmetric tensor becomes part of the gravitational multiplet. Of course one can go to the Einstein frame, by applying an interpolating dilatation and S ˇ?ν is not altered by this step supersymmetry transformation. The supersymmetry variation of B ˇ because B?ν is inert under the interpolating transformations. On the other hand the Einstein ˇ?ν is gravitinos are dilatino dependent functions of the original string gravitinos, so when δB written in terms of the Einstein frame variables a (spurious) dilatino dependence gets induced. One might interprete the resulting Einstein frame con?guration as an on-shell heterotic vectortensor multiplet, but this con?guration is clearly not selected by string theory.

2

N=2 superconformal supergravity coupled to vector and hypermultiplets

In this section we discuss some basic facts concerning d = 4 N = 2 superconformal supergravity and its couplings to vector and hypermultiplets. Our discussion will be rather brief, as we intend to use the standard superconformal vector and hypermultiplet theories merely as a starting point for our construction of the antisymmetric tensor version of the low-energy e?ective action for heterotic strings. Symplectic transformations are discussed in some more detail though, because as far as we know they never received a full treatment in the superconformal framework. The interested reader can ?nd more details about the superconformal approach to N = 2 supergravity in the original articles [11, 12, 13]. For more recent texts we refer to [15] and also to the last article of [9] (which contains a comprehensive list of conventions). The Weyl multiplet is a central object in the superconformal multiplet calculus as it contains i , the gauge the gravitational degrees of freedom. It consists of the vierbeins e? a , the gravitinos ψ? ?elds b? , A? , V? ij — which gauge dilatations, chiral U (1) and SU (2) transformations—and a set + of matter ?elds: a selfdual tensor T?ν ij which is antisymmetric in its SU (2) indices, a real scalar D and a doublet of chiral fermions χi . As such the Weyl multiplet forms the basic representation of (a deformed version of) the superconformal algebra, which closes o?-shell. This algebra consists of general coordinate, local Lorentz, chiral U (1) and SU (2) transformations, dilatations, special conformal and Q- and S -supersymmetry transformations. For future reference we list the transformation laws for the vierbeins and the gravitinos δe? a = ? ?i γ a ψ?i + h.c. ? ΛD e? a ,
1 1 i

i i . δψ? = 2D? ?i ? 4 σ · T ? ij γ? ?j ? γ? η i ? 2 ΛD + 2 ΛU(1) ψ?

(2.1)

Here ?i , η i , ΛD and ΛU(1) are parameters for Q- and S -supersymmetry transformations, dilatations and chiral U (1) transformations respectively. General coordinate, local Lorentz and SU (2) transformations (with parameter Λi j ) are not explicitly given because these can automatically be inferred from the index structure of the ?elds that are being transformed. Here and in what follows a derivative D? stands for a covariant derivative with respect to local Lorentz, dilatation, 4

chiral U (1), SU (2) and gauge transformations. Since the parameter ?i carries non-trivial Weyl and chiral U (1) weights one has that
ab D? ?i = ?? ? 2 ω? σab + 2 b? + 2 A? ?i + 2 V? i j ?j .

1

1

i

1

(2.2)

ab is the (dependent) gauge ?eld for local Lorentz transformations. Here ω? Next one introduces a set of vector multiplets (labeled by an index I ) which can be consistently coupled to the Weyl multiplet. Each vector multiplet contains a complex scalar X I , a I , a real SU (2) triplet of scalars Y I and a doublet of chiral gauginos ?I . vector gauge ?eld W? ij i The lagrangian describing the most general couplings of vector multiplets to superconformal supergravity was given in [13] in formula (3.9). We will use this lagrangian several times in what follows and denote it by e?1 Lvector . In order to clarify our notations we list the bosonic terms: I ijJ ? J ? 1 NIJ Yij ? 1 R + D + NIJ D? X I D ? X Y e?1 Lvector = XN X 6 8 1 1 i ? +I +?ν J +I +?ν ij + ? 8 XNI F?ν Tij ε + 64 XN X T?νij εij ?8F IJ F?ν F

2

+ h.c. (2.3)

+fermionic terms .

This lagrangian depends on a holomorphic prepotential F (X ) which is homogeneous of second degree in the X I ’s.5 Derivatives of the prepotential with respect to the scalars X I are denoted by def ? I =X ? J NJI . FI , FIJ , · · ·, and NIJ = ?Im FIJ . Sometimes we omit contracted I indices, e.g. XN Furthermore we use abelian vectors only. This is a necessity for those vectors we want to dualise at some point. Although we could keep (some of) the others non-abelian we choose not to do so for simplicity. The ?eld strengths, their duals and (anti)selfdual parts are given by
I I F?ν = 2?[? Wν ] I ? ?ν I = 1 e?1 ε?νλσ Fλσ F 2

1 ±I I I ??ν F?ν = 2 F?ν ±F .

(2.4)

+I with F ?I . The vector We take ε0123 = i such that complex conjugation interchanges F?ν ?ν multiplet ?elds transform as follows:

δX I δ?iI
I δW? I δYij

= ? ?i ?iI + ΛD ? iΛU(1) X I ? I T ? kl εkl + Y I ?j + 2X I ηi + 3 ΛD ? i ΛU(1) ? I ? ?I ? 1 X = 2D / X I ?i + εij σ ?ν ?j F i ?ν ij ?ν 4 2 2 I ij I i j I ? ? = ? ?i γ? ?j ε + 2X ? ψ? εij + h.c. + ?? θ
I = 2? ?(i D / ?I ?(k D / ?l) I + 2ΛD Yij . j ) + 2εik εjl ?

(2.5)

The derivatives D? are covariant with respect to all the superconformal (and possibly also gauge) symmetries. The covariant ?eld strengths for the vectors are given by
j I I ? iI ? I ?i j ??ν F = 2?[? Wν ] + ? γ[? ψν ] εij ? X ψ? ψν εij + h.c. .

(2.6)

Finally one adds a set of hypermultiplets consisting of r quaternions Ai α and 2r chiral fermions ζ α , where α = 1, · · · , 2r . The scalar ?elds Ai α satisfy a reality constraint given in [13] which
Note that compared to [13] we have rescaled the prepotential F (X ) → 2iF (X ) as is common practice in the recent literature. We also changed the sign of the Ricci scalar R such that under ?nite Weyl rescalings g?ν ′ e?2ΛD R′ = = e?2ΛD g?ν R + 6 2ΛD ? 6 ?? ΛD ? ? ΛD .
5

5

implies that they describe 4r real degrees of freedom. In order to obtain a fully o?-shell description for the hypermultiplets, one must add a set of 4r auxiliary scalars. These auxiliary degrees of freedom completely decouple from the other ?elds, so we choose to integrate them out right from the start. Having done so we get the following transformation rules for the hypermultiplets ?α ?i + 2ραβ εij ζ ?β ?j + ΛD Ai α δAi α = 2ζ δζ α = D / Ai α ?i + Ai α η i + 2 ΛD ? 2 ΛU(1) ζ α .
3 i

(2.7)

The superconformal action describing the hypermultiplet couplings to the Weyl multiplet is given in [13] in formula (3.29). We list its bosonic part: e?1 Lhyper =
1 α i 1 β α ? i β 2 Ai A β dα ? 3 R + D ? D? Ai D A β dα

+fermionic terms . For conventions concerning the constant matrices ραβ and dα β we refer to [13].

(2.8)

2.1

Symplectic transformations

The superconformal vector multiplet theories we just described can be acted upon with symplectic transformations. In general these transformations relate a given theory (characterised by a prepotential F (X )) to other vector multiplet theories which are classically equivalent to the ?rst one, in the sense that their equations of motion and Bianchi identities are transformed into eachother. Symplectic transformations can be introduced in the following way. First of all one requires that they act linearly on the equations of motion and the Bianchi identities that are satis?ed by the ?eld strengths for the vectors [16]:
?I ? F +I = 0 ? ? F?ν ?ν + ? ? G? ?ν I ? G?ν I = 0

Bianchi identities Equations of motion (2.9)

δ e Lvector G? ?ν I = ?4i δF ?ν ?I
def ?I and G? This can be accomplished by putting F?ν ?ν I into a vector transforming as ?I F?ν G? ?ν I ?I ′ F?ν ′ G? ?ν I

?1

?→

=

UIJ WIJ

Z IJ VI J

?J F?ν G? ?ν J

,

(2.10)

where U I J , Z IJ , WIJ , VI J are constant real matrices. These matrices cannot be chosen arbitrarily ′ because one must ensure that the G? ?ν I result from varying the new vector multiplet theory ?I ′ . This implies [12] that the transformation must be with respect to the new ?eld strengths F?ν symplectic, i.e. U Z 0 0 UT W T (2.11) = T T ? 0 ? 0 W V Z V and that the scalars X I and the FI must form a symplectic vector too: XI FI ?→ XI ′ FI ′ = UIJ WIJ Z IJ VI J XJ FJ . (2.12)

In the present superconformal setup one has to impose an additional restriction on the concept of symplectic transformations. The transformed theory only makes sense as a superconformal 6

theory when it is based on a new prepotential, say F ′ (X ′ ). But this presupposes that all the transformed scalars X I ′ are independent variables, which excludes some a priori interesting symplectic transformations6 . The symplectic transformation which transforms the perturbative heterotic ST U theory into its stringy analog is a particular example of a transformation which leads to new scalars which are not all independent. In section 3.2 we verify in which sense this forces us to exceed the o?-shell superconformal framework7 . Note that only a subset of the full symplectic group maps the original theory onto itself. These so-called duality invariances [12] are characterised by the fact that F ′ (X ′ ) = F (X ′ )8 . This implies that the FI ′ = WIJ X J + VI J FJ (X ) are simply given by FI (X ′ ). Sometimes the duality invariances are also called proper symmetries, while the other symplectic transformations which really change the form of the prepotential are called symplectic reparametrisations [17] or pseudo-symmetries [18]. Let us now proceed with an explicit description of the symplectic properties of the lagrangian (2.3) including the fermionic terms and the Weyl multiplet auxiliaries. Of course many of these symplectic properties overlap with those that have been studied previously in the Poincar? e case [12, 19, 7], so we will concentrate on the ingredients that are speci?c to the superconformal setup. See also [20] for a (purely bosonic) treatment of symplectic transformations in the presence of a Weyl multiplet background. The symplectic vectors that are relevant for (2.3) can be derived starting from the basic symplectic vector (X I , FI ) by requiring consistency with supersymmetry. If one computes succesive supersymmetry variations of the basic symplectic vector, one gets the following results9 : δQ δQ XI FI ?iI FIJ ?iJ = ? ?i ?iI FIJ ?iJ XI FI + εij σ ?ν ?j
?I ??ν F ?? G 1 ? kl T?ν εkl ?4

= 2γ ? ?i D? + ?j
I Yij Zij I

?νI

?I X ?I F

i ? 1 ? ij K ij ? ? def ? ?J ?J G ?νI = FIJ F?ν + 2 XNI T?ν εij ? 4 FIJK ?i σ?ν ?j ε def 1 ? J ?K . Zij I = FIJ Y J ? FIJK ?
ij

2

(i

j)

(2.13)

? ? ) automatically de?ne new consistent symplectic ? ?I , G The expressions (?iI , FIJ ?iJ ) and (F ?ν ?ν I ?I , G ?I , G? ) we introduced in (2.9) ??ν ? ? ) is equivalent to the vector (F?ν vectors. Note that (F ?ν I ?ν I and (2.10), because the di?erence between the two is itself symplectic. However, the would-be vector (Yij I , Zij I ) is not automatically consistent. The reason is that the auxiliary ?elds Yij I satisfy a reality condition, viz. Yij I εjk = εij Y jk I , whereas the Zij I apparently don’t. But one can verify that the Zij I do satisfy a similar reality condition if one imposes the Yij I equations of motion: i ? ? J ?K + i F ? kJ ?lK εik εjl = 0 . ? (2.14) NIJ Yij J ? 4 FIJK ? i j 4 IJK
It is well-known that one can circumvent this last restriction if one goes on-shell. The point is that the Poincar? e theories for N = 2 vector multiplets can be formulated in terms of symplectic sections (X I , FI ) [14], which need not be derived from a prepotential. The restriction on the admissable symplectic transformations then evaporates accordingly [7]. 7 In fact we will show that one can stay quite close to the original superconformal setup. It su?ces to go partly + on-shell by eliminating the T?ν ij auxiliary ?eld. 8 Remark that F (X ) is not an invariant function because F ′ (X ′ ) = F (X ). 9 For a related discussion, see [21] in which symplectic transformations are de?ned on entire N = 2 chiral super?elds.
6

7

We henceforth assume that these equations of motion have been enforced. The Yij I ’s must then be viewed as particular dependent expressions quadratic in the gauginos ?I i or their complex conjugates. Nevertheless we will ?nd it useful to keep on using the shorthand Yij I in what follows. It is clear from equation (2.13) that the symplectic vectors (X I , FI ) and (?iI , FIJ ?iJ ) transform into other symplectic vectors under supersymmetry. This fact holds in general although ? ? ) only transforms ? ?I , G one has to be careful. One may verify that the symplectic vector (F ?ν ?ν I into other symplectic vectors provided one uses the equations of motion for the gauginos ?I i. At this point, however, we prefer not to impose the latter equations of motion, because they are not needed in order to guarantee the consistency of the symplectic vectors we just de?ned. Therefore we simply use the symplectic vectors of (2.13) as they stand. The resulting symplectic transformation rules for the vector multiplet ?elds make sense at a lagrangian level, and the lagrangian (2.3) — with dependent Yij I , but all the other Weyl multiplet auxiliaries still present as independent degrees of freedom— turns out to be symplectically invariant, up to a familiar +I G?ν + ) term. Im(F?ν I When checking the behaviour of the lagrangian (2.3) under symplectic transformations it su?ces to concentrate on those terms that would vanish if the auxiliary ?elds were eliminated and the superconformal gauge choices were imposed. The symplectic properties of the other terms are known in advance as they are not changed by the transition to the Poincar? e theory for which a complete treatment has already appeared elsewhere [7, 12]. One ?nds that the vector multiplet lagrangian can be written as
+I ?ν + + h.c. + invariant terms , GI e?1 Lvector = ? 8 F?ν

i

(2.15)

which is completely analogous to the Poincar? e result. In principle there could have been additional non-invariant terms, which then would have to vanish in the Poincar? e limit, but such + terms do not appear. As an illustration, we treat the T?νij terms explicitly and show that they can be nicely absorbed into symplectic invariant combinations. We start with the invariant
i I ?+ ? +I ?ν + ij 32 X G?ν I ? FI F?ν Tij ε + h.c. 1 i + +I ?IJK ? ? kJ σ?ν ?lK εkl T ?ν + εij + h.c. (2.16) ??ν = 64 XN X T?νkl εkl ? 4XNI F ? 2 XI F ij
+ which contains all the O(T?νij )2 terms of (2.3). After taking into account the terms linear in + +I G?ν + term of (2.15) one is left over with T?νij which appear in (2.16) and in the F?ν I

1 j ? ?i j ?ν + ? iI 24 XNI ? γ? ψν + XN X ψ? ψν Tij + h.c. 1 ?ν + kl ij I ? ij ?I ? 16 XNI ? ε + h.c. Tkl i γ? ψνj ε ? X ψ?i ψνj ε i ?IJK ? ? iJ σ?ν ?jK T ?ν + + h.c. ? 64 X I F ij

(2.17)

+ The ?rst line of (2.17) is invariant by itself. The second line conspires with suitable T?νij independent terms in (2.3) to form the invariant

i + ?I ij ij ??i ψνj εij ? F ?ν +I FIJ ? ? ?J ? 8 G?ν ?i γ? ψνj εij ? X I ψ i γ? ψνj ε ? FI ψ?i ψνj ε I

+ h.c. (2.18)

+I G?ν + term of (2.15) already contain all the F I One can check that (2.16), (2.18) and the F?ν ?ν I terms of the lagrangian apart from

i ?ν +I ? ? iJ σ?ν ?jK εij + h.c. F FIJK ? 32

(2.19)

8

But the sum of (2.19) and the last line of (2.17) transforms into itself up to 4-fermi terms (which can be further analysed in the same way as in the Poincar? e case). To ?nish our discussion of symplectic transformations we introduce a useful piece of terminology [7]. A given symplectic transformation is said to be of the “semi-classical” type when Z IJ = 0 (in a well chosen symplectic basis). In that case the full set of Bianchi identities is left invariant, which implies that the vectors transform linearly into themselves. For Z IJ = 0 one makes a further distinction between the WIJ = 0 case which is called “classical”, and the genuine “semi-classical” case WIJ = 0. It follows from (2.10), (2.11) and (2.15) that the lagrangian (2.3) is left invariant under semi-classical transformations up to a topological term: e?1 Lvector
semi?class. ?1

?→

e

i I ? ?νJ . [U T W ]IJ F Lvector ? 8 F?ν

(2.20)

2.2

The transition to the Poincar? e theory

The o?-shell theories for N = 2 vector and hypermultiplets coupled to superconformal supergravity contain some redundant variables which do not describe true physical degrees of freedom. These redundant variables can be eliminated by going on-shell and by reducing the superconformal symmetry algebra to the super Poincar? e algebra. As has been explained in [13] there exists a well-de?ned procedure to perform this step, and we now recall its most important ingredients. As a starting point one considers the sum of the vector multiplet action (2.3) and the hypermultiplet action (2.8). The total number of hypermultiplets one introduces is equal to r = (Nh + 1), where Nh denotes the number of physical hypermultiplets one wants to obtain in the ?nal theory. The extra hypermultiplet is a so-called compensating multiplet which plays an important r? ole in the whole Poincar? e reduction. In a moment we will sketch how the degrees of freedom of this compensating hypermultiplet can be eliminated. In the vector multiplet sector there are some compensating degrees of freedom too: one complex scalar and its fermionic partner are unphysical and as such they will be removed in the process of going to the Poincar? e theory. This implies that the number I always runs over (Nv + 1) values, where Nv counts the (complex) dimension of the vector multiplet moduli space. + The Weyl multiplet ?elds D, χi , A? , V? ij , and T?ν ij are auxiliary ?elds. D and χi are Lagrange multipliers which enforce the following constraints on the matter sector of the theory: ? + 1 Ai α Ai β dα β = 0 XN X 2 I ? XNI ?i + 2Ai α ζβ dα β = 0 .

(2.21)

One can use these constraints to eliminate several component ?elds of the compensating hypermultiplet, i.e. one real component of the scalar and all the components of the associated + fermion. The auxiliary ?elds A? , V? ij , and T?ν ij appear quadratically in the action so they can be solved for by imposing their own equations of motion. In the sequel we will explicitly need + the A? , V? ij and T?ν ij equations of motion which are given by ? A? = XN X ? 2 Ak α Ak β dα β V?i j
+ ij XN X T?ν ij ε

i ? ?? X I + h.c. ? i NIJ ? ?α γ? ζβ dα β ? iI γ? ?J ? iζ XNI ? i 2 8

1

?? Aj β dα β ? Aj β ? ?? Ai α dα β = ? Ai α ?
1 1 jJ kJ ?I ?I + 4 δi j NIJ ? ? 2 NIJ ? i γ? ? k γ? ? ?α σ?ν ζ γ . (2.22) ? +I + i X I F ?IJK ? ? iJ σ?ν ?jK εij ? 8 dα β ρβγ ζ = 4XNI F ?ν 2

In order to derive the equation (2.22) we used the constraints (2.21). We also neglected any ?? we mean a derivative which b? dependence, because we will put b? = 0 in a moment. By ? 9

is covariant with respect to local Lorentz, Q-supersymmetry and gauge transformations10 . One easily veri?es that the equations (2.22) are symplectically invariant, see for instance (2.16). The superconformal algebra can be broken down to the super Poincar? e algebra by imposing several gauge choices. First one breaks the special conformal symmetry by putting the dilatation gauge ?eld b? = 0. Next one uses the dilatation symmetry to bring the R term in the action into a conventional form. It is common practice in the N = 2 supergravity literature to go to 1 the Einstein frame for the metric, i.e. e?1 L = 2 R + more. Later on in the heterotic string case (section 5.4) we will not follow this common practice, but rather choose an alternative gauge which immediately leads to the string frame. Nevertheless we present the standard Einstein frame results here, so that the reader may compare to them in section 5.4. Given the fact that the D equation of motion (2.21) has already been imposed, one ?nds that the dilatation gauge choice leading to the Einstein frame reads ? = 1. XN X The corresponding standard S -supersymmetry gauge choice reads XNI ?iI = 0 . (2.24) (2.23)

This S -gauge choice is chosen such that the dilatation gauge choice (2.23) is invariant under Q-supersymmetry transformations. Moreover one may verify that (2.24) removes many of the mixed gravitino-gaugino propagators. It is important to realise that the condition (2.24) itself is not Q-supersymmetric invariant. This shows that the Poincar? e sypersymmetry transformations —which by de?nition leave the various gauge choices invariant— are in fact composed out of a Q-supersymmetry part and a compensating S -supersymmetry part: δ(?) = δQ (?) + δS (η (?))
1 ? jI γ? ?J ? 1 δj i ? ? kI γ? ?J ηi (?) = γ ? ?j 4 NIJ ? i k 2 ?α γ? ζβ dα β ? εij σ ?ν ?j dα β ραγ ζ ?β σ?ν ζγ . ?δ j i ζ

(2.25)

In order to ?nish the whole Poincar? e reduction one has to ?x the chiral U (1) and SU (2) symmetries. The U (1) gauge freedom can be used to further restrict the scalar ?elds X I . If one introduces special coordinates XI def Z I = ?i 0 (2.26) X one derives from eq.(2.23) that ? )?1 , |X 0 |2 = (ZN Z (2.27) so in this case the dilatation gauge choice has e?ectively ?xed the length of the scalar X 0 . One then uses the U (1) symmetry to ?x the phase of X 0 as well. A convenient choice is
1 ? )? 2 Z I . X I = (ZN Z

(2.28)

This expresses the (dependent) scalars X I in terms of the (independent) Z I . It should be noted that the previous formula and the subsequent ones are also valid when the Z I are not just special coordinates, but rather holomorphic sections Z I (z A ) of a symplectic bundle over the special K¨ ahler manifold SK which is de?ned by the vector multiplet theory [22]. The z A with A = (1, · · · , Nv ) are arbitrary coordinates on SK. The relation (2.28) is not Q-supersymmetry
?? di?ers from the one given in [13] in that our ? ?? is covariant with respect to QOur convention for ? supersymmetry as well.
10

10

invariant, which implies that a compensating U (1) transformation must be included in order to obtain the correct Poincar? e supersymmetry transformation rules. The precise form of this compensating transformation can be deduced from the fact that δ(?)X I = ? ?i ?iI ? iΛU(1) X I ? J 1 ZN 1 i J ? )? 2 ? = (ZN Z ?i ΛiI ? 2 X I ? ? Λi + h.c. , ZN Z where the fermions ΛiI are de?ned by δ(?)Z I = ? ?i ΛiI . From (2.29) one immediately reads o? that ΛU (1) (?) = ?I i ? I i ZN i I ? ? Λi + h.c. 2 ZN Z
I ΛI i ?Z

(2.29)

(2.30)

1 ? )? 2 = (ZN Z

? J ΛJ ZN i ? ZN Z

.

(2.31)

Remark that the Z I (and thus any K¨ ahler coordinates z A ) are de?ned to be chiral ?elds, in the sense that they only transform into fermions of a de?nite handedness under Poincar? e 11 I supersymmetry . The scalars X are not chiral under Poincar? e supersymmetry because of the compensating U (1) transformation. The SU (2) symmetry, ?nally, can be used to remove the last 3 degrees of freedom of the compensating hypermultiplet scalar. A similar reasoning as the one followed for the chiral U (1) gauge ?xing leads to12 Ai α = ζα = ?2 α δi s Bs C (B ) s = 1, 2 C (B ) = Bs α B s β dα β

2B s β dβ γ ξ γ ?2 ξ α ? Bs α C (B ) C (B ) (2.32)

j s Λi j = 4? ?i δs B α dα β ξ β C (B )?1 ? h.c.; traceless

where the ξ α are de?ned by ?α ?i + 2ραβ εij ξ ?β ?j δi s . δBs α = 2ξ (2.33)

3

The vector multiplet e?ective action for perturbative heterotic strings on K3 × T 2

In this section we concentrate on a particular subset of all possible N = 2 vector multiplet theories, namely those that arise from type II strings on K3 -?bered Calabi-Yau manifolds, or from heterotic strings on K3 × T 2 . We begin with type II strings because the vector multiplet sector of their low-energy e?ective action is relatively easy to analyse. This is so because the type II dilaton is contained in the hypermultiplet sector of the theory13 . The fact that the vector
As usual we use Weyl fermions for which the position of the SU (2) index also indicates the chirality. δ j (Ak k ? Ak k ) and Aj i = (Aj i )? . To clarify our notation Ai j ? h.c.; traceless = Ai j ? Aj i ? 1 2 i 13 One must be careful with this statement, though. As has been explained in [23] the type II dilaton is naturally described by the real part of the sum of a compensating N = 2 vector and tensor multiplet. Only after imposing an Einstein frame dilatation gauge choice the type II dilaton appears to be sitting in a physical tensor multiplet which can be converted into a hypermultiplet for practical reasons. So it is only in an Einstein frame that the type II dilaton can be identi?ed with one of the hypermoduli.
12 11

11

multiplet sector and in particular the prepotential of the type II e?ective action do not depend on the hypermoduli implies that they are completely ?xed at the tree level and don’t receive any perturbative or non-perturbative corrections. The prepotential which is relevant for type IIA strings compacti?ed on a generic Calabi-Yau manifold Y (and for type IIB strings on the ? ) is given by [24] mirror manifold Y FtypeII (X 0 , X A ) = i(X 0 )2 ftypeII (Z A ) ζ (3) 1 1 χ(Y ) + 3 ftypeII (Z A ) = dABC Z A Z B Z C ? 3 6 16π 8π A X def Z A = ?i 0 . X

?2πdA Z A nr d1 ,...,dn+1 Li3 e
d1 ,...,dn+1

(3.1)

Here the index A runs from 1, ..., Nv = n + 1 (n ≥ 1). The special coordinates Z A are identi?ed with the complexi?ed K¨ ahler moduli of Y , the dABC are the Calabi-Yau triple intersection numbers, χ(Y ) is the Euler characteristic and the rational instanton numbers nr d1 ,...,dn+1 count the number of rational curves of multi degree (d1 , ..., dn+1 ) on Y . When Y is a K3 -?bration over IP1 [3, 4] there is one distinguished modulus, the K¨ ahler modulus of the IP1 base, which we call S . We may choose a basis such that Z A = (S, Z A ) A = 2, ..., n + 1 . (3.2)

The intersection numbers of the K3 -?bered Calabi-Yau manifold then satisfy d111 = 0 d11A = 0 d1AB = ηAB , (3.3)

where the n × n “metric” ηAB is of signature (1, n ? 1).14 Notice that the index I – which runs over all the vector multiplets — takes the values 0, 1, A. As it will be relevant further on, we introduce a (n + 2) × (n + 2) metric ηIJ of signature (2, n), which is de?ned as ηIJ 0 1 0 ? ? 0 ?. =? 1 0 0 0 ηAB
? ?

(3.4)

3.1

Identi?cation of the dilaton-axion complex

Thanks to the type II-heterotic duality hypothesis – which has been tested very succesfully in many circumstances, see for instance [1, 2, 3, 4, 25], the nice review [26] and references therein — one can view the prepotential implied by the relations (3.1), (3.2), (3.3) as a prepotential describing heterotic strings on K3 × T 2 . Of course it is crucial to make contact with the results which have been obtained in the past by direct computations in the heterotic string picture [5, 27, 28]. In order to do so, one identi?es the modulus S with the heterotic dilaton-axion complex: X 1 def (3.5) S = ?i 0 = φ ? ia . X Here φ is a dilaton-like ?eld which is closely related to the heterotic string loop counting parameter, while a is an axion-like ?eld15 . The heterotic prepotential thus satis?es the following
It should be mentioned that one may want to perform some additional linear rede?nitions of the K¨ ahler moduli (3.2) and the corresponding scalars X α . Consider e.g. the type II model based on the Calabi-Yau space W P1,1,2,8,12 (24) [1]. This model contains 3 K¨ ahler moduli Z 1 = S, Z 2 , Z 3 with intersection numbers d123 = 1, d133 = d223 = 2, d233 = 4, d333 = 8. It is advantageous to de?ne Z 2 = T ? U, Z 3 = U where T and U can be identi?ed with the standard moduli of the T 2 in the dual heterotic picture. In this way ηT U = 1, ηT T = ηU U = 0. 15 Henceforth we simply call φ the dilaton and a the axion, even though they di?er from the true SO(2, n) invariant dilaton φinv and axion ainv which will be discussed later.
14

12

expansion in the string coupling constant S : Fhet (X ) = ?
∞ 1 X1 A B (0) 0 A η X X + F ( X , X ) + F (k) (X 0 , X A ) e?2πkS . AB 2 X0 k =1
1

(3.6)

X A B corresponding to At the tree level one recovers the well-known prepotential ? 1 2 X 0 ηAB X X the special K¨ ahler manifold SU (1, 1) SO(2, n) ? (3.7) U (1) SO(2) × SO(n)

which describes the classical vector moduli space for heterotic strings. This manifold is the only special K¨ ahler manifold having a direct product structure [29]. This re?ects the fact that (in the Einstein frame) the heterotic dilaton has no tree-level couplings to the other vector moduli. When n ≥ 2 the tree-level prepotential can be brought into a standard form by performing some linear rede?nitions of the special coordinates Z A to ?nd Ftree = ?
n+1 1 X1 A B 0 2 η X X = i ( X ) ST U ? S φi φi . AB 2 X0 i=4

(3.8)

The ?elds T and U are the moduli of the T 2 and the φi are possible Wilson line moduli. The n = 1 case can be viewed as degenerate case for which the di?erence T ? U has been frozen to a zero value. Due to the result (3.8) we say that the prepotential (3.6) leads to the heterotic vector multiplet e?ective action in the ST U basis. The other terms in (3.6) can be “explained” by noting that they are the only possible ones allowed by the quantised Peccei-Quinn symmetry under which the heterotic theory is invariant. This symmetry maps P.Q. (3.9) S ?→ S ? ic c∈ , and leaves the other moduli untouched. In order to fully understand the consequences of this particular symmetry one needs some extra knowledge which will be provided in section 4.1 where we study generic Peccei-Quinn invariant models. At present it su?ces to mention that in the case at hand the prepotential has to transform as follows16 Fhet (X ) ?→ Fhet (X ) ? 2 c ηAB X A X B .
P.Q.

1

(3.10)

Since the tree level part of the prepotential already saturates the latter equation all the other terms must be Peccei-Quinn invariant by themselves which directly leads to (3.6). In this way one has proven a powerful non-renormalisation theorem which states that perturbatively there is just the tree-level contribution and the one-loop term F (0) (X 0 , X A ). The Peccei-Quinn symmetry is continuous at the perturbative level. This indicates that the axion a describes the same physical ˇ?ν which is familiar from the the standard degrees of freedom as the dual antisymmetric tensor B world-sheet formulation of heterotic strings. As usual this continuous symmetry is broken to its discrete subgroup at the full quantum level due to space-time instanton e?ects. These give rise to the non-perturbative F (k) (X 0 , X A ) exp[?2πkS ] terms. The heterotic string is invariant under a SO(2, n) T-duality group [30], which is expected to survive at the non-perturbative level because it can be viewed as a discrete gauge symmetry [31]. The SO(2, n) transformations can be embedded into the symplectic group Sp(2(n + 2); )
16 This follows from the equations (4.3) and (4.11) (appropiately applied to the heterotic string example) and 1 the fact that F (X ) = 2 X I FI . At ?rst sight the reader might be surprised by the fact that the prepotential is not invariant under the Peccei-Quinn symmetry even though this symmetry corresponds to a duality invariance. The point is that F ′ (X ′ ) = F (X ′ ) but = F (X ).

13

and thus naturally act on These SO(2, n) symplectic which is di?erent from the ˇI, F ˇI ) symplectic vector (X in the following way [7]:
? ˇ0 X ? X ˇ1 ? ? X A ˇ ? ? ˇ ? F0 ? ˇ1 ? F

the vector multiplets contained in the low-energy e?ective action. transformations are most easily written down in a symplectic basis ST U basis. This new basis is completely speci?ed by the “stringy” def het (X ) which is related to the ST U symplectic vector (X I , FI = δFδX ) I
? ?

ˇA F

? ? ? ? ? ? ? ? ?=? ? ? ? ? ? ?

1 0 0 0 0 0 0 0 δA B 0 0 0 0 ?1 0 0 0 0

0 0 0 1 0 0

0 0 1 0 0 0 0 0 0 0 0 δA B

?? ?? ?? ?? ?? ?? ?? ?? ??

X0 X1 XA F0 F1 FA

?

? ? ? ? ?, ? ? ?

(3.11)

In the stringy basis the SO(2, n) transformations are of the semi-classical type, and are given by ˇJ ˇ I SO(2,n) X 0 UIJ X , (3.12) ?→ ˇJ ˇI (U ?1 )T I K ΛKJ (U ?1 )T I J F F where [U T η U ]IJ = ηIJ ΛIJ = symmetric, real . (3.13) The matrices ΛIJ are absent at the string tree-level. They must be introduced at the one-loop level and also non-perturbatively to accomodate for the monodromies generated by encircling the singularities in the quantum moduli space. See e.g. [27, 6] for more details concerning this point. The relations (3.11) - (3.13) have some important consequences. First of all one veri?es that


ˇI = ?iS Xη ˇ I+ F
k =0

e?2πkS ?I F (k) (X 0 , X A )

(3.14)

where the ?I stand for partial functional derivatives with respect to the original scalars X I . Note that we have conveniently included the the one-loop term (k = 0) into the instanton sum. In the I = 1 version of the equation (3.14) the one-loop and non-perturbative contributions ˇ1 how the dilaton-axion ?eld S behaves clearly vanish, so one can derive from the variation of F under the T-dualities. This yields S ?→ S +
k =0 SO(2,n) ∞

i

ˇ Λ U ?1 ]1 [X ?I F (k) (U ?1 )I 1 ?2πkS e + i . ˇ U ?1 ]1 ˇ U ?1 ]1 [Xη [Xη

(3.15)

In other words, the N = 2 special coordinate S is only invariant under SO(2, n) transformations in the classical limit, when the F (0) (X 0 , X A ), F (k>0) (X 0 , X A ) and ΛIJ dependent terms vanish. From the moment on that one-loop and non-perturbative e?ects are taken into account S is no longer inert under SO(2, n) transformations, and even ceases to be single-valued due to the non-trivial monodromies. Therefore it is best to perform a change of coordinates on the K¨ ahler manifold which amongst others trades the special coordinate S for a single-valued and SO(2, n) invariant alternative, which we call Shol . This ?eld Shol was introduced in [5] at the perturbative level and in [6] non-perturbatively, and it turns out to be a complicated holomorphic function of the moduli S and Z A . Therefore Shol is not a N = 2 special coordinate. In the non-perturbative case one de?nes i ˇIJ + L . η IJ F (3.16) Shol = (n + 2) 14

In order to properly understand this last formula a few more remarks must be added. Equation (3.11) implies that ˇ I = (X 0 , X ˇ 1, X A) X ˇ 1 = ? 1 ηAB X X X 2 X0
A B

+

2πik (k) 0 A ?2πkS F (X , X ) e . X0 k =1



(3.17)

ˇ I in general de?ne a set of independent variables. However, As has been emphasized in [6] the X this is no longer true in the perturbative regime, when the instanton terms are supressed. In ˇ 1 is a function of (X 0 , X A ) only, which is re?ected in the constraint that case X


ˇ X ˇ = Xη
k =1

perturb. 4πik F (k) (X 0 , X A ) e?2πkS ?→ 0 .

(3.18)

ˇ I are really independent, which implies that So it is only for ?nite S that the stringy scalars X only in that case one can de?ne a stringy prepotential and derivatives thereof ˇIF ˇI ˇ (X ˇ) = 1X F 2 ˇI ? ˇJ F ˇIJ = ? ˇ. F (3.19)

In the perturbative case a prepotential doesn’t exist, so the abstract expression (3.16) doesn’t make sense there. Nevertheless one can work out (3.16) in terms of the functions F (k) (X 0 , X A ) and then take the perturbative limit. This procedure leads to the perturbative expression for Shol as it was given in [5]: Shol ?→ S +
perturb.

i (0) η AB FAB + L(0) . (n + 2)

(3.20)

The function L and its semi-classical limit L(0) are necessary to cancel the in?nities contained ˇIJ and η AB F (0) respectively. Moreover, one must impose L → L ? η IJ ΛIJ in order to in η IJ F AB keep Shol invariant under the monodromies. Although Shol clearly is a natural coordinate on the special K¨ ahler manifold, it is still not describing the true heterotic dilaton φinv . Perturbatively one ?nds [32] that φinv
perturb.

=

φ+

? I F (X ) + h.c. iX I , ˇ ˇ X ? 2Xη

(0)

(3.21)

where the second term is the so-called Green-Schwarz term. At this point one may want to introduce yet another complex scalar, called Sinv , which contains the invariant dilaton φinv as its real part. We propose the following de?nition, which makes sense at the full non-perturbative level and reduces to (3.21) semi-classically: Sinv = i ˇ ˇI X ?I F def ? iM = φinv ? iainv . ˇ ˇ ? Xη X ˇ ˇ ΛX ? X , ˇ ˇ X ? Xη (3.22)

Here M is a real function transforming as M →M+ (3.23)

which ensures that Sinv is monodromy invariant. Note that Sinv is a non-holomorphic function of the special coordinates S, Z A so one cannot use it as a prefered K¨ ahler coordinate. However, all the physical couplings in the e?ective action should be expanded in terms of the true dilaton-axion complex Sinv in order to properly identify the perturbative and non-perturbative contributions. 15

3.2

The “stringy” vector ?elds

ˇI, F ˇI ) in terms of which the T-duality In (3.11) we introduced the stringy symplectic vector (X transformations acquire a simple form. Of course one can also write the T-dualities in terms of the ST U symplectic vector (X I , FI ) in which case — in the notation of (2.12) — a Z IJ = 0 I transform into their “duals” term is generated. This means that some of the ?eld strengths F?ν I themselves transform in a G?νI under the T-dualities, which implies that the ST U vectors W? I are not the most natural non-local way. This state of a?airs indicates that the ST U vectors W? 1 for a objects to work with. As was explained in [7] one may proceed by trading the vector W? 1 via a duality transformation. In this way one gets the so-called stringy vector ˇ? new vector W ?elds I 0 ˇ1 A ˇ? W = (W? , W? , W? ). (3.24) By construction these satisfy
I 0 A ˇI =F ˇ?ν 2?[? W = (F?ν , G?ν 1 , F?ν ), ν]

(3.25)

I transform just linearly into eachother ˇ? from which one easily derives that the stringy vectors W under the T-dualities. Having established that the stringy variables are the most natural ones to work with, we must still explain how one constructs the stringy version of the vector multiplet lagrangian. At the non-perturbative level everything is straightforward because one may e?ectively implement the 1 by applying the symplectic transformation (3.11). This duality transformation on the vector W? ˇ (X ˇ ) of (3.19) which can be inserted into the general superconformal leads to the prepotential F action (2.3). In the perturbative case a sensible stringy prepotential doesn’t exist because the ˇ I are not independent. As a result the superconformal action formula (2.3) cannot be used as X it stands. In what follows we will explore in which sense the superconformal approach fails to deal with the perturbative e?ective action in the stringy basis. In particular we will show that one can actually stay pretty close to the familiar superconformal ideas and still obtain the desired theory. We take the superconformal action in the ST U basis as a starting point and explicitly dualise 1 . In doing so we will clearly see which ingredient of superconformal supergravity is the vector W? incompatible with the dualisation procedure. The crucial point is that one is forced to eliminate + the auxiliary ?eld T?ν ij contained in the Weyl multiplet in course of the dualisation. As a result the Weyl multiplet is no longer realised o?-shell. Below we exhibit the results of the necessary computations in rather detail, amongst others because it sets the stage for the other duality transformation we intend to perform, namely the dualisation of the heterotic axion a. Both dualisations share a lot of common features, as will become clear in section 5. We consider the lagrangian (2.3) with as prepotential the perturbative expression

F (X I ) = ?

1 X1 ηAB X A X B + F (0) (X 0 , X A ). 2 X0

(3.26)

From now on we will always work in the perturbative regime, which from a supergravity point ˇ I and ? ˇ I are then given by of view is the most interesting one. The stringy variables X i ˇ I = (X 0 , X ˇ 1, X A) X ˇ I = (?0 , ? ˇ 1 , ?A ) ? i i i i 1 ηAB X A X B ˇ 1 def X = F1 = ? 2 X0 1 ˇ X 0 XA B I ˇ 1 def ? ? . i = F1I ?i = ? 0 ?i ? ηAB X X0 i 16

(3.27)

1 can be dualised by treating the ?eld strength F 1 as an independent variable on The vector W? ?ν ˇ 1 . This leads to which a Bianchi identity has been imposed by means of a lagrange multiplier W ? the lagrangian i 1 ˇ 1. ?λ W (3.28) e?1 Lvector + 4 e?1 ??νλσ F?ν σ 1 dependent terms in the lagrangian: For future use we list the F?ν

e?1 Lvector = e?1 Lvector

1 =0 F?ν

i +1 ? ? ?ν +I 1 ? ? iI ?ν jJ i ?ν + ij ? 4 F?ν ε ? 4F F1I F ? 2 XN1 Tij 1IJ ? σ ? εij + h.c. i 1 ˇ ij ij ?1 ˇ1 ? + 8 e?1 ??νλσ F?ν ? + h.c. (3.29) i γλ ψσj ε ? X ψλi ψσj ε
1 . This is a direct consequence of the fact that Note that (3.29) depends at most linearly on F?ν 1 the prepotential (3.26) is at most linear in X . The action (3.28) —which still contains all the auxiliary ?elds of the Weyl multiplet— is most suitable for determining the supersymmetry ˇ 1 . The supersymmetry transformation rules for the transformation of the Lagrange multiplier W ? 1 which is replaced by original ST U vector multiplet ?elds are as in (2.5), except for δW? j 1 ? 1? ?i γν ] ?j1 εij + 2X ?i ψν δF?ν = 2?[? ? ] εij + h.c. .

(3.30)

ˇ 1 -independent part of the lagrangian is necessarily of the following form The variation of the W ?
1 + total derivatives , δ Lvector = ? 4 ??νλσ O? ?ν Fλσ

i

(3.31)

otherwise one would not regain invariance if the Lagrange multiplier would be eliminated again. Here O? stands for an a priori unknown expression whose precise form can be determined by 1 in the Lagrange multiplier term of (3.28) generates a direct computation. Transforming F?ν ˇ 1 be equal to O? one obtains invariance. This total derivative, so if one lets the variation of W ? reasoning yields 1 j ˇ ˇ? ˇ j1 εij + 2X ? 1? δW =? ?i γ? ? ?i ψ? εij + h.c. , (3.32)
0 , δW A and δ W ˇ 1 nicely rotate into eachother under SO(2, n) transformations, as Note that δW? ? ? expected. 1 . From (3.29) one derives In order to ?nish the dualisation we eliminate the auxiliary ?eld F?ν 1 that the equation of motion enforced by F?ν reads

? I ˇ ˇ ? IF ˇ+ ˇ ˇ ? + ij ? iI ˇ jJ 4Xη ?ν ? Xη X T?ν ij ε ? ηIJ ? σ?ν ? εij = 0 ,

(3.33)

? ˇ +1 where the covariant ?eld strength F ?ν is de?ned in the same spirit as in (2.6). Remark that (3.33) is saturated at the string tree level in the sense that it doesn’t receive any loop corrections. 1 Moreover it is manifestly SO(2, n) invariant. However, the most important property of the F?ν 1 equation of motion is that it doesn’t determine the value of F?ν itself, at least as long as one + + treats T?ν ij as an independent auxiliary ?eld. But in fact T?ν ij cannot be kept as an independent + variable, because (3.33) must be viewed as a constraint determining T?ν ij as a function of the + 17 stringy ?eld strengths, scalars and gauginos . As a result the T?ν ij equation of motion (2.22) must be imposed as well. In the present case this equation of motion can be worked out, yielding
+ +1 +0 ˇ ˇ Z ? F?ν ? ?ν Zη + iSF ? iφX 0 T?ν ij εij

ˇ +I ηIJ Re Z ˇ J + fermions2 + F (0) terms + hypermultiplet terms , = 4φ F ?ν
17

(3.34)

ˇ ˇ X ? can be inverted, otherwise the R term in the action would vanish. The factor Xη

17

1 ?nally becomes a dependent expression of the remaining ˇ I = ?i X 0 . In this way F?ν where Z X 1 receives loop and hypermultiplet physical ?elds. Remark that the actual expression for F?ν dependent contributions. To ?nish our construction of the vector multiplet theory in the stringy basis we discuss its symplectic properties. We start from the lagrangian (3.28), impose the two equations of motion we just mentioned and call the resulting lagrangian e?1 Lstringy . In fact the expression ˇ ?ν +1 = G1 ?ν + . Moreover we de?ne that (3.33) is nothing but a nice way of writing that F ?ν + ?ν +1 ˇ ˇ ?ν +1 and G ˇ 1 ?ν + are by de?nition given G1 = ?F . The symplectic transformations of F ?ν + ?ν +1 18 by the transformations of G1 and ?F respectively . One may verify that with these de?nitions the stringy action is symplectically invariant apart from the non-invariant term +I ?ν + +1 ˇ ?ν +1 +I ˇ ?ν + ˇ?ν GI + 4 F?ν F + h.c. = ? 8 F GI + h.c. . ? 8 F?ν

ˇI

i

i

i

(3.35)

In particular this implies that the stringy lagrangian transforms as follows under SO(2, n) transformations: SO(2,n) i I ? ˇ?ν ˇ ?ν J . F (3.36) e?1 Lstringy ?→ e?1 Lstringy ? 8 ΛIJ F

4

Construction of generic antisymmetric tensor theories

In this section we take a general point of view and study arbitrary vector multiplet theories containing a Peccei-Quinn symmetry. We impose just one restriction on this class of theories, namely we demand that there exists a set of special coordinates on which the Peccei-Quinn transformation acts in a standard way. By this we mean that there exists one distinguished special coordinate, henceforth called S , which shifts under the Peccei-Quinn symmetry by an imaginary constant, whereas the other special coordinates Z A are invariant under it19 . Given this ansatz we are able to completely characterise the Peccei-Quinn invariant vector multiplet theories and we show that they precisely comprise the cases discussed in [9] plus the perturbative heterotic string case which was still missing there. For all these theories we can obtain an antisymmetric tensor version by dualising the axion a = ?ImS . This dualisation can be performed in a way which is to a large extent model independent and as such it explains why the resulting antisymmetric tensor theories show some universal behaviour. We will see for instance that they are all characterised by a similar gauge structure. They contain a particular U (1) gauge symmetry, with parameter z , under which the antisymmetric tensor ?eld B?ν and an appropriately de?ned vector gauge ?eld V? transform in the following way:
(z ) V? → zV?

B?ν

(z ) → zB?ν .

(4.1)

Here the ?elds V? (z ) and B?ν (z ) stand for some (complicated) functions of the independent ?elds in our superconformal theory which will be speci?ed below. In the treatment of [9] this z gauge symmetry coincides with the central charge transformation which is necessary for o?-shell closure of the supersymmetry algebra. At a later stage we verify whether or not one may dualise the vector V? and we ?nd that the di?erent antisymmetric tensor theories behave di?erently in this respect. The so-called
+ +1 Notice that F?ν as a dependent ?eld still transforms in the naive way due to the fact that the T?ν ij equation of motion is symplectically invariant. 19 The ?elds S , Z A are the obvious generalisations of the variables we encounter in the description of perturbative heterotic strings. The interpretation of S as a string coupling constant is of course not valid in the general case. 18

18

non-linear vector-tensor multiplet theory of [9] doesn’t allow for this dualisation. The linear vector-tensor theory [9] can be dualised. This last theory is known to contain two abelian background vector ?elds, one of which gauges the central charge transformation. Under the duality the role of both background vectors gets interchanged. In the heterotic case, ?nally, the dualisation removes the complete central charge-like structure. We will see later that in this last ˇ 1 we encountered before. So the stringy case the dual of V? is nothing but the stringy vector W ? vectors play at least two important roles. We saw in section 3.2 that they make the SO(2, n) invariance of the heterotic vector multiplet theory as manifest as possible, and at present we ?nd that they also considerably simplify the gauge algebra acting on the heterotic vector and tensor gauge ?elds.

4.1

Peccei-Quinn invariant vector multiplet theories

The aim of this subsection is to characterise the set of Peccei-Quinn invariant N = 2 vector multiplet theories. We demand that any theory in this set contains some special coordinate moduli (S, Z A ) transforming in the following way under a continuous Peccei-Quinn symmetry: S ZA
def

=

?i

def

=

X 1 P.Q. ?→ S ? ic X0 X A P.Q. ?i 0 ?→ Z A , X

(4.2)

Here c is an arbitrary real constant. In order to understand how the Peccei-Quinn symmetry acts on a the full vector multiplet theory (so not only on the scalars) we embed the Peccei-Quinn transformation into the symplectic group Sp(2(n + 2), IR)
?

UI

J

Z IJ VI J
P.Q.

WIJ

? ? ? ? =? ? ? ?

1 0 0 c 1 0 0 0 δA B c c W11 + W1 ? 2 W1B + WB W00 ?2 c W11 W1B 2 W11 + W1 c W + W W W A 1A AB 2 1A

0 0 0 0 0 0 0 0 0 1 ?c 0 0 1 0 0 0 δA B

?

? ? ? ? ? . (4.3) ? ? ?

This particular symplectic transformation is a crucial object in the construction of antisymmetric tensor theories, so we will comment on some of its characteristic features. First we note that Z IJ = 0, so the symplectic transformation is of the semi-classical type. In particular this means I just linearly into eachother. Note that the Peccei-Quinn symmetry transforms the vectors W? that the submatrices U I J and VI J = (U ?1 )T I J are completely ?xed by the transformations (4.2) of the special coordinate moduli and by the de?ning symplectic relation (2.11), so they are valid for all cases we are interested in. The submatrix WIJ is model dependent. In equation (4.3) the most general matrix WIJ has been given which is consistent with (2.11) (provided WAB is taken to be symmetric), but in fact there are additional restrictions. In any case the various entries of WIJ must be real constants. They may depend on the parameter c but not for instance on any of the moduli ?elds. The additional restrictions we just mentioned stem from the fact that the symplectic transformation (4.3) must generate a duality invariance of the theory and not just a symplectic reparametrisation. This implies that FI (X 0 , X 1 , X A ) ?→ FI (X 0 , X 1 + cX 0 , X A ) ,
P.Q.

(4.4)

so when a suitable prepotential F (X ) is given the explicit form of WIJ can be read o? easily. Nicely enough the reverse procedure is possible as well. As we are going to explain now, one 19

can selfconsistently solve for the most general matrix WIJ leading to a duality invariance. The most general prepotential F (X ) can then be reconstructed from the knowledge of WIJ . We recall that the lagrangian (2.3) is Peccei-Quinn invariant up to a topological term
I ? ?ν J e?1 Lvector ?→ e?1 Lvector ? 8 (U T W )IJ F?ν F , P.Q.

i

(4.5)

thanks to the fact that the transformation (4.3) is semi-classical. In terms of the standard vector multiplet variables the non-invariance of the lagrangian may have di?erent sources, because not 1 , ?1 and Y 1 transform only the axion a shifts under Peccei-Quinn transformations, but also W? i ij due to the equation (4.3). The situation becomes much more transparant though, if one performs a change of variables and rather works with a set of variables which — apart from the axion itself— are all invariant under Peccei-Quinn. Given equation (4.3) one easily veri?es that the following variables do the job: a , φ = Re S , X 0 , X A X1 0 def ?i 1 A ? , ?0 ? ? λi = i i , ?i 2X 0 X0 i
1 0 0 A V? = W? ? aW? , W? , W? 1 0 0 A Zij = Yij ? aYij , Yij , Yij def def

(4.6)

When the lagrangian is expressed in terms of (4.6) there is only one source for the non-invariance term in (4.5): it must come from terms in the lagrangian in which the axion a appears undi?erentiated. As an example we concentrate on the term
I ? ?ν J A ? ?ν B F = ? 8 WAB (c)F?ν F + ··· ? 8 (U T W )IJ F?ν

i

i

(4.7)

This term should arise upon varying
i A ? ?ν B e?1 Lvector = ? 8 ReFAB F?ν F + ···

(4.8) (4.9)

from which we derive that the following relation must be valid: ReFAB (a → a + c) = ReFAB (a) + WAB (c) It is then clear that WAB (c) = ?c ηAB ReFAB (a) = ?a ηAB + a independent terms , (4.10) where ηAB is a real symmetric constant not depending on the parameter c. In the same spirit one can determine the other entries in WIJ by writing (4.5) in terms of the Peccei-Quinn invari0 , 2? V , F A and by reconstructing the undi?erentiated a terms in the ant ?eld strengths F?ν [? ν ] ?ν lagrangian which lead to (4.5). The result is WIJ
c c η11 η0B ? 4 η1B η00 ? c6 η11 η01 ? 2 ? ? c 1 = ?c ? η01 + 2 η11 ?, η11 2 η1B 1 c η1A η η η0A + 4 AB 2 1A

?

2

?

(4.11)

where all ηIJ are constants. This result can be integrated back to yield the prepotential20 F (X ) = ? 1 (X 1 )2 1 X1 1 (X 1 )3 A η ? η X ? ηAB X A X B + F (0) (X 0 , X A ) 11 1A 6 X0 4 X0 2 X0 1 ? (X 1 )2 η01 + X 1 X 0 η00 + 2X 1 X A η0A 2

(4.12)

In order to make contact to the perturbative heterotic prepotential given in (3.26) all entries of ηIJ apart from ηAB must be put equal to zero. Therefore the ηIJ which is used throughout this section doesn’t coincide with the SO(2, n) metric de?ned in (3.4). Note that all ηIJ ’s occuring outside section 4 refer to the SO(2, n) metric and not to the matrix ηIJ we just de?ned.

20

20

Note that the η0I terms in the prepotential are quadratic polynomials with real coe?cients. These just add total divergence terms to the action [13] so from now on we put η0I = 0 for simplicity. As such we precisely obtain the theories discussed in [9] and on top of that the perturbative heterotic string case η11 = 0, η1A = 0 which was excluded in a fully o?-shell superconformal context. We see no sign of any other vector multiplet theories which might have a dual antisymmetric tensor description. Remark that the prepotential (4.12) has a natural type II interpretation. For type II strings on generic Calabi-Yau manifolds there is a Peccei-Quinn symmmetry for every modulus Z A [33, 19]. One may pick any value for A and identify the corresponding modulus with S . In the large S limit the prepotential (3.1) coincides with the 1 prepotential we just found, provided we take d111 = η11 , d11A = 2 η1A , d1AB = ηAB .

4.2

Dualising the axion

All the Peccei-Quinn invariant theories we just speci?ed can be dualised into antisymmetric tensor theories. In order to check that this is indeed possible, it su?ces to show that the vector multiplet lagrangians for these theories can be brought into a form such that the axion a appears only via its “?eld strength” ?? a (up to total derivative terms in the lagrangian). The existence of the variables (4.6) is very important in this respect. When the lagrangian (2.3) is written in terms of the standard vector multiplet variables one sees undi?erentiated a dependencies occurring at various places. By going to the new variables one e?ectively absorbs most of these unwanted a terms. Only in the gauge sector some undi?erentiated a terms remain. As we remarked before these left-over a terms can be reconstructed on the base of equation (4.5). They read
i A ? ?νB ? ?ν ? ?νA + ηAB F?ν F 8 a η11 F?ν (V )F (V ) + η1A F?ν (V )F 1 0 ? ?νA 0 ? ?ν 0 ? ?ν 0 + 1 η1A F?ν + a2 η11 F?ν (V )F F F + 3 a3 η11 F?ν 2

,

(4.13)

where the gauge covariant ?eld strength F?ν (V ) is de?ned as F?ν (V ) = 2?[? Vν ] ? 2W[0 ? ?ν ] a . Equation (4.13) can be rewritten as
A A B ?λ Vσ + ηAB Wν ?λ Wσ + e?1 total derivative . (4.15) ? 4 e?1 ??νλσ ?? a η11 Vν ?λ Vσ + η1A Wν

(4.14)

i

After dropping the total derivative we may dualise the axion by replacing the “?eld strength” ?? a everywhere in the lagrangian by an auxiliary vector V? (z ) and by adding a Lagrange multiplier term i ?1 ?νλσ (z ) V? ?ν Bλσ . (4.16) 4e ? In principle the auxiliary ?eld V? (z ) can then be eliminated (possibly together with some other auxiliaries) in order to obtain an antisymmetric tensor theory. This turns V? (z ) into a dependent expression of the physical ?elds. In section 5.3 we will explicitly discuss the elimination of V? (z ) in the perturbative heterotic context.

4.3

The gauge structure of the antisymmetric tensor theories

It is interesting to see how a central-charge-like gauge structure, which is a crucial ingredient in the o?-shell construction of [9], arises in the present context. The gauge transformations of

21

I = ? θ I and the rede?nition (4.6). the vector ?elds are easily determined starting from δgauge W? ? De?ning z = θ 0 , y = θ 1 ? aθ 0 one gets: (z ) δgauge V? = ?? y + zV? 0 δgauge W? = ?? z A δgauge W? = ?? θ A .

(4.17)

Notice that the z -gauge transformation maps the vector V? into V? (z ) which explains why V? (z ) 0 is a distinguished one appears in the gauge covariant ?eld strength F?ν (V ). The vector ?eld W? in that it gauges this z -gauge transformation. The gauge transformations of the antisymmetric tensor ?eld B?ν can most easily be determined from the lagrangian
(z ) e?1 Lvector + 4 e?1 ??νλσ V? ?ν Bλσ , (z )

i

(4.18)

in which V? and the other auxiliaries are kept as independent non-propagating degrees of freedom. The B?ν independent part of this lagrangian is not invariant under gauge transformations. First there are the F?ν (V ) dependent terms. Due to the fact that δgauge F?ν (V ) = 2z?[? Vν ] (z ) they give a non-zero contribution. Secondly there is the non-total-derivative part of (4.15) (with ?? a replaced by V? (z ) ) which contains explicit gauge ?elds. Note that all these non-invariances are proportional to ?[? Vν ] (z ) , so they can be cancelled by a suitable choice of the gauge variation of the antisymmetric tensor ?eld. In this way one ?nds that δgauge B?ν
B (z ) = 2?[? Λν ] + η11 y ?[? Vν ] + η1A θ A ?[? Vν ] + ηAB θ A ?[? Wν ] + z B?ν

(z ) ??ν ? η11 V[? V (z ) ? η1A W A V (z ) B?ν = 4iA [? ν ] ν]

A?ν

=

δ e?1 Lvector δF ?ν 1

1 →F F?ν ?ν (V ), FIJ →FIJ (a=0)

,

(4.19)

where Λ? is a parameter for tensor gauge transformations. Under the z -gauge transformation the antisymmetric tensor is mapped into B?ν (z ) . This is a complicated function which does not only contain the ?eld strengths but also scalars, gauginos, gravitinos etcetera. The gauge covariant ?eld strength for B?ν reads
A B 0 H?νλ = ?[? Bνλ] ? η11 V[? ?ν Vλ] ? η1A W[A ? ?ν Vλ] ? ηAB W[? ?ν Wλ] ? W[? Bνλ] . (z )

(4.20)

The constants ηIJ which were previously introduced in order to specify the prepotential (4.12) turn out to be directly related to the various Chern-Simons couplings of the antisymmetric tensor. It was already remarked in [9] that a cubic in S prepotential leads to a Chern-Simons coupling which is quadratic in the vector V? and that the quadratic in S prepotential leads to a Chern-Simons coupling which is linear in V? . Here we see that the Chern-Simons couplings depend on the background vector multiplet ?elds only, when the prepotential is just linear in S . There is a last point concerning the gauge structure of the antisymmetric tensor theories which deserves to be investigated. When we specialise to the heterotic string case we see that we have constructed an antisymmetric tensor theory containing the vector V? which is directly 1 . We already discussed in section 3.2 that in the vector multiplet related to the ST U vector W? 1 for its dual, the version of the perturbative heterotic theory one may bene?t from trading W? 1 . One may ask a similar question in the present context, and verify what are ˇ? stringy vector W the e?ects of dualising the vector V? . First of all we note that we have to exclude the η11 = 0 case, because otherwise the theory contains explicit V? terms (see for instance (4.15)), which prevent us from performing the dualisation we have in mind. In the other cases there is no obstruction, so one may replace the ?eld strength 2?[? Vν ] by an auxiliary ?eld C?ν and add the Lagrange multiplier term i ?1 ?νλσ d e ? C?ν ?λ Vσ (4.21) 4 22

d can be ?xed in the standard way by adopting δ The gauge transformation of V? gauge C?ν = ( z ) 2?[? {zVν ] } and checking the gauge invariance of the theory. One ?nds that d (z ) = ?? y d ? 2 η1A θ A V? δgauge V? .

1

(4.22)

But now we have a extra possibility which crucially hinges on the fact that C?ν is not a total derivative. We can introduce an extra gauge transformation δextra C?ν = ?2z?[? Vν ] (z ) and still maintain the gauge invariance of the full theory provided we add a compensating δextra B?ν . This yields a modi?ed
A B δgauge B?ν = 2?[? Λν ] + 2z?[? Vνd] + ηAB θ A ?[? Wν ] + 2 η1A θ C?ν

1

(4.23)

d What have we gained by this whole operation? First we look at the η1A = 0 case. The vector V? A . In going transforms under a central charge-like transformation gauged by the vector ? 2 η1A W? from (4.19) to (4.23) the complicated expression B?ν (z ) has been replaced by the ?eld strength 2?[? Vνd ] and conversely the ?eld strength 2?[? Vν ] has been replaced by C?ν . After imposing the equation of motion for C?ν , this ?eld becomes a dependent expression of the physical ?elds and A as such it can be viewed as a dual B?ν (η1A θ ) . So under the duality transformation the theory 0 and ? 1 η W A has has been mapped onto a similar one in which the role of the vectors W? 2 1A ? been interchanged21 . In the η1A = 0 case the whole central charge-like structure disappears and the Chern-Simons couplings are of a completely conventional form. We will see later that in d can be identi?ed with the stringy vector W 1 . The vector ?elds W 0 , V d , W A then ˇ? that case V? ? ? ? appear on an equal footing, re?ecting the underlying SO(2, n) invariance of the heterotic string theory. The resulting antisymmetric tensor theory will be presented in the next section.

1

5
5.1

The antisymmetric tensor e?ective action for heterotic strings
The SO (2, n) invariant antisymmetric tensor theory

The results of the preceding section for the particular case of perturbative strings can be summarised as follows. One starts from the vector multiplet lagrangian (2.3) in the ST U basis and introduces the new Peccei-Quinn invariant variables (4.6). Then one subtracts the total d Lagrange multiplier terms given in (4.16) derivative term of (4.15) and adds the B?ν and V? and (4.21) respectively. The order in which the various dualisations are carried out should not matter so one can equally start from the stringy vector multiplet theory described in section 1 has already been traded for its dual W 1 and afterwards dualise ˇ? 3.2 in which the vector W? the axion. Let us sketch the di?erent steps one takes in this second scenario and verify that it indeed leads to the same theory as in section 4.3. The advantage of ?rst dualising the vector
As an example we may take A = 2, η12 = 2. The vector-tensor theory is then really mapped onto itself under the duality. This can be checked by going back to vector multiplet langauge and verifying that the symplectic transformation ? ? 0 0 1 0 0 0 ? 0 0 0 0 1 0 ? ? ? U I J Z IJ ? ?1 0 0 0 0 0 ? =? (4.24) ?, J 0 0 0 0 1 ? ? 0 WIJ VI dual ? 0 ?1 0 0 0 0 ? 0 0 0 ?1 0 0
21

corresponds to a duality invariance.

23

1 is that one can more easily keep track of the SO (2, n) invariance of the ?nal antisymmetric W? tensor theory. Our starting point is the lagrangian (3.28), or rather e?1 Lstringy which is obtained from +1 and T + . This lagrangian transforms in a simple way it by eliminating the auxiliaries F?ν ?ν ij under SO(2, n) transformations, see equation (3.36). As before one introduces the Peccei-Quinn ′ which is de?ned as the invariant variables φ, λi , Zij given in (4.6), as well as a new ?eld C?ν 1 : Peccei-Quinn invariant part contained in the dependent ?eld F?ν ′ 1 0 C?ν = F?ν ? aF?ν . def

(5.1)

I are automatically Peccei-Quinn invariant as can be veri?ed by rewriting ˇ? The stringy vectors W the symplectic transformation (4.3) in the stringy basis. Therefore the stringy vectors need not be rede?ned and as a result no central charge-like gauge transformations appear. The PecceiQuinn argument can be repeated, mutatis mutandis, in order to compute the explicit axion dependence of the lagrangian. One ?nds that the only undi?erentiated axion term reads

i ? ˇI F ˇ ?νJ a η F 8 inv IJ ?ν

(5.2)

which can be rewritten as ˇ I ?λ W ˇ J + e?1 total derivative . ? 4 e?1 ??νλσ ?? ainv ηIJ W ν σ
i

(5.3)

Note that in these formulae we introduced the SO(2, n) invariant axion ainv of (3.22) which is related to a by (0) ? I 1 FI X + h.c. ainv = a ? + M (0) (5.4) ˇ ˇ ? 2 Xη X The use of ainv is just a matter of convenience. It guarantees that the total derivative term in (z ) (5.3) is SO(2, n) invariant. At this point we drop the total derivative, replace ?? ainv by V? inv ˇ?ν . This leads to our ?nal antisymmetric and add the SO(2, n) invariant Lagrange multiplier B tensor lagrangian
(z ) i ˇλσ . e?1 Ltensor = e?1 Lstringy ? total derivative of (5.3) + 4 ??νλσ V? inv ?ν B

(5.5)

This lagrangian clearly transforms under the T-dualities as
SO(2,n) i I ? ˇ?ν ˇ ?ν J . e?1 Ltensor ?→ e?1 Ltensor ? 8 ΛIJ F F

(5.6)

One can verify that the lagrangian we just speci?ed equals the one discussed in section 4.3 provided we identify22
d ˇ 1 def W ? = V?

ˇ?ν def ˇ B = B?ν + W[0 ? Wν ]

(5.8)

d as given in (4.22) is consistent with the transformation Note that the gauge transformation of V? ˇ 1 . Taking into account the result (4.23) one ?nds the following gauge transformations for of W ? the antisymmetric tensor theory:

ˇI δgauge W ? ˇ δgauge B?ν
22

ˇI = ?? θ ˇI ?[? W ˇ ν ] + ηIJ θ ˇJ. = 2?[? Λ
ν]

(5.9)

To be completely precise: both lagrangians di?er at the one-loop level by the gauge invariant total derivative term (0) ? I + h.c. 1 FI X i I J ˇλσ ? ηIJ W ˇν ˇσ . (5.7) ?λ W ? M (0) ?ν B ? e?1 ??νλσ ?? ˇ 4 2 ˇ X ? Xη

24

5.2

The supersymmetry transformation of the antisymmetric tensor

ˇ?ν can be determined The supersymmetry transformation of the antisymmetric tensor ?eld B from the lagrangian (5.5). As input one takes the supersymmetry variations of the background ˇ 1 in (3.32). Taking into account vector multiplet ?elds as given in (2.5) and the variation of W ? the rede?nitions (4.6) one ?nds that δφ δV? δλi
(z )

= ? ?i λi + h.c. = i?? {? ?i λi } + h.c. = ? (z ) ?i ? D / φ ? iV / ? 1 2X 0 i ? kl ε ′ ? iφF 0 + i φX ? 0 T?ν ??ν ??ν εij σ ?ν ?j C kl 2 2X 0 0 ?j λ 0 ?j ? 1 λ ? j 0 iZij + φYij j i ? ?j + ?i ? X0 (5.10)

with
′ ??ν C ′ ?i[? γν ] (2X 0 λj + φ?0 )εij + iφX 0 ψ ?i? ψνj εij + h.c. = C?ν ? iψ j

i ?i (z ) (z ) ?? V = V? ? 2ψ ? λi + h.c. .

(5.11)

ˇ?ν it su?ces to check only those terms in the In order to ?nd the supersymmetry variation of B variation of the lagrangian (5.5) which would vanish if the Bianchi identity ??νλσ ?? Vν(z ) = 0 (5.12)

would be imposed. There are two di?erent ways to obtain V? (z ) terms in the variation of the lagrangian. In the ?rst place there is the V? (z ) dependence of the lagrangian itself e?1 Ltensor = e?1 Ltensor
V? =0
(z )

i (z ) I J ˇλσ ? ηIJ W ˇν ˇσ ?ν B ?λ W + 4 e?1 ??νλσ V?

i (z ) ˇ ˇ ˇ I ? i V (z ) η ? ? iI γ ? ? ˇ J + h.c. ? XηI D ? X ? 2 V? IJ i 8 ? i (z ) ˇ ˇ ˇ ?i γλ ψσi + h.c. , ? iI γ ν γ ? ψνi + i e?1 ??νλσ V (z ) Xη ˇ X ?ψ + 4 V? XηI ? ? ν 8

(5.13)

from which the relevant contributions to δe?1 Ltensor can be derived in a straightforward way. ? (z ) ?i + · · ·. Taking everySecondly there are the λi terms which vary into V? (z ) due to δλi = ?iV / thing together one ?nds that the antisymmetric tensor action is invariant under supersymmetry if one de?nes ˇ? ˇ?ν = ?2? ˇ I? ˇ iI ? ηIJ W ˇI ? ˇ J ij ˇ I ?i ψν ]j εij ? 2Xη ˇ X ? δB ?i σ?ν Xη ?i γ[? ψν ]i + h.c. (5.14) [? ?i γν ] ?j ε + 2X ? ˇ?ν doesn’t transThe S -supersymmetry invariance of the theory is automatically guaranteed so B ˇ form under S -supersymmetry. The covariant ?eld strength for B?ν can be determined from the equations (5.9) and (5.14): ˇ ˇ ?i γλ ψσ]i + h.c. , ˇ X ?ψ ? νλσ = ?[ν B ˇ I? ? iI σ[νλ ψσ]i ? 1 Xη ˇλσ] ? ηIJ W ˇ I ?λ W ˇ J ? Xη H [ν [ν σ] 2 ? ?νλ is de?ned as while the dual of H
i ? ? def ? νλσ H = 2 e?1 ??νλσ H

(5.15)

(5.16)

The bosonic part of these ?eld strengths is denoted by dropping the “hat”. 25

ˇ?ν transforms only into background Weyl multiplet and vector It is important to notice that B multiplet ?elds, and not for instance into the dilatini λi . One can easily understand why this is the case. At this particular stage of our computation the ?elds φ, V? (z ) and λi are the only remnants of the original vector multiplet based on the scalar X 1 . They appear at most linearly in the lagrangian e?1 Ltensor , which immediately follows from the linear X 1 dependence of the prepotential (3.26). The ?elds φ, V? (z ) and λi also transform at most linearly into eachother, so e?1 Ltensor can only transform into
i ?1 ?νλσ (z ) V? ?ν background ?elds 4e ?
λσ

.

(5.17)

For the other antisymmetric tensor theories, namely the η11 = 0 or η1A = 0 cases described in section 4 the situation is quite di?erent. The supersymmetry variation of the tensor B?ν as it ˇ?ν , and one ?nds that was de?ned in section 4.2 can be computed much in the same way as δB δB?ν = ?4 2η11 φ + η1A ReZ A |X 0 |2 ? ?i σ?ν λi + h.c. + · · · . (5.18)

The latter transformation law automatically coincides with what was found in [9]. It then su?ces to add an independent auxiliary scalar φ(z ) to recover the o?-shell vector-tensor multiplets of [9].

5.3

(z ) Eliminating the auxiliary ?eld V?

ˇ?ν we proceed by eliminating Having determined the Q-supersymmetry transformation rule for B ( z ) the auxiliary ?eld V? . Before actually doing so we brie?y recall what happens in the vectortensor multiplet cases of [9]. There one can eliminate V? (z ) by imposing its own equation of motion, which is of the following type:
(z ) V? ? H? + · · · .

(5.19)

A relation like (5.19) is also what we expect to ?nd in the present case because it expresses what (z ) duality is all about: V? was ?rst introduced as the ?eld strength for the axion and it should ˇ?ν after the dualisation. However, in the become equal to the dual of the ?eld strength for B heterotic theory a relation like (5.19) cannot be recovered in one go. Given (5.13) one readily veri?es that the equation of motion enforced by the auxiliary V? (z ) reads ˇ ˇ ? iI γ? ? ˇJ ? ? = iXη ˇ I D? X ? I + i ηIJ ? H i + h.c. 4 I ˇ ˇ ˇ ?? X ˇ XA ? ? + iXη ˇ I? ? + i ηIJ ? ? iI γ? ? ˇJ = Xη i + h.c. , 4
(z )

(5.20)

which cannot be used to solve for V? itself. Instead the r? ole of (5.20) is to constrain the auxiliary Weyl multiplet ?eld A? , such that it becomes a dependent expression involving the dual ?eld strength H? . Of course, the A? equation of motion (2.22) must be imposed as well ˇ ˇ X ? A? = φ Xη
1 ˇ ˇ ˇ ? V (z ) ? i φ Xη ˇ I (?? ? b? )X ? I + h.c. Xη X ? 2 2

+fermions2 + F (0) terms + hypermultiplet terms ,

(5.21)

and this relation ?nally determines the value of V? (z ) as a function of A? such that altogether equation (5.19) is indeed ful?lled. The fact that we have to eliminate V? (z ) and A? both at the same time is of course not completely innocent. During the axion dualisation process we loose (another) part of the Weyl multiplet. As we already said, this is di?erent in the η11 = 0 or η1A = 0 cases of section 4 where the Weyl multiplet is not a?ected by the dualisation. This 26

seems to be a crucial property which guarantees the existence of the o?-shell vector-tensor multiplets of [9]. We also note the di?erent character of the relations (5.20) and (5.21). The former just depends on the tree-level part of the theory, whereas the latter receives one-loop and hypermultiplet dependent corrections. Both relations are SO(2, n) invariant. We would like to stress that the duality transformations we have encountered so far, i.e. the one yielding the stringy vectors on the one hand and the one yielding the heterotic antisymmetric tensor ?eld on the other hand, are very similar in nature. In both cases the Weyl multiplet cannot be preserved during the dualisation process such that the ?nal theories are far from being realised o?-shell. This is in line with the obervations in [7] and [9] that the o?-shell superconformal constructions that are known today are not applicable in case of the stringy vectors or the heterotic antisymmetric tensor theory. Of course it cannot really be excluded that there might exist alternative yet unexplored o?-shell superconformal theories, which would reduce —after a straightforward elimination (in the sense of not involving any duality transformations) of some well-chosen auxiliary ?elds— to the on-shell stringy vector or antisymmetric tensor theories we have described so far. We don’t know what such o?-shell theories would have to look like, if they exist at all.

5.4

Implementing the superconformal gauge choices

So far we described the various ingredients of the heterotic antisymmetric tensor theory. AcI do not transform ˇ?ν and the vectors W ˇ? cording to the equations (5.14) and (3.32) the tensor B under supersymmetry into the scalar φ and the fermions λi but rather into the background ˇ I and ? ˇ I . In the present on-shell situation it is in fact quite arti?cial to call X ˇ I and ? ˇI ?elds X i i “background” ?elds because due to the relations (3.33) and (5.20) the gravitational, vector and antisymmetric tensor variables interfere with eachother such that one can no longer tell which multiplet serves as a background for the others. Due to the on-shellness there is also no more ˇ I , λi and ? ˇ I as independent degrees of freedom. In reason to keep on using all the variables φ, X i order to reduce the number of matter degrees of freedom then, we go to the Poincar? e version of the heterotic antisymmetric tensor theory. The general strategy for going from a superconformal to a Poincar? e theory has been discussed in section 2.2, so we can simply apply the general rules to the case at hand. As before the lagrange multipliers D and χi enforce the constraints (2.21) which can be used to restrict the hypermultiplet variables. When these constraints are taken into account the Einstein term in the action reads
1 ˇ ? R = 1 φXη ˇ X ? + i F (0) X ? I ? h.c. XN X 2 2 2 I 1 ˇ ˇ X ?R R = 2 φinv Xη

(5.22)

Remark that this term does not depend on the one-loop part of the theory. It is the relation between the true dilaton φinv and the ?eld φ = ReS which is subject to loop-corrections and (0) one may view the appearance of FI in (5.22) as (yet another) indication that φ should not be identi?ed with the true string loop counting parameter. In fact one might use the result (5.22) as a purely supergravity de?nition of what the true dilaton ?eld is: the ?eld φinv is the only scalar ?eld which 1) makes the Einstein term (and hence any sensible dilatation gauge choice) loop independent, and 2) reduces to φ when loop-e?ects are neglected. Note that the requirement of SO(2, n) invariance is not strong enough as a criterion to select the true dilaton because it would yield both φinv and φhol = Re Shol as possible candidates. If we would impose the standard Einstein-frame dilatation gauge at this point (as well as the ˇ I ?elds would become loop-independent functions of usual U (1) gauge) we would ?nd that the X ˇ I ,23 multiplied by a common dilaton factor. But it is straightforward to the special coordinates Z
23

ˇ I = (?i, ? 1 iZ A ηAB Z B , Z A ) with Z

2

27

see that this dilaton dependence can be completely removed by choosing an alternative dilatation gauge, i.e. ˇ ˇ X ? =1 Xη (5.23) Nicely enough this last gauge choice directly leads to the heterotic e?ective action in the string frame, because one may identify ?2? φinv = e (5.24) ? where the expectation value of e equals the string coupling constant gs .24 The S -supersymmetry gauge choice which keeps equation (5.23) invariant under Q-supersymmetry reads ˇ ? I? ˇ iI = 0 . Xη (5.25)

As usual the S -supersymmetry gauge choice itself is not Q-supersymmetric invariant. As a ?rst step in the computation of the compensating S -transformation we note that ˇ iI δ? ? ?I ? 1 X ? kl I j ˇ ˇ I ?i + εij σ ?ν ?j F ˇ ? I T?ν ˇij = 2D /X εkl + Y ? ?ν 4 ˇ iI , ˇ I ηi ? i ΛU(1) ? +2X
2

(5.26)

ˇ 0 = Y 0 and Y ˇ A = Y A while Y ˇ 1 is ?xed by the constraint where by de?nition Y ij ij ij ij ij ˇ ˇ ? IY ˇ I = 1 ηIJ ? ? kI ? ˇ lJ εik εjl . Xη (5.27) ij 2 Using this result the compensating S -supersymmetry transformation can be determined to be
?I 1 i ? ? ˇ ˇ ? IF ? kI ? ˇ lJ εik εjl ˇ ?ν ? 1 ?j ηIJ ? / ?i ? 2 εij σ ?ν ?j Xη ηi (?) = ? 2 H 4 1 j ˇ ˇ ? jI γ? ? ˇJ ? kI ˇ J + 4 γ ? ?j ηIJ ? i ? δ i ηIJ ? γ? ?k .

(5.28)

+ In order to obtain the equations (5.26) and (5.28) the dependent expressions for T?ν ij (3.33) and A? (5.20) were freely used. As we already discussed it is only thanks to the fact that we dualised 1 and the axion a that we could make the dependent expressions for T + the ST U vector W? ?ν ij and A? —and hence also ηi (?)— completely loop and hypermultiplet independent. Note that ˇ?ν I and H? dependence of various supersymmetry ηi (?) depends on the gauge ?elds, so the F transformation laws changes by going to the Poincar? e theory. ˇ I , the Fixing the U (1) gauge leads to the following expressions for the dependent scalars X I ˇ and the compensating U (1) transformation respectively gauginos ? i

ˇI X ˇI ? i

1 ˇ ˇ Z ? )? 2 Z ˇI = (Zη 1 ˇ ˇ Z ? )? 2 = (Zη

ˇ ? ˇJ I ZηJ Λi ˇI ˇ Λ ? Z i ˇ ˇ Z ? Zη (5.29)

ΛU(1) (?) =

? I? ˇ ?Λ i Zη i + h.c. , 2 Zη ˇ ˇ Z ?
i I

ˇ

ˇI . ˇ I are de?ned as the fermionic partners of the special coordinates Z where as usual the Λ i Now we concentrate on the hypermultiplet variables. Given the conditions (2.21), (5.23) and (5.25) we have that
1 α i A A β dα β 2 i

= ?e

?2?

Ai α ζβ dα β = e
24

?2?

?i
2?

(5.30)
factor which

Later on we will see that any one-loop correction to our theory will be weighted by a relative e proves that the latter identi?cation is indeed the correct one.

28

where the SO(2, n) invariant dilatini ?i are de?ned as ?i = ? 2 e
def

1

2?

ˇ ? 1 i ˇJ ˇ ˇ ? K F (0) (Z ) ? h.c. ? I F (0) ? h.c. ? ZηJ Z ? ? 2? Z λi ? 4 ? i (Zη Z ) 2 e K IJ ˇ ˇ ? Zη Z

.

(5.31)

The string-frame dilatation gauge choice is directly responsible for the dilaton-dilatini dependence at the right hand side of (5.30). The ? and ?i dependence would have been moved from the hypermultiplet sector to the vector multiplet sector if we would have imposed the alternative Einstein frame dilatation gauge. In the present string frame we may split the hypervariables into a dilaton-dilatini part and the rest: Ai α = e ζα = e Fixing the SU(2) gauge then leads to A′ i α = ζ′
α ??

A′ i α ζ ′ ? A′ i α ?i
α

??

(5.32)

?2 α δi s Bs C (B )

s = 1, 2

C (B ) = Bs α B s β dα β

=

2B s β dβ γ ξ γ ?2 ξ α ? Bs α C (B ) C (B ) (5.33)

Λi j

= 2? ?i ?j + 4? ?i δj s B s α dα β ξ β C (B )?1 ? h.c.; traceless

α and ξ α are the physical hypervariables. Special coordinates on the quaternionic where Bs manifold are de?ned by splitting the index α into the values 1, 2 and the rest, and by putting

Bs α=1,2 = δs 1,2 In that case C (B ) 1,2 ?? Ai = e δi 1,2 ?2

ξ 1,2 = 0 .

(5.34)

α β s C (B ) 1,2 ?? ?? 1,2 B α d β ξ ζ = ?e ?1,2 ? 2e δs . ?2 C (B )

(5.35)

which indicates that after the Poincar? e reduction to the string frame the compensating hypermultiplet (Ai 1,2 , ζ 1,2 ) describes the dilaton and dilatini degrees of freedom. This is the N = 2 analog of the N = 1 statement [23, 34] that in N = 1 d = 4 heterotic string e?ective actions the dilaton can be viewed as sitting in the real part of a compensating chiral + antichiral multiplet. It remains to integrate out the auxiliary ?eld V? ij . When writing the second equation of (2.22) in a manifestly SO(2, n) invariant way, and taking into account the various constraints which have been imposed on the vector and hypermultiplet variables we get that ?? A′ j β dα β ? A′ j β ? ?? A′ i α dα β V?i j = ? A′ i α ? ? 2 ηIJ ? e
1
2?

ImFIJ

(0)cov

ˇ ˇ ? I γ? ? ˇ kJ ? I γ? ? ˇ jJ ? 1 δi j ? ? k i 2

(5.36)

I equation of motion (2.14) this equation is the only one —out Remark that together with the Yij of all the equations of motion or gauge choices we have imposed so far on the antisymmetric

29

tensor theory— which does involve the one-loop part of the theory. In (5.36) the following covariantised second derivative matrix appears
(0)cov FIJ (0) (0) ? K ˇ ?K Xη FK X (I FJ )K ? h.c. X = ? ηIJ ?2 ˇ ˇ ˇ X ? ˇ X ? Xη Xη ˇ ˇ (I Xη ? J) ˇ (I Xη ˇ J) Xη Xη (0) ? K ? K F (0) ? h.c. X ?L + 2 F X ? h.c. . +X K KL ˇ ˇ ˇ X ˇ X ? )2 ? )2 (Xη (Xη (0) FIJ

(5.37)

This (non-holomorphic!) function FIJ (0)cov is a natural object in terms of which many oneloop properties of the heterotic e?ective action can be expressed25 . As can be veri?ed by a straightforward but tedious computation FIJ (0)cov transforms as a tensor under the SO(2, n) transformations (apart from a non-trivial monodromy transformation): FIJ
(0)cov SO(2,n)

?→

FKL

(0)cov

+ ΛKL (U ?1 )K I (U ?1 )L J ? ηIJ

ˇ ? ˇ ΛX X . ˇ ˇ X ? Xη

(5.38)

At this point we have ?nished the construction of the Poincar? e version of the antisymmetric version of the heterotic e?ective action and we may now present the supersymmetry transformation rules for all the ?elds, as well as the (bosonic part) of the action. The supersymmetry transformation rules read δe? a = ? ?i γ a ψ?i + h.c.
I ? i ˇ ?? ?i + V? i j ?j + Xη ?? X ˇ I? ? I ?i ? iσ?ν H ? ν ?i + εij γ ν ?j Xη ˇ IF ˇ ?ν + · · · δψ? = 2?

ˇ?ν δB ˇI δZ I ˇ? δW ˇI δ? i

ˇI ?[? W ˇI ? ˇJ ˇ I ?i ψν ]j εij + h.c. + 2?[? Λ ˇ ν ] + ηIJ θ ˇJ = ? 2? ?i γ[? ψν ]i + ηIJ W [? ?i γν ] ?j + 2X ? ν] ˇI = ?i Λ i =
j ˇ ˇI ˇ jI εij + 2X ?I? ? ?i γ? ? εij + h.c. + ?? θ ?i ψ? I J ? ? ˇ ˇ ?? X ?? X ˇI ? X ˇ I Xη ? J? ˇ J ?i + εij σ ?ν ?j F ˇ ? 2Re (X ˇ I Xη ? J )F ˇ ˇI j = 2 ? ?ν ?ν + Yij ? + · · ·

δ? = ?i ?i + h.c. δ?i = δBs α = δζ ′
α I 1 ? j ′ α ? ′j ?ν j ˇ β ?? + i H ? ?i + 1 V ? IF ˇ + ··· / / ? + A ε σ ? Xη ? / A d ? + ? / i j i ij α j β ?ν 2 2 2 ?α ?i + 2ραβ εij ξ ?β ?j δi s 2ξ

=

? A′ i α + A′ j α A′ j β dβ γ ? ? A′ i γ ?i + · · · ? / /

(5.39)

Let us discuss a few aspects of the equation (5.39). Almost all transformation laws are completely i , δ? ˇ I and δ?i , because these saturated at the string tree level. The only exceptions are δψ? i I i ˇ terms. As we already said the latter generate an F (0) dependence which contain V? j and Y ij is of higher order in the fermions. The · · · stand for other higher-order fermion terms which are not a?ected by string loop e?ects. These · · · terms are not particularly interesting, and
25

The following identities are handy in explicit computations: FIJ
(0) ? K ˇ J = F (0) ? Xη ˇ I FK X X I ˇ ˇ X ? Xη I (0)cov ˇ J (0) ˇ X FIJ X =F (0)cov

ˇ ? I F (0)cov X ˇJ = 0 . X IJ

30

in any case they can be reconstructed on the base of the results we presented before. It is ˇ?ν is still completely decoupled particularly interesting to see that the antisymmetric tensor B from the dilaton-part of the theory. Rather then transforming into ?i the antisymmetric tensor i , so in the string frame B ˇ?ν has become part of the gravitational transforms into the gravitinos ψ? multiplet. The gravitinos themselves transform back into the antisymmetric tensor, thanks to the ˇ I are H? independent because there is a cancellation compensating ηi (?) transformation. The δ? i between two contributions coming from the Q- and the S -supersymmetry sectors respectively. The variation of the dilatini ?i is most easily obtained by varying the left hand side of the second line of (5.30), although it can also be computed directly from (5.10) and (5.31). Note that the complete V? i j and ηi (?) dependence of δζ α feeds into δ?i and not into δζ ′ α . As a result the I and loop-independent as expected. ˇ?ν latter is H? , F The antisymmetric tensor lagrangian reads e?1 Ltensor = e
1 ? 2 R + 2? ? ?? ? + ?2? 1 ˇ +I F ˇ ?ν +J N F +e 8 IJ ?ν 1
(0)cov ?2?

1 ? ? ˇI ˇ ? J + ?α β ? ? Bs α ?? B s β ?? Z ?? Z 4 H H? + GI J

+ h.c. (5.40)

ˇ ?I ? 2 H ? ?? Z with GI J ? = ?α β = NIJ =

ˇJ FIJ Z 1 + 4 H ? ?? M (0) + h.c. ˇ ˇ ? Zη Z

(0)cov ˇ ? I Zη ˇ J Zη ηIJ ? h.c. 2? iF + 2 IJ ? e ˇ ˇ ˇ 2 ˇ ? ˇ ? ˇ ? (Zη Z ) Zη Z Zη Z 4 2 dα β ? (B r γ dγ α )(Br δ dδ β ) C (B ) C (B )2 ˇ ˇ (I Zη ? J) Zη ? (0)cov + M (0) ηIJ e2? ηIJ ? 4 ?i F IJ ˇ ˇ ? Zη Z

(5.41)

This lagrangian is manifestly SO(2, n) invariant except for the NIJ term which generates a shift in the θ -angles according to (5.6). The dilaton almost completely decouples from the other ?elds ?2? except from the familiar e prefactor appearing at the string tree level. The antisymmetric tensor ?eld also largely decouples from the rest, although it starts to interact with the vector ˇ I at one loop. Of course these facts were already known from string theory, multiplet moduli Z but here we see that we can reproduce them on the base of N = 2 supersymmetry only. We want to draw attention to the fact that the dilaton kinetic term (with the characteristic +2 prefactor ) is entirely generated by the compensating hypermultiplet contribution to the ?D? Ai α D ? Ai β dα β term in the original superconformal action. Had we imposed the Einstein frame dilatation gauge, then the dilaton dynamics (with a -1 prefactor) would have come from the compensating vector ?J. multiplet part of NIJ D? X I D ? X In this article we have chosen a string frame dilatation gauge because then the metric g?ν automatically coincides with the string metric. Given the string frame results (5.39) and (5.40) one can of course immediately read o? what the Einstein-frame theory would look like. It su?ces to perform an interpolating dilatation and S -supersymmetry transformation on the various ?elds, which implies that ? e? a = e e? a (E ) ˇI Λ i
? ˇ I (E ) = e 2Λ i i (E ) + γ (E ) ?i (E ) i = e 2 ψ? ψ? ? ? ?i = e 2 ?i (E )

?

?

?

(5.42)

ξα =

? ? e 2 ξ α (E )

?i = 31

? e 2 ?i (E )

I , ?, B α are left invariant. In this way one ?nds that ˇ?ν , Z ˇI , W ˇ? while B s

ˇ?ν δB

= ?e

2?

4? ?i σ?ν ?i ? 2? ?i γ[? ψν ]i

(E )

ˇI ? ˇJ ˇ I ?i ψν ]j ? ηIJ W [? ?i γν ] ?j + 2X ?

(E )

εij + h.c. (5.43)

ˇI ?[? W ˇ ν ] + ηIJ θ ˇJ +2?[? Λ ν] where
1 ˇ ˇ I (E ) = e? (Zη ˇ Z ? )? 2 Z ˇI X 1 ˇ ˇ I (E ) = e? (Zη ˇ Z ? )? 2 ? i

ˇ ? ˇ J (E ) ˇ I ?i (E ) ˇ I (E ) ? Z ˇ I ZηJ Λi + 2Z Λ i ˇ ˇ ? Zη Z

(5.44)

So in the Einstein-frame the antisymmetric tensor ?eld ?nally does transform into the dilatini, and one ultimately recovers what might be called the (on-shell) heterotic vector-tensor multiplet. Of course one is really looking at is a spurious dilatino dependence which has to correct for the fact that one is not working with the true string gravitinos.

6

Conclusions

In this paper we presented the antisymmetric tensor version of the low-energy e?ective action for perturbative heterotic strings on K3 × T 2 . In fact we took a slightly broader point of view and showed that the complete set of antisymmetric tensor theories is quite restricted, and that in the dual vector multiplet picture they all lead to a theory characterised by one of the standard prepotentials given in (4.12). Contact between the vector multiplet and the antisymmetric tensor multiplet pictures can be made by performing the change of variables (4.6), which immediately leads to the appearance of (central charge-like) shift symmetries. Out of the general class of antisymmetric tensor theories the one relevant for heterotic strings is then singled out as the only one for which these shift symmetries appear to be inessential. In fact one can obtain a completely conventional gauge structure for the heterotic e?ective action by using the so-called stringy vector ?elds instead of the usual ST U vectors. As such the stringy vectors play a double r? ole because they were already known to make the SO(2, n) invariance manifest at a lagrangian level. Another remarkable feature of the heterotic antisymmetric tensor theory is that it seems to resist an o?-shell description. Within the superconformal setup which we used, this on? ij and A shell character comes about when one is forced to eliminate the auxiliary ?elds T?ν ? contained in the Weyl multiplet. On the other hand we have seen that the dependent expressions ? ij and A we obtain, are relatively simple and in particular do not depend on the onefor T?ν ? loop part of the heterotic theory. We want to emphasize that the loop-independent expressions ˇ I and the antisymmetric for the auxiliary ?elds only arise when one uses the stringy vectors W ? ˇ?ν as the fundamental variables. This at the same time explains why the ?nal tensor ?eld B ˇ?ν ˇ I and B Poincar? e supersymmetric e?ective action is considerably simpli?ed by going to the W ? formulation. Several times in this article we found examples where N = 2 supersymmetry considerations are su?cient to unique select the most natural variables for the heterotic e?ective action. The heterotic dilaton φinv for instance arises as the only SO(2, n) invariant generalisation of φ = Re S which makes the Einstein term in the superconformal theory loop independent. Choosing a string frame formulation for the Poincar? e theory also has a very clear supergravity interpretation: it ˇ I dilaton indecorresponds to imposing a dilatation gauge which makes the stringy scalars X pendent. As such one forces the dilaton and dilatini to sit in a compensating hypermultiplet, which —after three bosonic degrees of freedom have been removed by an SU (2) gauge choice— 32

indeed describes one bosonic and 8 fermionic degrees of freedom. When using the natural heterotic variables we discussed before one gets a ?nal theory in which 1) the SO(2, n) symmetry is manifestly realised, 2) the supersymmetry transformation laws are loop independent apart from some higher order fermionic corrections and 3) the speci?c couplings of the dilaton and the antisymmetric tensor (as we know them from string theory) are easily reproduced. Acknowledgement I would like to thank P. Claus, B. de Wit, M. Faux, B. Kleijn and P. Termonia for numerous discussions concerning the vector-tensor multiplet in its various appearances. I also would like to thank the Belgian IISN for ?nancial support.

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