Contents

ENSL-00147449


GAUGING HIDDEN SYMMETRIES

[2ex] IN TWO DIMENSIONS


Henning Samtleben and Martin Weidner


Laboratoire de Physique

[-.6ex] Ecole Normale Supérieure Lyon

[-.6ex] 46, allée d’Italie

[-.6ex] F-69364 Lyon, CEDEX 07, France

Department of Economics

[-.6ex] University of Southern California

[-.6ex] 3620 S. Vermont Ave. KAP 300

[-.6ex] Los Angeles, CA 90089, U.S.A.


henning.samtleben@ens-lyon.fr, mweidner@usc.edu



ABSTRACT

[3ex]

We initiate the systematic construction of gauged matter-coupled supergravity theories in two dimensions. Subgroups of the affine global symmetry group of toroidally compactified supergravity can be gauged by coupling vector fields with minimal couplings and a particular topological term. The gauge groups typically include hidden symmetries that are not among the target-space isometries of the ungauged theory. The gaugings constructed in this paper are described group-theoretically in terms of a constant embedding tensor subject to a number of constraints which parametrizes the different theories and entirely encodes the gauged Lagrangian. The prime example is the bosonic sector of the maximally supersymmetric theory whose ungauged version admits an affine global symmetry algebra. The various parameters (related to higher-dimensional -form fluxes, geometric and non-geometric fluxes, etc.) which characterize the possible gaugings, combine into an embedding tensor transforming in the basic representation of . This yields an infinite-dimensional class of maximally supersymmetric theories in two dimensions. We work out and discuss several examples of higher-dimensional origin which can be systematically analyzed using the different gradings of .

May 2007

1 Introduction

One of the most intriguing features of extended supergravity theories is the exceptional global symmetry structure they exhibit upon dimensional reduction [1]. Eleven-dimensional supergravity when compactified on a -torus gives rise to an -dimensional maximal supergravity with the exceptional global symmetry group and Abelian gauge group , where is the dimension of some (typically irreducible) representation of in which the vector fields transform. The only known supersymmetric deformations of these theories are the so-called gaugings in which a (typically non-Abelian) subgroup of is promoted to a local gauge group by coupling its generators to a subset of the vector fields. The resulting theories exhibit interesting properties such as mass-terms for the fermion fields and a scalar potential that provides masses for the scalar fields and may support de Sitter and Anti-de Sitter ground states of the theory [2]. Recently, gauged supergravities have attracted particular interest in the context of non-geometric and flux compactifications [3] where they describe the resulting low-energy effective theories and in particular allow to compute the effective scalar potentials induced by particular flux configurations.

A systematic approach to the construction of gauged supergravity theories has been set up with the group-theoretical framework of [4, 5]. Gaugings are defined by a constant embedding tensor that transforms in a particular representation of the global symmetry group . It is subject to a number of constraints and entirely parametrizes the gauged Lagrangian. E.g. in the context of flux compactifications, all possible higher-dimensional (-form, geometrical, and non-geometrical) flux components whose presence in the compactification induces a deformation of the low-dimensional theory can be identified among the components of the embedding tensor. Once the universal form of the gauged Lagrangian is known for generic embedding tensor, this reduces the construction of any particular example to a simple group-theoretical exercise.111Still, the explicit calculation of the various couplings from the closed formulas may pose a considerable task. Moreover, since the embedding tensor combines the flux components of various higher-dimensional origin into a single multiplet of the U-duality group , this formulation allows to directly identify the transformation behavior of particular flux components under the action of the duality groups. In particular, this allows to straightforwardly extend the analysis of the effective theories beyond the region in which the parameters have a simple perturbative or geometric interpretation.

Gaugings of two-dimensional supergravity () have not been studied systematically so far. Yet, this case is particularly interesting, as the global symmetry algebra of the ungauged maximal theory is the infinite-dimensional , the affine extension of the exceptional algebra , and the resulting structures are extremely rich. The realization of the affine symmetry on the physical fields requires the introduction of an infinite tower of dual scalar fields, defined on-shell by a set of first order differential equations. Consequently, these symmetries act nonlinearly, nonlocally and are symmetries of the equations of motion only. As a generic feature of two-dimensional gravity theories, the infinite-dimensional global symmetry algebra is a manifestation of the underlying integrable structure of the theory [6, 7, 8, 9, 10]. In view of the above discussion one may expect that the various parameters characterizing the different higher-dimensional compactifications join into a single infinite-dimensional multiplet of the affine algebra which accordingly parametrizes the generic gauged Lagrangian in two dimensions. We confirm this picture in the present paper. The corresponding multiplet is the basic representation of .

Apart from its intriguing mathematical structure, there are two features of two-dimensional supergravity which render the construction of gaugings somewhat more subtle than in higher dimensions. First, the overwhelming part of the affine symmetries present in the two-dimensional ungauged theory, is of the hidden type and in particular on-shell. Only the zero-modes of the affine algebra are realized as target-space isometries of the two-dimensional scalar sigma-model and thus as off-shell symmetries of the Lagrangian. In contrast, the action of all higher modes of the algebra is nonlinear, nonlocal and on-shell as described above. Gauging such symmetries is a nontrivial task. Second, in two dimensions there are no propagating vector fields that could be naturally used to gauge these symmetries.

It turns out that both these problems have a very natural common solution: introducing a set of vector fields that couple with a particular topological term in the Lagrangian allows to gauge arbitrary subgroups of the affine symmetry group. The resulting gauge groups generically include former on-shell symmetries and thus extend beyond the target-space isometries of the ungauged Lagrangian. The construction in fact is reminiscent of the four-dimensional case where global symmetries that are only on-shell realized can be gauged upon simultaneous introduction of magnetic vector and two-form tensor fields which couple with topological terms [11, 12].

The structure emerging in two dimensions is the following. In addition to the original physical fields, the Lagrangian of the gauged theory carries vector fields in a highest weight representation of . In addition, a finite subset of the tower of dual scalar fields enters the Lagrangian, with their defining first-order equations arising as genuine equations of motion. The gauging is completely characterized by a constant embedding tensor in the conjugate vector representation and subject to a quadratic consistency constraint. The local gauge algebra is a generically infinite-dimensional subalgebra of . The result is a Lagrangian that features scalars and vector fields in infinite-dimensional representations of the affine . However, for every particular choice of the embedding tensor only a finite subset of these fields enters the Lagrangian and only a finite-dimensional part of the gauge algebra is realized at the level of the Lagrangian (with its infinite-dimensional part exclusively acting on dual scalar fields that do not show up in the Lagrangian). We illustrate these structures with several examples for the maximal () theory for which the symmetry algebra is and vector fields and embedding tensor transform in the basic representation and its conjugate, respectively.

In addition to the standard minimal couplings within covariant derivatives and the new topological term, the gauging induces a scalar potential whose explicit form is usually determined by supersymmetry. It is specific to two dimensions that in absence of such a potential, the gauging merely induces a reformulation of the original theory. I.e. the field equations imply vanishing field strengths, such that the only nontrivial effect of the newly introduced vector fields is due to global obstructions. In absence of such, the theory reduces to the original one. On the other hand, integrating out the vector fields in this case leads to an equivalent (T-dual) formulation of the original theory in terms of a different set of scalar fields. This procedure is well-known from the study of non-Abelian T-duality [13, 14, 15], however the results here go beyond the standard expressions, as the gaugings generically include non-target-space isometries. In contrast, in presence of a scalar potential, as is standard in supersymmetric theories, the gaugings constitute genuine deformations of the original theory.

It is worth to stress that although the construction we present in this paper is worked out for a very particular class of two-dimensional models — the coset space sigma-models coupled to dilaton gravity as the typical class of models obtained by dimensional reduction of supergravity theories — it is by far not limited to this class. The entire construction extends straightforwardly to the gauging of hidden symmetries in arbitrary two-dimensional integrable field theories.

The paper is organized as follows. In section 2, we give a brief review of the ungauged two-dimensional supergravity theories and their global symmetry structure. In particular, we give closed formulas for the action of the affine symmetry on the physical fields. In section 3, we proceed to gauge subalgebras of the affine global symmetry by introducing vector fields in a highest weight representation of and coupling them with a particular topological term. We present the full bosonic Lagrangian which is entirely parametrized in terms of an embedding tensor transforming under in the conjugate vector field representation and subject to a single quadratic constraint. In section 4, we discuss various ways of gauge fixing part of the local symmetries by eliminating some of the redundant fields from the Lagrangian. In particular, we show that in absence of a scalar potential the presented construction leads to an equivalent (T-dual) version of the ungauged theory whereas a scalar potential leads to genuinely inequivalent deformations of the original theory. Finally, in section 5 we study various examples of gaugings of the maximal () two-dimensional supergravity. Among the infinitely many components of the embedding tensor, we identify several solutions to the quadratic constraint and discuss their higher-dimensional origin. The various gradings of provide a systematic scheme for this analysis.

2 Ungauged theory and affine symmetry algebra

The class of theories we are going to study in this paper are two-dimensional coset space sigma models coupled to dilaton gravity. These models arise from dimensional reduction of higher-dimensional gravities: pure Einstein gravity in four space-time dimensions gives rise to the coset space while e.g. the bosonic sector of eleven-dimensional supergravity leads to the particular coset space . In this chapter we briefly review the Lagrangian for these theories, their integrability structure, and as a consequence of the latter the realization of the infinite-dimensional on-shell symmetry , cf. [16, 17, 18] for detailed accounts.

2.1 Lagrangian

To define the Lagrangian of the theory we employ the decomposition of the Lie algebra into its compact part  and the orthogonal non-compact complement . For the theories under consideration this is a symmetric space decomposition, i.e. the commutators are of the form

(2.1)

We denote by the generators of and indicate by subscripts the projection onto the subspaces and , i.e. for it is

(2.2)

In addition, it is useful to introduce the following involution on algebra elements

(2.3)

The () bosonic degrees of freedom of the theory are described by a group element of which transforms under global transformations from the left and local transformations from the right, i.e. the theory is invariant under

(2.4)

It is sometimes convenient to fix the local freedom by restricting to a particular set of representatives of the coset , on which the global then acts as

(2.5)

where depends on in order to preserve the class of representatives. This defines the nonlinear realization of on the coset space .

The -invariant scalar currents are defined by

(2.6)

The current is a composite connection for the local gauge invariance, i.e. it appears in covariant derivatives of all quantities that transform under , in particular

(2.7)

The integrability conditions for (2.6) are then given by

(2.8)

The two-dimensional Lagrangian takes the form

(2.9)

In addition to the scalar current it contains the dilaton field and the conformal factor . The latter originates from the two-dimensional metric which has been brought into conformal gauge , such that space-time indices in (2.9) are contracted with the flat Minkowski metric . The only remnant of two-dimensional gravity is the first term descending from the two-dimensional (dilaton coupled) Einstein-Hilbert term in conformal gauge. The Lagrangian (2.9) is manifestly invariant under the symmetry (2.4). It is straightforward to derive the equations of motion which take the form222Our space-time conventions are , ; i.e. , .

(2.10)

where we have introduced light-cone coordinates . In addition, the theory comes with two first order (Virasoro) constraints

(2.11)

which might equally be obtained from the Lagrangian before the fixing of conformal gauge. It is straightforward to check that these first order constraints are compatible as a consequence of the equations of motion for and and moreover imply the second order equation for the conformal factor .

2.2 Global symmetry and dual potentials

It is well known — starting from the work of Geroch on dimensionally reduced Einstein gravity [6, 19, 20] — that the global symmetry algebra of the coset space sigma model (2.9) is not only the algebra of target-space isometries , but half of its affine extension  [21]. We denote the generators of by and those of by , . The latter close into the algebra [22]

(2.12)

where and are the structure constants and the Cartan-Killing form of , respectively, and denotes the central extension of the affine algebra. In addition to and we will find the Witt-Virasoro generator to be crucial for the construction of this paper. It obeys

(2.13)

The central extension commutes with both and . We denote by the algebra spanned by .

To define the action of on the fields , and that enter the Lagrangian (2.9) we need to introduce an infinite hierarchy of dual potentials. These are additional scalar fields that are defined as nonlocal functions of (and ), but whose definition is only consistent if one invokes the equations of motion. Therefore is only realized as an on-shell symmetry on (2.10).

To start with, the dilaton is a free field, such that it gives rise to the definition

(2.14)

of its dual . Obviously, the dual of gives back . More interesting are the nonlinear equations of motion for that can be rewritten as a conservation law for the current . This allows the definition of the first dual potential

(2.15)

which is valued and according to (2.4) transforms in the adjoint representation of the global . Integrability of these equations is ensured by . From the point of view of higher-dimensional supergravity theories, equations (2.15) constitute nothing but a particular case of the general on-shell duality between forms and forms (, ). In two dimensions however, these equations are just the starting point for an infinite hierarchy of dual potentials of which the next members , are defined by

(2.16)

Again, integrability of these equations is guaranteed by the field equations and the defining equation (2.15) of the lower dual potentials. A convenient way to encode the definition of all dual potentials (and the action of the affine symmetry) is the linear system [7, 8] which we will describe in the next subsection. In order make the symmetry structure more transparent we will restrict the discussion in the present subsection to the lowest few dual potentials and to the action of the lowest few affine symmetry generators .

We identify the zero-modes with the generators of the off-shell symmetry . These zero-mode symmetries do not mix the original scalars and the dual potentials of different levels, i.e.  transforms according to (2.4) and all the () transform in the adjoint representation. The fields , , and are left invariant by .

The dual potentials , are defined by (2.14)–(2.16) only up to constant shifts , . The generators in corresponding to these shift symmetries are and (), i.e.

(2.17)

where and denote the action of and , respectively, and  . Since the definition of the dual potentials also involves and lower dual potentials, it follows that and also act nontrivially on the higher dual potentials (), e.g.

etc. (2.18)

None of the shift symmetries and () acts on the physical fields , or . So far we have thus not introduced any new physical symmetry. The crucial point about the symmetry structure of the model is the existence of another infinite family of symmetry generators (). Their action on the physical fields is expressed in terms of the dual potentials and thus nonlinear and nonlocal in terms of the original fields. For the lowest generators, this action is given by

The field is left invariant while the action on the dual potentials and on the conformal factor follows from (2.11), (2.15). We find for example

(2.20)

One can easily check that the symmetries defined in (2.17) and (LABEL:NonLocalSym) indeed close according to the algebra (2.12). In particular, it follows that the central extension acts exclusively on the conformal factor [21]:

(2.21)

In order to define all dual potentials () and describe the action of all symmetry generators in closed form we will in the following introduce the linear system [7, 8] showing the classical integrability of the theory.

2.3 The linear system

A compact way to encode the infinite family of dual potentials and the action of the full symmetry algebra  is the definition of a one-parameter family of group-valued matrices according to the linear system [7, 8, 16]

(2.22)

where is a scalar function

(2.23)

of the constant spectral parameter which labels the family. As is a double-valued function of we will in the following restrict to the branch , i.e. in particular

(2.24)

around .

It is straightforward to verify that the compatibility of (2.22) is equivalent to (2.8) and the equations of motion (2.10):

Expanding around

(2.26)

defines the infinite series of dual potentials . In particular, the expansion of (2.22) around reproduces (2.15), (2.16). For later use we also give the linear system in light-cone coordinates

(2.27)

Using the matrix , the action of the symmetry algebra  can be expressed in closed form. To this end, we parametrize the loop algebra of by a spectral parameter  and identify the generators with . Elements of are represented by -valued functions , meromorphic in the spectral parameter plane. In terms of , the action on the physical fields , can be given in closed form as

(2.28)

Here we have defined the dressed parameter333 For notational simplicity we use here and in the following the notation , even though by definition globally is a function of and thus on the double covering of the complex -plane. We will however be mainly interested in its local expansion around on the sheet (2.24).

(2.29)

with the split according to (2.2). In addition, we have introduced the notation

(2.30)

for an arbitrary function of the spectral parameter . The path is chosen such that only the residual at is picked up. For definiteness we will treat the functions in these expressions as formal power series with almost all equal to zero. Some useful relations for calculating with these objects are collected in appendix A.

It is straightforward to check that the transformations (2.28) leave the equations of motion invariant. Since the solution of the linear system (2.22) explicitly enters the transformation, this is in general not a symmetry of the Lagrangian but only an on-shell symmetry of the equations of motion (2.10). This will be of importance later on. Moreover, it is straightforward to check, that the algebra of transformations (2.28) closes according to (2.12). Relation (A.6) is crucial to verify the action (2.21) of the central extension.

The group-theoretical structure of the symmetry (2.28) becomes more transparent if we consider its extension to and thereby to the full tower of dual potentials [9]:

(2.31)

in the above notation. This action may be rewritten as

and thus takes the form of an infinite-dimensional analogue of the nonlinear realization (2.5), in which the left action of parametrizing is accompanied by a right action of in order to preserve a particular class of coset representatives. The algebra is the infinite-dimensional analogue of in (2.5), i.e. the maximal compact subalgebra of , and is defined as the algebra of -valued functions , satisfying [16]444 Note that .

(2.33)

We shall see in the following that the particular set of coset representatives starring in (2.3) are the functions regular around in accordance with the expansion (2.26).

For illustration, let us evaluate equation (2.3) for the particular transformation ,  ,  . Expanding both sides around , it follows directly from (A.5) that for positive values of , vanishes, such that the transformation merely amounts to a shift of the dual potentials in the expansion (2.26); for this reproduces (2.17), (2.18). These transformations do not act on the physical fields present in the Lagrangian (2.9). For a transformation with negative on the other hand the second term in (2.3) no longer vanishes but precisely restores the regularity of at that has been destroyed by the first term [24]. These transformations describe the nonlinear and nonlocal on-shell symmetries on the physical fields and the dual potentials which leave the equations of motion and the linear system (2.27) invariant. They are commonly referred to as hidden symmetries, for one recovers (LABEL:NonLocalSym). Finally, for one recovers the action (2.4) of the finite algebra acting as an off-shell symmetry on all the fields. Here, the local freedom in (2.4) has been fixed such that .

To summarize, the negative modes , act as nonlocal on-shell symmetries whereas the positive modes , act as shift symmetries on the dual potentials. Only the zero-modes are realized as off-shell symmetries on the physical fields of the Lagrangian (2.9).

In addition to the affine symmetry algebra described above, a Witt-Virasoro algebra can be realized on the fields [23] which essentially acts as conformal transformation on the inverse spectral parameter . From these generators we will in the following only need

(2.34)

which acts only on the dual dilaton and the dual potentials according to equations (2.17), (2.18)

(2.35)

The pair and which extends the loop algebra of to turns out to be crucial for our construction of the gauged theory in section 3. The distinguished role of in this construction — as opposed to all the other Virasoro generators that can be realized following [23] — stems from its action on the dual dilaton (2.17). The gaugings we are mainly interested in will carry a scalar potential whose presence in particular deforms the free field equation (2.10) of by some source terms . The only way to maintain a meaningful version of the dual dilaton equation (2.14) in this case is by gauging its shift symmetry while imposing . We shall see that this indeed appears very natural in the subsequent construction.

In the following we will parametrize a general algebra element of with a collective label for the generators of as

with . The commutator between two such algebra elements takes the form

(2.37)

where we use the notation in order to distinguish the general algebra commutator from the simple matrix commutators .

Let us finally mention, that the symmetry algebra is equipped with an invariant inner product , given by

(2.38)

Note that this invariant form differs from the standard one by the shift of in the grading. This is precisely consistent with the use of rather than in the pairing with the central extension .

2.4 Structure of the duality equations

For the following it turns out the be important to analyze in more detail the structure of the duality equations (2.14) and (2.22) which have been used to define the dual fields and . Let us for the moment consider these dual fields as a priori independent fields and the duality equations as their first order equations of motion relating them to the physical fields and . In particular, we may define the -valued current as

(2.39)

which is a particular combination of the duality equations, i.e. on-shell we have . Under a generic symmetry transformation the constituents of transform according to (2.28), (2.31), and (2.35) and some lengthy computation shows that altogether transforms as

(2.40)

in light-cone coordinates. In order not to overburden the notation here, all spectral parameter dependent functions within the brackets depend on the parameter which is integrated over, whereas all functions outside depend on the spectral parameter . In slight abuse of notation, the commutators represent the full commutator (2.37) however without the central term .555Inclusion of this term would presumably require the extension of by a -valued term proportional to the Virasoro constraints (2.11). This is in accordance with the generalized linear system proposed in [25]. For the purpose of this paper however this would complicate things unnecessarily. In particular, (2.40) shows that transforms homogeneously under — consistent with the fact that vanishes on-shell. This current will play an important role in the following.

3 Gauging subgroups of the affine symmetry

In the previous section we have reviewed how the equations of motion of the ungauged two-dimensional theory are invariant under an infinite algebra of symmetry transformations. The symmetry action on the physical fields (2.28) is defined in terms of the matrix which in turn is defined as a solution of the linear system (2.22). As a result, the global symmetry is nonlinearly and nonlocally realized on the physical fields.

We will now attempt to gauge part of the global symmetry (2.28), i.e. turn a subalgebra of into a local symmetry of the theory. This is rather straightforward for subalgebras of , as is the off-shell symmetry algebra of the Lagrangian. In fact, since is already the off-shell symmetry of the three-dimensional ancestor of the theory, the corresponding gaugings are simply obtained by dimensional reduction of the three-dimensional gauged supergravities [4, 26]. The gauging of generic subalgebras of is much more intricate, as their action explicitly contains the matrix which is defined only on-shell as a nonlocal functional of the physical fields. This is the main subject of this paper. The problem is analogous to the one faced in four dimensions when trying to gauge arbitrary subgroups of the scalar isometry group – not restricting to triangular symplectic embeddings – which has been solved only recently [11, 12]. We will follow a similar approach here.

As a key point in the construction we will introduce the dual scalars and as independent fields on the Lagrangian level. The duality equations (2.39) relating them to the original fields will naturally emerge as first order equations of motion. Specifically, the field equations obtained by varying the Lagrangian with respect to the newly introduced gauge fields of the theory turn out to be proportional to the current introduced in section 2.4 which combines the duality equations.

3.1 Gauge fields and embedding tensor

In order to construct the gauged theory, we make use of the formalism of the embedding tensor, introduced to describe the gaugings of supergravity in higher dimensions [4, 5]. Its main feature is the description of the possible gaugings in a formulation manifestly covariant under the global symmetry of the ungauged theory. As a first step we need to introduce vector fields in order to realize the covariant derivatives corresponding to the local symmetry. In contrast to higher dimensions where the vector fields come in some well-defined representation of the global symmetry group of the ungauged theory, in two dimensions these fields do not represent propagating degrees of freedom and are absent in the ungauged theory.666Also in three dimensions it is most convenient to start from a formulation of the ungauged theory in which no vector fields are present [4, 26]. In contrast to the present case, however, the vector fields in three dimensions are dual to the scalar fields and thus naturally come in the adjoint representation of the scalar isometry group. We will hence start by introducing a set of vector fields transforming in some a priori undetermined representation (labeled by indices ) of the algebra .

An arbitrary gauging then is described by an embedding tensor that defines the generators

(3.1)

of the subalgebra of which is promoted to a local symmetry by introducing covariant derivatives

(3.2)

with a gauge coupling constant .777The coupling constant always comes homogeneous with the embedding tensor and could simply be absorbed by rescaling . We will keep it explicitly to have the deformation more transparent. The way appears within these derivatives shows that under it naturally transforms in the tensor product of two infinite-dimensional representations. Gauge invariance immediately imposes the quadratic constraint (or embedding equation)

(3.3)

on , where denote the structure constants of the algebra (2.12), (2.13), and are the generators of in the representation of the vector fields. Equivalently, this constraint takes the form

(3.4)

with “structure constants”888 We have put quotation marks here because according to this definition the constants are not antisymmetric in the first two indices, but only after further multiplication with a generator . Manifest antisymmetrization on the other hand defines objects that do no longer satisfy the Jacobi identities. Analogous structures arise in higher-dimensional gauged supergravity theories [27]. . We will impose further constraints on in the sequel.

It will sometimes be convenient to expand the covariant derivatives (3.2) according to (LABEL:SymParamD2) as

(3.5)

with the projected vector fields

(3.6)

While the appearance of the infinite sums (over and over ) in the definition of (and thus the appearance of an infinite number of vector fields) looks potentially worrisome, we will eventually impose constraints on such that only a finite subset of vector fields enters the Lagrangian.

Explicitly, the action of the covariant derivative on the various scalars reads999 Comparing (3.7) to (2.6) one notices that does not depend on the coupling constant . This is due to our particular gauge choice in equation (2.28).

(3.7)

with .

3.2 The Lagrangian

As a first step towards introducing the local symmetry on the level of the Lagrangian, we consider the covariantized version of (2.9)

(3.8)

with covariant derivatives according to (3.7). Obviously, (3.8) cannot be the full answer since the equations of motion for the newly introduced vector fields will pose unwanted (and in general inconsistent) first order relations among the scalar fields. Likewise, according to (3.7) the now carry the dual potentials and which are to be considered as independent fields. Variation with respect to these fields then gives rise to even stranger constraints.

Remarkably, all these problems can be cured by adding to the Lagrangian what we will refer to as a topological term101010We call this term topological as after relaxing conformal gauge it does not depend on the two-dimensional metric.

which is made such that the vector field equations of motion precisely yield (a projection of) the covariantized version of the duality equations (2.14), (2.22). Explicitly, the variation of the Lagrangian with respect to the vector fields reads

(3.10)

where is the properly covariantized version of the -valued current defined in (2.39) above. It contains the covariantized versions of the duality equations (2.14) and (2.22) that render dual to and dual to , respectively. As vector field equations in the gauged theory we thus find a -projection of  :

(3.11)

In the limit back to the ungauged theory these equations consistently decouple.

The fact that the higher order terms of (3.11) can be consistently integrated to the variation (3.10) is nontrivial and puts quite severe constraints on the construction. Namely, it requires the following constraint

(3.12)

on the variation with respect to the projected vector fields. Fortunately, this condition translates directly into the covariant constraint

(3.13)

for the embedding tensor . For consistency, this constraint must thus be imposed together with the quadratic constraint (3.3) ensuring gauge invariance. As in higher-dimensional gaugings [5], we expect that the latter constraint (3.13) should eventually be a consequence of (3.3). This is one motivation for the ansatz

(3.14)

for the embedding tensor parametrized by a single conjugate vector . In terms of representations this means that does not take arbitrary values in the tensor product of the coadjoint and the conjugate vector field representation, but only in the conjugate vector field representation contained in this tensor product. This is the analogue of the linear representation constraint that is typically imposed on the embedding tensor in higher dimensions [4, 5]. Indeed, it is straightforward to verify that the ansatz (3.14) reduces the quadratic constraints (3.3) and (3.13) to the same constraint for :

(3.15)

Further support for the ansatz (3.14) comes from the fact that all the examples of gauged theories in two dimensions (presently known to us) turn out to be described by an embedding tensor of this particular form. In particular, in all examples originating by dimensional reduction from a higher-dimensional gauged theory, the constraint (3.14) is a consequence of the corresponding linear constraint in higher dimensions. We will come back to this in section 5. This shows that (3.14) describes an important class of if not all the two-dimensional gaugings.

It is useful to give the projected vector fields (3.5) using (3.14)