Higher Derivative Extension of 6D Chiral Gauged Supergravity

Six-dimensional (1,0) supersymmetric gauged Einstein-Maxwell supergravity is extended by the inclusion of a supersymmetric Riemann tensor squared invariant. Both the original model as well as the Riemann tensor squared invariant are formulated off-shell and consequently the total action is off-shell invariant without modification of the supersymmetry transformation rules. In this formulation, superconformal techniques, in which the dilaton Weyl multiplet plays a crucial role, are used. It is found that the gauging of the U(1) R-symmetry in the presence of the higher-order derivative terms does not modify the positive exponential in the dilaton potential. Moreover, the supersymmetric Minkowski(4) x S^2 compactification of the original model, without the higher-order derivatives, is remarkably left intact. It is shown that the model also admits non-supersymmetric vacuum solutions that are direct product spaces involving de Sitter spacetimes and negative curvature internal spaces.


Introduction
Higher-order curvature terms in supergravity theories are of considerable importance for different reasons. They can be considered as higher-order correction terms (in α ) to an effective supergravity Lagrangian of a (compactified) string theory (see, e.g., [1]). These Lagrangians are supersymmetric only order by order in the perturbation parameter α . On the other hand off-shell formulations for different curvature squared invariants in 4, 5 and 6 dimensions have been constructed in [2,3,4,5,6,7]. These invariants, added to a pure off-shell supergravity theory, are exactly supersymmetric and can be considered in their own right. The off-shell nature of these theories implies that they contain auxiliary fields. It is well-known that, when adding higher derivative terms to the Lagrangian, the auxiliary fields become propagating. Hence, the elimination of these auxiliary fields becomes much harder since their field equations are not algebraic anymore. Assuming that the dimensionful parameter in front of the higher derivative part of the Lagrangian is very small, one can solve the auxiliary field equations perturbatively and eliminate these fields order by order in the small parameter. It remains an open question if and how the on-shell Lagrangian obtained in this way is related to the compactified string Lagrangian, which does not contain any auxiliary fields to begin with. 1 Theories containing higher-order curvature terms can provide corrections to black hole entropies [9,10,11] and can source higher-order effects in the AdS/CFT correspondence [12,13]. When considering these theories as toy models on their own they can be compactified to lower dimensions. A particular case to consider is the compactification to three dimensions [8]. A particular feature of three dimensions is that D = 3 gravitons are non-propagating when only considering 2-derivative Lagrangians. Instead, the addition of higher-derivative terms can turn these non-propagating modes into propagating massive graviton modes, see e.g. [14] and references therein. These theories can then be regarded as simple toy models to study quantum gravity.
In this paper we study higher-order corrections to a six-dimensional (1, 0) supersymmetric U (1) R gauged Einstein-Maxwell supergravity theory, usually referred to as the Salam-Sezgin model [15], which is a special case of a Sp(n) × Sp(1) R gauged matter-coupled supergravity theory that was first obtained in [16]. We shall refer to this more general case as 6D chiral gauged supergravity as well. An intriguing feature of the Salam-Sezgin model is that it allows a compactification over S 2 to a four-dimensional Minkowski spacetime while retaining half of the supersymmetry [15]. One of the purposes of this work is to investigate whether this feature survives after the addition of higher-order derivative corrections. To facilitate the addition of such higher-order corrections to the model we will first construct its off-shell formulation. It turns out that this is only possible for the dual formulation of the model where the 2-form potentialB has been replaced by a dual 2-form potential B [17,18]. This has the effect that the curvature of the original 2-form potential no longer contains a Maxwell-Chern-Simons term, but that instead a term of the form B ∧ F ∧ F , where F is the Maxwell field strength, appears in the Lagrangian.
To construct the off-shell formulation we will make use of the superconformal tensor calculus. As a first step we will review the construction of off-shell minimal D = 6 supergravity [19,20]. In this construction one makes use of the dilaton Weyl multiplet (obtained by coupling the regular Weyl multiplet to a tensor multiplet) coupled to a linear multiplet as compensator. After fixing the conformal symmetries, this theory still has a remaining U (1) R-symmetry which is gauged by an auxiliary vector V µ . We will couple this 'pure' theory to an Abelian vector multiplet and show that after solving for the auxiliary V µ , the gauging proceeds via the vector W µ of the Abelian vector multiplet.
After constructing the off-shell formulation of the gauged (1, 0) supergravity theory, we investigate its deformation by an off-shell curvature squared invariant [2,3]. To construct this invariant it is essential to make use of the dilaton Weyl multiplet. We review the construction of this higher-derivative term and add it to the off-shell (1, 0) supergravity theory. Next, we study the gauging procedure in the presence of the Riemann tensor squared invariant.
As a first step towards understanding the properties of the higher-derivative extension of the model we perform a systematic search for vacuum solutions. We construct both supersymmetric as well as non-supersymmetric solutions. For one particular supersymmetric solution, namely six-dimensional Minkowski spacetime, we calculate the fluctuations around this background and show how these fluctuations fit into supermultiplets.
This paper is organized as follows. In section 2 we review the off-shell version of the (1, 0) supergravity model [19,20] and describe its gauging. In Section 3, we introduce an alternative off-shell formulation of the model in view of the fact that it is best suited for the addition of the Riemann tensor squared invariant [2]. In section 4 we discuss the construction of the Riemann tensor squared invariant and arrive at the total Lagrangian for the higherderivative extended 6D chiral gauged supergravity theory. In Section 5, we investigate the vacuum solutions of this model. We summarize and comment further on our results and on some interesting open problems in the Conclusions section. Throughout the paper we follow the notation given in Appendix A of [20].
2 Off-shell Gauged (1, 0) Supergravity In this section we present an off-shell version of the dual formulation [17,18] of the Salam-Sezgin model [15,16]. In the first subsection we give the off-shell Lagrangian of pure supergravity plus a tensor multiplet as constructed in [19,20]. In the next subsection we couple a vector multiplet to this theory and show that the resulting Einstein-Maxwell model leads to a non-trivial U (1) gauge symmetry that is not gauged by an auxiliary vector field. In the last subsection we show that after eliminating the auxiliary fields one ends up with a Lagrangian in which the U (1) gauge symmetry is effectively gauged by the physical vector of the vector multiplet. We furthermore show that, after dualizing the 2-form potential into a dual 2-form potential, this Einstein-Maxwell model is nothing else than the original Salam-Sezgin model.

Off-shell Poincaré Action
The off-shell (1, 0) supergravity action has been constructed by means of a superconformal tensor calculus in which the off-shell so-called dilaton Weyl multiplet with independent fields and Weyl weights (−1, −1/2, 0, 0, 0, 5/2, 2), respectively, is coupled to an off-shell linear multiplet consisting of the fields with Weyl weights (0, 4, 9/2), respectively. The fields (ψ i µ , ψ i , ϕ i ) are symplectic Majorana-Weyl spinors labelled by a Sp(1) R doublet index, the fields B and E are two-and four-forms with tensor gauge symmetries, respectively, b µ is the dilatation gauge field and L ij are three real scalars. An appropriate set of gauge choices for obtaining off-shell supergravity with the Einstein-Hilbert term, namely L = eR + · · · , is given by which fixes the dilatations, conformal boost and special supersymmetry transformations. Moreover, the first of the gauge choices in (2.3) breaks Sp(1) R down to U (1) R . This set of gauge choices leads to an off-shell multiplet containing 48 + 48 degrees of freedom described by the fields [19] (see Table 5 of [20]) The field V µ is the gauge field of the surviving U (1) R gauge symmetry. It arises in the decomposition where the traceless part V ij µ has no gauge symmetry. A superconformal tensor calculus method was employed in [19] where the bosonic action was given, and a procedure for obtaining the full action was provided. This full action, including the quartic terms, was constructed in [20]. The Lagrangian up to quartic fermion terms is given by [19,20] The indication L = 1 in the left-hand side indicates all the gauge choices (2.3). Here we have defined the field strength for the 2-form potential and the dual of the field strength for the 4-form potentials as follows 3 The U (1) R covariant derivatives D µ (ω) and the full SU(2) covariant derivatives D µ (ω) are given by where ω µab is the standard torsion-free connection. Note that the symmetric traceless field V µ ij , occurring in the decomposition (2.5), is absent in the covariant derivative of the gravitino [20]. This is a consequence of having broken the SU(2) symmetry present in the dilaton Weyl multiplet by the gauge choices (2.3). In the above formula, and throughout the paper the spin connection ω µab is the standard one associated with the Christoffel symbol, and as such, it does not depend on fermionic or bosonic torsion. The supersymmetry transformations, up to cubic fermion terms, are obtained from Sec. 2 of [20]: where D µ (ω) i is defined as in (2.9), R µν i (Q) is the gravitino curvature and η i is the effective contribution from the S-supersymmetry in the superconformal algebra: The latter equation gives the compensating special supersymmetry transformation parameter in the gauge ϕ i = 0, as can be read off from eq. (3.37) of [19]. Note that the U (1) R part of V ij µ has dropped out in this expression. The surviving U (1) R symmetry of the Lagrangian L R is gauged by the auxiliary gauge field V µ , which acts as follows 4 with λ being the parameter of the gauged symmetry.

Coupling to an Off-shell Vector Multiplet
We now wish to introduce a gauge multiplet, whose vector is not auxiliary, to gauge the U (1) R-symmetry. The present gauging by V µ , discussed in the previous subsection, is undesirable since V µ has no standard kinetic term. In fact, we will show in subsection 2.3 that the gauge symmetry becomes trivial after solving the 4-form potential in terms of a scalar field. To obtain this non-trivial gauging we follow [19] and add to L R the kinetic terms for an abelian vector multiplet L V . The multiplet consists of the fields (W µ , Y ij , Ω i ), being a physical gauge field, an auxiliary SU(2) triplet, and a physical fermion. They transform under dilatations with Weyl weights (0, 2, 3/2), respectively. We add the coupling gL VL of the vector multiplet to the compensating linear multiplet. Prior to fixing any of the conformal symmetries, these Lagrangians, up to quartic fermion terms, are given by [19] 16) 4 The U(1)R is the subgroup of the full SU (2), under which the gravitino transforms as where λ ij is traceless. A similar formula holds for ψ i .
where D µ (ω)Ω i is defined as in (2.10). This action has the full SU(2) symmetry. The coupling of the vector multiplet to supergravity is then achieved by considering the Lagrangian where as before 'L = 1' refers to the set of gauges given in (2.3). This formula, up to quartic fermion terms, yields the result The action corresponding to the Lagrangian L 1 is invariant under the supersymmetry transformations (2.11) supplemented by the supersymmetry transformations of the components of the off-shell vector multiplet. The transformations of the latter are given up to cubic fermion terms by [19] δW µ = −¯ γ µ Ω , where η is as defined in (2.13). The Lagrangian L 1 also has a manifest U (1) R × U (1) symmetry with transformations parametrized by λ and η δV µ = ∂ µ λ , δW µ = ∂ µ η, where (λ , η) are the parameters of the U (1) R , U (1) symmetry, respectively.

Elimination of Auxiliary Fields
We consider Lagrangian L 1 given in (2.18), and begin by writing down the field equations for the auxiliary fields Y ij , V ij µ , V µ , E µνρσ : The elimination of Y ij in (2.18) by means of (2.21) gives a positive definite potential 1 4 g 2 σ −1 and the elimination of V ij µ by means of (2.22) gives only quartic fermion terms in the action. Next, (2.24) implies that locally we can write for some scalar field φ transforming under the U (1) R × U (1) transformations (2.20) as The terms in (2.25) can be rearranged to write with the covariant derivative of the scalar field defined as Using (2.27) to eliminate E µ in the Lagrangian (2.18) amounts to dualization of the 4-form potential E µνρσ related to E µ as in (2.8) 5 . The shift symmetry (2.26) can be used to eliminate the scalar field φ, by setting it to a constant φ 0 . This in turn implies a compensating λ = −gη/ √ 2 transformation, leading to an unbroken U (1) symmetry. Eliminating φ in this way, (2.23) and (2.25) imply Using this equation and (2.23) in the terms involving E µ in the action gives rise to only quartic fermion terms. The use of (2.25) in the fermionic kinetic terms, however, has the effect of replacing V µ by −gW µ / √ 2, up to quartic fermion terms in the action. Thus, altogether, the elimination of all the auxiliary fields yields, up to quartic fermion terms, the following Lagrangian: This Lagrangian has the on-shell supersymmetry given, up to cubic fermion terms, by the transformation rules for (e a µ , ψ i µ , B µν , ψ i , σ) in (2.11), and for (W µ , Ω i ) in (2.19), with the replacements The last substitution is due to the fact that the elimination of V ij µ and E µ in (2.13) gives rise to quadratic fermion terms only. These results agree with the Lagrangian obtained in [17] by direct application of the Noether procedure based on the on-shell closed supersymmetry transformations.
A dual formulation in which the field equation and Bianchi identity for the 2-form potential are interchanged is easily obtained by adding a Lagrange multiplier term Treating F µνρ (B) as an independent field in L + ∆L, its field equation can be used back in the action, yielding This Lagrangian has the on-shell supersymmetry given, up to cubic fermion terms, by the transformation rules for (e a µ , ψ i µ , B µν , ψ i , σ) in (2.11), and for (W µ , Ω i ) in (2.19), with the replacements These results agree with [15,16,17], after taking into account the fact that some of the fermions are to be redefined by scaling them with a suitable power of the scalar field σ.

An Alternative Off-Shell Formulation
Starting from a superconformal coupling of the dilaton Weyl multiplet to the compensating linear multiplet, we made the set of gauge choices (2.3) which led to an off-shell Poincaré supergravity with field content (2.4). If we do not insist on the canonical Einstein-Hilbert term in the action, there exists a natural alternative set of gauge choices given by which fix the dilatations, conformal boost and special supersymmetry, and lead to an alternative off-shell Poincaré multiplet consisting of the fields Compared to the previous multiplet given in (2.4) σ and ψ i are replaced by L and ϕ i , and therefore this multiplet again has 48 + 48 off-shell degrees of freedom. It turns out that this formulation of the off-shell Poincaré multiplet is very convenient in the construction of the only known off-shell higher derivative invariant in D = 6, which is a supersymmetric completion of the Riemann tensor squared [2]. What makes the gauge choice (3.1) very useful in this construction is that it furnishes a map between the off-shell supersymmetry transformations of the Yang-Mills and Poincaré multiplets. We shall review this construction in the next section. Here we shall focus on coupling a vector multiplet to this alternative Poincaré supermultiplet. This amounts to seeking an expression for Starting from (2.15) and (2.16), it is straightforward to obtain L V and gL V L in the gauge (3.1). To construct the Einstein-Hilbert Lagrangian in this gauge, on the other hand, we first restore superconformal invariance 6 by performing suitable field redefinitions in (2.6). This is achieved by replacing the fields that transform under dilatations and special supersymmetry by which are invariant under dilatations and special supersymmetry, as can be checked by using the transformation rules given in [19]. Next, we impose the gauge choices (3.1). Thus, we construct the Lagrangian where L R is the Lagrangian given in (2.6) with the field redefinitions (3.3) performed, such that the superconformal invariance is restored, and σ = 1 refers to all the gauge choices of (3.1). A summary of the different gauge conditions and what parts of the superconformal Lagrangian they affect can be found in table 1.
, up to quartic fermion terms, gives rise to the following expression: where E µ is not an independent field but rather the dual of the field strength for the four-form potential, see (2.8), the derivative D µ (ω)ψ ν is U (1) covariant as in (2.9), and the derivatives D µ (ω)ϕ and D µ (ω)Ω are SU(2) covariant as in (2.10). The off-shell supersymmetry transformations for this Lagrangian are to be obtained from those of the dilaton Weyl multiplet upon fixing the gauges (3.1). It is important to note that the field redefinitions (3.3) are not to be performed in this process since these transformations are independent of the linear multiplet fields that were used to impose the gauge choices (2.3). In obtaining these transformations, the compensating transformations required to maintain the gauge (3.1) must also be incorporated. These are a compensating special supersymmetry transformation and a compensating (traceless) SU(2) transformation with parameters given by (up to cubic fermion terms) is the supersymmetry transformation of the traceless part of L ij . Note that the prime stands for 'traceless', i.e. S ij δ ij = 0. These compensating transformations can be obtained from the transformation rules for ψ i and L ij given in [19]. Thus, using the supersymmetry transformation rules for the dilaton Weyl multiplet provided in [19,20], the gauge conditions (3.1) and the compensating transformations with parameters given in (3.6), we find that the supersymmetry transformations of the off-shell Poincaré multiplet, up to cubic fermion terms, take the form It is instructive to write out the λ parameter in components: The supersymmetry transformations of the off-shell vector multiplet are (up to cubic fermion terms) (3.10) To keep the notation relatively simple we did not use the explicit expression for λ ij in the above transformation rules. Remember that it is given in (3.6).
Considering the Lagrangian (3.5) by itself, that is, without any higher derivative extension, all the auxiliary fields, namely (V ij µ , E µνρσ , Y ij ) can be eliminated, thereby arriving at the on-shell formulation. Computations similar to those described in detail in section 2.3 imply that the on-shell Lagrangian, up to quartic fermion terms, is obtained from (3.5) by the following substitutions: The on-shell supersymmetry transformations, up to cubic fermion terms, are obtained from (3.8) and (3.10) by making these substitutions, and dropping the transformation rules for the auxiliary fields (E µνρσ , V ij µ , Y ij ).

Inclusion of the R µνab R µνab invariant
In this section we add an off-shell supersymmetric Riemann tensor squared term to the Lagrangian L 2 , defined in (3.4), which we constructed in the gauge (3.1). This gauge gave rise to an alternative off-shell formulation of the Poincaré multiplet. In the first subsection we begin with a review of the construction of the Riemann squared invariant [2]. In the second subsection we consider the total Lagrangian and briefly discuss the gauging procedure and the elimination of auxiliary fields.

Construction of the R µνab R µνab invariant
To begin with, we shall review a map between the Yang-Mills supermultiplet and a set of fields in the alternative Poincaré multiplet discussed in the previous section. We follow the discussion in [3]. This map can be used, together with an expression for the superconformal action for the Yang-Mills multiplet given in [19], to write down a supersymmetric Riemann tensor squared action. We will describe this in detail below.
In establishing the map between the Yang-Mills and Poincaré multiplets, it is important to consider the full supersymmetry transformations, including the cubic fermion terms which have been omitted so far. In particular, this means that we need to keep track of the complete spin connection, containing the fermionic torsion terms. This is due to the fact that, while the fermionic torsion gave rise to only quartic fermion terms in the Lagrangians considered above, in the case of the Riemann tensor square invariant under consideration in this section, the same fermionic torsion will contribute to terms that are bilinear in the fermion terms. We shall show this explicitly below. In the following, we shall need the (full) supersymmetry transformation rules only for the fields (e a µ , ψ µ , V ij µ , B µν ), and the Yang-Mills multiplet fields (W I µ , Ω I , Y ijI ), where I labels the adjoint representation of the Yang-Mills gauge group. We begin with the supersymmetry transformation rules of (e a µ , ψ µ , V ij µ , B µν ) in the gauge (3.1). Up to cubic fermions the transformation rules are already given in (3.8). In this section we will, however, keep the complete SU(2) symmetry, i.e. we do not impose L ij = 1 √ 2 Lδ ij . In this way we do not need to accommodate for the compensating SU(2) transformations proportional to λ in (3.8). 8 The full version of the supersymmetry transformations is given by [2] δe µ a = 1 2¯ γ a ψ µ , where the fermionic torsion and the different supercovariant objects are defined as In this section we only want to establish a map between the Poincaré multiplet and the Yang-Mills multiplet and propose an R 2 -invariant based on the action for the Yang-Mills multiplet. Both actions are invariant under the SU(2) R-symmetry. To prove the validity of this map, we need the full nonlinear SUSY transformation rules. After we construct the action we can still impose the gauge L ij = 1 √ 2 Lδ ij . This will not affect the R 2 -invariant. It modifies the supersymmetry transformation rules with SU(2) compensating transformations, which leave the action separately invariant. The resulting transformations are those given already in (3.8).
Next, we consider the following transformations [3] δ where F µν ij (V) and R µν ab ( ω − ) are the supercovariant curvatures of V µ ij and ω − µ ab , respectively: We now compare the above transformation rules with those of the N = (1, 0), D = 6 vector multiplet [19] δW µ I = −¯ γ µ Ω I , where We observe that the transformation rules (4.3) and (4.5) become identical by making the following identifications: Using this observation we can now easily write down a supersymmetric R 2 -action using the superconformal invariant exact action formula for the Yang-Mills multiplet constructed in [19]. In the gauge (3.1) and up to quartic fermions, the Lagrangian becomes Using the map (4.7) in this formula produces the result for the supersymmetrized Riemann tensor squared action. In presenting the results up to quartic fermion terms, it is useful to note the following simplification in the torsionful spin connection where ω µ ab is the standard torsion-free connection. The map (4.7) applied to the action formula (4.8) then yields, up to quartic fermion terms, the result 9 where 11) and the torsionful modification of the Christoffel symbol Γ ρ µν± is defined as This completes the construction of the supersymmetric R 2 -invariant.

The Total Gauged R + R 2 Supergravity Lagrangian
We now want to discuss what the influence is of these R 2 -terms on the gauging procedure described in section 2.2. The Lagrangian we consider is the following (4.13) 9 To obtain (4.10) we used −L V . Note also that F µν with L 2 given in (3.5) and L R 2 given in (4.10) and with M an arbitrary mass parameter. Recall that L 2 has been obtained as a sum of off-shell supersymmetric Lagrangians L R , L V and L V L and that L R 2 is off-shell supersymmetric as well. Thus all four parts of the total Lagrangian we consider are completely off-shell supersymmetric. So their sum, the total Lagrangian, is still off-shell supersymmetric. In particular, the bosonic part of this total Lagrangian, which will be the starting point of the next section, takes the form where we have defined the complex vector fields and field strengths The part of the total Lagrangian containing the fermions is given in (3.5) and (4.10). None of the auxiliary fields have been eliminated so far, and the Lagrangian still possesses the U (1) R × U (1) symmetry. The field equations for the auxiliary fields Z µ and V µ are not algebraic anymore and therefore they become propagating. The auxiliary fields (Y ij , E µνρσ ), on the other hand, still have algebraic field equations. Their elimination, as well as the breaking of U (1) R × U (1) down to a single U (1) will be discussed in the next section. At this point one may pursue two different lines of thought. The first is to consider the theory as a toy model in its own right and consider M 2 as an arbitrary (not necessarily large) parameter of the theory. The other is to think of |M 2 | as being large compared to a cut-off Λ in the momentum squared. In that case the theory is to be treated as an effective field theory that describes phenomena with external momenta not exceeding √ Λ. Furthermore, the curvature-squared term is a correction term of order Λ/|M 2 |. 10 In this case we can compare the theory with an effective (up to curvature squared terms) string theory Lagrangian compactified to 6 dimensions. In the next section we will only focus on the first line of thought. Let us however briefly comment on the elimination of the Z µ and V µ . For Λ/|M 2 | 1, one particular consequence of eliminating the auxiliary fields up to order Λ/|M 2 | is that which, upon substitution back into the Lagrangian (4.14), and trivial elimination of the other auxiliary fields, gives We observe that g 2 = 2M 2 is a critical coupling at which the Maxwell kinetic term drops out. However, this is a regime for large coupling constant, and as such it falls outside the regime of perturbative validity. We shall nonetheless examine further what happens for this coupling in the next section where we study the field equations in more detail. Another property of this Lagrangian is that the dualization of the 2-form potential by adding the Lagrange multiplier term (2.33) and integrating over F (B), gives a dualized field strength of the form (2.35) which now contains also a Lorentz Chern-Simons term.
In the Lagrangian (4.14) presented above, the Einstein-Hilbert term is not in a canonical frame. The metric can be rescaled appropriately to obtain the canonical Einstein-Hilbert action, still remaining in the formulation in terms of the off-shell Poincaré supermultiplet displayed in (3.2). Alternatively, we can employ the off-shell Poincaré multiplet that results from the gauge choices (2.3) by following the following procedure. Since the Lagrangian L 1 given in (2.18) is already formulated in the desired supermultiplet formulation, we need to only construct L R 2 in the same gauge. This can be done as follows. Firstly, we restore the superconformal invariance (again modulo the conformal boosts which do not affect the final result) in (4.10) by going over to hatted fields defined by e µ a = σ 1/2 e µ a , , Next, we impose the gauge conditions listed in (2.3) and add the result to (2.18) to obtain the full R + R 2 theory in this gauge. This straightforward computation will not be carried out here since we shall be working in the gauge (3.1) which leads to the result (4.13) for the total Lagrangian.

Vacuum Solutions
The purpose of this section is to investigate the different supersymmetric and non-supersymmetric vacuum solutions of the R 2 -extended Salam-Sezgin model discussed in the previous section. In the first subsection we present the bosonic field equations of this model. In the following three subsections we investigate vacuum solutions with no fluxes, 2-form fluxes and 3-form fluxes, respectively. In the last subsection we compute the spectrum of the theory around six dimensional Minkowski spacetime.

Bosonic Field Equations
For the purpose of finding the vacuum solutions, it is convenient to eliminate the auxiliary fields as much as possible. Prior to adding the Riemann tensor squared invariant, we saw that the auxiliary fields (E µνρσ , V ij µ , V µ , Y ij ) can all be eliminated by using their field equations. However, upon the addition of the Riemann tensor squared invariant, while we can still eliminate (Y ij , E µνρσ ), we can no longer eliminate (V ij µ , V µ ) since they acquire kinetic terms. Thus, we shall proceed with the elimination of (Y ij , E µνρσ ) only. The relation readily follows from (3.5), while the E µνρσ field equation gives This implies that we can locally write for some scalar φ, which inherits the shift gauge symmetry transformations (2.26). This symmetry is readily fixed by setting φ equal to a constant, thereby arriving at the field equation Taking into account (5.1) and (5.4), we find the following bosonic field equations for the propagating fields in the theory (4.14): where E µ is the U (1) invariant vector field determined in terms of the vector fields W µ and V µ as in (5.4). The fact that E µ is divergence free follows from (5.8), and separately from (5.9). We have also defined where F ± (B) = (F (B) ±F (B))/2 withF µνρ = − 1 6 e −1 ε µνρσλτ F σλτ . We have simplified the Einstein equation by using (5.4) and the L field equation (5.6). We have also used the definitions Thus, we have Given the vielbein postulate ∂ µ e a ν + ω µ± ab e νb − Γ ρ ∓µν e a ρ = 0 (5.14) with ω µ± ab and Γ ρ ±µν defined in (4.9) and (4.12), respectively, it follows that The occurrence of covariant derivatives with and without bosonic torsion in the quantity S µν is due to the following manipulation: (

5.16)
A partial integration in the first term is then responsible for the occurrence of ∇ in the expression for S µν . Another useful variational formula takes the form The field equations for the abelian vector fields W µ and V µ have an intricate structure. Suitable combinations of these fields describe a gauge field X µ and a gauge invariant Proca field Y µ given by The field equations (5.8) and (5.9) can then be written as for 2M 2 − g 2 = 0, and X µν , Y µν given by In the special case that M 2 = g 2 /2, the left hand side of the field equations (5.19) and (5.20) can no longer be diagonalized. As we saw earlier, this is a critical point at which the coefficient of the kinetic term for the Maxwell vector field vanishes to lowest order in 1/M 2 when the auxiliary vector field V µ is eliminated to the same order.

Vacuum Solutions Without Fluxes
If g = 0, the field equations do not admit a single constant curvature 6D spacetime solution for any value of the constant curvature, with or without supersymmetry. In particular, Minkowski spacetime is not a solution as can be readily seen from the equation R = g 2 L 0 , where L = L 0 is a non-vanishing constant and all other fields are set equal to zero. If g 2 = 0, on the other hand, setting L equal to a constant and all the other fields equal to zero yields Minkowski 6 as a supersymmetric solution.
Next, let the six dimensional spacetime be a direct product of constant curvature spaces M 1 × M 2 , with dimensions d 1 and d 2 . We find that solutions exist with with all the other fields vanishing. Here L 0 is an arbitrary non-vanishing positive constant, and the numbers (n 1 , n 2 , n 3 ) are given in Table 2. Note that here we are using the coordinates (x µ , y r ).  There are also solutions involving a product of three 2-dimensional constant curvature spaces, whose curvature constants, allowed to vanish as well, are chosen properly. In all these solutions, and those tabulated above, M 2 is fixed in terms of g 2 , and all solutions are non-supersymmetric.

Vacuum Solutions With 2-Form Flux
Next, let us consider a spacetime M 4 ×M 2 , which is a direct product of two constant curvature spaces and turn on the fluxes produced by F (W ) and F (V) on M 2 . We set L equal to a positive non-vanishing constant and the remaining fields equal to zero. In particular, from (5.4) it follows that V µ = −gW µ / √ 2. Using this information, we can make the following Ansatz for the non-vanishing fields: where a, b, c, L 0 are constants, g 2 = det g rs , we have used the coordinates (x µ , y r ) and ε 12 = ε 12 = 1. Using this ansatz we find the following solutions. One of them is a direct product of 4D Minkowski spacetime with a 2-sphere, given by Remarkably, this is precisely the supersymmetric Salam-Sezgin solution for any value of M 2 ! For this solution, the integrability condition for the Killing spinor equation δ ψμ = 0 is The vanishing of δ ϕ i follows trivially, and, using (5.24) and (5.26), it follows that δ Ω i = 0 as well. So the only independent condition on the Killing spinor is given by (5.26). It implies N = 1 supersymmetry in Minkowski 4 . Indeed, using the Majorana spinors η 1 and η 2 defined in footnote 11, the condition (5.26) turns into iγ * η 1 = ±η 2 . The other solutions are given by 27) 11 We decompose the 6D Dirac matrices as Γ µ = γ µ ⊗ 1, Γ 4 = γ * ⊗ σ 1 and Γ 5 = γ * ⊗ σ 2 . Then Γ * = γ * ⊗ σ 3 . This defines 4-dimensional spinors Ai = γ * (σ 3  and therefore they describe the following spaces: where H 2 is a 2-hyperboloid. For the special case of M 2 = g 2 /2, there exists the following solution Mink 4 × S 2 : for any value of c, which contains as a special case the solution (5.24) for c = ±gL 0 / √ 2, and the first entry in Table 2 for c = 0. Of all the solutions with the 2-flux turned on, the only supersymmetric one is the one given in (5.24).

Vacuum Solutions With 3-Form Flux
We shall take the 6D spacetime to be a direct product of two three-dimensional constant curvature spaces M 1 × M 2 with coordinates (x µ , y r ), set L = L 0 , turn on the 3-form flux and set the remaining fields equal to zero. Thus we have where g 1 = det g µν and g 2 = det g rs . From (5.13) we get If we set g 2 = 0, then all the terms that depend on M 2 vanish since the curvatures defined above vanish due to the non-vanishing (parallelizing) torsion. This gives the known AdS 3 ×S 3 solution AdS 3 × S 3 : This solution preserves full supersymmetry. Indeed the integrability condition for the existence of Killing spinors requires that the torsionful curvatures vanish, and this is the case with the 3-form fluxes as given in (5.34). As a consequence, all the contributions of the Riemann tensor squared invariant to the field equations vanish in this case. Next, we seek solutions with g 2 = 0 and nonvanishing 3-form flux. To bring the field equations to a manageable form, we shall supplement the Ansatz (5.32) with a further condition and introduce some notation Finding the most general such solution yields rather complicated relations among the parameters. However, we have managed to find the following relatively simple and intriguing solutions: for arbitrary c 2 > 0. This solution corresponds to dS 3 ×S 3 for 0 < c 2 < g 2 L 0 6 and to AdS 3 ×S 3 for c 2 > g 2 L 0 6 . Another solution is given by , where the + solution corresponds to dS 3 × S 3 for g 2 12 < M 2 < g 2 6 and the − solution corresponds to AdS 3 × S 3 for M 2 > 11g 2 36 and to dS 3 × H 3 for M 2 < g 2 12 . These solutions are non-supersymmetric.

Spectrum in Minkowski Spacetime
Setting g 2 = 0, and expanding around 6D Minkowski spacetime, we define the linearized fluctuations g µν = η µν + h µν , L = L 0 + φ . Since all the other background fields are vanishing, we find that the linearized Einstein and L field equations take the form Note that we have not imposed any gauge conditions yet. Using (5.40) in the trace of (5.39), we find To simplify Einstein's equation, however, it is convenient to impose the gauge condition We shall assume that M 2 = 0. Then it follows from (5.41) that either (2 − M 2 L 0 )φ = 0 or 2φ = 0. In the first case, defining χ ≡ 2φ, it follows from (5.41), (5.43) and (5.44) that there is one massive scalar obeying (2 − M 2 L 0 )χ = 0. In the latter case, 2φ = 0 and it follows from (5.44) that 2h = 0 as well. However, the solution of 2h = 0 can be gauged away by the residual general coordinate transformations that preserve the gauge condition (5.42). Thus, we are left with a massless scalar described by 2φ = 0. Turning to Einstein's equation, using the gauge condition (5.42), and the field equations obeyed by h and φ, it becomes This equation, when (2 − M 2 L 0 )φ = 0, reduces to (2 − M 2 L 0 ) 2h µν = 0, describing a massless graviton and a massive graviton with mass M √ L 0 , one of which, depending on the overall sign in the action, has the wrong sign kinetic term. If 2φ = 0, then we have In this case, the solution of 2φ = 0 is to be substituted to the right hand side of this equation and treated as a given external source. Note that the gravitational field does not appear as a source in the field equation for the scalar φ, and there is no diagonalization problem here. Thus, the equation (5.45) again describes a massless and ghost massive graviton of mass M √ L 0 . The remaining field equations in the usual Lorentz gauges take the form where the notation for the fluctuations is self explanatory. These equations describe a massless vector and 2-form potential, a massive ghostly 2-form potential and three massive ghostly vectors. Next, we examine the linearized fermion field equations. Imposing the gauge condition γ µ ψ µ = 0, and defining 12 ψ i ≡ ∂ µ ψ i µ , a straightforward manipulation of the fermion field equations gives where ψ µ ≡ ψ µ − 2 −1 ∂ µ ∂ ν ψ ν , i.e. ∂ µ ψ µ = 0. From (5.48), it follows that / ∂ (2 − M 2 L 0 ) ϕ = 0. Therefore, altogether we have a massless gravitino, tensorino ϕ and gaugino together with a massive gravitino and tensorino, both with mass M √ L 0 . In summary, the full spectrum consists of the massless Maxwell multiplet with fields (a µ , Ω), the (reducible) massless 16+16 supergravity multiplet with fields (h µν , b µν , φ, ψ µ , ϕ) and a massive 40+40 supergravity multiplet of ghosts with fields (h µν , b µν , z µ , v µ , φ, ψ µ , ϕ), all with the same mass, M √ L 0 , as expected.

Conclusions
Our main goal in this paper has been the study of the R-symmetry gauging in the presence of higher derivative corrections to Poincaré supergravity and its consequences for the vacuum solutions. To this end, we first studied the gauging of the U(1) R-symmetry of N = (1, 0), D = 6 supergravity in the off-shell formulation. The off-shell Poincaré supergravity theory already has a local U (1) R symmetry but it is gauged by an auxiliary vector field which is not dynamical. We performed the gauging that employs a dynamical gauge field by coupling the model to an off-shell vector multiplet equipped with its own U (1) symmetry.
Then, we showed that this model has a shift symmetry which can be fixed, thereby breaking U (1) R × U (1) down to a diagonal U (1) diag R . As a result the auxiliary vector gets related to the vector coming from the Maxwell multiplet, and the on-shell model obtained in this manner agrees with the dual formulation [16] of the gauged Einstein-Maxwell supergravity constructed long ago [15,16].
Next, we added a curvature squared supersymmetric invariant, with the Riemann tensor squared as its leading term, to the off-shell model and studied its influence on the gauging procedure. This invariant causes the auxiliary fields to become 'propagating' and to mix with the physical fields. A particular combination of the physical vector and the auxiliary vector gauges the symmetry and another combination describes a massive vector field inert under U (1). We can, however, put a small parameter in front of the curvature squared part of the Lagrangian and consider it as a higher-order correction term. Then the auxiliary fields can be eliminated order by order and the gauging proceeds again via the vector field residing in the Maxwell multiplet. Treating the higher derivative extension either way, we have seen that the positive definite potential that arises in the minimal model does not get modified.
Chiral gauged supergravity in six dimensions is known to admit a (supersymmetric) chiral Minkowski 4 × S 2 compactification, while it does not admit a six-dimensional Minkowski or (anti) de Sitter spacetime as a solution, regardless of supersymmetry [15]. We have shown that the inclusion of the Riemann tensor squared invariant remarkably leaves the supersymmetric Minkowski 4 × S 2 solution intact. We have also found new solutions in which the spacetime and the internal spaces may have positive or negative curvature constants. It is noteworthy that de Sitter spacetime solutions exist, avoiding a no go theorem that exists for ten dimensional supergravities 13 [21,22]. While the spectrum in the 2-sphere compactification remains to be determined, we have found that the spectrum of the ungauged theory in six dimensional Minkowski spacetime, not surprisingly, has a ghostly massive spin two multiplet in addition to a massless supergravity and a Maxwell vector multiplet.
Given that the (1, 0) supergravity theory in six dimensions is the most supersymmetric and highest-dimensional supergravity model that admits an off-shell formulation, and that it admits an exactly supersymmetric higher derivative extension, it is worthwhile to study this model further. The coupling of Yang-Mills and hypermultiplets would be useful. In particular, a possible modification of the quaternionic Kähler geometry, and consequences for the compactification would be interesting to determine. The model without such couplings har-bors many anomalies. It is important to study the gravitational, gauge and mixed anomalies in the matter-coupled version of the higher derivative extended theory. The Green-Schwarz anomaly counterterm that involves the gravitational Chern-Simons term arises as part of the Riemann tensor squared invariant. However, the presence of the Riemann tensor squared term raises the question with regard to the presence of ghosts in the spectrum, defined in the presence of a suitable vacuum solution. Indeed, dealing with the ghost problem is of great importance for this model to have applications to model building, and it remains to be investigated. In particular, the consequences of the higher derivative extension for the braneworld scenarios put forward in [23] where 3-branes are inserted at singular points of the 2-dimensional internal space, would be worthwhile to explore.
Various properties of the model we have studied here would naturally be affected by the presence of an additional higher-derivative supersymmetric invariant. In five dimensions, for example, it is known that a Weyl tensor squared invariant exists, in addition to the Riemann tensor squared invariant, which can be obtained from a circle reduction of the one studied here. However, whether the Weyl tensor squared or another combination of the curvature squared terms can be supersymmetrized in six dimensions is an open problem. If such invariants exist, not only would they be useful in avoiding the ghost problem, they would also play a significant role in a possible embedding of these theories, albeit in the ungauged setting, to the string theory low energy effective action. For a preliminary discussion of this problem, in the context of the Riemann tensor squared model we already have, see [8].
The embedding of the higher-derivative extended model to string theory might also provide new grounds for testing the conjectured connection between microscopic and macroscopic black hole entropy. The use of off-shell supersymmetric Riemann tensor squared extended N = 2, D = 4 supergravity in this respect has been illustrated in [24]. The existence of static, rotationally symmetric black hole solutions that are N = 2 supersymmetric and that approach Minkowski spacetime at spatial infinity and Bertotti-Robinson spacetime at the horizon play a significant role in the work of [24]. It is notoriously difficult to find exact black hole solutions of higher-derivative gravities. Black hole solutions of the ungauged (1, 0) 6D supergravity have been found in [25] and there exists an exact string solution of the theory we have studied in this paper [26]. Nevertheless, black hole solutions in the presence of gauging and/or a higher-derivative extension remains an open and challenging problem.