Supersymmetric Backgrounds and Black Holes in ${\cal N}=(1,1)$ Cosmological New Massive Supergravity

Using an off-shell Killing spinor analysis we perform a systematic investigation of the supersymmetric background and black hole solutions of the ${\cal N}=(1,1)$ Cosmological New Massive Gravity model. The solutions with a null Killing vector are the same pp-wave solutions that one finds in the ${\cal N}=1$ model but we find new solutions with a time-like Killing vector that are absent in the ${\cal N}=1$ case. An example of such a solution is a Lifshitz spacetime. We also consider the supersymmetry properties of the so-called rotating hairy BTZ black holes and logarithmic black holes in an $AdS_3$ background. Furthermore, we show that under certain assumptions there is no supersymmetric Lifshitz black hole solution.

Supergravity in three dimensions has a long history. Especially in its configuration with minimal, N = 1, supersymmetry the theory has been established a long time ago both in on-shell and off-shell formulations as well as in the superconformal framework [1][2][3][4][5].
Despite of all the achievements in the supersymmetric constructions of the theory, the three-dimensional Poincaré (super)gravity by itself is of not much physical interest as the field equations of the theory imply that the spacetime curvature is zero, hence no physical degrees of freedom propagate.
The Poincaré theory can be supplemented with a parity-breaking Lorentz-Chern-Simons term. This combination is known as Topologically Massive Gravity (TMG) [6], and leads to a non-trivial dynamics of the gravitational field describing a helicity +2 or −2 state.
The N = 1 supersymmetric completion of TMG was constructed in [7,8], and the supersymmetric background and black hole solutions of this supersymmetric theory were studied in [9]. In a subsequent development, a parity-preserving higher derivative extension of three-dimensional gravity, known as New Massive Gravity (NMG), was constructed [10].
Similar to TMG, the NMG theory also provides dynamics to the three-dimensional gravity theory corresponding in this case to two states of helicity +2 and −2. The supersymmetric background configurations of both N = 1 TMG and N = 1 NMG are severely restricted due to the spinor structure of the N = 1 supersymmetry, which allows only planar-wave type solutions with a null Killing vector as well as maximally supersymmetric AdS 3 and Minkowski backgrounds [11,12].
In a recent paper some of us have formulated all four-derivative extension of the threedimensional N = (1, 1) off-shell cosmological Poincaré supergravity theory [13]. For a discussion of this construction (and more) from a superspace point of view, see [14]. Extending the N = 1 theory with more supersymmetry cannot affect the dynamics of the Poincaré supergravity theory. It merely extends the size of the N = 1 Poincaré multiplet, consisting of a dreibein e µ a , a gravitino ψ µ and a scalar A with an additional gravitino, an auxiliary vector V µ and a pseudo-scalar B. As we will show in this paper, the merit of the N = (1, 1) theory is that the spinors of the theory are Dirac instead of Majorana spinors, which allows a larger variety of supersymmetric background solutions than in the N = 1 case [15,16].
The main aim of this paper is to study the supersymmetric backgrounds as well as black hole solutions of the N = (1, 1) cosmological NMG, or shortly N = 1 CNMG, theory [13] using the off-shell Killing spinor analysis. The power of the off-shell analysis is reflected by the fact that, once the conditions on the possible field configurations are obtained by using the off-shell supersymmetry transformation rules, one can use them to study the solutions of any model which respects the same set of transformation rules. This might include higher derivative corrections and/or matter couplings.
We begin our study in section 2 with a brief review of N = (1, 1) CNMG and its off-shell transformation rules. As a typical property of off-shell supergravity theories, the auxiliary fields of the theory start to propagate 1 when the Poincaré supergravity is extended with higher-order curvature terms. Assuming that the supersymmetric theory admits at least one Killing spinor, we present the Killing spinor equation and its integrability condition.
We then review the implications of the existence of an off-shell Killing spinor as presented in [15]. The existence of such a spinor imposes numerous algebraic as well as differential identities on the metric, the vector V µ and the scalars A and B of the theory. These identities are the backbone of our analysis that we present in the remainder of the paper.
The Killing spinor equation in particular implies that the background solutions can be put into two categories depending on whether the Killing vector that is formed out of the Killing spinors is null or timelike.
In section 3, we investigate the solutions that admit a null Killing vector. In this case, the analysis for finding supersymmetric solutions simply reduces to the one corresponding to the N = 1 theory [11] as the vector V µ and the pseudo-scalar B are set to zero due to the algebraic and differential constraints, which are consequences of the existence of a Killing spinor. The solution are, therefore, of the pp-wave type, like in the N = 1 case.
In the timelike case, which we present in section 4, the supersymmetric solutions are categorized according to the values of the components V a of the vector in a flat basis. We find that the N = (1, 1) CNMG theory allows all solutions of N = (1, 1) TMG [15] with shifted parameters. Furthermore, we find additional AdS 2 × R and Lifshitz solutions. These extra solutions are possible because of the fact that in N = (1, 1) TMG, the vector equation gives rise to a second order algebraic equation for the components V a , whereas in the case of N = (1, 1) CNMG, the resulting equation is cubic, allowing more solutions.
In section 5, we investigate the supersymmetric black holes with AdS 3 and Lifshitz backgrounds. We first show that the rotating hairy BTZ black hole of [17], which is a generalization of the well-known BTZ black hole [18] obtained by introducing a gravitational hair parameter, and the logarithmic black hole of [19] are solutions of the N = (1, 1) CNMG theory for the extremal cases. This is also true for the rotating BTZ black hole which can 1 There are exceptional cases where the auxiliary fields do not propagate such as in N = 1 CNMG [11].
be obtained by setting the hair parameter b to zero. Given the Lifshitz solution, we then analyze whether we can find an extremal Lifshitz black hole. As the theory is ungauged, one can hope to saturate the BPS bound with the massive vector hair V µ . As opposed to the pseudo-supersymmetry analysis of the Einstein-Weyl theory in four dimensional N = 1 supergravity [20], this is not the case in N = (1, 1) CNMG. Furthermore, we will show that a simple rotating black hole ansatz fails to satisfy the Killing spinor equation and the field equations of the theory simultaneously.
Finally, in section 6, we present our conclusions and discuss further directions.
The field content of the N = (1, 1) supergravity theory consists of the dreibein e µ a , the gravitino ψ µ , a complex scalar S, and a vector V µ . The model we shall study is a particular combination of supersymmetric invariants up to dimension four [13] that leads to a model that, when expanded around a supersymmetric AdS 3 vacuum, propagates only helicity ± 2 and ± 3/2 states with AdS energies that respect perturbative unitarity. This model is called cosmological New Massive Gravity (CNMG). Here we focus on the bosonic part of the supersymmetric CNMG Lagrangian which is given by where (σ, M, m 2 ) are arbitrary real constants and we have defined S = A + iB. The action corresponding to this Lagrangian is invariant under the following off-shell supersymmetry transformation rules 2 δe µ a = 1 2ǭ γ a ψ µ + h.c. , whereǫ = ǫ ⋆ , ω is the super-covariant spin-connection and 3) 2 In this paper, we follow the conventions of [15], with the only difference being that the S we are using here is replaced by S → −Z.
The transformation rules (2.2) are off-shell as the algebra closes on these fields without imposing the field equations corresponding to the Lagrangian (2.1).
In order to determine the supersymmetric backgrounds allowed by a model with the transformation rules (2.2), one considers the Killing spinor equation Any Killing spinor ǫ satisfying this equation must also satisfy the integrability condition Considering the field equations for A, B, V µ and g µν , it can be seen that for cosmological Poincaré supergravity, i.e. m → ∞, A, B and V µ can be eliminated algebraically. In this case, the integrability condition (2.5) reduces to which implies that the maximally supersymmetric background is either Minkowski with M = 0, or AdS 3 with radius 1/M 2 . More solutions, with less supersymmetry, can be obtained by imposing projection conditions on ǫ. Note that even with the higher derivative contributions, the maximally supersymmetric solution is still given by the same background solution with a shifted value of the cosmological constant. The reason for this is that the expectation value of A receives a contribution from the higher derivative corrections whereas B and V µ do not and, therefore, can still be set to zero.
In the case of cosmological Poincaré supergravity, the auxiliary fields can be eliminated from the theory, resulting in an on-shell supergravity theory with the field content (e µ a , ψ µ ).
However, with the higher derivative contributions added, the massive vector and the real scalars become dynamical and hence cannot be solved algebraically. These 'auxiliary' fields play a crucial role in determining the supersymmetric backgrounds allowed by the CNMG Now that we have clarified the maximally supersymmetric backgrounds, let us proceed to the case where we have at least one unbroken supersymmetry. In order to do so, we will briefly review the implications of an off-shell Killing spinor following the discussion of [15]. From the symmetries of the gamma matrices, one finds the following identities for a commuting Killing spinor ǫǭ Thus, non-vanishing spinor bilinears can be defined as follows where f is a real function and K µ (L µ ) is a real (complex) vector. Using the Fierz identities for commuting spinors, one can show that The first equation implies that the vector is either null or timelike. Using the Killing spinor equation (2.4) one finds that showing that K µ is a Killing vector. Finally, we may derive the following differential identities following from the Killing spinor equation (2.4) We refer to [15] for readers interested in the derivation of these Killing spinor identities and of other implications of the existence of a Killing spinor. 6 We first consider the case that the function f introduced in eq. (2.9) is zero everywhere, i.e. f = 0. This implies that K µ is a null Killing vector. The case that f = 0 will be discussed in the next section. In our conventions, a Majorana spinor field has all real components.
This being said, the first spinor bilinear equation in (2.9) leads to a Dirac spinor ǫ that is proportional to a real spinor ǫ 0 up to a phase factor characterized by an angle θ [15], The above equation implies that L µ = e iθ K µ . Taking this into account, the differential equation (2.12) reads Contracting this equation with K µ we find that The same equation also implies that K ∧ dK = 0, i.e. K is hypersurface orthogonal. Thus, there exist functions u and P of the three-dimensional spacetime coordinates such that 3) implies that that K is tangent to affinely parameterized geodesics in the surface of constant P . One can, then, choose coordinates (u, v, x) such that v is an affine parameter along these geodesics, i.e.
By virtue of our choice for K µ the metric components further simplify to where P = P (u, x) since we demand the null direction to be along the v direction. All these choices yield a metric of the following generic form where x i = (x, u). Without loss of generality, this metric can be cast in the following form by a coordinate transformation [9,11] with |g| = P . With these results in hand, the auxiliary fields of the theory should satisfy the following constraints [15] In the next subsection we will investigate the solutions of CNMG under the assumption that f = 0.

The General Solution
To find the general solution with f = 0 we set S to be a constant, to be precise we set The u component of this equation reads where we have used that ε xuv = 1. Provided that the function P (u, x) is nowhere vanishing, it is straightforward to integrate the first (vector) equation in (3.9) and obtain for arbitrary c(u). From the second (scalar) equation in (3.9) we deduce that which, upon using eq. (3.12), yields where P (u) is an arbitrary function of u. We may set P (u) to unity without loss of generality [11]. Using eqs. (3.13) and (3.14) in the vector field equation (3.12), we deduce that c(u) = 0 and θ(u, x) = nπ. In order to fix n we use to the trace of the gravity equation and find that θ(u, x) = π.
We thus find that the metric (3.8) takes the following final form This is the general form of a pp-wave metric. Taking the limit l → ∞ gives rise to the pp-wave in a Minkowski background. Setting l = 1 and substituting A = −1, B = 0, into the metric field equation, we find that Q(x, u) satisfies the following differential equation where the prime denotes a derivative with respect to x. The most general solution of this differential equation is given by where the functions C i (u) , i = 1, · · · , 4, are arbitrary functions of u. We note that this expression for Q(x, u) matches with that of [12,22]. It differs, however, with the expression given in [11]. This is due to the fact that the off-diagonal coupling of gravity to the scalar A was included in the supersymmetric New Massive Gravity model studied in [11], whereas such a term is absent in our case, see eq. (2.1).
The solution for Q(x, u) given in (3.17) has a redundancy [9]. To make this redundancy manifest we consider the following coordinate transformation where a( u) and b( u) are arbitrary functions of u and the prime denotes a derivative with respect to u. By choosing the function a( u) and b( u) such that the differential equations are satisfied the functions C 3 and C 4 can be set to zero. This implies that, without loss of generality, we may set C 3 = C 4 = 0. In addition to this, we get There are two special values of parameters which must be handled separately. These are the cases σm 2 = ± 1 2 . The reason for this is that for the σm 2 = 1 2 case the function C 1 degenerates with C 2 whereas for the σm 2 = − 1 2 case the function C 1 degenerates with C 4 while the function C 2 degenerates with C 3 . Therefore, we solve the field equation (3.16) for these special cases, and display the solutions Q(x, u) for these special values of the parameters explicitly: Here D i (u) , i = 1, . . . , 4, are arbitrary functions of u. Setting D 3 = D 4 = 0, we are led to the following cases: These pp-wave solutions coincide with the solutions of N = 1 CNMG [12]. Having found the most general solutions for the null case, we will continue in the next subsection with determining the amount of supersymmetry that these solutions preserve by working out the Killing spinor equation (2.4).

Killing Spinor Analysis
In order to construct the Killing spinors for the pp-wave metric (3.15) we introduce the following orthonormal frame [9] e 0 = e 2x l −β dv, where Q(u, x) = e 2β(u,x) . It follows that the components of the spin-connection are given by We make the following choice of the γ matrices where σ i 's are the standard Pauli matrices. With this choice the Killing spinor equation Decomposing a Dirac spinor into two Majorana spinors as ǫ = ξ + iζ, i.e.
we find the following equations for the components The first two equations are uniquely solved by ξ 1 = ξ 2 = 0. For the last two equations, the solution for a generic function β(u, x) is given by There is an additional solution for the special case that β = x. It is given by This solution corresponds to the first case given in eq. (3.22) with D 1 (u) = 0 and D 2 (u) = 1.
There is, however, a problem with this solution. One must choose σm 2 = − 1 2 for this solution and this conflicts with the condition imposed on this pp-wave solution when we classified the different solutions in the previous subsection. Therefore, we conclude that the pp-wave Killing spinor equation is uniquely solved by We conclude this section by noting that when D 1 = D 2 = 0, the metric reduces to The Killing spinor equation then turns into Making use of the fact that that the ξ and ζ equations are decoupled from each other, we find the following four independent solutions: Therefore, the AdS 3 solution has a supersymmetry enhancement with four Killing spinors.

The Timelike Killing Vector
In this section, we shall consider the case that f = 0 and hence that K is a timelike Killing vector field. Introducing a coordinate t such that K µ ∂ µ = ∂ t , the metric can be written as [15] ds 2 = −e 2ϕ(x,y) (dt + B α (x, y) dx α ) 2 + e 2λ(x,y) (dx 2 + dy 2 ) , where λ(x, y) and ϕ(x, y) are arbitrary functions and B α (α = x, y) is a vector with two components. The Dreibein corresponding to this metric is naturally written as where we have defined f ≡ e ϕ and W α = e 2ϕ−λ B α . We write µ = (t, α) for the curved indices and a = (0, i) for the flat ones, respectively. We also require that all functions occurring in the metric (4.2) are independent of the coordinate t. Taking all these things into account, the components of the spin connection ω abc in the flat basis read as follows, Following [15], it can be shown that the existence of a timelike Killing spinor leads to the following relations between the auxiliary fields V µ , S and the metric functions where c(x, y) is an arbitrary time-independent real function and z = x + iy denotes the complex coordinates.
At this stage we have paved the way for constructing supersymmetric background solutions by exploiting the Killing spinor identities. Making an ansatz for the vector field V µ we can now solve eqs. (4.4)-(4.7) and determine the metric functions λ and ϕ. Following the same logic in [15], we now look for solutions for with the following field configuration Note that, by setting V 2 = c = 0, we can solve for λ and ϕ using eqs. (4.5) and (4.6) and their integrability conditions. Furthermore, B y can be set to zero by a gauge choice. As a result, we obtain the following differential equations for the functions ϕ, λ and B x e −λ ∂ y ϕ = V 1 + Λ, (4.10) with ∂ x ϕ = ∂ x λ = 0.

Classification of Supersymmetric Background Solutions
In this subsection, we first integrate the differential equations (4.10)-(4.12) depending on the different values of the vector field components V a , which yields the metric functions λ and ϕ. Next, we impose the field equations and determine the couplings. The results for the different cases are given in three subsubsections. For the convenience of the reader, we have summarized all supersymmetric background solutions allowed by the theory (2.1) in Table 1. Thus, the metric becomes which describes the round AdS 3 spacetime with l = − 1 Λ , see Table 1.

The case
(4.16) The vector and the scalar field equation lead to the following subclasses A, B and C which we describe below.
With this choice of parameters the metric reads ds 2 = −e 4Λy dt 2 + dx 2 + dy 2 . (4.17) After a simple coordinate transformation y = log r 2Λ , x = x ′ 2Λ the metric is brought into the following form which is AdS 2 × R. This background also appeared in the bosonic version of NMG, although given in different coordinates [23,24].
the metric (4.19) can be put into the more familiar form [21] which is null warped AdS 3 .
Using these values for the parameters and fixing the value of V 0 we deduce from the vector equation that Using the components of the vector equation, we find From eq. (4.30) it is straightforward to see that this subclass has two different branches, i.e. V 0 = 0 and V 1 = εV 0 with ε 2 = 1. We will discuss these two branches as separate cases A and B below, With this choice of parameters the vector equation gives rise to The parameter M can be solved by using the field equation for A as follows, Plugging in the metric functions, we obtain the following expression for the metric Performing the coordinate transformation [15] this metric can be written as follows This is the metric of a AdS 3 pp-wave. Note that the limit V 1 → Λ is well defined and gives rise to the minus null warped AdS 3 metric of eq. (4.21), as expected.
The final spacetime we consider appears for V 0 = 0. Rather than solving the vector equation for V 1 as we did in the previous cases, we set V 1 = α + 1 α − 1 using eq. (4.29). The field equations further imply that whose solution is given by .
Here, we would like to restrict our attention to α < 0, as α will be the minus of the Lifshitz exponent, thus giving rise to spacetimes with positive Lifshitz exponent Provided that the vector field components are chosen as discussed, we obtain the Lifshitz where l L is the Lifshitz radius which is defined as (4.38) We have redefined t as t → (V 1 − Λ) 2α+2 t. Note that in the limit V 1 → 0 one obtains the round AdS 3 metric given in eq. (4.15). Taking y = 1 r gives the metric in the standard form where l 2 L and V 1 are given in terms of α and Λ as 3 As shown in [15], all the supersymmetric backgrounds that we have found in this section except the AdS 3 metric preserve 1/4 of the supersymmetries.
In this section, we discuss the supersymmetry aspects of black hole solutions of CNMG in a AdS 3 or Lifshitz background. The existence of a Killing spinor is highly restricted due to the global requirement that the angular coordinate φ should be periodic. As shown in [15], the spacelike squashed AdS 3 solution can be interpreted as an extremal black hole upon making a coordinate transformation. Therefore, in this section we will discuss three specific cases of black hole solutions. We start our discussion in subsection 5.1 with a generalization of the BTZ black hole, and show that the periodicity condition implies the extremality of the black hole. In the next subsection we investigate the 'logarithmic' black hole given in [19], and show that, the logarithmic black hole is also supersymmetric. Finally, in a third subsection we investigate the possible black holes in a Lifshitz background.

The Rotating Hairy BTZ Black Hole and its Killing Spinors
The CNMG Lagrangian (2.1) admits the following rotating black hole solution [17] where N , N φ and F are functions of the radial coordinate r, given by where we have set the AdS 3 radius l = 1. Here Ξ := 1 − J 2 /M 2 , and the rotation parameter J /M is bounded in terms of the AdS radius according to The parameter b is the gravitational hair, and for b = 0 one recovers the BTZ black hole [18].
Since we impose the global requirement that φ should be periodic, i.e. 0 ≤ φ ≤ 2π, the vacuum of the BTZ black hole with gravitational hair, defined by M = J = b = 0, admits only two Killing spinors. In order to see that, we consider the Killing spinor equations (3.38).
Since the equations for ξ 1 and ζ 2 enforce exponential solutions for ξ 1 and ζ 2 , we cannot find a solution for ξ 2 and ζ 1 that is periodic in φ. Therefore, finding a periodic solution requires setting ξ 1 = ζ 2 = 0. This implies that only two of the solutions of equations (3.38) are valid.
Introducing the following orthonormal frame for the metric e 0 = N F dt , e 1 = rdφ + rN φ dt , e 2 = F −1 dr , (5.5) the spin-connection components are given by and hence the Killing spinor equation reads Decomposing the Dirac spinor into two Majorana spinors like in eq. (3.29), we obtain the following equations From these equations it follows that for the generic case not all the dφ components can be set to zero, which is the requirement for finding a periodic Killing spinor. Therefore, we turn our attention to the extremal solutions with M = |J |. For this case we find the following Killing spinors that are periodic in φ Using these expressions in the Killing spinor equation (2.4), we find that the Killing spinors of the logarithmic black hole are given by ii. ε = −1 This result may be somewhat surprising considering the expectation that the only existing supersymmetric black hole in an AdS 3 background is an extremal BTZ black hole [12].
However, unlinke the rotating BTZ black hole, the "logarithmic" black hole does not have a non-extremal limit J = M . Thus, one cannot recover a static, non-supersymmetric black hole from the J → 0 limit of the "logarithmic" black hole. Therefore, this particular case evades the argument presented in [12]. 4

Searching For a Supersymmetric Lifshitz Black Hole
In this section, we briefly present our attempts to find a supersymmetric Lifshitz black hole.
Following [20], we first try to saturate the BPS bound using the vector field V µ , since it can, in principle, contribute as a massive vector hair. In order to do so, we consider the following metric ansatz where the functions a and f depend on the coordinate r only. With this ansatz for the metric, one can show that the Killing spinor equation imposes the following constraint of these functions As we wish to find a solution for f which has a double root at r = r 0 , which is a necessary condition for an extremal black hole, we need to be able to eliminate the f terms in the vector equation. Using the fact that the Killing spinor constraint (5.19) can be cast into the following form the vector equation can be written as Alternatively, one may try to start with a rotating Lifshitz black hole using the following metric ansatz where F (r) and G(r) are arbitrary functions that depend on the coordinate r only. In this case the Killing spinor equation constrains the function F (r) to be of the form where a is a constant. Furthermore, the vector equation constraints the function G(r) via the following differential equation The result of this subsection is somewhat expected, considering the fact that for the only rotating Lifshitz solution known to us [25], the couplings are determined by using a stationary Lifshitz spacetime which has a rotation term. This is not allowed by the given matter configuration of the N = (1, 1) CNMG theory.
Finally, we would like to comment that as our attempts to find a supersymmetric Lifshitz black hole has failed with the parity-even theory under our consideration (2.1), one may consider to modify the CNMG by adding a parity violating Lorentz-Chern-Simons term, which gives rise to the so-called N = (1, 1) Generaized Massive Gravity (SGMG) [13]. In that case, we found that the vector equation is modified in such a way that the Lifshitz background is no longer a solution with the field configuration given in (4.8).

Conclusions
Using the off-shell Killing spinor analysis, we have investigated in this work the supersym- vanish. Therefore, the solution is of the pp-wave type which preserves 1/4 of the supersymmetries. In the AdS 3 limit, there is a supersymmetry enhancement, and the AdS 3 solution is maximally supersymmetric.
As a second case, in section 4 we investigated the case that the Killing vector is taken to be timelike. In particular, we did consider a special class of solutions in which the pseudo-scalar B vanishes. In that case all the supersymmetric solutions can be classified in terms of the components V a of the massive vector in the flat basis. A subclass of these solutions, with different parameters, are also solutions of the supersymmetric TMG model, see Table 1. In addition to these solutions, we found that the N = (1, 1) CNMG model possesses Lifshitz and AdS 2 × R solutions. All these background solutions preserve 1/4 of the superymmetries.
Next, in section 5 we investigated three cases of black hole solutions in a AdS 3 or Lifshitz background. In the case of AdS 3 , we studied the rotating hairy BTZ black hole in subsection 5.1 and the logarithmic black hole in subsection 5.2. We found that in general the rotating hairy BTZ black hole is not supersymmetric due to the fact that the periodicity condition on the φ coordinate and the periodic Killing spinors only arise when the black hole is extremal. In the case of the logarithmic black hole, we found that only the extremal black hole solution exists, which is supersymmetric by its own nature. Finally, we analyzed the conditions for the existence of a supersymmetric Lifshitz black hole, and showed that it does not exist given the field configuration of the N = (1, 1) CNMG model.
There are numerous directions one can consider for future study. An intriguing problem is to find a supersymmetric Lifshitz black hole. Although our trials with the current model has failed, it is natural to consider different approaches. For instance, one could saturate the BPS bound with a U (1) charge. This can be achieved by coupling the N = (1, 1) CNMG model to an off-shell vector multiplet and repeat the analysis presented in this paper.
Finally, we would like to mention that the same procedure that we presented in this paper can be applied to the N = (2, 0) CNMG model. This model has a different field content consisting of two auxiliary vectors and a real scalar as well as the graviton and the gravitino.
Given that the N = (2, 0) theory with matter couplings has new supersymmetric solutions [26], we would expect that the N = (2, 0) CNMG model exhibits different supersymmetric solutions. Therefore, it would be interesting to see what the consequences of the different field content is for the supersymmetric solutions of the model.