Nonlinear realisation approach to topologically massive supergravity

We develop a nonlinear realisation approach to topologically massive supergravity in three dimensions, with and without a cosmological term. It is a natural generalisation of a similar construction for ${\cal N}=1$ supergravity in four dimensions, which was recently proposed by one of us. At the heart of both formulations is the nonlinear realisation approach to gravity which was given by Volkov and Soroka fifty years ago in the context of spontaneously broken local supersymmetry. In our setting, the action for cosmological topologically massive supergravity is invariant under two different local supersymmetries. One of them acts on the Goldstino, while the other supersymmetry leaves the Goldstino invariant. The former can be used to gauge away the Goldstino, and then the resulting action coincides with that given in the literature.


Introduction
The method of nonlinear realisations of groups (also known as the coset construction), which was systematically developed by Coleman, Wess and Zumino [1,2] (see also [3,4]), is the mathematical formalism to construct phenomenological Lagrangians describing the low-energy dynamics of Goldstone fields in theories with spontaneously broken symmetry.This method was extended to spacetime symmetries by Volkov [5] (see also [6]), although the case of spontaneously broken conformal symmetry had been studied earlier [7][8][9][10].In modern applications of the method of nonlinear realisations, an important role is played by the inverse Higgs mechanism discovered by Ivanov and Ogievetsky [11].An interesting interpretation of this mechanism was given in [12].
The formalism of nonlinear realisations can also be used to construct gauge theories, including those describing gravity and its matter couplings.The importance of nonlinear realisations for gravity was realised fifty years ago by Volkov and Soroka [13,14] (for related developments see [15,16]). 1 These authors gauged the N -extended super-Poincaré group in four dimensions (4D) and proposed a super-Higgs mechanism by constructing the N = 1 supergravity action with nonlinearly realised local supersymmetry (see [22] for a review and [23] for a critical analysis of the Volkov-Soroka construction and modern developments).Restricting their analysis to the N = 0 case results in the nonlinear realisation approach to gravity, which corresponds to the coset space ISL(2, C)/SL(2, C) in the formulation of [14]. 2 A review of this construction is given in appendix A. The theory is described by a vielbein, an independent Lorentz connection and a vector Goldstone field V a .There are two types of gauge transformations with vector-like parameters, the general coordinate transformations and the local Poincaré translations.The latter gauge freedom acts on the Goldstone field by the rule V ′a = V a + b a and, therefore, it can be fixed by imposing the condition V a = 0.As a result, one arrives at the first-order formulation for gravity [18].The Goldstone field in this setting is a compensator.In the terminology of [12], V a is an unphysical Goldstone boson describing purely gauge degrees of freedom.
Within the Volkov-Soroka approach to spontaneously broken local supersymmetry [13,14], there are two Goldstone fields, the vector field V a and a spinor field (ψ α , ψ α).The latter is the Goldstone field for supersymmetry transformations 3 [24,25].It is called the Goldstino.While V a is an unphysical Goldstone boson, the Goldstino is in general a genuine Goldstone field for it triggers spontaneous breakdown of the local supersymmetry.In the gauge ψ α = 0, the gravitino becomes massive.A natural question is the following.Is it possible to have a dynamical system such that ψ α turns into an unphysical Goldstone field?The positive answer was given in [26] where it was shown that, for specially chosen parameters of the theory, the Volkov-Soroka action is invariant under two different local supersymmetries.One of them is present for arbitrary values of the parameters and acts on the Goldstino, while the other supersymmetry emerges only in a special case and leaves the Goldstino invariant.The former can be used to gauge away the Goldstino, and then the resulting action coincides with that proposed by Deser and Zumino for consistent supergravity in the first-order formalism [27]. 4n this paper we will extend the construction of [26] to the case of 3D N = 1 supergravity [29,30], with and without a cosmological term, and then the obtained results will be generalised to topologically massive N = 1 supergravity [31] and its cosmological extension [32].
It should be pointed out that the literature on simple supergravity in three dimensions is immense.In particular, superfield approaches to N = 1 supergravity-matter systems were developed, e.g., in [33][34][35][36][37][38].The N = 1 supersymmetric Lorentz Chern-Simons term [31], which is at the heart of (cosmological) topologically massive supergravity [31,32], has been interpreted as the action for 3D N = 1 conformal supergravity [39]. 5uperfield formulations for N = 1 conformal supergravity were derived in [41][42][43] (somewhat incomplete results had appeared earlier in [34][35][36]).The Chern-Simons formulation for N = 1 anti-de Sitter (AdS) supergravity was proposed in [44].The super-Higgs effect for N = 1 supergravity was first described in [45].The Hamiltonian form of (topologically) massive N = 1 supergravity was constructed in [46,47].This paper is organised as follows.In section 2 we present a 3D analogue of the Volkov-Soroka construction.Using this framework, we demonstrate in section 3 that the action for pure N = 1 Poincaré supergravity (3.1) is invariant under two different local supersymmetries.One of them is present for an arbitrary relative coefficient between the two terms in (3.1) and acts on the Goldstino, while the other supersymmetry emerges only in a special case and leaves the Goldstino invariant.The former can be used to gauge away the Goldstino, and then the resulting action coincides with the standard action for Poincaré supergravity in the first-order formalism.In subsection 3.2 we show that the same formalism of nonlinearly realised local supersymmetry can be used to describe AdS supergravity, however the second local supersymmetry has to be deformed.In section 4 we generalise the analysis of section 3 to topologically massive supergravity and its cosmological extension.The main body of the paper is accompanied by three technical appendices.Appendix A reviews the nonlinear realisation approach to 4D gravity.In appendix B we collect the key formulae of the 3D two-component spinor formalism.Finally, appendix C derives the first Bianchi identity.
2 The Volkov-Soroka approach in three dimensions Let P(3|N ) be the three-dimensional N -extended super-Poincaré group.Any element where M ∈ SL(2, R), R ∈ SO(N ), η = (η I β ), η 2 := η α I η αI , and b is defined in (B.3b).The SL(2, R) invariant spinor metric ε = (ε αβ ) = −(ε βα ) and its inverse ε −1 = (ε αβ ) = −(ε βα ) are defined in appendix B. The group element s(b, η) is labelled by three bosonic real parameters b a and 2N fermionic real parameters η I α = η α I ≡ η α I .Let us introduce Goldstone fields Z A (x) = (X a (x), Θ α I (x)) for spacetime translations (X a ) and supersymmetry transformations (Θ α I ).They parametrise the homogeneous space (N -extended Minkowski superspace) according to the rule: A gauge super-Poincaré transformation acts as with g = sh.This is equivalent to the following transformations of the Goldstone fields: and Introduce a connection A = dx m A m taking its values in the super-Poincaré algebra, and possessing the gauge transformation law Here the one-form Ω α β is related to the Lorentz connection Ω ab = dx m Ω m ab = −Ω ba as As in the first-order formalism to gravity, the Lorentz connection is an independent field and may be expressed in terms of the other fields by requiring it to be on-shell.The oneform e αβ is the spinor counterpart of the dreibein e a = dx m e m a .The fermionic one-forms ψ I β describe N gravitini.Finally, the one-form r IJ = −r JI is the SO(N ) gauge field.
It should be pointed out that our parametrisation of the super-Poincaré algebra follows [43] and differs from [48].Under an infinitesimal Lorentz transformation a two-component spinor ψ α transforms as where the Lorentz parameters λ ab , λ a and λ αβ are related to each other according to the rules (B.14), (B.15) and (B.16).
Associated with S and A is the different connection with gauge transformation law for an arbitrary gauge parameter g = sh.This connection is the main object in the Volkov-Soroka construction.Direct calculations give the explicit form of A where we have defined and D denotes the covariant derivative, ) Equation (2.12) is equivalent to the following gauge transformation laws: ) It is worth pointing out that the supersymmetric one-forms E a and Ψ I β transform as tensors with respect to the Lorentz and SO(N ) gauge groups.
Under a supersymmetry transformation, g = s(0, η), one can use the Goldstone field transformations (2.5a) and (2.5b) to deduce the local supersymmetry transformation laws of the gravitini and the dreibein In the infinitesimal case, these supersymmetry transformation laws take the form These should be accompanied by the supersymmetry transformations of the Goldstone fields A local Poincaré translation is given by g = s(b, 0).It acts on the Goldstone vector field X a and the dreibein e a as follows while leaving the Goldstini and gravitini inert.
The curvature tensor is found through Direct calculations give where R = (R α β ) is the Lorentz curvature, F = (F IJ ) is the Yang-Mills field strength, are the gravitino field strengths, and is the supersymmetric torsion tensor.In vector notation, the torsion tensor reads The Lorentz curvature tensor with spinor (R α β ) and vector (R a b ) indices has the form Using the above results, one can construct a locally supersymmetric action.With the notation E = det(E m a ), gauge-invariant functionals include the following: • The Einstein-Hilbert action • The Rarita-Schwinger action (2.28) • The cosmological term • The mass term In contrast to the 4D case, the mass term is invariant under the entire R-symmetry group SO(N ).Making use of the SO(N ) connection r and the corresponding field strength F , we can construct standard Chern-Simons and Yang-Mills actions.We will not use them.
In the N = 1 case, a linear combination of the above functionals gives an action for spontaneously broken supergravity.

Second local supersymmetry
In the remainder of this paper our discussion is restricted to the N = 1 case for simplicity.If N > 1, it is necessary to take into account the SO(N ) connection.An extension of our approach to the N = 2 case will be studied elsewhere.

Poincaré supergravity
Each of the functionals (2.27)-(2.30) is invariant under the local supersymmetry transformation (2.19).We are going to show that a special linear combination of the actions (2.27) and (2.28) possesses a second local supersymmetry described by the parameter ǫ = (ǫ α ).This combination is Then (3.1) turns into the action for pure N = 1 supergravity without a cosmological term [29].
Under the second supersymmetry, the composite fields E a and Ψ α are postulated to transform as The Goldstone fields are required to be inert under this transformation, The elementary fields ψ α and e a transform as follows: The dependence on δ ǫ Ω in (3.3c) and (3.3d) is such that the composite fields Ψ α and E a remain unchanged when the connection gets the displacement Ω → Ω + δ ǫ Ω.As will be shown, the transformation law of Ω will be determined by demanding the action (3.1) to be invariant under this new local supersymmetry (3.3).
We now compute variations of the two terms in the action (3.1).Denote δ (1) ǫ for variations with respect to the transformations (3.3a) and δ (2) ǫ for variations with respect to the Lorentz connection.Computing the δ (1) ǫ variation of the Einstein-Hilbert action (2.27) gives δ (1)  ǫ Computing the δ (1) ǫ variation of the Rarita-Schwinger action (2.28) gives where we have used the relations As a result, computing the δ ǫ variation of the action (3.1), we observe that the curvature contributions (3.4) and (3.5) precisely cancel each other, δ (1)  ǫ S SG = δ (1)  ǫ (S EH − 2S RS ) = δ (1)  ǫ S EH − 2δ (1)  ǫ S RS = 0 . (3.7) Next, we vary the action (3.1) with respect to the Lorentz connection Ω ab .We give the Lorentz connection a small disturbance Ω → Ω + δ ǫ Ω, with δ ǫ Ω to be determined below, and assume that the elementary fields ψ α and e a also acquire δ ǫ Ω-dependent variations given in (3.3c) and (3.3d).For the Einstein-Hilbert action we get the variation δ (2)  ǫ The Rarita-Schwinger action variation is Hence, the variation of the total action (3.1) with respect to the Lorentz connection is Combining the results (3.7) and (3.10), we end up with This variation vanishes if δ ǫ Ω bc = 0, which differs from the case of N = 1 supergravity in four dimensions considered in [26].
Alternatively, we can work with a composite connection obtained by imposing the constraint In the case of vanishing Goldstone fields, X a = 0 and Θ α = 0, one can uniquely solve (3.12) for the connection giving its well-known expression in terms of the dreibein and gravitino, Ω = Ω(e, ψ).
It is a simple observation that (3.12) is the equation of motion for the Lorentz connection Ω.If this equation holds, the explicit form of the variation δ ǫ Ω is irrelevant when computing δ ǫ S SG .Thus the Volkov-Soroka approach allows one to naturally arrive at the 1.5 formalism [49,50].

Anti-de Sitter supergravity
In order to describe a supersymmetric extension of gravity with a cosmological term the second supersymmetry transformation (3.3) has to be deformed.
Let us alter the Ψ α transformation (3.3a) in the following way while keeping the E a and Goldstone field transformations the same, as given by the equations (3.3a) and (3.3b).Here m is a constant real parameter.The elementary fields ψ α and e a pick up an additional term proportional to m: ) Let us now add to the action (3.1) a supersymmetric cosmological term We will show that the resulting additional variation for the action (3.1) due to the term proportional to m in (3.14) combined with the total variation of the action (3.16) does not contribute to the already established variation (3.11) if we require certain conditions.
First we compute the additional variation of the action (3.1), where we have denoted δ for the variation due to the additional term − 1 2 m(ǫγ a ) α E a appearing in (3.14).The total variation of the action (3.16) under the transformations (3.3a) and (3.14) respectively reads 7

.18)
7 There is no connection variation contribution from this action.
Combining all variations (3.11), (3.17) and (3.18), we end up with where we have denoted In the unitary gauge (3.2), the action (3.20) coincides with that proposed by Howe and Tucker to describe AdS supergravity [29].
Alternatively, we can deal with a composite connection obtained by imposing the constraint (3.12), which makes the variation (3.19) vanish.In the reminder of this paper, we will work with the condition (3.12), which will be necessary for our consideration of (cosmological) topologically massive supergravity theories in section 4. Requiring the constraint (3.12) to be invariant under the transformations ) we can determine a non-trivial variation of the connection.In particular, one finds which has the unique solution for the dual connection Ω ma := where is the Hodge dual of the gravitino field strength When m = 0, this transformation law is compatible with δ ǫ Ω ma = 0 since this variation vanishes when Ψ is on-shell, DΨ = 0 [31].

Topologically massive supergravity
A unique feature of three dimensions is the existence of Chern-Simons terms that can be used to define topologically massive couplings [51][52][53][54][55].

Conformal supergravity
Here we study a generalisation of the N = 1 supersymmetric Lorentz Chern-Simons action [31] which involves the Goldstone fields X a and Θ α .We consider the action where is the Lorentz Chern-Simons term, and the fermionic Chern-Simons term.The latter involves the gravitino field strength (3.26) and its Hodge dual (3.25).In the unitary gauge (3.2), the functional (4.1) coincides with the N = 1 supersymmetric Lorentz Chern-Simons action [31] which is also known as the action for N = 1 conformal supergravity [39].
We endeavour to demonstrate that the action (4.1) is invariant under the local supersymmetry transformations (3.22a), (3.22b) and (3.24).The elementary fields ψ α and e a transform according to (3.15a) and (3.15b).As before, it is assumed that the δ ǫ Ωdependence in these transformation laws is such that the composite fields Ψ α and E a remain unchanged when the connection is perturbed Ω → Ω + δ ǫ Ω.
Once this has been achieved, we can couple the action (4.1) to the AdS supergravity action (3.20) giving a generalisation of cosmological topologically massive supergravity proposed in [32].However, we first consider topologically massive supergravity without a cosmological term [31] by restricting to the case m = 0.
Let us compute variations of the action (4.1) in parts, beginning with the variation of the Lorentz Chern-Simons term (4.2), where G ab = R ab − 1 2 η ab R is the Einstein tensor, R ab = R c acb is the Ricci tensor and R = −2η ab G ab = η ab R ab is the Ricci scalar.Varying the fermionic Chern-Simons term (4.3) with respect to Ψ gives where we have used the second relation in (3.6).Combining the variations (4.4) and (4.5) results in the cancellation of the Ricci scalar curvature terms leaving Let us introduce the Hodge dual of the antisymmetric part R [ab] of the Ricci tensor so that the combined variation (4.6) takes the form With some algebraic manipulations, this combination can be brought to the simplified form of a single term involving (4.7) which upon inserting the relation (C.5) and substituting the identity (B.7a) becomes Next we vary the action (4.3) with respect to the composite field E a .This variation reads (4.11) The final variation to be computed is the variation of S FCS (4.3) with respect to the Lorentz connection.Direct calculations give (4.12) In order to show that the total variation of the action (4.1) vanishes, we will need to perform systematic Fierz rearrangements on the individual terms contained within the variations of (4.10) and (4.12) such that all terms have products of the form (FF)(ǫΨ), potentially with gamma matrices wedged between the fields.Note that the variation (4.11) is already in the desired form and so will not require a Fierz rearrangement of its terms.After a series of tedious calculations guided by the use of the Fierz rearrangement rule for two-component spinors (B.11), we achieve the desired forms of the variations (4.10) and (4.12): Summing all variations (4.13a), (4.13b) and (4.11) gives The combination in curly brackets can be rewritten in the equivalent form Now if we consider the first cycled combination in curly brackets we notice that X acd is totally antisymmetric in its indices and therefore Similarly, applying the same trick for the second cycled term in curly brackets leads to the same result.As a consequence of these observations, we cancel the term in (4.15) proportional to (ǫγ d Ψ d ).After additional cancellations within the combination (4.15), it reduces to three remaining terms, This combination may be shown to be identically zero.
As a result, we have demonstrated that the conformal supergravity action (4.1) is invariant under the local supersymmetry transformations (3.22a), (3.22b) and (3.24).This implies that the action is also invariant under the second local supersymmetry, with κ and µ being coupling constants.In the unitary gauge (3.2), the functional (4.19) turns into the action for N = 1 topologically massive supergravity originally constructed in [31].

Cosmological topologically massive supergravity
We now incorporate the supersymmetric cosmological term (3.16) by demonstrating that the additional variations of the action (4.1) arising from the m-dependent transformation terms in (3.22a) and (3.24) keep the action stationary.The first contribution comes from varying the Lorentz Chern-Simons action (4.2), The variation of the fermionic Chern-Simons action (4.3) resulting from the m-dependent transformation term for the field Ψ reads Combining these two variations results in the cancellation of the terms proportional to G ab leaving where the additional terms have arised from a Fierz rearrangement of the second term in (4.20).Finally, the m-dependent transformation term for the Lorentz connection gives us the contribution Following the strategy used in the m = 0 case, we perform systematic Fierz rearrangments on the individual terms contained within the variation (4.23) such that all terms have products of the form (ǫF)(ΨΨ), potentially with gamma matrices wedged between the fields.After applying the Fierz rearrangement rule (B.11) on all terms, we arrive at the desired form of the variation (4.23), Summing the variations (4.22) and (4.24) gives The combination of these three terms may be shown to be identically zero, and therefore Finally we arrive at We have demonstrated that the action is invariant under the second local supersymmetry given by (3.22a), (3.22b) and (3.24), with κ and µ coupling constants.In the unitary gauge (3.2), the functional (4.28) turns into the action for N = 1 cosmological topologically massive supergravity originally constructed in [32].

Conclusion
In this paper we have developed a nonlinear realisation approach to (cosmological) topologically massive N = 1 supergravity in three dimensions.In addition to the supergravity multiplet, the action involves the Goldstone fields X a and Θ α which are purely gauge degrees of freedom with respect to the local super-Poincaré translations generated by the parameters (b a , η α ).The action is invariant under two different local supersymmetries.One of them acts on the Goldstino, while the other supersymmetry leaves the Goldstino invariant.The former can be used to gauge away the Goldstino, and then the resulting action coincides with that given in the literature [32].
There is a remarkable feature of uniqueness in the proposed approach to N = 1 supergravity.The explicit structure of the first local supersymmetry (2.19) is uniquely determined by the coset construction under consideration.In the case of topologically massive supergravity (4.19), the structure of the second local supersymmetry (3.3) is modelled on the first one, eq.(2.19).In the case of cosmological topologically massive supergravity, the second supersymmetry is deformed by m-dependent contributions.
The action for cosmological topologically massive supergravity, eq.(4.28), involves two different functionals, which are separately invariant under the two local supersymmetry transformations.The first functional S AdS = S SG + S super-cosm is a combination of four terms with fixed relative coefficients.The second functional S CSG is a combination of two terms with fixed relative coefficients.Changing at least one of the relative coefficients breaks explicitly the second local supersymmetry, and then the resulting action describes a model for spontaneously broken N = 1 supergravity.
In principle, our construction, which is a natural application of the ideas pioneered by Volkov and Soroka [13,14], may be generalised to include more general models for massive N = 1 supergravity constructed in [58,59] and recast in the superspace setting of [60].That would require a further deformation of the second local supersymmetry transformation.Of course, the massive supergravity theories of [58][59][60] were constructed using off-shell supergravity techniques, and our nonlinear realisation approach to supergravity is not a competitor to the off-shell methods, simply due to the fact that the second local supersymmetry is on-shell.It is still quite remarkable that the structure of N = 1 Poincaré supergravity is uniquely determined by applying the formalism of nonlinear realisations.
The Volkov-Soroka approach was also inspirational for a recent work [61] in which the minimal massive gravity theory of [62] was shown to be a particular case of a more general 'minimal massive gravity' arising upon spontaneous breaking of a local symmetry in a Chern-Simons gravity based on a Hietarinta or Maxwell algebra.It would be interesting to extend the construction of [61] to the supersymmetric case.The (γ a ) αβ and (γ a ) αβ are invariant tensors of the Lorentz group SO 0 (2, 1).They can be used to convert any three-vector V a into symmetric second-rank spinors As is known, the invariance properties of (γ a ) αβ and (γ a ) αβ follow from the isomorphism SO 0 (2, 1) ∼ = SL(2, R)/Z 2 which is defined by associating with a group element M ∈ SL(2, R) the linear transformation on the vector space of symmetric real 2 × 2 matrices V In the 3D case, the spinor indices are lowered and raised using the SL(2, R) invariant spinor metric ε = (ε αβ ) = −(ε βα ) and its inverse ε −1 = (ε αβ ) = −(ε βα ), which are normalised by ε 12 = −ε 12 = 1.The rules for lowering and raising the spinor indices are: By construction, the γ-matrices (B.2a) and (B.2b) are real and symmetric.
Properties of the 4D relativistic Pauli matrices imply analogous properties of the 3D γ-matrices.In particular, for the Dirac matrices In three dimensions, any vector F a can be equivalently realised as a symmetric secondrank spinor F αβ = F βα or as an antisymmetric second-rank tensor F ab = −F ba .The former realisation is obtained using the gamma-matrices: The symmetric spinor F αβ is defined in terms of F ab as follows We emphasise that the three algebraic objects F a , F ab and F αβ are equivalent to each other.The corresponding inner products are related to each other as follows: More details can be found in [38].

C The first Bianchi identity
The first Bianchi identity is given by Requiring the supersymmetric torsion to vanish gives