Four dimensional topological supergravities from transgression field theory

In this work, we propose a four-dimensional gauged Wess-Zumino-Witten model, obtained as a dimensional reduction from a transgression field theory invariant under the $\mathcal{N}=1$ Poincar\'{e} supergroup. For this purpose, we consider that the two gauge connections on which the transgression action principle depends are given by linear and non-linear realizations of the gauge group respectively. The field content of the resulting four-dimensional theory is given by the gauge fields of the linear connection, in addition to a set of scalar and spinor multiplets in the same representation of the gauge supergroup, which in turn, correspond to the coordinates of the coset space between the gauge group and the five-dimensional Lorentz group. We then decompose the action in terms of four-dimensional quantities and derive the corresponding equations of motion. We extend our analysis to the non- and ultra- relativistic regime.


Introduction
In the last decades, several gravitational theories have been introduced as alternatives to General Relativity.The most general theory that can be formulated in arbitrary dimensions and fulfill the fundamental requirements of invariance under diffeomorphisms, lead to equations of motion of second degree in the metric tensor and preserve the conservation law of the energymomentum tensor is known as the Lovelock theory [1,2].The Lovelock Lagrangian density is a sum over all the possible combinations between the Lorentz curvature depending on the spin connection, and the vielbein that codifies the metricity.In three and four dimensions, the Lovelock theory reproduces General Relativity with positive, negative, or zero cosmological constant depending on the values of its arbitrary constants [3][4][5][6].A case of special interest is obtained when the constants of the sum in the Lovelock Lagrangian are fixed such that the theory presents the maximum number of degrees of freedom [7].In that case, the three-dimensional Lovelock Lagrangian becomes proportional to the Chern-Simons (CS) three-form of the AdS group [8,9], which has a topological origin and is, up to boundary terms, invariant under the gauge transformations induced by the AdS group.Such a feature allows us to formulate General Relativity as a topological gauge theory in three dimensions.However, the same does not occur in four dimensions, since CS forms exist only in odd dimensions.Moreover, the CS form of the AdS group does not lead to General Relativity in dimensions higher than three in any regime.With the purpose of formulating even-dimensional gravity theories, especially in four dimensions, A. H.
Chamseddine proposed a topological gravity depending on the same variables than the Lovelock theory, in addition to a scalar multiplet in the same representation of the gauge group [10][11][12].
From a mathematical point of view, CS forms appear in the study of topological invariant densities that only exist in even dimension and, as a consequence of the Poincaré lemma, allow the existence odd-dimensional secondary forms that inherit their gauge invariance properties.These forms known as transgression forms, become CS forms as locally defined particular cases [13][14][15][16][17][18].Thus, both transgression and CS forms are often used as Lagrangian densities for odd-dimensional gauge theories.In contrast to CS theories, transgression field theories depend on two gauge connections and their Lagrangian densities are globally defined differential forms that, as well as the topological densities in which they originate, are fully gauge invariant.In addition to be useful in the construction of Lagrangian densities, transgression forms can naturally induce a dimensional reduction that allows to formulate even-dimensional gauge invariant theories without the need of breaking the gauge covariance.These are known gauged Wess-Zumino-Witten (gWZW) models [19,20] and, as well as the even-dimensional topological gravity proposed by Chamseddine, include a scalar multiplet in the gauge group representation as part of the fundamental field content.Indeed, it was shown in [19][20][21] that Chamsedine's topological gravities can be obtained as gWZW models of the Poincaré group in arbitrary even dimensions.Furthermore, in ref. [20], these results were generalized to the Maxwell algebra and the Poincaré superalgebra in three dimensions, obtaining fully gauge invariant (1 + 1)-dimensional theories for gravity and supergravity respectively.Moreover, in refs.[19,20], it was studied the relation between four-dimensional gWZW models and general relativity.
The existence of abstract even-dimensional gWZW models motivates us to study a fourdimensional gauge invariant supergravity theory that emerges from considering a supersymmetric extension of the five-dimensional Poincaré supergroup as gauge group.Moreover, due to the wellknown relation between the Poincaré group and the Galilei and Carroll groups [22], we aim the purpose of studying the non-and ultra-relativistic regimes of the resulting theory and thus to obtain the corresponding superalgebras and gWZW action principles.This paper is organized as follows: In section 2, we consider a brief introduction to the SW-GN formalism and the gWZW model.In section 3, we study the non-linear realization of the five-dimensional Poincaré superalgebra.In section 4 and 5, we derive the four-dimensional gWZW action invariant under the aforementioned Poincaré supergroup, and study the dynamics of the resulting theory.In section 6, we consider the non-relativistic limit of the Poincaré superalgebra and derive the corresponding gWZW non-relativistic action principle.In section 7 we perform the same analysis for the ultra-relativistic limit of the theory.Section 8 contains our final conclusions.

Non-linear realizations and gWZW actions
In this section we briefly review the Stelle-West-Grignani-Nardelli (SW-GN) formalism and the gauged Wess-Zumino-Witten (gWZW) models.

The SW-GN formalism
The Stelle-West-Grignani-Nardelli formalism makes use of non-linear realizations of Lie groups in the construction of gauge invariant action principles [23,24].Thus, the gauge symmetry of a physical theory can be extended from a stability group, to a higher-dimensional group that contains it as a subgroup.Let us consider a Lie group G with Lie algebra G, and a subgroup H ⊂ G with Lie algebra H as stability subgroup.We denote as {V i } dim H i=1 to the basis of H and as {T l } dim G−dim H l=1 the to the set of generators of the remaining subspace.We assume that the basis can be chosen such that the generators T l form a representation of the stability subgroup.
Therefore, the Lie products between the vectors of the introduced basis satisfy [V, T ] ∽ T , i.e. these products are linear combinations of T l .An arbitrary group element g can be decomposed in terms the generators of the subgroup and the remaining subspace as where h ∈ H is a group element defined by the action of the group on the zero-forms ξ l which, in turn, play the role of coordinates that parametrize the coset space G/H.From eq. (2.1), it follows that the action of an arbitrary element g 0 ∈ G on e ξ l T l can be also split as Eq. (2.2) allows to obtain the non-linear functions ξ ′ = ξ ′ (g, ξ) and h 1 = h 1 (g, ξ).By considering that the transformation law ξ → ξ ′ is described by the variation δ, and by choosing the group element g 0 such that (g 0 − 1) is infinitesimal, eq.(2.2) leads to [25][26][27] e −ξ l T l (g 0 − 1) e ξ l T l − e −ξ l T l δe Let us now consider the case in which g 0 = h 0 belongs to the stability subgroup.In this case, eq.(2.2) becomes and since the Lie product [V i , T l ] is proportional to T l , one gets h 0 = h 1 and the transformation law becomes linear: On the other hand, if we consider g 0 = e ξ l 0 T l , eq. (2.2) becomes which is a non-linear transformation law for ξ.
Let us now consider a one-form gauge connection A taking values on G and an action principle with gauge invariance under the transformations of the stability subgroup H but not under those along the generators of the coset space.Under the action of an arbitrary group element g, the gauge connection transforms as [23,24,28,29] We split µ into its contributions belonging in h and the coset space as A = a + ρ, with a = a l T l and ρ = ρ i V i .Moreover, we introduce a group element z = exp ξ l A l and define the non-linear gauge connection The functional form of A z is given by a large gauge transformation of A that non-linearly depends on the zero-forms ξ l and their derivatives.However, in the SW-GN formalism A z is interpreted as the fundamental field of a gauge theory and therefore, both A and ξ will change under the action of the gauge group.As before, we split the contributions to the non-linear connection as It is possible to prove that, under the transformation δ generated by the action of the group, the transformation laws for p and v are given by i.e., when acting with a group element belonging to the coset space, the non-linear one-forms p and v transform as a tensor and as a connection respectively.These transformations are linear but the group element is now a function of the parameters h 1 = h 1 (ξ 0 , ξ).From the transformations laws in eq. ( 2.10), it follows that the non-linear gauge connection transforms in the same way that under the action of the stability subgroup and the coset space.Therefore, an action principle defined as a functional of A whose gauge symmetry is described by the stability subgroup, becomes invariant under the entire group G when A is replaced by A z .The original non-invariance of the action principle is thus compensated by the transformation law of the gauge parameters ξ.

gWZW models
Let us consider two independent gauge connections A 1 and A 2 evaluated in the same gauge algebra.The transgression (2n + 1)-form corresponding to both gauge connections is defined as where denotes the symmetrized trace along the generators of the Lie algebra, and F t is the gauge curvature associated to the homotopic gauge connection . Transgression forms are globally defined and fully invariant under the transformations of the gauge group.
CS forms emerge as particular cases of transgression forms, by locally setting one of the gauge connections as vanishing.Thus, the CS form corresponding to a gauge connection A is locally defined as where the homotopic gauge connection takes form A t = tA.Furthermore, by applying the Cartan homotopy formula, it is possible to prove that a general transgression form can be written in terms of two CS forms and a total derivative, as follows [14,15,30] The 2n-form inside the exterior derivative is explicitly given as the following integral: where F st is the gauge curvature associated to the homotopic gauge field which depends on two parameters t and s taking values between 0 and 1.For details on the use of the Cartan homotopy formula and the homotopy operator in this context, see refs.[14,15,17,31].
Let us now consider two gauge connections A and A z , related by the gauge transformation , where z = exp (ξ) is an element of the gauge group and ξ a zero-form mutiplet in the same representation of the Lie algebra.From eq. (2.19), it follows that the transgression form associated to A and A z can be written in terms of the difference between their corresponding CS forms.Let us now introduce a homotopic gauge field A t = tA which takes values between 0 and A, as the parameter t takes values between 0 and 1.The transformed connection, obtained from A t , denoted by (A t ) z , and its corresponding gauge curvature (F t ) z are given by1 These homotopic quantities verify (2.17) By applying the Cartan homotopy formula once again, it is possible to prove that the CS forms corresponding to the pure gauge connection z −1 dz and the transformed gauge connections A z are related by the following equation with k 01 = 1 0 ℓ t , where ℓ t is the homotopy operator defined by the following action on A t and F t : By directly applying eq. ( 2. 19), one finds that the first term in the right side of (2.18) is given by so that, eq.(2.18) allows to write the difference between two CS forms related by means of a gauge connection as where we introduce Notice that the first term in the r.h.s. of eq.(2.21) is the CS form corresponding to the pure gauge connection z −1 dz, which is explicitly given by Thus, by virtue of eq.(2.13), it is possible to write down the transgression form corresponding to A and A z in terms of the pure gauge connection and a total derivative Given a gauge group, the so-called gWZW action is defined as the boundary action that appears from the transgression action in accordance with the Stoke's theorem Since the transgression Lagrangian is odd-dimensional, the gWZW action principle is always evendimensional.As it happens in the SW-GN formalism, the zero-forms ξ are not longer interpreted as the parameters of a symmetry transformation but as physical fields with a topological origin.
However, in contrast with the gauge invariant action principles that are obtained in the SW-NG formalism, gWZW action principles are not exclusively functionals of the non-linear gauge fields, but of the linear ones and the zero-form multiplets.
where the metric signature is chosen as η AB = diag (−, +, +, +, +).This superalgebra allows an inner invariant rank-3 product with the following components: We gauge the algebra by considering a one-form gauge connection A with non-vanishing gauge curvature F =dA + 1 2 [A, A], to whose components we denote The components of the gauge curvature are given explicitly by where D denotes the covariant derivative defined with respect to the five-dimensional spin connection ω AB .By using the subspace separation procedure (see refs.[15,17]), it is possible to write down the five-dimensional CS Lagrangian in a convenient way: where κ is a constant.

Non-linear realization
In order to perform the dimensional reduction, it is necessary to introduce a non-linear realization of the gauge supergroup.We therefore consider a second gauge connection A z related with A by means of the following large gauge transformation Here, z is an element of the gauge group, specifically in the coset space G 5 /SO (4, 1).Taking in account the following decomposition of the Poincaré superalgebra we can express A z by using the following gauge group element where ϕ and φ A are zero-forms, and where χ and χα are Dirac spinors zero-forms.In order to explicitly write down for A z in terms of the components of A and the parameters of the gauge transformation, we use the following identities [32] e with the notation X ∧ Y = [X, Y ] and where δ is any variation.By directly and successively applying in the (anti)commutation relations of the gauge algebra into eq.(3.6), we obtain the following transformed gauge field where each component is given by The non-linear realization of the gauge algebra allows the construction of invariant action principles.In fact, the five-dimensional standard supergravity theory, whose Lagrangian functional includes the Einstein-Hilbert and Rarita-Schwinger terms becomes invariant under the Poincaré superalgebra when one identifies the non-linear field V A as the fünfbein field associated to the metric tensor of the supergravity theory, and Ψ α and Ψα as the spin 3/2 fields.

Four-dimensional Poincaré supergravity
In order to write down the transgression form depending on both gauge connections A and A z , let us recall (3.8).By inspection of the (anti)commutation relations and invariant tensors of the gauge algebra, it follows that, in this case, the pure gauge connection z −1 dz has no components along the Lorentz rotation generators.Therefore, the pure gauge contribution to the transgression form, vanishes for any gauge parameter lying in u (4|1) /so (4, 1) As a consequence, the transgression form Q A z ←A is always exact and, according eq.(2.24), can be written as To find an explicit expression of this transgression, let us first consider eq.(2.13) with the following choice of gauge connections: A←ω − dQ (4) Notice that we now choose the intermediate connection as Ā = ω.A direct calculation shows that the difference between both transgression forms in the r.h.s. of eq. ( 4.3) is given by where letters carrying no index inside the trace are vectors of the Lie superalgebra, i.e., R = 1 2 R AB J AB , ψ = Qα ψ α , ψ = ψα Q α , χ = Qα χ α and χ = χα Q α .By using the Bianchi identities, we have that the second derivatives of the fermion zero-forms can be written in terms of the Lorentz curvature as Then, by integrating by parts and using the properties of the gamma matrices in five dimensions, we find that the difference between both transgression forms is given by the following total derivative On the other hand, the the boundary term in the r.h.s. of eq. ( 4.3) can be obtained from eq.
(2.14) by setting n = 2, A 2 = A z , A 1 = A and Ā = ω.Consequently, the homotopic gauge field becomes Then, by plugging in eqs.(4.6) and (4.7) into (4.3),we finally obtain an explicit expression for the transgression form in terms of a total derivative Therefore, the four-dimensional induced action is given by This action in analogue to the one found in ref. [33] for (1 + 1)-dimensional supergravity.It is invariant under the transformations of the five-dimensional Poincaré supergroup and it can be interpreted as a supersymmetric extension of the topological gravity introduced in ref. [10], and alternatively found as a gWZW action in ref. [20] for the bosonic case.

Decomposition of the action
The obtained gWZW action from eq. (4.9) is four-dimensional.However, it is a functional of the original five-dimensional field content of the transgression field theory.When gauging Poincaré or AdS supergroups, the gauge field associated to the translation operator is usually identified as the vielbein of the corresponding supergravity theory, and therefore it is considered that it carries the information about the metric in the resulting field equations.Notice that, at this point, we have not yet introduced a notion of metricity in the supergravity theory.Moreover, the field h A has been removed from the functional in the dimensional reduction process and it is not longer present in the action principle in eq.(4.9).This is a common feature of gWZW models originated in the gauging of space-time symmetries, and allows us to identify the some components of the five-dimensional spin connection as vierbein in a four-dimensional supergravity theory.Thus, the original gauge invariance under the five-dimensional Lorentz group is now interpreted as invariance under the four-dimensional de Sitter group.We therefore decompose the index A = (a, 4) with a = 0, 1, 2, 3, and rename ω a4 = −ω 4a = e a as vierbein one-form.The five-dimensional Lorentz curvature is also decomposed, as follows: Consequently, we split the action principle into its bosonic and fermionic sectors as S = S B + S F , being S F the contribution depending on spinor fields in the r.h.s. of eq.(4.9).In terms of the previous decomposition, the bosonic sector of the action is given by In the same way, the fermionic sector of the action is given in terms of the four-dimensional quantities as follows where the five-dimensional Lorentz covariant derivatives are given in terms of the four-dimensional ones according to From now on, we will denote as L G to the bosonic Lagrangian four-form, and identify to the fermionic contribution as a matter Lagrangian, i.e., Although we hold the writing in terms of the five-dimensional indices for convenience, it is important to recall that they describe a four-dimensional theory with ω AB packing the spin connection and vielbein forms, while R AB contains the Lorentz curvature and torsion.In these terms, we introduce a generalized spin form Σ AB , such that the variation of L M with respect to ω AB is given by with * the Hodge dual operator and k a dimensional constant.The components of the generalized spin form are split into the four dimensional spin form and energy-momentum forms, as follows Therefore, the field equations can be written as The variation of L G with respect to ω AB is given by On the other hand, the field variation of the matter Lagrangian is given by where we have used the identities By integrating by parts and plugging in the Bianchi identities we obtain Finally we have an expression for the dual spin form The field equations coming from the variation of the action with respect to ω AB are therefore given by ε AB = 0 with ( 5.11) or, equivalently in components, the field equations related with the independent variations δe a and δω ab are given by ) where D denotes the covariant derivative defined with respect to the four dimensional spin connection ω ab .

Chern-Simons theory
Let us now consider a non-relativistic contraction of the gauge algebra (3.1).We split Lorentz index A in the space-time components as A = (0, I) with I = 1, . . ., 4.Moreover, we rename and perform the following rescaling on the Poincaré superalgebra generators as in [34,35] When taking the limit λ → ∞, the superalgebra (anti)commutation relations become 2) The non-relativistic limit of the Poincaré superalgebra (3.1) reproduces a supersymmetric extension of the Galilei algebra [36].We now introduce a one-form gauge connection A and the corresponding gauge curvature F , to whose components we denote The components of the new gauge curvature are explicitly given by T = dτ , where D ω is the covariant derivative with respect to the spatial spin connection ω IJ .
At the level of the invariant tensor, one can check that the non-relativistic limit of the nonvanishing components (3.2) reproduces with the convention ǫ 0IJKL = ǫ IJKL .Then, the five-dimensional CS Lagrangian takes the form Note that, although the non-relativistic limit of the Poincaré superalgebra is a supersymmetric extension of the Galilei algebra, the CS Lagrangian does not lead to supergravity.This is a consequence of the fact that, in this limit, the invariant tensor of the algebra only carries nonzero components in the bosonic sector.As we shall see in the next section, the Carrollian limit preserves supergravity.In a future work, it would be interesting to explore the existence of a non-relativistic counterpart of Poincaré supergravity that preserves supersymmetry.

gWZW model
Let us now consider the non-relativistic limit of the gWZW action principle obtained in section 4. As we did in the relativistic case, in order to construct the transgression field theory, we introduce a secondary gauge connection A z related with A by means of a gauge transformation.
We denote the components of A z and the supergroup parameters as follows: The four dimensional action is reduced to where we rename φ 0 = φ.We now perform a second index decomposition; the spatial index of the five-dimensional theory is split as I = (i, 4) where i = 1, 2, 3 is the spatial index of the non-relativistic four-dimensional theory.We also decompose the non-relativistic spin connection and curvature as follows: with where R ij = dω ij + ω i k ω kj is the so (3) curvature.Since h A is not longer present in the theory, we interpret e i as spatial vielbein.In this way, the Lagrangian density is written as As it happens in the non-relativistic five-dimensional CS theory, the resulting Lagrangian density is purely bosonic.

Chern-Simons theory
Let us now consider the ultra relativistic limit of the five-dimensional Poincaré superalgebra [22,36,37].With this purpose, we consider again the space-time splitting of the Lorentz index A in the Poincaré superalgebra.We rename and rescale the generators as The resulting ultra-relativistic superalgebra is obtained by taking the limit λ → ∞, and is given by the following supersymmetric extension of the Carroll algebra [36] in five dimensions Following the procedure used in the non-relativistic case, we now introduce a one-form gauge connection A and the corresponding gauge curvature F =dA+A 2 .Since the ultra-relativistic algebra has the same number of generators as its non-relativistic analogue, we denote the components of A and F as they are given in eqs.(6.3) and (6.4) respectively.In this case, the components of the gauge curvature are given by T = dτ + ω J h J − 2 ψα Γ 0 α β ψ α , T I = D ω h I , 3) The CS Lagrangian invariant under the transformation of this algebra is given by In contrast with the non-relativistic Lagrangian, the invariant tensor still carry non-vanishing components in the fermionic sector after taking the limit and, as a consequence, the supergravity is preserved in the ultra-relativistic regime.

gWZW model
Let us now consider the ultra-relativistic limit of the four dimensional gWZW Lagrangian.
As before, we introduce a group element z and a non-linear gauge field A z , obtained from A through a large gauge transformation.We denote to the components of z and A z as in eq.(6.8) respectively.In this case, the four-dimensional Lagrangian is reduced to As before, we shall decompose the five-dimensional spatial index, following the structure and change of notation of eqs.(6.10) and (6.11).In these terms, the four-dimensional ultra-relativistic gWZW Lagrangian is given by with the convention ǫ 1234 = 1 and φ 4 = ρ.

Concluding remarks
In this article, we have obtained a four-dimensional theory for supergravity.The construction of the action makes use of the five-dimensional N = 1 Poincaré supergroup as gauge group, a one form gauge connection evaluated on it, and a transformed gauge connection in which the gauge parameter is evaluated in the coset space resulting between the five-dimensional Poincaré

3 N = 1
Poincaré supergravity3.1 Chern-Simons supergravityThe construction of a four-dimensional gWZW model requires the five-dimensional CS action as starting point.We first consider the N = 1 supersymmetric extension of the five-dimensional Poincaré algebra u (4|1), which is spanned by the set of generators J AB , P A , K, Q α , Qα where Qα and Q α are independent Dirac spinors.Capital latin letters denote five-dimensional Lorentz indices taking values as A = 0, . . .4, while Greek letters denote spinor indices taking values as α = 1, . . . 4. In the chosen basis, the (anti)conmutation relations between the introduced generators are given by[9,15,17]

3 ǫ
abcd e a e b e c Dφ d +ǫ abcd R ab R cd − 2R ab e c e d + e a e b e c e d φ 4 + iR ab R ab ϕ + 2iT a e a dϕ .(4.11) superalgebra and its Lorentz algebra.The existence of such transformed connection is enough to formulate five-dimensional standard supergravity as a gauge invariant theory of the Poincaré supergroup by means of the SW-GN formalism.This is carried out by interpreting the new connection as the fundamental field of a supergravity theory (instead of an equivalent gauge through Fondecyt grant 1211219.E.R. acknowledges financial support from ANID through SIA grant No. SA77210097 and Fondecyt grant No. 11220486 and 1231133.P.C. and E.R. would like to thank to the Dirección de Investigación and Vicerectoría de Investigación of the Universidad Católica de la Santísima Concepción, Chile, for their constant support.S.S. acknowledges financial support from Universidad de Tarapacá, Chile.