In general terms, duality consists of two descriptions of a physical system through different degrees of freedom. There are different kinds of dualities and they have been extremely useful to uncover the underlying dynamics of strong coupling gauge theories in various dimensions and those of the diverse string theories [1].

In some cases this relation involves the inversion of the coupling constant in such a way that non-perturbative phenomena of the original degrees of freedom can be mapped to a perturbative theory of the dual degrees of freedom. This duality is termed S-duality and it will be the subject of the present article in the context of gravitational theories. Here we will not intend to give an exhaustive, complete and detailed overview on this subject, which is out of our scope. Thus for a general review, see for instance [2, 3] and references therein.

The paradigmatic example satisfying the property of duality is the Maxwell theory with magnetic charges and magnetic currents. At the present time a lot has been understood about duality in gauge theories. The most relevant examples are the N = 4 supersymmetric gauge theories in four dimensions, where the Montonen-Olive duality [4] was proved [5], and the Seiberg-Witten duality for N = 2 theories (see for instance, [6]). Further, a duality that relates a gravitational theory (or a string theory) and a conformal gauge theory, as in the case of gauge/gravity correspondence, has also been studied with quite detail [7]. However, much of the general work on duality has been done using the Feynman functional integral and it has not been considered as a well established mathematical result. This has motivated many mathematicians and mathematical physicists to work on rigorous proofs to support these results.

The case of duality between two gravitational theories is the so called gravitational duality. At the present time this duality has not been exhaustively understood and it is a conjectured symmetry existing in some gravity theories or in the gravitational sector of some higher-dimensional supergravity or superstring theories; some advances have been reported in the literature [2, 3]. In fact, the reach of this analysis is not comparable with those obtained in supersymmetric Yang-Mills theories or superstring theories. In these latter theories the power of strong/weak duality and T-duality in superstring theories has allowed to compute many non-trivial observables carrying much information on the system [8].

Very recently new advances in duality have been done in the context of condensed matter systems, see for instance [9]. There is a conjecture asserting that a fermionic system coupled to a Chern-Simons field is dual to a Chern-Simons gauged Wilson-Fisher bosonic theory. This conjecture has been proved for the case of negative mass deformation of the fermionic theory [9].

In Dijkgraaf et al. [10] the so called topological M-theory has been proposed. This is a gravity theory of three-forms on a seven-dimensional manifold of G₂-holonomy, which has been regarded as a master system from which it is possible to obtain, by dimensional reduction, the different form theories of gravity in lower dimensions, six, four and three. Among the known theories of gravity are: In six dimensions, on a Calabi-Yau manifold, there are two theories of gravity describing the moduli of complex structures and the Kähler cone, i.e., the so called Kodaira-Spencer gravity [11], and the Kähler gravity [12] respectively. Further, in four dimensions there is Plebański gravity [13] and in three dimensions Chern-Simons gravity [14–16].

The action principle for topological M-theory is Hitchin's functional defined as the integration of the volume form in seven dimensions, constructed with the G₂-holonomy invariant forms or calibrations [17]. Volume functionals in six dimensions also have been constructed by Hitchin [18]. These actions reflect the possibility of constructing volumes in terms of the Kähler structure or the complex structure of the underlying six-manifold. In Dijkgraaf et al. [10] it was showed that these six-dimensional Hitchin's functionals are related to the topological string theories of A and B types. Also topological M-theory has been proposed as a master system to derive all the form gravity theories in dimensions lower or equal to seven [10]. This theory has motivated recent work in exploring some new relations between Hitchin's functionals in seven and six-dimensions and form gravity theories in four and three dimensions [19–21].

In the present paper we give first a general overview of the case of non-abelian dualities in various gravity theories developed by the authors in Garcia-Compean et al. [22–27]. These works were done motivated from Ganor and Sonnenschein [28], Mohammedi [29], Lozano [30], and Kehagias [31], which were based on the Rocek and Verlinde duality given in Rocek and Verlinde [32], and in the Buscher duality algorithm [33, 34].

A version of linearized gravitational duality was proposed in Nieto [35] based on our results mentioned in the previous paragraph. Recently this subject has been intensively studied with very interesting results, see for instance [36, 37], and references therein. It is not our purpose to overview this subject here.

Finally, as a new development in this direction, we study the gravitational duality of Hitchin's functional in seven dimensions and its relationship to the gravitational duality algorithm in the corresponding six-dimensional theories.

In section 2 we give a brief review on the work performed by our group regarding non-abelian gravitational duality. In section 3 we focus on the 3 dimensional Chern-Simons gravity where we describe in detail the duality algorithm. Section 4 is devoted to study the gravitational duality in the topological M-theory [10], which is described by using the Hitchin's functionals [17] in seven-dimensional manifold with G₂-holonomy. Further, the six-dimensional Hitchin's functionals [18] are discussed. We find the corresponding dual actions by using the duality algorithm. Moreover, using a relation of Hitchin's functionals in six and seven dimensions, given via the Hamiltonian flow, we prove that the duality in the six-dimensional model is obtained from the duality in the seven dimensional theory. Thus the parent action in the six-dimensional theory follows from the corresponding action in the seven-dimensional case. Finally in section 5 we give our final remarks.

In this section we overview the idea of gravitational duality. There are many conceptions of this duality. We are going to give a path to our work. We will not intend to give a historical and detailed view on the subject here. For a review about work done on the different visions of gravitational duality and the development in early stages, see [2, 3].

First of all, we review some aspects from Garcia-Compean et al. [22]. This work was inspired by the work in Ganor and Sonnenschein [28], where a dual action for non-supersymmetric pure Yang-Mills theories is given. We start from the partition function

where Fμνa(A)=∂μAνa-∂νAμa+gfbc aAμbAνc. After integration of the auxiliary field G one get the original Yang-Mills Lagrangian LYM=-14gFaμν(A)Fμνa(A). In order to get the dual action, one can integrate with respect to the gauge field A, from which follows [28]

where Mabμν=fab cGcμν. This is an action of the Freedman-Townsend type [38]. Note that in these computations it was not necessary to relate Gaμν to Gμνa by any specific metric, the only condition is invariance under the required symmetries. This formulation can be generalized for a Lagrangian L(F), which could also depend on other fields. Consider the associated partition function

Proceeding as in the case of (1), after integration of A the partition function becomes

For example, if L(F)=θ2πFaμνFμνa, where Faμν=𝔤abμνρσFρσb, is defined through some metric 𝔤. After integration of F the partition function becomes

This formulation has been applied in Garcia-Compean et al. [22] to topological gravity, considering the Pontryagin and Euler topological invariants LP=θGP2π ∫εμνρσδabcdRμν ab(ω)Rρσcd(ω) and LE=θGE2π∫ εμνρσεabcdRμν ab(ω)Rρσ cd(ω), where a, b, c, d are Minkowski indices, ωμab is the spin connection and δabcd= 12(δacδbd-δadδbc). Thus the gauge group is SO(3, 1). In this case in (5) we take Gabμν=12εμνρσ ηab,cdGρσcd, respectively Gabμν=14εμνρσ εabcdGρσcd, where ηab,cd=12(ηacηbd−ηadηbc). −η_adη_bc). In fact, the equivalence of SO(3, 1) with SL(2, C) × SL(2, C) leads to the decomposition into self-dual and anti-self-dual metrics δabcd=Π  ab(+) cd+Π  ab(−) cd and εab  cd=2i(Π  ab(+) cd−Π  ab(−) cd).. Hence it is enough to consider L(±)=θG(±)2π∫ εμνρσR μν(±)ab (ω)Rρσab(±)(ω), where G μν(±)ab=Π(±)abcd G μν(±)cd and G ab(±)μν=12εμνρσ ηacηbdG ρσ(±)cd.

For the MacDowell-Mansouri theory worked out in Garcia-Compean et al. [23] we have considered the classically equivalent formulation S=∫d4xεμνρσεabcd (τ+Fμν+abFρσ+cd-τ-Fμν-abFρσ-cd), hence it is sufficient to consider the self-dual action

where Fμνab(ω,e)=∂μωνab-∂νωμab+ωμa cωνc b-λ2(eμaeνb-eνbeμa) is the ISO(1, 3) field strength. In this case the partition function is given by

where the * corresponds to the Hodge dual. In this expression, the integration must be made considering the redundancy of the self-dual integration variables, as can be seen from the following identities: u+abυab+=-4u0i+υ0i+, *Gab+μνωμ+acωνc+b=4*Gij+μνω0iμ+ω0jν+ and *Gab+μνeμaeνb=*Gij+μνeμieνj-iεijk*Gij+μνeμ 0eνk. Thus, integrating successively the quadratic terms in Fμν+0i, ωμ+0i and eμi, we get

where the integration over De stands for integration over eμ 0, and G(0)μν=G+ρσ-1ijεiklεjmn*Gkl+ρμ*Gmn+σν. After integration of eμ 0, it follows the dual partition function of (7)

The MacDowell-Mansouri gravity can be generalized to supergravity. This is done simply by promoting the gauge group SO(3,2) to the supergroup OSp(1|4). This theory was also studied in the context of gravitational duality and a parent action was constructed and its dual action was given in Garcia-Compean et al. [24].

For 3D Chern-Simons theories this approach simplifies considerably. In this case we have L_CS = Tr (AH(A)), where H(A) is a (D−1) form. We exemplify it for D = 3, and following [25], consider the parent action IP=i Tr ∫(αAF+βGF+γGdA+ηGA2), i.e.

where α, β, γ and η are constants. We show that independently of the integration order in the partition function, the resulting dual action is the same (self-dual). In the next section we make it explicitly since it is an important example in order to compare with the dual actions of Hitchin's functionals in seven and six dimensions (see section 4).

If we integrate first over G in the partition function

we get

from which, after integration over F gives

Now, if we integrate over F in (11), we get

Integrating now over A and after the change Gia→-Gia it yields

and, otherwise, integrating over G gives (13).

Finally, the integration over A and then over F can be performed if we observe that the parent action (10) can be rewritten, after partial integration for the third term and simple algebraic manipulations, as

where Mabij= εijkfabcGkc, A~ia= Aia+2αηMij-1ab(Fbj+γ2αεjkl∂kGlb), and F~ai=Fai+γ2αεijk(∂jGka+βηαγfabcGjbGkc). Thus, the integration over A and F in the partition function (11), leads to the integration of the first two gaussian terms in the partition function of (16), whose contributions cancel up to a factor, following (17).

Therefore, the parent action (10) leads to the dual partition functions (13) and (17). The dependence of the coupling constants in the determinants in these partition functions can be eliminated by rescalings, after which both actions coincide with

where constant factors have been discarded and for (17) a parity transformation G → −G must be made. Note that apparently the partition functions (13) and (17) have inverted coupling constants, but after the preceding rescalings, they have the same dependence on the coupling constants. This is a characteristic of 3 dimensional Chern-Simons.

Gravity in three dimensions is also described as a gauge theory. This is a Chern-Simons theory with gauge group being SO(2, 2), ISO(1, 3) and SO(1, 3) according if the cosmological constant takes negative, zero o positive values. These theories also do admit a dual gravitational description and this was described in Garcia-Compean et al. [25]. This will be reviewed in the next section. The generalization to Chern-Simons supergravity is also obtained by promoting for instance the group SO(2, 2) to the supergroup Osp(2, 2|1). The analysis of the gravitational duality was performed in Garcia-Compean et al. [26].

Finally for the Plebański formulation, which is also known as of the BF-type, it is somewhat different. In order to analyze it, we consider the original complex version with a SU(2) symmetry and an action [13] I=∫εμνρσ[1κΣμνiFρσi(ω)+ϕijΣμνiΣρσj]d4x, where ϕ is a traceless Lagrange multiplier matrix. The solution of the constraints εμνρσ(ΣμνiΣρσj-13δijΣμνkΣρσk)=0 is Σμνi=12(eμ 0eνi+i2εjkieμjeνk), which substituted into the action gives the Palatini action. Following the previous analysis, we consider the parent action

First we observe that the quadratic form in Σ, the last term, cannot be integrated in combination with the first term to give a F² term, because ϕ is traceless, and the first term is contracted with δ_ij. Furthermore, similar to the Chern-Simons case, the integrations over F and G, and over F and Σ, give all of them the Plebański action

However, the integration over ω, F and G in (18) gives

where Mijμν=εij k*Σkμν.

In the present section we explain the details of gravitational duality for the Chern-Simons theory in three dimensions. This subject has been discussed in the previous section for a general Chern-Simons theory. However here we intend to exhibit the details corresponding to gravitational Chern-Simons.

2+1 gravity dimensions is a theory that has played a very important role as a toy model of four dimensional general relativity at the classical and quantum levels. In Witten [15] 2+1 gravity was described in terms of the standard and exotic actions. In Garcia-Compean et al. [26] we showed that both actions correspond to the self-dual and anti-self-dual of the Chern-Simons actions with respect the Lorentz gauge group. The given Lie algebras g of the gauge groups have three generators constructed with four capital Latin letters A, B, C, D = 0, …, 3 and correspond to g = so(3, 1) and g = so(2, 2) for λ > 0 and λ < 0 respectively. The Lie algebra g is generated by M_AB, which satisfy [MAB,MCD]=if  ABCDEFMEF, where f  ABCDEF are its corresponding structure constants.

Consider the non-abelian Chern-Simons action

where g is the Chern-Simons coupling, H=dA+23A∧A and Tr is a quadratic form of g such that it satisfies Tr(M_ABM_CD) = η_(ACη_BD). The gauge field A=AiABMABdxi and the H-field H=HijABMABdxi∧dxj. In local coordinates we have

The duality algorithm require to propose a parent action, which in [26] has been proposed as

where

The duality algorithm allows to recover the original action (22) after an integration with respect the Lagrange multipliers B and G. In fact, in the preceding section we have presented a somewhat more general parent action, although the details of the computations in [25] are the same, in such a way that the resulting actions, with a = −g/4π, b = 1 and c = 1, are

Whereas the dual one is

In Chern-Simons gravity Newton's gravitational constant G_N is related to the Chern-Simons coupling constant g [16] i.e.

In the context of Abelian Chern-Simons theory, this duality was previously working out by Balchandran [39]. In particular, it was shown that in the Abelian case, the consistency of the dual theory requires from periodic boundary conditions for the dual fields BiAB. More recently this duality symmetry has been explored in the context of supersymmetric Chern-Simons QED [40]. Finally a more exhaustive analysis was carried out in Witten [41].

In this section we will discuss the so called topological M-theory [10]. This is a theory in a seven-dimensional manifold X with G₂-holonomy and stable real 3 and 4-forms Φ and G = ⋆Φ. The Hitchin's functionals [17, 18] describing the volume of the seven-manifold X in terms of the stable forms are given by

and

The critical points of these actions determine special geometric structures on X. For instance the variation of Equation (28) determines a metric with G₂-holonomy which can be constructed through solutions of equations of motion

Another form we can see that this is true is as follows. If one takes Φ to be an exact form: Φ = dB with B being a two form on X. Then immediately one has dΦ = 0. Then the variation of the volume (28) is given by

Taking into account that Φ = dB we have

Then we have that Equation (30) are fulfilled. Similarly this procedure can ba carried out for Equation (29).

A similar procedure can be implemented for Equation (29). It is easy to see that the corresponding equations of motion are

For seven manifolds which are the global product X = M × I, where I is the finite interval. The stable forms in the 7-dimensional theory induce stable forms in the 6-dimensional theory through a Hamiltonian flow. This determines the form of Φ and ⋆Φ in terms of the 6-dimensional stable real 3-form ρ and 2-form k [10]. This is given by

where σ=12k∧k. In terms of stable forms ρ and k the Hitchin's action is written as [10]

where

and

Variations of (36) and (37) with respect to ρ and k respectively lead to the following equations of motion

The first equation implies the existence of a closed holomorphic and invariant (0, 3) form Ω on M with dΩ = 0 and the existence of a Kähler form k on M. That means that M is a Calabi-Yau manifold.

In the present article we give an overview of some of our results regarding gravitational duality in some gravity theories [22–27]. In these papers we found some explicit dual actions to some specific theories of gravity. The duality procedure was implemented from the Rocek-Verlinde non-abelian duality algorithm applied to Yang-Mills theories [28–31].

In section 2, we reviewed the gravitation duality in a unified framework that contains the cases of topological gravity, MacDowell-Mansouri gravity, Chern-Simons gravity and BF-gravity. In order to present a complete case, in section 3 we overview in more detail the corresponding gravitational duality to Chern-Simons gravity in 2+1 dimensions.

In section 4 we give a new contribution to the subject. We apply for the first time the duality algorithm to the Hitchin's volume functionals in seven and six dimensions. Hitchin's functional in seven dimensions is the starting point to define topological M-theory [10]. In the present article we find the dual action which is written in terms of the dual degrees of freedom Λ and it is observed to be self-dual since it has the same form as the original action. Furthermore the dual action has inverted the coupling constant. Moreover the dual theory has the same equations of motion (50) than the original theory. Thus they have the same dynamics and both theories are classically equivalent.

The duality algorithm was also implemented for the Hitchin's functionals in six dimensions. In this case we have two volume functionals V_H(ρ) and V_S(σ). We have found the dual actions for such functionals given by expressions (63) and (68). Moreover we showed that the corresponding parent giving rise to these actions given by expressions (55) and (56) follows, under certain sensible conditions, from the parent action of the topological M-theory (39). Thus we find that the duality algorithm in six dimensions come from the duality procedure of the underlying seven-dimensional theory. It is expected that this connection can be carried out to other duality procedures in form theories of gravity in lower dimensions. In a future work we expect to report our results in the search of this web of dualities from M-theories to lower dimensions including the four and three dimensions. It would be interesting to investigate if the recent results [19–21], will be of some relevance in this analysis.

All authors listed have made a substantial, direct and intellectual contribution to the work, and approved it for publication.