Hidden symmetries in minimal five-dimensional supergravity

We study the hidden symmetries arising in the dimensional reduction of d=5, N=2 supergravity to three dimensions. Extending previous partial results for the bosonic part, we give a derivation that includes fermionic terms, shedding light on the appearance of the local hidden symmetry SO(4) in the reduction.


Introduction
Since their role in the greater scheme of M-theory was postulated [1], there has been intense renewed interest in the hidden symmetries of supergravity [2][3][4][5]. While most work has, understandably, centered on the maximal elevendimensional theory and its various dimensional reductions, with recent work ranging from the gauging of subgroups of the global exceptional groups [6][7][8] to the identification of new vacua [9,10] to possibilities for enlarging the hidden symmetries even further [11], there have also been results of a more general scope; notably the work showing in the most systematic manner yet how the hidden symmetries arise in successive dimensional reduction [12,13] has been generalized to other dimensional reductions to three dimensions, [14].
The present work, whose results form part of the thesis [15], is concerned with the hidden symmetry arising from the reduction to three dimensions of (minimal) five-dimensional supergravity [4,16]. Since its inception, that model, in many ways a "little brother" to the eleven-dimensional theory, has been used time and again to learn more about its higher-dimensional kin. Recent examples include toy models of the M5 brane [17], cosmological models [18], and methods developed for the study of the U-dualities of M-theory [19,20]. Concerning the hidden symmetry in question, there are so far only partial results, namely a construction [21] of the bosonic part of the model using a decomposition of G 2(+2) with respect to SL(3, R) and a corresponding construction by Cremmer et al. as part of the aforementioned more general study of reductions to three dimensions [14]. What has been missing, so far, is a complete analysis, notably one that includes the fermionic sector. The latter is interesting not only as another data point in an area for which, in contrast to the bosons, no systematic scheme yet exists, namely the relationship between the local extended symmetry and the dimensional reduction of the spinors, but also for another reason: In maximal supergravity, hidden symmetries have been successfully "lifted" to eleven dimensions [22][23][24], and recent work has uncovered tantalizing hints of "exceptional geometric structures" associated with these liftings [25,26]. The fermionic sector plays a crucial part in this type of lifting, and the results of this paper are thus a prerequisite for a search for such "exceptional geometry" in five-dimensional gravity. The present paper contains a construction of the g 2(+2) /so(4) target model in three dimensions and then proceeds to a dimensional reduction of five-dimensional supergravity, including the fermionic sector, in which the emergence of the hidden symmetry is shown.
2 The g 2(+2) -model in (2 + 1) dimensions In this part, we construct the g 2(+2) /so(4)-supergravity in 2 + 1 spacetime dimensions. For the sigma-model part we use the conventions 1 (as well as some general formulae) of Marcus and Schwarz [27]. The basic fields of our model will be, firstly, scalars ϕ i parametrizing the coset; as detailed in [27], they occur in the Lagrangian in the form of a Lie-algebra valued field P µ and composite connection coefficients Q µ ; secondly, a dreibein field e µ α ; finally, fermionic superpartners of these bosonic fields: a spin-1/2 field χ and a spin-3/2 (gravitino) field Ψ, respectively. To match degrees of freedom, we need an N = 4 extended supersymmetry so that, in addition to Lorentz symmetry, the gravitino transforms non-trivially under an additional R-symmetry.
Before the actual construction, we need to assign the different fields to their proper representations with respect to the internal symmetries involved, notably the local so(4) symmetry of our model. In parallel with the e 8(+8) -case, one might think that the so(4) R-symmetry would have to be identified with the local so(4) symmetry from the coset construction; however, the situation is more complicated. If we assign the supersymmetry parameter ǫ (and hence the gravitino Ψ) to the vector representation of the R-symmetry so(4), then from a general analysis of the supercharges as in [28] we must conclude that regarding the representations possible for the (massless) matter fields, one chiral subgroup, henceforth denoted so(3) F , acts only on fermionic degrees on freedom, the other, so(3) B , only on bosons. On the other hand, from the group theory literature [29] we know that the proper coset decomposition for the 14 representation of the g 2 is 14 = (1, 3) + (3, 1) + (4, 2) (with the usual convention of denoting representations by their dimensions, and with the tupels on the right hand side referring to the two chiral so(3) components), so in particular, the coset scalars transform non-trivially under both of the local so(3). To resolve the problem, we need to introduce a third algebra, which we shall call so(3) 2 . This symmetry included, there is indeed an assignment to representations of so(3) F × so(3) B × so(3) 2 that is consistent with the above requirements as well as with the form of the supersymmetry transformations (schematically, δ S Ψ ∼ Dǫ; δ S χ ∼ ϕǫ; δ S ϕ ∼ǭχ), namely with Ψ/ǫ in the representation (2 F , 2 B , 1 2 ), the ϕ transforming as (1 F , 2 B , 4 2 ), and the χ as Next, we need the algebra for the coset decomposition. We denote the indices of the two chiral components so(3) B and so(3) 2 of the maximal compact g 2 subalgebra so(4) byā,b, . . . andȧ,ḃ, . . ., respectively. Decomposed with respect to representations of that subalgebra, an algebra element can be written as a contraction of coefficients with generators E, with the part in so(3) 2 given as M˙a˙bE˙bȧ, the so(3) B as NābEāb and the non-compact part as Yā˙a˙b˙cEā˙a˙b˙c. The commutators can be found in the usual way, by decomposing tensor products and imposing Jacobi identities. They are the usual matrix commutators for the M˙a˙b and Nāb among themselves plus [M, N] = 0 as well as Next, for the realization of the different so(3)-representations. Denoting the fundamental indices of so(3) F by i, j, . . ., the assignment of representations leads to an index structure φā˙a˙b˙c for the scalars, Ψ iā for gravitino and supersymmetry parameter, and χ iȧḃċ for the matter fermions. The action of infinitesimal so(3)-transformations is fixed by linearity and by demanding for each two such transformations To ensure consistency, both the so(3) F coefficients and the fields inherit the symplectic reality condition, On the fermionic side, this makes fully contracted products of (anticommuting) spinors symmetric, with Clifford conjugation as their adjoint, e.g.
After these preparations, we can find the Lagrangian. We restrict ourselves to the terms that are necessary for the comparison with the dimensional reduced theory, omitting quartic or higher fermionic terms in the Lagrangian. Starting with the fields' standard kinetic terms and supersymmetry variations, supersymmetry demands the inclusion of a Noether term and fixes ambiguities of the relative constants, with the resulting Lagrangian invariant under the supersymmetry variations where Σ S is the highly non-linear expression Σ S = − tanh(1/2 · ad ϕ ) ϕ that is a consequence of describing the variation of group-valued objects in terms of algebra-valued objects [27, sect. 2]. The coset-specific properties come into play in that the variation of the Rarita-Schwinger term contains [D µ (Q), D ν (Q)], to be cancelled by a term proportional to [P µ , P ν ] arising from the variation of the Noether term with respect to χ.

The dimensional reduction from five to three dimensions
The stage is now set for the dimensional reduction from which we mean to recover the model found in the previous section. Starting point is the minimal N = 2 supergravity in five dimensions [4,16]. We choose a "mostly minus" metric with the gravitini symplectic- where the supercovariant derivativeD is defined bŷ The initial steps of the dimensional reduction to three dimensions are fairly generic [2,4,5,30]: Exploiting part of the Lorentz gauge freedom, certain vielbein components can be gauged to zero, allowing the decomposition where ∆ = det e m a is a Weyl scaling factor. Here and in the following, curved indices are decomposed in the manner M = (µ, m), with µ a three-dimensional space-time index and m an index in the two-dimensional internal space; flat indices are split analogously as A = (α, a). We adopt the convention of splitting fields such as F or the gravitino starting from their (all-lowered) flat-indexed form; curved-index versions are then obtained by application of the component vielbeins e ′ µ α and e m a .
The dimensional split of the anholonomy coefficients leads to non-zero expressions using ∆-independent primed coefficients defined as where G n µν := ∂ µ B ν n − ∂ ν B µ n is the field strength of the Kaluza-Klein vector field.
As for the five-dimensional gravitino, and adopting the convention to suppress (five-and three-dimensional) space-time spinor indices, the first decomposition is of Ψ i A into a three-dimensional spin-3/2 fermion Ψ iā α and a spin-1/2 fermion Ψ iā a .ā can be promoted to an su(2) ≃ so(3)-index: As far as internal spinor indices are concerned, the spinor product's adjoint is Hermitian conjugation, with an invariance group U(2). The symplectic reality condition restricts this to SU(2). However, there are problems: This decomposition leads both to a non-standard form for the threebein's supersymmetry variations and to mixed first-derivative terms between spin-3/2 and spin-1/2 degrees of freedom. The remedy is a redefinition Ψ ′iā ; with this field as the three-dimensional gravitino and e ′ µ β as vielbein, the dimensional reduction reproduces both the gravitino kinetic term (with the mixed terms now absent) and the vielbein supersymmetry variation given in (2) and (3), respectively. From this, it is tempting to identify the gravitino's index a as an index of so(3) B . However, matters are more complicated, which can be seen by considering the spin-1/2 fields. To start with, they have the index structure Ψ iā a , whereā can be promoted to an so(3) index, as before. However, from the construction in section 2 we know that the matter fermions transform non-trivially only with respect to so(3) 2 and so(3) F . This apparent problem can be resolved once it is realized that in similar situations, notably in the E 8 case, dimensional reduction leads to models in which the enhanced local symmetry is gauge-fixed, with expressions e.g. for the P µ and Q µ that are not explicitly covariant under that symmetry. Our case is analogous in that, apparently, the dimensionally reduced model "sees" only the diagonal so(3) subgroup of so(3) B × so(3) F . To restore the enhanced symmetry, that diagonal group needs to be disentangled into its so(3) B × so(3) F parts, using the model developed in section 2 as a guide. This promotes theā index of Ψ ′iā µ to an so(3) B index, and the internal spinor index of the spin-1/2 fields to an so(3) 2 index. For the latter, the kinetic term should have the same simple form as shown in (2). This can be achieved by exploiting the freedom to re-define Ψ iȧ a → Ψ iȧ a + (Γ aΓ c )˙aċΨ iċ c . All in all, new matter fermion fields defined as have the required properties. It can be checked directly that the fermions thus defined have inherited the correct symplectic reality condition. With these preparations, the terms quadratic in Ψ ′ that are obtained from the five-dimensional Rarita-Schwinger term combine into the three-dimensional enhanced Rarita-Schwinger term with gauge-fixed connection coefficients while the corresponding connection coefficients in the kinetic term of the spin-1/2 fermions turn out to be The mixed term containing both Ψ ′ and χ, compared with the Noether term in the three-dimensional target lagrangian, yield By their index structure, it can be checked directly that these objects transform under the proper representations of so(3) 2 × so(3) B ; by using the reality conditions for theΓ, that they satisfy the proper reality conditions.
To complete the match, we present three consistency checks. From (3), it follows that Q νāb must occur in the new gravitino's supersymmetry variations, while one should also be able to read off (P µ )āḋėḟ from the matter fermion's supersymmetry variation. Both expressions agree with those derived above. The final check is the match of the dimensionally reduced bosonic Lagrangian with the sigma model kinetic term in terms of the P µ )āḋ˙eḟ of eq. (14); the same check used in [21] for the g 2 -construction in terms of sl(3) representations. The sigma-model term is while the reduction of the bosonic terms gives the three-dimensional Einstein-Hilbert term plus That there is no match comes as no surprise, as it is a general feature of hidden symmetries to become manifest only upon dualization of appropriate dimensionally-reduced p-forms. In this case, the objects that allow dualization are the Kaluza-Klein field strength G m µν , dual to two scalars ξ m , and a composite "field strength"F µν := ∆ −2 F ′ µν − G m µν A m , tailor-made to fulfill the Bianchi identity and dual to a scalar ϕ. After dualization, the original Lagrangian (15) plus the constraint terms becomes which is the same as the P 2 -Lagrangian (14) upon substitution of the dualized entities. This completes our cross-checks.
There are a number of possible directions for extending the present results. The possibility of exploring "exceptional geometries" has already been mentioned; another interesting question would be to what happens to the hidden symmetry if the R-symmetry or some subgroup thereof is gauged [31,32] (making contact with recent, more general, studies of the possible gaugings in three dimensions [6]) or to study the case of compactification on AdS 3 × S 2 , for which the spectrum has already been worked out in [33].