On Dimensional Reduction of Magical Supergravity Theories

We prove, by a direct dimensional reduction and an explicit construction of the group manifold, that the nonlinear sigma model of the dimensionally reduced three-dimensional A = R magical supergravity is F4(+4)/(USp(6)xSU(2)). This serves as a basis for the solution generating technique in this supergravity as well as allows to give the Lie algebraic characterizations to some of the parameters and functions in the original D = 5 Lagrangian. Generalizations to other magical supergravities are also discussed.


I. INTRODUCTION
One of the common features of dimensionally reduced supergravity theories is that they contain a noncompact scalar coset sigma model in their Lagrangians. Perhaps the most famous example is the E 7(+7) /SU (8) coset in D = 4 N = 8 supergravity [1] obtained by a dimensional reduction of the eleven-dimensionally supergravity [2] to four dimensions. It is always the case that the global symmetry of the nonlinear sigma model is a symmetry of the whole supergravity system including the fermionic sector. A reduction of the elevendimensional supergravity to an intermediate dimension from 5 to 10 also yields an E-series symmetry [3,4], whose discrete subgroup is nowadays understood as a U-duality [5] of Mtheory or type-II string theories. It is also known that the symmetry is enhanced to E 8 or much larger (infinite-dimensional) upon reduction to three or lower dimensions [6][7][8][9].
The E-series is a token of lower-dimensional M/typeIIA/typeIIB theories upon toroidal compactifications. The D-series, on the other hand, is known to appear as a similar symmetry group of the non-linear sigma model of the dimensionally reduced NS-NS sector supergravity, whose discrete subgroup is a T-duality of the toroidally compactified string theory [12]. It is also very well known that the A-series is a symmetry of dimensionally reduced pure gravity [10,11]. The B-series may be obtained as a reduction of the NS-NS sector coupled to an odd number of vector fields [12,13], and G 2(+2) has been shown to be the symmetry of the dimensionally reduced D = 5 minimal supergravity to three dimensions [14]. So what about the remaining simple Lie algebras?
As for F 4 , many years ago it was anticipated that F 4(+4) /(USp(6) × SU(2)) should be the sigma model of the dimensionally reduced D = 5 magical supergravity of the simplest kind, reduced down to three dimensions [16] 1 2 . Although the appearance of this particular quaternionic manifold has been justified on various grounds and is now believed to be true, a direct proof by performing a dimensional reduction of the supergravity and comparing to the construction to the coset group manifold seems to have never appeared in print. The aim of this letter is to fill this gap.
The direct proof of the F 4(+4) /(USp(6) × SU(2)) coset structure has the following benefits: 1 Among the C-series, which is also missing in the above description, Sp(6, R)/U (3) (Sp(6) = C 3 ) has also been shown to appear [15] as a scalar coset of the same magical supergravity reduced to four dimensions. 2 See [17,18] for the gaugings of the three-dimensional magical supergravities.
(1) The direct dimensional reduction and the explicit construction of the coset sigma model enable us to find the precise relationship between the various components of the fivedimensional supergravity fields and the relevant group elements. This allows us to use the F 4(+4) global symmetry to generate a new supergravity solution from some known seed solution. Such a solution-generating technique utilizing the three-or four-dimensional global symmetry has been very powerful in deriving, for instance, the five-dimensional black hole solutions in five-dimensional minimal supergravity [19].
(2) By the above relationship between the supergravity fields and the group manifold one can also give group theoretical characterizations to some of the parameters and functions in the original magical supergravity Lagrangians. For example, as we show below, the F F A coupling constants C IJK are identified as the structure constants of the commutation relations between generators both belonging to one of the "Jordan pair" in the decomposition [22] of the quasi-conformal algebra of the relevant Jordan algebra. We will also find a Lie algebraic characterization of the functions of the scalars In fact, the procedure of the dimensional reduction itself is common to all the magical supergravity theories; the only difference is the range of the values of the indices of the vector and scalar fields. Although the three-dimensional duality Lie algebras also allow a common decomposition in terms of the relevant Jordan algebras [15,16,[20][21][22], in this letter we will work out in particular the F 4(+4) case in detail. We expect, however, a similar identification or a characterization of the coupling constants and scalar metric functions may be done in other magical supergravities.

II. DIMENSIONAL REDUCTION OF D = 5 MAGICAL SUPERGRAVITY
The magical supergravities are D = 5 N = 2 Einstein-Maxwell supergravities whose scalars of the vector multiplets constitute a coset sigma model with a symmetry group being a simple Lie group [15]. There exist four such theories, each of which is associated with one of the four division algebras A = R, C, H, O and a rank-3 Jordan algebra J A 3 associated with it. One of the characteristic features of these theories is that their five-dimensional Lagrangians as well as their dimensional reductions to four and three dimensions universally contain scalar sigma models of the forms [15,22]: where Aut(J A 3 ), Str 0 (J A 3 ), Mö(J A 3 ) and qConf(J A 3 ) are respectively the automorphicm group, the reduced structure group, the superstructure group and the quasi-conformal group of the Jordan algebra J A 3 . denotes the corresponding compact form. There supergravity theories have been dubbed "magical" [16] because these groups are precisely the elements of the "magic square" (see [16] and references therein), each Lie algebra L A,A ′ of which allows the decomposition The magical supergravity corresponding to the division algebra A has n = 3(1+dimA)−1 vector multiplets. Keeping only the bosonic terms, the Lagrangian is given by where E (5) is the determinant of the fünfbein, and R (5) is the scalar curvature in D = 5.
To reduce the dimensions to D = 3, we set the fünfbein and its inverse as To dualize A ′I µ and B m µ fields, we introduce Lagrange multipliers Using the equations of motion for F (3)I µν and B m µν , we obtain the dualized LagrangianL ≡ L + L Lag.mult. : III. The real form F 4(+4) of the exceptional Lie algebra F 4 is decomposed into a sum of representations of the Lie algebra of a maximal subgroup SL(3, R) × SL(3, R) as In spite of the notation, they are represented by real matrices. Later we will identify the first SL(3, R) as the global symmetry group arising from the reduction of the gravity sector from five to three dimensions, and the second one as the numerator group of the coset sigma-model scalars already existing in five dimensions. To distinguish them we call the first simply SL(3, R) while the second SL(3, R).
Their commutations relations are [Êãb,Êcd] = δc bÊãd − δãdÊcb, We also introduce additional generators E I i , E * i I (i = 1, 2, 3, I = 1, . . . , 6) transforming respectively as (3,6), (3,6) under SL(3, R) ⊕ SL(3, R): [Êãb, Tãb J I and Tãb I J are respectively the6 and 6 representation matrices of SL(3, R). In fact, in the present choice of the basis of the generators the structure constants satisfȳ Finally we set the commutation relations among two of these generators as where C IJK = C IJK are symmetric with respect to any permutation of indices, and Their actual values in the present basis are C 456 = +2, otherwise 0. One may verify that the commutations relations (9)- (19) close and generate the whole F 4(+4) Lie algebra.
• The remaining eight generators that do not belong to any of the above turn out to generate another SL(3, R) algebra, SL(3, R).
• Finally, one can verify that these six pairs of 3 and3 respectively transform as6 and 6 under SL(3, R).
In terms of the SL(3, R) × SL(3, R) decomposition, the whole F 4(+4) generators are classified into H and K, of which F 4(+4) is a direct sum: H consists of "compact" generators: The Killing bilinear form on H is negative definite. It turns out that the independent 3 + 3 + 18 = 24 generators of H generate USp(6) ⊕ SU (2). The generators of this factorized On the other hand, K is spanned by all the "noncompact" generators: The 52 − 28 = 24 generators of K parametrize the "physical" degrees of freedom of the F 4(+4) /(USp(6) × SU(2)) nonlinear sigma model. (2)) is a symmetric space for which we denote the Cartan involution as τ : τ (H) = −H, τ (K) = +K.
As usual, to construct a coset nonlinear sigma model, we define some group element V and Then the Lagrangian is given, up to a constant, by In order to reproduce the dimensionally reduced Lagrangian (7) of the magical supergravity, we take 3 V grav.
where we have taken the zweibein for the reduced dimensions to be in the upper-triangular For V − we take Then a straightforward calculation yields where are respectively the6 and 6 representation matrices of the SL(3, R) group elementẽ −1 (54).
We can give some Lie algebraic characterizations to various geometrical quantities defined in the supergravity Lagrangian: • C IJK 's are the structure constants of the commutation relations between generators both belonging to (3,6). In particular I = 1, . . . We note that the structures we found here are very similar to the dimensionally reduced eleven-dimensional supergravity or the D = 5 minimal supergravity to three dimensions [14,23,24], whose sigma models are respectively E 8(+8) /SO (16) and G 2(+2) /SO (4).
In all the magical supergravity theories, the number of the original scalars (= n) is always one less than the number of the abelian gauge fields. In the simplest magical case considered in this letter, this is the number of the dimension of the symmetric tensor representation, which is 6. In fact, for the other three magical cases, we can also find representations of the numerator group of the coset whose dimensions are pricisely one more than the dimensions of the coset of the respective theories [22]: • J C 3 magical:  (1, 8)).
The dimension of the five-dimensional scalar coset is so the index I runs from 1 to 9. This agrees with the fact that the direct product representation (3, 3) or (3,3) is nine-dimensional.
• J H 3 magical: In view of this common structure of decompositions (known as the decomposition of the quasi-conformal algebra of the Jordan algebra in terms of the super-Ehlers' algebra [22] ), we expect the same characterization for C IJK or • a IJ and • a IJ will be possible for the other three magical supergravity theories. To show this the realizations worked out in [20] will be useful. Work along this line is in progress.