New coset matrix for D=6 self-dual supergravity

Toroidal reduction of minimal six-dimensional supergravity, minimal five-dimensional supergravity and four-dimensional Einstein-Maxwell gravity to three dimensions gives rise to a sequence of cosets $O(4,3)/(O(4)\times O(3))\supset G_{2(2)}/(SU(2)\times SU(2))\supset SU(2,1)/S(U(2)\times U(1))$ which are invariant subspaces of each other. The known matrix representations of these cosets, however, are not suitable to realize these embeddings which could be useful for solution generation. We construct a new representation of the largest coset in terms of $7\times 7$ real symmetric matrices and show how to select invariant subspaces corresponding to lower cosets by algebraic constraints. The new matrix representative may be also directly applied to minimal five-dimensional supergravity. Due to full O(4,3) covariance it is simpler than the one derived by us previously for the coset $G_{2(2)}/(SU(2)\times SU(2))$.


Introduction
The remarkable sequence of groups O(7) ⊃ G 2 ⊃ SU (3) attracted attention in particle physics long ago. In a seminal paper Gunaydin and Gürsey [1] have given an extensive discussion of their properties, representations and applications to model building. Within the Lie algebra of O (7), the subalgebras G 2 and SU (3) form rather sophisticated closed structures which were explicitly given in [1] in terms of rotation generators of O (7).
The purpose of the present paper is to construct a new matrix representative of the coset O(4, 3)/(O(4) × O(3)) which allows for simple truncation to subspaces corresponding to MSG5 and EM4 theories. This is based again on the 7 × 7 representation, but with different parametrization of moduli. The new matrix is much simpler than the corresponding HSMS matrix and can be truncated to lower cosets by imposition of purely algebraic constraints. Our derivation is based on the direct toroidal reduction of MSG6 to three dimensions, explicit determination of target space isometry generators and subsequent exponentiation of the Borel subalgebra. We then extract the generators of the G 2(2) and SU (2, 1) subgroups of O(4, 3) and derive algebraic constraints selecting the corresponding invariant subspaces of the coset O(4, 3)/(O(4) × O(3)) on which they act transitively.

D = 6 minimal supergravity
The bosonic action of six-dimensional minimal supergravity contains the metric and self-dual three-form field whereĜμνλ ≡ 3Ĉ [μν,λ ] , with subsidiary condition which has to be imposed after variation of the action. 1 The action (2.1) is a lowest-dimensional member of the even-dimensional sequence of actions containing self-dual form fields, the largest representative of which is the IIB ten-dimensional supergravity. Somewhat unexpectedly, this action, being compactified on a circle, turns out to be non-locally dual to the truncated five-dimensional heterotic string effective action [17][18][19] which belongs to another sequence of the string actions. This can be hinted from the fact that the D = 5 heterotic string effective action truncated to the one-vector case gives rise to the D = 3 O(4, 3)/(O(4)×O (3)) coset theory (a particular case of the Sen's coset where d is the number of compactified dimensions and p is the number of vector fields in the initial dimension [21]). Meanwhile the generic oxidation of the O(4, 3)/(O(4) × O(3)) coset has apart from the regular oxidation point D = 5 (which is the above heterotic effective action) also an anomalous six-dimensional oxidation point [3] which is just minimal D = 6 self-dual supergravity. This leads to a non-local duality between the two theories which can be made explicit as follows.
Denoting the coordinates xμ = (x µ , z) and assuming existence of the Killing vector ∂ z , we decompose the metric and the two-form potential as where α 2 = 1/24. The field equations are then equivalent to those derived from the fivedimensional action As in some other supergravity actions involving self-dual form fields, the quadratic action of the type (2.1) does not imply the self-duality condition (2.2), moreover it is zero, if self-duality is imposed in the action itself. One needs extra fields to construct a consistent action for chiral forms in a Lorentz-covariant way. We thank Dmitri Sorokin for drawing our attention to the references [20] where such an action for D = 6 minimal supergravity was presented. Here we deal with classical equations of motion, so it will be sufficient to impose the condition (2.2) by hand after variation is performed. The dimensional reduction of the full action [20] is more involved, but this does not change the results on the classical level. This is a heterotic string type effective action [19,21] with one vector and one antisymmetric second rank tensor fields. Note that the Maxwell field F µν in this action originates from the sixdimensional three-form, while the five-dimensional three-form H µνρ is obtained by dualisation of the Kaluza-Klein two-form. Therefore the relation between the six and five-dimensional metrics and matter fields is non-local. Due to this duality one can reduce the six-dimensional action (2.1) (which is the subject of the present paper) to three dimensions along two different compactification schemes. The first consists in using the well-studied compactification of the corresponding five-dimensional heterotic string action (2.5) along the lines of [19,21]. The second, suggested in the present paper, consists in direct compactification of the initial six-dimensional action (2.1) on a three-torus.
The first way which we briefly sketch here gives Sen's type representation for the coset matrix [21]. Splitting the coordinates as z a , x µ = x i , a = 1, 2, i = 1, 2, 3, with z a along the compactified dimensions, we parameterize the metric and the matter fields as The three-dimensional reduced action can be presented in terms of the matrix sigma model According to [21], the coset matrix M is constructed in three steps: first one defines of the matrix M of non-dualized moduli, then dualisation of three-dimensional vectors to scalar potentials is performed, and finally the matrix M is constructed in terms of M and the new scalars. To built the moduli matrix M one arranges the five vector fields in a column matrix A A i (A = 1, ..., 5), with the field strengths where L is the 5 × 5 matrix written in block form The 2-form B i j is actually fixed by the gauge condition The 5 × 5 moduli matrix M AB then reads, in block form, 14) The next step involves dualisation of the three-dimensional vector fields according to Now, using the result of [21] it is straightforward to write down the 7 × 7 matrix M in a block form: (2.17) Thus, in principle, the Sen's matrix can be also used in the case of D = 6 minimal supergravity not belonging to the sequence of the heterotic string effective actions. But disadvantage of such an approach, apart from relative complexity of the matrix (2.17), lies in the fact that the variables of the five-dimensional heterotic action in terms of which this representation is written, are still non-trivially related to the initial six-dimensional variables. Another desired feature which can be demanded from the coset representation of the D = 6 theory is the possibility of its simple truncation to five-dimensional minimal supergravity whose D = 3 coset (3)). This can be achieved using the direct toroidal compactification of D = 6 minimal supergravity to three dimensions. Before doing this, we briefly review the coset structure of five-dimensional minimal supergravity. In both cases we will use the technique applied in [5] which consists in i) obtaining an explicit form form of the target space metric, ii) identifying its isometry algebra, iii) exponentiating the Borel subalgebra to get suitable matrix representation. Though technically different, it is conceptually the same construction as used by Maharana-Schwarz and Sen [18,21].

D = 5 minimal supergravity
The action of MSG5 reads with F = dA. We compactify on a two-torus using where a, b = 0, 1 and τ ≡ |detλ |. The ν = i components of the Maxwell-Chern-Simons equations allow to dualize the vector magnetic potential A i to a scalar magnetic potential µ defined by Similarly, the µ = i, ν = a components of the Einstein equations are integrated by where G b = da b , and ω a is the 'twist' or gravimagnetic two-potential. The D = 3 sigma model is then obtained with eight target space coordinates Φ A = {λ ab , ω a , ψ a , µ} and metric (3.7) This space has 14 Killing vectors which were determined in terms of these variables in [5,16]. Nine manifest infinitesimal symmetries (or generalised gauge transformations), grouped according to their transformations under GL(2R) (the group of linear transformations in the (z 1 , z 2 ) plane) into the quadruplet (the generators of the gl(2, R) subalgebra), the doublet and the singlet associated with the the three cyclic 'magnetic' coordinates: 9) and the doublet generating infinitesimal gauge transformations of the ψ a The five remaining, non trivial infinitesimal isometries L a , P a and T closing the Lie algebra g 2(2) are more complicated, their full expression is given in [16]. The L a , M a b and N a generate the vacuum subalgebra sl (3, R). Assuming a spacelike two-torus, the target space (3.7) is identified as the coset space G 2(2) /(SU (2) × SU (2)). The 7× 7 symmetric matrix representative of the coset obtained by exponentiation of the Borel subalgebra [5,16] exhibits a highly nonlinear dependence on the moduli. Its structure is quite different from that of the Sen matrix  1, 2, 3). The six-dimensional metric and 3-form may be parameterized by ) and the 10 remaining components ofĜ related to these by self-duality. PutB The remaining Einstein equations then lead to the gravitating sigma model with target space metric where The dimension of this target space is twelve: six components of the symmetric matrix λ ab and two triplets ω a , χ a . In Appendix A we check that it admits 21 Killing vectors generating the Lie algebra o(4, 3). These include nine  (4, R). Rather than using the Maison parametrization [22] of sl(4, R) in terms of 4 × 4 matrices (which presumably would lead to a representation of o(4, 3) in terms of 8 × 8 matrices), we use the representation of o (3,3) in terms of 6 × 6 matrices. These are then promoted to 7 × 7 matrices by the addition of a row and a column, and completed by six 7 × 7 matrices R a and P a closing the algebra o(4, 3). The 7 × 7 coset matrix representative is then constructed in a standard fashion as M = N T ηN , (4.8) where N is obtained by exponentiating a suitable Borel subalgebra of 0(4, 3), and η is a suitably chosen constant matrix. The resulting coset representative is, in block form, where˜denotes the anti-transposition, i.e. transposition relative to the anti-(or minor) diagonal, and One can check that the target space metric (4.6) can be expressed as In the case of a Lorentzian six-dimensional space E 6 with signature (− + + + ++) and an Euclidean reduced three-space (so that one of the Killing vectors of E 6 is timelike), the symmetric target space T of metric (4.6) is the coset G/H = O(4, 3)/O(2, 2) × O (2,1). H is the isotropy group leaving invariant any given point of the target space, which may be chosen to be the point at infinity of T . Thus it is relevant to examine the various possible asymptotic behaviors for asymptotically flat six-dimensional configurations.
Minkowski asymptotics. For an asymptotically Minkowskian metric, or for a metric which is asymptotically the product of a four-dimensional black hole by a 2-torus, with x 1 the time coordinate, the asymptotic coset representative is (4.12) This asymptotic behavior is preserved by the nine Killing vectors (with the first three pure gauge), satisfying the commutation relations with η 1 = −1, η 2 = η 3 = +1. The combinations generate three commuting copies of the Lie algebra of O(2, 1), We thus recover the isotropy subgroup Black string asymptotics. The static Myers-Perry (Tangherlini) six-dimensional black string (the product of a five-dimensional black hole by a circle) is This has four commuting Killing vectors. Reduction relative e.g. to ∂ 1 = ∂ t , ∂ 2 = ∂ η and ∂ 3 = ∂ ζ leads to leading to with c a constant of integration. Computation of the asymptotic behavior of the lower right-hand side 3 × 3 block in (4.9) gives 20) which is equal to the asymptotic behavior of the upper left-hand side block for the value c = −m/2. In this case, This asymptotic behavior is preserved by the nine Killing vectors X 0 , X − , Y 0 , Y − and Z − are pure gauge. The first three generate an SL(2, R), 23) or symbolically [X , X ] = X . The full algebra (with commutators such as [X 0 ,Y 0 ] and [X ± ,Y ± ] vanishing) can be split, as in the case of Minkowski asymptotics, into three commuting sl(2, R) = O(2, 1) generated by the combinations has only three commuting Killing vectors ∂ t , ∂ ζ and ∂ ϕ . Reduction relative to these vectors leads to The resulting matrix M has no regular limit at spatial infinity. As in the well-known case of fourdimensional Einstein-Maxwell theory reduced relative to the azimuthal Killing vector [23], new solutions can be generated from this by O(4, 3) transformations, but these will be always nonasymptotically flat.

Invariant subspaces
The new matrix (4.9) parameterizes the twelve-dimensional coset space of MSG6 theory. It may be also applied as a representative of the embedded eight-dimensional coset G 2(2) /(SU (2) × SU (2) corresponding to MSG5 and four-dimensional coset SU (2, 1)/S(U (2) × U (1)) corresponding to EM4. These may be selected by purely algebraic constraints on the potentials. To find these constraints one has to consider dimensional reductions and consistent truncations which relate these theories to MSG6.

D = 5 minimal supergravity
The coset G 2(2) /(SU (2) × SU (2) is a totally geodesic subspace of the coset O(4, 3)/(O(4) × O(3)), so MSG6 compactified on a circle can be consistently truncated to MSG5. Indeed, it can be checked [24] that the equations of motion following from (2.5) are consistent with the constraints in which case they reduce to those of MDG5, provided the two-form field F µν is rescaled by Note that, in view of the second relation in (2.6), the second contraint (5.1) is equivalent to identification of the Maxwell two-form and the Kaluza-Klein two-form of the reduced theory (2.5), in which case the dilaton can be consistently set to zero. We must now identify the corresponding constraints in terms of the target space variables. Inspecting the definitions of the three-dimensional target space variables (4.1)-(4.5) one finds Since the G 2 sector arises as a consistent truncation of the five-dimensional reduction of the original six-dimensional model, it is not surprising that in the reduction to three dimensions the threecovariance is broken down to two-covariance. Actually, knowing this (and making some educated guesses) is enough to find the two g 2 subalgebras generated by the Killing vectors preserving the constraints (5.3). In the two-covariant notation of Sect. 4 of [5], their generators are related to those of o(4, 3) by with a, b = 1, 2. It is easy to check that these combinations satisfy the commutation relations (92)-(97) of [5].
Conversely, comparing the covariant three-dimensional reductions of MSG5 and MSG6, we see that any solution (ds 2 (5) , A (5) ) of MSG5 with two commuting Killing vectors can be oxidized to a solution of MSG6 with three commuting Killing vectors given by It follows that, given a solution of MSG5 with two commuting Killing vectors, one can generate from this a new solution by going through the following steps: 1) oxidize the seed solution to a solution of MSG6 by (5.5); 2) construct its coset representative (4.9); 3) transform this, by the action of an O(4, 3) transformation P generated by the generators of the second column of (5.4); 4) extract from M ′ the new solution of MSG6; 5) reduce this to five dimensions by (5.5). In view of the simplicity of the matrix representation (4.9) compared to that previously known for MSG5, this procedure might be easier to implement than direct generation by G 2 (2) transformations.
The generators preserving both five-dimensional Myers-Perry (or black string) asymptotics and G 2 truncation are generating two commuting copies of sl(2, R) = o (2, 1). The non-trivial generators are Clearly, all the generators (4.22) preserving black string asymptotics are linear combinations of the four non-trivial generators F + , G +(+) , G +(−) , and one of the G 0(±) , together with gauge transformations. These four generators applied to a black string will do the same job as the corresponding G 2 generators (and in particular, preserve the black string condition λ 33 = 1), but in a simpler fashion. The non-vanishing commutators between these four generators are

D = 4 Einstein-Maxwell
To identify the constraints selecting the SU (2, 1)/S(U (2) × U (1)) subspace of the G 2 coset, one must first compactify MSG5 on a circle [5], since this subcoset corresponds to four-dimensional Einstein-Maxwell theory. Assuming the existence of a space-like Killing vector ∂ z , we parametrize the five-dimensional metric and Maxwell field by 2 4 , (5.10) where and F * is the four-dimensional Hodge dual of F, describes an Einstein theory with two coupled abelian gauge fields F and G, a dilaton φ and an axion κ. The field equations in terms of the four-dimensional variables read  (4) . (5.18) After reduction to three dimensions, this leads to the constraints We find that these constraints are preserved by the eight infinitesimal transformations (5.20) From the commutation relations given in [16], we find that the K M (M = 1, ..., 8) generate the Lie algebra of SU (2, 1), which may be put in the Cartan-Weyl form [25], with (5.21)

Outlook
The main result of this paper is the new representative (4.9) of the coset O(4, 3)/(O(4) × O(3)) which is a symmetric 7×7 matrix, given in block form. This matrix is substantially simpler than the matrix (2.17) constructed by Sen's method. Moreover, it also looks simpler than the G 2 matrix for the coset G 2(2) /(SU (2) × SU (2)) constructed by us previously [5] and used for solution generation in [5,13]. The reason is that constraining to the subspace G 2(2) /(SU (2) × SU (2)) one loses the O(4, 3) covariance which simplifies the underlying matrix structure. After truncation to vacuum five-dimensional gravity (χ a = 0), the new matrix leads to a 7 × 7 matrix representative of the coset SL(3, R)/O(2, 1) which is different from that resulting from the truncation to vacuum gravity (ψ a = µ = 0) of our previous G 2 matrix [5]. It is therefore expected that imposing the constraints ( 3)) one should obtain a 7 × 7 representative of the G2 coset different from our previous one. This new G2 matrix could also be obtained as in [5,16] O(4, 3) transformations which do not belong to the G 2(2) subgroup, which may be chosen to have required asymptotic properties, as discussed at the end of section 4. The same arguments equally apply to the SU (2, 1)/S(U (2) ×U (1)) subspace of the G2 coset.

A Isometry algebra of the metric (4.6)
Fifteen obvious Killing vectors are: generating linear transformations in the three-Killing vector space, generating translations of the "magnetic" coordinates ω a , and generating gauge transformations of the χ a . Their commutation relations are Three more vectors L a are needed to complete the algebra sl(4, R) of the vacuum sector: Adding to the known form of the sl(4, R) for 6D Einstein the information from (1.11), (the omitted terms are of order 0 in ω a ). Assuming that the full Lie algebra is O (4, 3), it must close with the three remaining generators P a defined by [R a , L b ] = ε abc P c , (1.14) leading to and obeying the commutation relations The degrees of the various fields can be found from their commutators with Tr(M); This leads to The resulting coset representative is related to its inverse by