Throughout this section we writeG=G2(2)for the simply connected noncompact form of the Lie group of typeG2.
5.2.1 Description of the Lie algebra of G2(2)
We now seek to describe the Lie algebragof the Lie groupG. We follow Ch. 5, Section 1.2 of [106].
The Lie algebra gadmits a Z3-grading
g=V +sl(V) +V∗,
where the subspaces have degree (1,0,−1) ∈ Z3 respectively under the grading, and
V =R3. Note that sl(V) =sl(3,R). The Lie brackets are given by
[A, x] = Ax, [A, ξ] =−ATξ,
[x, y] = −2x×y,
[ξ, η] = 2ξ×η,
[x, ξ] = 3x⊗ξ−ξ(x)Id∈sl(V)⊂gl(V)∼=V ⊗V∗, (5.11)
for all x, y∈V,ξ, η ∈V∗ and A∈sl(V). The cross products are defined by
x×y = det(x, y,·)∈V∗, ξ×η= det−1(ξ, η,·)∈V∗∗=V,
where det−1 ∈ ∧3V is the inverse of det∈ ∧3V∗. We shall denote by (e
i) = (e1, e2, e3) the standard basis of V, by (ei) its dual basis and by eji the endomorphism ei⊗ej of
V. Then, e.g. e1×e2 =e3.
The Cartan subalgebra a ⊂g consisting of all diagonal matrices in sl(V) is given by a= ( X i λieii X i λi= 0 ) .
5.2.2 The solvable Iwasawa subgroup L⊂G
Choose a basis
H1 =e11−e33, H2=e11−2e22+e33,
of the Cartan subalgebra a ⊂ g. Then, following Section 2.4.3 we can construct a maximal nilpotent subalgebra n⊂g
n= span{e1, e2, e3, e12, e31, e32} ⊂g,
as the sum of the positive root spaces ofa. We then definel=a⊕n⊂g, which is the2 solvable Lie algebra (Iwasawa subalgebra) appearing in the Iwasawa decomposition of g, as described in Section 2.4.4. Define a basis (V1, . . . ,V8) oflby V1=−3e31, V2 = 1 2(e 1 1−e33), V3= 1 √ 3e 2, V 4= 1 2√3(e 1 1−2e22+e33), V5= √ 3e32, V6 =e3, V7 :=− √ 3e21, V8 :=−e1. (5.12)
Using the relations (5.11) we can show that the Iwasawa subalgebra l ⊂ g with this basis has precisely the same nontrivial brackets (5.8) as the basis (Ta) of the Lie algebra
obtained from each of the three dimensional reductions.
We have seen so far then that the three reductions of pure five-dimensional super- gravity (SS, ST and TS) provide us with scalar manifolds M(SS), M(ST) and M(T S)
which can all be identified with the group manifoldLof an Iwasawa subgroup ofG2(2), parametrized by (x, σ, φ,φ, ζ˜ 0, ζ1,ζ˜0,ζ˜1). For each of these reductions the scalar mani- fold is equipped with a different left-invariant metricg(1,2) with signature
sign(g) = (−1,+,−1,+,−,−2,−2,−). (5.13)
Hence, for SS reduction the metric is positive definite, while for ST and TS reductions we obtain split-signature metrics, albeit with a different distribution of signs.
In the case of the Riemannian symmetric spaceG/SO4, we know thatSO4is a max-
2
imally compact subgroup of G, and so the Iwasawa decomposition is exact (Theorem 6.46 of [79]). In particular, this means that the Iwasawa subgroup L acts transitively on the symmetric spaceG/SO4. Hence, we can identifyglobally the Riemannian man- ifold (M(SS), g(SS)) obtained from SS reduction with the Riemannian symmetric space
G/SO4, or alternatively with the L-orbitL·o of the canonical base pointo∈G/SO4. For the split-signature case, we need to study the pseudo-Riemannian real form
G/(SL2·SL2), which we turn to next.
5.2.3 The symmetric space S =G2(2)/(SL2 ·SL2)
In order to describe the pseudo-Riemannian symmetric space appearing in the ST and TS reductions of five-dimensional supergravity, we introduce the Z2-grading g =
gev+godd of the Lie algebra gwith
gev = a+ span{e3, e3, e12, e12} ∼=sl2⊕sl2,
godd = span{e1, e2, e1, e2, e31, e32, e13, e23}.
That this is indeed a Z2-grading of g can be seen simply by using the Lie brackets (5.11). The twosl2 factors are generated by the sl2-triples
(h(1)= [e21, e12] =e11−e22, e(1) =e21, f(1)=e12), and (h(2)= [e3, e3] =−e11−e22+ 2e33, e(2)=e3, f(2)=e3). That is, h h(a),e(a) i = 2e(a), h h(a),f(a) i =−2f(a), h e(a),f(a) i =h(a), and X(1), X(2) = 0,for any X(a)∈sl(a) 2 .
The corresponding pseudo-Riemannian symmetric spaceS =G/Gev is
S= G2(2)
SL2·SL2
Proposition 6. The symmetric space S admits a G-invariant para-quaternionic- K¨ahler structure(g, Q), where the metricg is induced by a multiple of the Killing form. Hence (S, g, Q) is a para-quaternionic-K¨ahler manifold.
Proof: The proof is given as Proposition 1 of [37].
We saw earlier that the standard Iwasawa subgroup L⊂Gacts transitively on the Riemannian symmetric space G/SO4. However, the orbit L·o of the canonical base pointo∈S if not even open (in the topological sense). To see this we note that, given a subgroup U ⊂ G, the orbit U ·o ⊂ S = G/Gev of the canonical base point o ∈ S
is open iff gev+ Lie(U) = g, i.e. dim(U) = dim(G/Gev). For an Iwasawa subalgebra
l0 ⊂g, this is the case iffgev∩l0= 0.
Clearly, for the Iwasawa subalgebralin Section 5.2.2, we see l∩gev 6= 0, and so the
orbitL·o⊂S is not open. This means that we cannot globally describe the pseudo- Riemannian symmetric spaceS simply by picking some representative in the standard Iwasawa subgroupL⊂G, as was the case for the Riemannian symmetric spaceG/SO4. This is related to the failure of the Iwasawa decomposition to provide a global covering of G/H for H non-compact. However, it is still possible to find a decomposition of the form HL for an open subsetU ⊂G, thus providing a local parametrization of the symmetric space G/H. In this case, the orbitL·o of the canonical base point will be open.
This failure of the Iwasawa decomposition has important consequences for station- ary supergravity solutions, which in many applications can be described by cosetsG/H
forH non-compact [35]. In [107] it was shown that solutions with regular event hori- zons correspond to complete geodesics contained within an open orbit of the Iwasawa subgroup, whereas those not contained in a single orbit lift to singular spacetimes. In [108] it was argued that elements of G for which the Iwasawa decomposition fails to hold correspond to the so-called ‘active duality transformations’ mapping BPS to non-BPS solutions.
5.2.4 Open orbits in the symmetric space
The Iwasawa decomposition is not unique. In fact, given some Iwasawa subgroup
L ⊂ G, we can find a conjugate Iwasawa subgroup L0 = Ca(L) := aLa−1 for a ∈ G
with the same Lie algebra3. In the case whereL0 acts with open orbit inS, we obtain a left-invariant locally symmetric para-quaternionic-K¨ahler structure on L0 ∼=L induced from the symmetric para-quaternionic-K¨ahler structure onS.
Our strategy in this section is to look for conjugate Iwasawa subgroupsL0 =Ca(L)
such that the orbit M1 =L0·o is open in S, and then show that M1 is isometrically covered by one of the scalar manifoldsM(ST) orM(T S). We do this at the level of the algebra: we seeka∈G such that the conjugate Iwasawa subalgebral0 = Ada(l)⊂gis
transversal togev.
To proceed, we first pick some element a= exp(ξ)∈G, whereξ ∈g, and calculate
X0 = AdaX = eadξX for each X ∈ l. One could then compare a generic element of
l0 with a generic element ofgev to see whether there exist any non-trivial members of
their intersection. However, we already know thatgodd∩gev = 0. Hence, if we can find
some vector space isomorphismϕ :l→ godd, then l0 ∼=godd will be transversal to gev. To construct such a map, let us denote by π :g → godd the projection along gev, and by ϕ:l→godd the map
X7→π(X0).
If the vectorsϕ(Vb) are linearly independent, then ϕis a vector space isomorphism.
From the isomorphism ϕ : l → godd we compute the left-invariant metric g1 on
L∼=L0 from the scalar producth·,·i1 =ϕ∗h·,·iB onl, whereh·,·iB is the scalar product on godd obtained by restricting 18B togodd. Here B is the usual Killing form on g, as defined in (2.31). It may still be the case thatg1in the basis (Va) does not correspond to
eitherg(ST) org(T S), which are both diagonal with respect to this basis. However, the metrics could still be equivalent if they are related by the action of some automorphism ofL (up to a positive scale factor). This will lead us in the next section to analyse the automorphism group ofL, or equivalently of l(sinceL is connected).
3That is,l= Lie(L) andl0
In what follows we use the basis of ggiven by
b1 =e11−e22, b2=e22−e33, b3 =e21, b4=e31, b5 =e32, b6=e12, b7 =e13,
b8 =e23 , b9=e1 , b10=e2, b11=e3 , b12=e1 , b13=e2 , b14=e3. (5.14)
In this notation, l = span{b1, b2, b3, b4, b5, b9, b13, b14}, while godd we take to have the
basis (f1, . . . , f8) := (b9, b10, b12, b13, b4, b5, b7, b8).
For future use, we note that the non-trivial scalar productsh·,·iB between elements
of godd are given by
hf1, f3iB =hf2, f4iB = 3, hf6, f8iB=hf5, f7iB= 1, (5.15)
from which we can read off the corresponding Gram matrix representing h·,·iB:
G = 0 0 3 0 0 0 0 0 0 0 0 3 0 0 0 0 3 0 0 0 0 0 0 0 0 3 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 . (5.16)