1 Freudenthal Duality

We start and consider the following Lagrangian density in four dimensions, (cf., e.g., [1]):

(1)

describing Einstein gravity coupled to Maxwell (Abelian) vector fields and to a non-linear sigma model of scalar fields (with no potential); note that \(\mathcal {L}\) may -but does not necessarily need to - be conceived as the bosonic sector of \(D=4\) (ungauged) supergravity theory. Out of the Abelian two-form field strengths \(F^{\varLambda }\)’s, one can define their duals \(G_{\varLambda }\), and construct a symplectic vector:

(2)

We then consider the simplest solution of the equations of motion deriving from \(\mathcal {L}\), namely a static, spherically symmetric, asymptotically flat, dyonic extremal black hole with metric [2]

$$\begin{aligned} ds^{2}=-e^{2U(\tau )}dt^{2}+e^{-2U(\tau )}\left[ \frac{d\tau ^{2}}{\tau ^{4}} +\frac{1}{\tau ^{2}}\left( d\theta ^{2}+\sin \theta d\psi ^{2}\right) \right] , \end{aligned}$$
(3)

where \(\tau :=-1/r\). Thus, the two-form field strengths and their duals can be fluxed on the two-sphere at infinity \(S_{\infty }^{2}\) in such a background, respectively yielding the electric and magnetic charges of the black hole itself, which can be arranged in a symplectic vector \(\mathcal {Q}\):

$$\begin{aligned} p^{\varLambda }:= & {} \frac{1}{4\pi }\int _{S_{\infty }^{2}}F^{\varLambda }, \quad q_{\varLambda }:=\frac{1}{4\pi }\int _{S_{\infty }^{2}}G_{\varLambda }, \end{aligned}$$
(4)
$$\begin{aligned} \mathcal {Q}:= & {} \left( p^{\varLambda },q_{\varLambda }\right) ^{T}. \end{aligned}$$
(5)

Then, by exploiting the symmetries of the background (3), the Lagrangian (6) can be dimensionally reduced from \(D=4\) to \(D=1\), obtaining a 1-dimensional effective Lagrangian (\(^{\prime }:=d/d\tau \)) [3]:

$$\begin{aligned} \mathcal {L}_{D=1}=\left( U^{\prime }\right) ^{2}+g_{ij}\left( \varphi \right) \varphi ^{i\prime }\varphi ^{j\prime }+e^{2U}V_{BH}\left( \varphi , \mathcal {Q}\right) \end{aligned}$$
(6)

along with the Hamiltonian constraint [3]

$$\begin{aligned} \left( U^{\prime }\right) ^{2}+g_{ij}\left( \varphi \right) \varphi ^{i\prime }\varphi ^{j\prime }-e^{2U}V_{BH}\left( \varphi ,\mathcal {Q} \right) =0. \end{aligned}$$
(7)

The so-called “effective black hole potential” \(V_{BH}\) appearing in (6) and (7) is defined as [3]

$$\begin{aligned} V_{BH}\left( \varphi ,\mathcal {Q}\right) :=-\frac{1}{2}\mathcal {Q}^{T} \mathcal {M}\left( \varphi \right) \mathcal {Q}, \end{aligned}$$
(8)

in terms of the symplectic and symmetric matrix [1]

$$\begin{aligned} \mathcal {M}:= & {} \left( \begin{array}{cc} \mathbf {I} &{} -R \\ 0 &{} \mathbf {I} \end{array} \right) \left( \begin{array}{cc} I &{} 0 \\ 0 &{} I^{-1} \end{array} \right) \left( \begin{array}{cc} \mathbf {I} &{} 0 \\ -R &{} \mathbf {I} \end{array} \right) =\left( \begin{array}{ccc} I+RI^{-1}R &{} ~ &{} -RI^{-1} \\ ~ &{} ~ &{} ~ \\ -I^{-1}R &{} &{} I^{-1} \end{array} \right) , \end{aligned}$$
(9)
$$\begin{aligned} \mathcal {M}^{T}= & {} \mathcal {M};~\mathcal {M}\varOmega \mathcal {M}=\varOmega , \end{aligned}$$
(10)

where \(\mathbf {I}\) denotes the identity, and \(R\left( \varphi \right) \) and \( I\left( \varphi \right) \) are the scalar-dependent matrices occurring in (6); moreover, \(\varOmega \) stands for the symplectic metric (\(\varOmega ^{2}=-\mathbf {I}\)). Note that, regardless of the invertibility of \(R\left( \varphi \right) \) and as a consequence of the physical consistence of the kinetic vector matrix \(I\left( \varphi \right) \), \(\mathcal {M}\) is negative-definite; thus, the effective black hole potential (8) is positive-definite.

By virtue of the matrix \(\mathcal {M}\), one can introduce a (scalar-dependent) anti-involution \(\mathcal {S}\) in any Maxwell-Einstein-scalar theory described by (6) with a symplectic structure \(\varOmega \), as follows:

$$\begin{aligned} \mathcal {S}\left( \varphi \right):= & {} \varOmega \mathcal {M}\left( \varphi \right) . \end{aligned}$$
(11)

Indeed, by (10),

(12)

In turn, this allows to define an anti-involution on the dyonic charge vector \(\mathcal {Q}\), which has been called (scalar-dependent) Freudenthal duality [4,5,6]:

$$\begin{aligned} \mathbf {F}\left( \mathcal {Q};\varphi \right):= & {} -\mathcal {S}\left( \varphi \right) \mathcal {Q}; \end{aligned}$$
(13)
$$\begin{aligned} \mathbf {F}^{2}= & {} -\mathbf {I}, (\forall \left\{ \varphi \right\} ). \end{aligned}$$
(14)

By recalling (8) and (11), the action of \(\mathbf {F}\) on \(\mathcal {Q}\), defining the so-called (\(\varphi \)-dependent) Freudenthal dual of \(\mathcal {Q}\) itself, can be related to the symplectic gradient of the effective black hole potential \(V_{BH}\):

$$\begin{aligned} \mathbf{{F}}\left( \mathcal {Q};\varphi \right) =\varOmega \frac{\partial V_{BH}\left( \varphi ,\mathcal {Q}\right) }{\partial \mathcal {Q}}. \end{aligned}$$
(15)

Through the attractor mechanism [7], all this enjoys an interesting physical interpretation when evaluated at the (unique) event horizon of the extremal black hole (3) (denoted below by the subscript “H”); indeed

$$\begin{aligned} \partial _{\varphi }V_{BH}= & {} 0\Leftrightarrow \lim _{\tau \rightarrow -\infty }\varphi ^{i}\left( \tau \right) =\varphi _{H}^{i}\left( \mathcal {Q} \right) ; \end{aligned}$$
(16)
$$\begin{aligned} S_{BH}\left( \mathcal {Q}\right)= & {} \frac{A_{H}}{4}=\pi \left. V_{BH}\right| _{\partial _{\varphi }V_{BH}=0}=-\frac{\pi }{2}\mathcal {Q} ^{T}\mathcal {M}_{H}\left( \mathcal {Q}\right) \mathcal {Q}, \end{aligned}$$
(17)

where \(S_{BH}\) and \(A_{H}\) respectively denote the Bekenstein-Hawking entropy [8] and the area of the horizon of the extremal black hole, and the matrix horizon value \(\mathcal {M}_{H}\) is defined as

$$\begin{aligned} \mathcal {M}_{H}\left( \mathcal {Q}\right) :=\lim _{\tau \rightarrow -\infty } \mathcal {M}\left( \varphi \left( \tau \right) \right) . \end{aligned}$$
(18)

Correspondingly, one can define the (scalar-independent) horizon Freudenthal duality \(\mathbf{{F}}_{H}\) as the horizon limit of (13):

$$\begin{aligned} \widetilde{\mathcal {Q}}\equiv \mathbf{{F}}_{H}\left( \mathcal {Q}\right) :=\lim _{\tau \rightarrow -\infty }{} \mathbf{{F}}\left( \mathcal {Q};\varphi \left( \tau \right) \right) =-\varOmega \mathcal {M}_{H}\left( \mathcal {Q} \right) \mathcal {Q}=\frac{1}{\pi }\varOmega \frac{\partial S_{BH}\left( \mathcal {Q}\right) }{\partial \mathcal {Q}}. \end{aligned}$$
(19)

Remarkably, the (horizon) Freudenthal dual of \(\mathcal {Q}\) is nothing but (\(1/\pi \) times) the symplectic gradient of the Bekenstein-Hawking black hole entropy \(S_{BH}\); this latter, from dimensional considerations, is only constrained to be an homogeneous function of degree two in \(\mathcal {Q}\). As a result, \(\widetilde{\mathcal {Q}}=\widetilde{\mathcal {Q}}\left( \mathcal {Q} \right) \) is generally a complicated (non-linear) function, homogeneous of degree one in \(\mathcal {Q}\).

It can be proved that the entropy \(S_{BH}\) itself is invariant along the flow in the charge space \(\mathcal {Q}\) defined by the symplectic gradient (or, equivalently, by the horizon Freudenthal dual) of \(\mathcal {Q}\) itself:

(20)

It is here worth pointing out that this invariance is pretty remarkable: the (semi-classical) Bekenstein-Hawking entropy of an extremal black hole turns out to be invariant under a generally non-linear map acting on the black hole charges themselves, and corresponding to a symplectic gradient flow in their corresponding vector space.

For other applications and instances of Freudenthal duality, see [9,10,11,12,13,14].

2 Groups of Type \(E_{7}\)

The concept of Lie groups of type \( E _{7}\) was introduced in the 60s by Brown [15], and then later developed e.g. by [16,17,18,19,20].

Starting from a pair \((G,\mathbf {R})\) made of a Lie group G and its faithful representation \(\mathbf {R}\), the three axioms defining (\(G,\mathbf {R}\)) as a group of type \(E_{7}\) read as follows:

  1. 1.

    Existence of a unique symplectic invariant structure \(\varOmega \) in \( \mathbf {R}\):

    $$\begin{aligned} \exists !\varOmega \equiv \mathbf {1}\in \mathbf {R}\times _{a}\mathbf {R}, \end{aligned}$$
    (21)

    which then allows to define a symplectic product \(\left\langle \cdot ,\cdot \right\rangle \) among two vectors in the representation space \(\mathbf {R}\) itself:

    $$\begin{aligned} \left\langle \mathcal {Q}_{1},\mathcal {Q}_{2}\right\rangle :=\mathcal {Q}_{1}^{M}\mathcal {Q}_{2}^{N}\varOmega _{MN}=-\left\langle \mathcal {Q}_{2},\mathcal {Q}_{1}\right\rangle . \end{aligned}$$
    (22)
  2. 2.

    Existence of a unique rank-4 completely symmetric invariant tensor (K-tensor) in \(\mathbf {R}\):

    $$\begin{aligned} \exists !K\equiv \mathbf {1}\in \left( \mathbf {R}\times \mathbf {R}\times \mathbf {R}\times \mathbf {R}\right) _{s}, \end{aligned}$$
    (23)

    which then allows to define a degree-4 invariant polynomial \(I_{4}\) in \(\mathbf {R}\) itself:

    $$\begin{aligned} I_{4}:=K_{MNPQ}\mathcal {Q}^{M}\mathcal {Q}^{N}\mathcal {Q}^{P}\mathcal {Q}^{Q}. \end{aligned}$$
    (24)
  3. 3.

    Defining a triple map T in \(\mathbf {R}\) as

    $$\begin{aligned} T&:\mathbf {R\times R\times R\rightarrow R}; \end{aligned}$$
    (25)
    $$\begin{aligned} \left\langle T\left( \mathcal {Q}_{1},\mathcal {Q}_{2},\mathcal {Q}_{3}\right) ,\mathcal {Q}_{4}\right\rangle&:= K_{MNPQ}\mathcal {Q}_{1}^{M}\mathcal {Q}_{2}^{N}\mathcal {Q}_{3}^{P}\mathcal {Q}_{4}^{Q}, \end{aligned}$$
    (26)

    it holds that

    (27)

    This property makes a group of type \(E_{7}\) amenable to a description as an automorphism group of a Freudenthal triple system (or, equivalently, as the conformal groups of an underlying Jordan triple system).

All electric-magnetic duality (U-dualityFootnote 1) groups of \(\mathcal {N} \ge 2\)-extended \(D=4\) supergravity theories with symmetric scalar manifolds are of type \(E_{7}\). Among these, degenerate groups of type \(E_{7}\) are those in which the K-tensor is actually reducible, and thus \(I_{4}\) is the square of a quadratic invariant polynomial \(I_{2}\). In fact, in general, in theories with electric-magnetic duality groups of type \(E_{7}\) holds that

(28)

whereas in the case of degenerate groups of type \(E_{7}\) it holds that \(I_{4}\left( \mathcal {Q}\right) =\left( I_{2}\left( \mathcal {Q}\right) \right) ^{2}\), and therefore the latter formula simplifies to

$$\begin{aligned} S_{BH}=\pi \sqrt{\left| I_{4}\left( \mathcal {Q}\right) \right| }=\pi \left| I_{2}\left( \mathcal {Q}\right) \right| . \end{aligned}$$
(29)
Table 1 Simple, non-degenerate groups G related to Freudenthal triple systems \(\mathbf{{M}}\left( J_{3}\right) \) on simple rank-3 Jordan algebras \(J_{3}\). In general, \(G\cong Conf\left( J_{3}\right) \cong Aut\left( \mathbf{{M}}\left( J_{3}\right) \right) \) (see e.g. [23,24,25] for a recent introduction, and a list of Refs.). O, H, C and R respectively denote the four division algebras of octonions, quaternions, complex and real numbers, and \(O_{s}\), \(H_{s}\), \(C_{s}\) are the corresponding split forms. Note that the G related to split forms \(O_{s}\), \(H_{s}\), \(C_{s}\) is the maximally non-compact (split) real form of the corresponding compact Lie group. \(M_{1,2}\left( O \right) \) is the Jordan triple system generated by \(2\times 1\) vectors over O [26]. Note that the STU model, based on \(J_{3}= R\oplus R\oplus R\), has a semi-simple \(G_{4}\), but its triality symmetry [27] renders it “effectively simple”. The \(D=5\) uplift of the \(T^{3}\) model based on \(J_{3}=R\) is the pure \(\mathcal {N}=2\), \(D=5\) supergravity. \(J_{3}^{H}\) is related to both 8 and 24 supersymmetries, because the corresponding supergravity theories are “twin”, namely they share the very same bosonic sector [26, 28,29,30].
Full size table

Simple, non-degenerate groups of type \(E_{7}\) relevant to \(\mathcal {N} \ge 2\)-extended \(D=4\) supergravity theories with symmetric scalar manifolds are reported in Table 1.

Semi-simple, non-degenerate groups of type \(E_{7}\) of the same kind are given by \(G=SL(2,R)\times SO(2,n)\) and \(G=SL(2,R)\times SO(6,n)\), with \(\mathbf {R}=\left( \mathbf {2},\mathbf {2+n}\right) \) and \(\mathbf {R}=\left( \mathbf {2},\mathbf {6+n}\right) \), respectively relevant for \(\mathcal {N}=2\) and \(\mathcal {N}=4\) supergravity.

Moreover, degenerate (simple) groups of type \(E_{7}\) relevant to the same class of theories are \(G=U(1,n)\) and \(G=U(3,n)\), with complex fundamental representations \(\mathbf {R}=\mathbf {n+1}\) and \(\mathbf {R}=\mathbf {3+n}\), respectively relevant for \(\mathcal {N}=2\) and \(\mathcal {N}=3\) supergravity [19].

The classification of groups of type \(E_{7}\) is still an open problem, even if some progress have been recently made e.g. in [31] (in particular, cf. Table D therein).

In all the aforementioned cases, the scalar manifold is a symmetric cosets \(\frac{G}{H}\), where H is the maximal compact subgroup (with symmetric embedding) of G. Moreover, the K-tensor can generally be expressed as [20]

$$\begin{aligned} K_{MNPQ}=-\frac{n(2n+1)}{6d}\left[ t_{MN}^{\alpha }t_{\alpha \mid PQ}-\frac{d}{n\left( 2n+1\right) }\varOmega _{M(P}\varOmega _{Q)N}\right] , \end{aligned}$$
(30)

where \(\dim \mathbf {R}=2n\) and \(\dim G=d\), and \(t_{MN}^{\alpha }\) denotes the symplectic representation of the generators of G itself. Thus, the horizon Freudenthal duality can be expressed in terms of the K-tensor as follows [4]:

$$\begin{aligned} \mathbf{{F}}_{H}\left( \mathcal {Q}\right) _{M}\equiv \widetilde{\mathcal {Q}} _{M}=\frac{\partial \sqrt{\left| I_{4}\left( \mathcal {Q}\right) \right| }}{\partial \mathcal {Q}^{M}}=\epsilon \frac{2}{\sqrt{\left| I_{4}\left( \mathcal {Q}\right) \right| }}K_{MNPQ}\mathcal {Q}^{N}\mathcal {Q}^{P}\mathcal {Q}^{Q}, \end{aligned}$$
(31)

where \(\epsilon :=I_{4}/\left| I_{4}\right| \); note that the horizon Freudenthal dual of a given symplectic dyonic charge vector \(\mathcal {Q}\) is well defined only when \(\mathcal {Q}\) is such that \(I_{4}\left( \mathcal {Q} \right) \ne 0\). Consequently, the invariance (20) of the black hole entropy under the horizon Freudenthal duality can be recast as the invariance of \(I_{4}\) itself:

$$\begin{aligned} I_{4}\left( \mathcal {Q}\right) =I_{4}\left( \widetilde{\mathcal {Q}}\right) =I_{4}\left( \varOmega \frac{\partial \sqrt{\left| I_{4}\left( \mathcal {Q} \right) \right| }}{\partial \mathcal {Q}}\right) . \end{aligned}$$
(32)

In absence of “flat directions” at the attractor points (namely, of unstabilized scalar fields at the horizon of the black hole), and for \(I_{4}>0\), the expression of the matrix \(\mathcal {M}_{H}\left( \mathcal {Q}\right) \) at the horizon can be computed to read

$$\begin{aligned} \mathcal {M}_{H\mid MN}(\mathcal {Q})=-\frac{1}{\sqrt{I_{4}}}\left( 2 \widetilde{\mathcal {Q}}_{M}\widetilde{\mathcal {Q}}_{N}-6K_{MNPQ}\mathcal {Q} ^{P}\mathcal {Q}^{Q}+\mathcal {Q}_{M}\mathcal {Q}_{N}\right) , \end{aligned}$$
(33)

and it is invariant under horizon Freudenthal duality:

$$\begin{aligned} \mathbf{{F}}_{H}\left( \mathcal {M}_{H}\right) _{MN}:=\mathcal {M}_{H\mid MN}(\widetilde{\mathcal {Q}})=\mathcal {M}_{H\mid MN}(\mathcal {Q}). \end{aligned}$$
(34)

3 Duality Orbits, Rigid Special Kähler Geometry and Pre-homogeneous Vector Spaces

For \(I_{4}>0\), \(\mathcal {M}_{H}\left( \mathcal {Q}\right) \) given by (33) is one of the two possible solutions to the set of equations [32]

(35)

which describes symmetric, purely \(\mathcal {Q}\)-dependent structures at the horizon; they are symplectic or anti-symplectic, depending on whether \(I_{4}>0\) or \(I_{4}<0\), respectively. Since in the class of (super)gravity \(D=4\) theories we are discussing the sign of I4 separates the G-orbits (usually named duality orbits) of the representation space \(\mathbf {R}\) of charges into distinct classes, the symplectic or anti-symplectic nature of the solutions to the system (35) is G-invariant, and supported by the various duality orbits of G (in particular, by the so-called “large” orbits, for which \(I_{4}\) is non-vanishing).

One of the two possible solutions to the system (35) reads [32]

whose corresponding \(\mathbf{{F}}_{H}\left( M_{+}\right) _{MN}\) reads

$$\begin{aligned} \mathbf{{F}}_{H}\left( M_{+}\right) _{MN}:= & {} M_{+\mid MN}(\widetilde{\mathcal {Q}})=\epsilon M_{+\mid MN}(\mathcal {Q}). \end{aligned}$$

For \(\epsilon =+1\Leftrightarrow I_{4}>0\), it thus follows that

$$\begin{aligned} M_{+}(\mathcal {Q})=\mathcal {M}_{H}\left( \mathcal {Q}\right) , \end{aligned}$$
(36)

as anticipated.

On the other hand, the other solution to system (35) reads [32]

whose corresponding \(\mathbf{{F}}_{H}\left( M_{-}\right) _{MN}\) reads

$$\begin{aligned} \mathbf{{F}}_{H}\left( M_{-}\right) _{MN}:= & {} M_{-\mid MN}(\widetilde{\mathcal {Q}})=\epsilon M_{-\mid MN}(\mathcal {Q}). \end{aligned}$$

By recalling the definition of \(I_{4}\) (24), it is then immediate to realize that \(M_{-}\left( \mathcal {Q}\right) \) is the (opposite of the) Hessian matrix of (\(1/\pi \) times) the black hole entropy \(S_{BH}\):

$$\begin{aligned} M_{-\mid MN}\left( \mathcal {Q}\right) =-\partial _{M}\partial _{N}\sqrt{ \left| I_{4}\right| }=-\frac{1}{\pi }\partial _{M}\partial _{N}S_{BH}. \end{aligned}$$
(37)

The matrix \(M_{-}\left( \mathcal {Q}\right) \) is the (opposite of the) pseudo-Euclidean metric of a non-compact, rigid special pseudo-Kä hler manifold related to the duality orbit of the black hole electromagnetic charges (to which \(\mathcal {Q}\) belongs), which is an example of pre-homogeneous vector space (PVS) [33]. In turn, the nature of the rigid special manifold may be Kähler or pseudo-Kähler, depending on the existence of a U(1) or SO(1, 1) connection.Footnote 2

In order to clarify this statement, let us make two examples within maximal \(\mathcal {N}=8\), \(D=4\) supergravity. In this theory, the electric-magnetic duality group is \(G=E_{7(7)}\), and the representation in which the e.m. charges sit is its fundamental \(\mathbf {R}=\mathbf {56}\). The scalar manifold has rank-7 and it is the real symmetric cosetFootnote 3 \(G/H=E_{7(7)}/SU(8)\), with dimension 70.

Table 2 Non-generic, nor irregular PVS with simple G, of type 2 (in the complex ground field). To avoid discussing the finite groups appearing, the list presents the Lie algebra of the isotropy group rather than the isotropy group itself [37]. The interpretation (of suitable real, non-compact slices) in \(D=4\) theories of Einstein gravity is added; remaining cases will be investigated in a forthcoming publication
Full size table
  1. 1.

    The unique duality orbit determined by the G-invariant constraint \( I_{4}>0\) is the 55-dimensional non-symmetric coset

    $$\begin{aligned} \mathcal {O}_{I_{4}>0}=\frac{E_{7(7)}}{E_{6(2)}}. \end{aligned}$$
    (38)

    By customarily assigning positive (negative) signature to non-compact (compact) generators, the pseudo-Euclidean signature of \(\mathcal {O} _{I_{4}>0}\) is \(\left( n_{+},n_{-}\right) =\left( 30,25\right) \). In this case, \(M_{-}\left( \mathcal {Q}\right) \) given by (37) is the 56 -dimensional metric of the non-compact, rigid special pseudo-Kä hler non-symmetric manifold

    $$\begin{aligned} \mathbf{{O}}_{I_{4}>0}=\frac{E_{7(7)}}{E_{6(2)}}\times R^{+}, \end{aligned}$$
    (39)

    with signature \(\left( n_{+},n_{-}\right) =\left( 30,26\right) \), thus with character \(\chi :=n_{+}-n_{-}=4\). Through a conification procedure (amounting to modding outFootnote 4 \(C\cong SO(2)\times SO(1,1)\cong U(1)\times R^{+}\)), one can obtain the corresponding 54-dimensional non-compact, special pseudo-Kähler symmetric manifold

    $$\begin{aligned} \mathbf{{O}}_{I_{4}>0}/C\cong \widehat{\mathbf{{O}}}_{I_{4}>0}:=\frac{ E_{7(7)}}{E_{6(2)}\times U(1)}. \end{aligned}$$
    (40)
  2. 2.

    The unique duality orbit determined by the G-invariant constraint \( I_{4}<0\) is the 55-dimensional non-symmetric coset

    $$\begin{aligned} \mathcal {O}_{I_{4}<0}=\frac{E_{7(7)}}{E_{6(6)}}, \end{aligned}$$
    (41)

    with pseudo-Euclidean signature given by \(\left( n_{+},n_{-}\right) =\left( 28,27\right) \), thus with character \(\chi =0\). In this case, \(M_{-}\left( \mathcal {Q}\right) \) given by (37) is the 56-dimensional metric of the non-compact, rigid special pseudo-Kähler non-symmetric manifold

    $$\begin{aligned} \mathbf {O}_{I_{4}<0}=\frac{E_{7(7)}}{E_{6(6)}}\times R^{+}, \end{aligned}$$
    (42)

    with signature \(\left( n_{+},n_{-}\right) =\left( 28,28\right) \). Through a “pseudo-conification” procedure (amounting to modding out \( C_{s}\cong SO(1,1)\times SO(1,1)\cong R^{+}\times R^{+}\)), one can obtain the corresponding 54-dimensional non-compact, special pseudo-Kähler symmetric manifold

    $$\begin{aligned} \mathbf{{O}}_{I_{4}<0}/C_{s}\cong \widehat{\mathbf{{O}}} _{I_{4}<0}:=\frac{E_{7(7)}}{E_{6(6)}\times SO(1,1)}. \end{aligned}$$
    (43)

(39) and (42) are non-compact, real forms of \(\frac{E_{7}}{ E_{6}}\times GL(1)\), which is the type 29 in the classification of regular, pre-homogeneous vector spaces (PVS) worked out by Sato and Kimura in [37]. From its definition, a PVS is a finite-dimensional vector space V together with a subgroup G of GL(V), such that G has a Zariski open dense orbit in V (thus open and dense in V also in the standard topology). PVS are subdivided into two types (type 1 and type 2), according to whether there exists an homogeneous polynomial on V which is invariant under the semi-simple (reductive) part of G itself. For more details, see e.g. [33, 38, 39].

In the case of \(\frac{E_{7}}{E_{6}}\times GL(1)\), V is provided by the fundamental representation space \(\mathbf {R}=\mathbf {56}\) of \(G=E_{7}\), and there exists a quartic \(E_{7}\)-invariant polynomial \(I_{4}\) (24) in the \(\mathbf {56}\); \(H=E_{6}\) is the isotropy (stabilizer) group.

Amazingly, simple, non-degenerate groups of type \(E_{7}\) (relevant to \(D=4\) Einstein (super)gravities with symmetric scalar manifolds) almost saturate the list of irreducible PVS with unique G-invariant polynomial of degree 4 (cf. Table 2); in particular, the parameter n characterizing each PVS can be interpreted as the number of centers of the regular solution in the (super)gravity theory with electric-magnetic duality (U-duality) group given by G. This topic will be considered in detail in a forthcoming publication.