Observations on Integral and Continuous Uduality Orbits in Supergravity
Abstract
One would often like to know when two a priori distinct extremal black brane solutions are in fact related by Uduality. In the classical supergravity limit the answer for a large class of theories has been known for some time now. However, in the full quantum theory the Uduality group is broken to a discrete subgroup, a consequence of the DiracZwanzigerSchwinger charge quantization conditions. The question of Uduality orbits in this case is a nuanced matter. In the present work we address this issue in the context of supergravity in four, five and six dimensions. The purpose of this note is to present and clarify what is currently known about these orbits while at the same time filling in some of the details not yet appearing in the literature. For the continuous case we present the cascade of relationships existing between the orbits, generated as one descends from six to four dimensions, together with the corresponding implications for the associated moduli spaces. In addressing the discrete case we exploit the mathematical framework of integral Jordan algebras, the integral Freudenthal triple system and, in particular, the work of Krutelevich. The charge vector of the dyonic black string in is related to a twocharge reduced canonical form uniquely specified by a set of two arithmetic Uduality invariants. Similarly, the black hole (string) charge vectors in are equivalent to a threecharge canonical form, again uniquely fixed by a set of three arithmetic Uduality invariants. However, the situation in four dimensions is, perhaps predictably, less clear. While black holes preserving more than 1/8 of the supersymmetries may be fully classified by known arithmetic invariants, 1/8BPS and nonBPS black holes yield increasingly subtle orbit structures, which remain to be properly understood. However, for the very special subclass of projective black holes a complete classification is known. All projective black holes are related to a four or five charge canonical form determined uniquely by the set of known arithmetic Uduality invariants. Moreover, acts transitively on the charge vectors of projective black holes with a given leadingorder entropy.
type:
Imperial/TP/2010/mjd/1, CERNPHTH/2010040, SUITP10/07pacs:
11.25.w, 03.65.Ud, 04.70.Dyleron.borsten@imperial.ac.uk, duminda.dahanayake@imperial.ac.uk, m.duff@imperial.ac.uk, sergio.ferrara@cern.ch, marrani@lnf.infn.it, william.rubens06@imperial.ac.uk
1 Introduction
The extremal black brane solutions of supergravity have played, and continue to play, a key role in unravelling the nonperturbative aspects of Mtheory. Evidently, understanding the structure of these solutions is of utmost importance. In particular, one would like to know how such solutions are interrelated by the set of global symmetries collectively known as Uduality. The electric/magnetic charge vectors of the asymptotically flat brane solutions form irreducible Uduality representations as in Table 1.
10A  

10B  
9  
8  
7  
6  
5  
4 
In many relevant cases the macroscopic leadingorder black brane entropy is a function of these charges only, a result of the attractor mechanism [1, 2, 3, 4]. Consequently, an important question is whether two a priori distinct black brane charge configurations are in fact related by Uduality. Mathematically this amounts to determining the distinct charge vector orbits under Uduality. In the classical limit the answer for a large class of theories has been known for some time now [5, 6, 7, 8]. In particular, for the maximally supersymmetric theories, obtained by the toroidal compactification of supergravity, a complete classification of all orbits in all dimensions is known [5, 6]. However, in the full quantum theory the Uduality group is broken to a discrete subgroup, a consequence of the DiracZwanzigerSchwinger charge quantization conditions [9]. Consequently, the Uduality orbits are furnished with a further level structural complexity, which, in some cases, is of particular mathematical significance [10, 11]. However, the question of discrete Uduality orbits is not only interesting in its own right, it is also of physical importance with implications for a number of topics including the stringy origins of microscopic black hole entropy [12, 13, 14, 15, 16, 17, 18, 19, 20]. Moreover, following a conjecture of finiteness of supergravity [21], it has recently been observed that some of the orbits of should play an important role in counting microstates of this theory [20], even if it may differ from its superstring or Mtheory completion [22].
In the present work we address this issue in the context of supergravity in four, five and six dimensions. To this end we exploit the mathematical framework of integral Jordan algebras and the integral Freudenthal triple system, both of which have at their basis the ring of integral splitoctonions [23, 24, 25, 11]. To a large extent this work is a continuation of the analysis used in studying the recently introduced black hole Freudenthal duality [26], which in turn has its provenance in recently established connections relating black hole entropy in Mtheory to entanglement in quantum information theory [27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 26, 39, 40, 41, 42].
It is well known that the black holes and strings appearing in the maximally supersymmetric 6, 5 and 4dimensional classical theories are elegantly described by the exceptional Jordan algebras and the closely related Freudenthal triple system (FTS) [43, 44, 45, 5].
In particular, the black string charge vectors of supergravity may be represented as elements of the exceptional Jordan algebra of Hermitian matrices defined over the splitoctonions, denoted . See C.2 for details. The reduced structure group , defined as the set of invertible linear transformations preserving , the quadratic norm (179), is the Uduality group , under which the black string charges transform as the vector . Moreover, in this case the quadratic norm is nothing but , the singlet in , which determines the black string entropy
See e.g. [5, 46, 47], and Refs. therein. There are two distinct charge vector orbits under , one consisting of 1/2BPS states and one consisting of 1/4BPS states, distinguished respectively by the vanishing or not of [6, 47]. Equivalently, these orbits may be distinguished by the rank of the Jordan algebra element representing the charge vector, rank 1 states being 1/2BPS while rank 2 states are 1/4BPS. See section 2.1 for details.
Similarly, the black hole charge vectors of supergravity may be represented as elements of the exceptional Jordan algebra of Hermitian matrices defined over the splitoctonions, denoted [5]. See D.2 for details. The reduced structure group , defined as the set of invertible linear transformations preserving , the cubic norm (191), and the symmetric bilinear trace form (196), is the Uduality group , under which the black hole charges transform as the fundamental . Moreover, in this case the cubic norm is in fact , the singlet in , which determines the black hole entropy
There are three distinct charge vector orbits under , one 1/2BPS, one 1/4BPS and one 1/8BPS, distinguished by the vanishing or not of and its derivatives [48]. See section 3.1 for details. Again, these orbits may also be distinguished by the rank of the Jordan algebra element representing the charge vector. Rank 1 states are 1/2BPS, rank 2 states are 1/4BPS while rank 3 states are 1/8BPS [38, 26]. A directly analogous treatment goes through for the black string charges in , which transform as the contragredient of .
Finally, the black hole charge vectors of supergravity may be represented as elements of the Freudenthal triple system denoted . See E for details. The automorphism group , defined as the set of invertible linear transformations preserving , the quartic norm (225), and the antisymmetric bilinear form (224), is nothing but , the Uduality group under which the black hole charges transform as the fundamental . Moreover, the quartic norm is exactly , the unique quartic invariant, which again determines the black hole entropy [49]
This is the first example exhibiting a nonBPS orbit. In total there are five distinct charge vector orbits under , three of which have vanishing , one 1/2BPS, one 1/4BPS and one 1/8BPS, which are distinguished by the vanishing or not of the derivatives of [48]. The two orbits with nonvanishing are either 1/8BPS or nonBPS according to whether or respectively. Again, these orbits may also be distinguished by the rank of the FTS element representing the charge vector. States of rank 1, 2 and 3 are 1/2BPS, 1/4BPS and 1/8BPS respectively, all with vanishing . Rank 4 states are split into 1/8BPS and nonBPS as determined by the sign of . See section 4.1 for details. For further details concerning these orbits and their defining Uinvariant BPS conditions the reader is referred to the original works [48, 5, 6].
As one descends from to , via spacelike dimensional reductions, a series of relationships connecting these Uduality orbits is generated. The Uinvariant BPS conditions in dimensions are “embedded” in those of dimensions, as is best understood by decomposing the dimensional Uduality group with respect to the dimensional Uduality group. Taking care of whether or not the of charges of KaluzaKlein vector are vanishing, one is then able to understand how the various black brane solutions, their orbits and the associated moduli spaces are embedded under these spacelike reductions. Moreover, in this way we can also study the reverse situation and so understand which higher dimensional black brane solutions a given black brane in a given dimension may be “uplifted” to. It should be pointed out that the vanishing or not of the KaluzaKlein vector charges is crucial in discriminating the various possible uplifts. A general result holding throughout the present treatment in can be stated as follows: if the charges of the KaluzaKlein vector are not switched on, the supersymmetry preserving features of the solution are unaffected by the dimensional reduction. These results are presented for and in section 3.2 and section 4.2 respectively.
This summarizes the classification of the Uduality orbits for real valued charges. However, as previously emphasized, the charges are actually quantized and the Udualities are correspondingly broken to discrete subgroups as described in [9]. The integral charge vector orbits are a nuanced matter and a complete characterization is as yet not known. Despite these additional complications, the discrete orbit classification is made possible in certain cases by the introduction of new arithmetic Uduality invariants not appearing in the continuous case. These are typically given by the greatest common divisor (gcd) of Uduality representations built out of the basic charge vector representations [25, 11, 13, 14, 15, 16, 17, 18, 19, 26]. One purpose of this note is to present and clarify what is currently known about these discrete orbits while at the same time filling in some of the details not yet appearing in the literature.
Let us now briefly summarize the present situation. An important general observation is that, since the conditions separating the continuous orbits are manifestly invariant under the corresponding discrete Udualities, those states unrelated in the continuous cases remain unrelated in the discrete case. Consequently, the discrete orbits fall into disjoint sets corresponding directly to the orbits of the classical theory. A second important observation, emphasized in [47], is that the gcd of a Uduality representation, built out of the relevant basic charge vector representation, is only well defined if that representation is nonvanishing. In practice this means first computing which class of orbits as defined by the continuous analysis a given state lies in. This, in turn, determines the subset of the arithmetic invariants that are well defined for this particular state. It is this subset that is then to be used in specifying the particular discrete orbit to which the state belongs, the remaining arithmetic invariants being illdefined and contentless.
Beginning in the integral charge vectors of the dyonic black strings may be represented as elements of the integral Jordan algebra of Hermitian matrices defined over the ring of integral splitoctonions, denoted . See C.3 for details. The discrete Uduality group is given by the set of invertible linear transformations preserving the quadratic norm (179). An arbitrary charge vector is related to a twocharge reduced canonical form (12) uniquely specified by a set of two arithmetic Uduality invariants (11). The two orbits of the continuous case now form two disjoint countably infinite sets of discrete orbits which may be parametrized using the arithmetic invariants.
Similarly, the black hole charge vectors in may be represented as elements of the integral Jordan algebra of Hermitian matrices defined over the ring of integral splitoctonions, denoted . See D.3 for details. The discrete Uduality group is given by the set of invertible linear transformations preserving the cubic norm (191) and the trace bilinear form (196). An arbitrary charge vector is equivalent to a threecharge canonical form (78), again uniquely fixed by a set of three arithmetic Uduality invariants (77). The three orbits of the continuous case now form three disjoint classes of discrete orbits which may be parametrized using the arithmetic invariants [25, 26]. A directly analogous treatment goes through for the black string charges in , which transform as the contragredient of .
However, the situation in four dimensions is, perhaps predictably, less clear. The black hole charge vectors may be represented as elements of the integral FTS defined over . See E for details. The discrete Uduality group is given by the set of invertible linear transformations preserving the quartic norm (225) and the antisymmetric bilinear form (224). An arbitrary charge vector is equivalent to a fivecharge canonical form (138). However, this canonical form is not uniquely fixed by the known set of arithmetic Uduality invariants (137). Despite this, for particular subcases more can be said. Indeed, the classes of discrete orbits corresponding to the 1/2BPS and 1/4BPS continuous orbits may be completely classified using the known arithmetic invariants [11], as described in section 4.3. For those black hole preserving less than 1/4 of the supersymmetries the orbit structure becomes more complicated and the orbit classification is not known. However, even in this case a full classification is possible for the of projective black holes. See section 4.4 for details. The concept of projectivity was originally introduced in the in the numbertheoretic context of [10] where such elements are mapped to invertible ideal classes of quadratic rings. This notion was later generalized by Krutelevich in [11] with a view to understanding orbit structure. Indeed, acts transitively on the set of projective black holes of a given quartic norm. Moreover, they are equivalent to a simplified four or five charge canonical form (144) depending on whether the quartic norm is even or odd respectively [11, 26].
This note is organized as follows. In section 2 we begin by recalling the continuous Uduality orbits in , emphasizing the Jordan algebraic perspective, before presenting in detail the corresponding discrete treatment. In section 3 the same treatment is applied to continuous Uduality black hole charge orbits in . Subsequently, the intricate web of relations connecting the orbits and moduli spaces of the 6dimensional theory to those of the 5dimensional theory are presented. This analysis is concluded with corresponding discrete Uduality treatment of the integral black hole charge orbits. The same continuous analysis is undertaken for in section 4, completing the cascade of relationships between the orbits and moduli spaces in 6, 5 and 4dimensions. This is followed by the discrete Uduality treatment of the integral black hole charge orbits. We conclude with a summary of open questions. For the most part the technical details are relegated to the appendices in an effort to avoid an oppressive number of formal definitions in the main body of the text. In A we present the minimal background necessary to introduce the ring of integral splitoctonions underlying this analysis. In C we describe the continuous and integral quadratic Jordan algebras together with their application to black strings in . In D we describe the continuous and integral cubic Jordan algebras together with their application to black holes (strings) in . In E we describe the continuous and integral FTS, defined over the Jordan algebra, together with its application to black holes in .
2 Black strings in
2.1 Uduality orbits of
In the classical supergravity limit the electric/magnetic black string charges form an vector ( throughout). Under the vector breaks as
(1) 
where the singlets lie in the NSNS sector and correspond to a fundamental string and an NS5brane, while the is made up of RR charges. In this basis the charges may be conveniently represented as an element of the Jordan algebra of splitoctonionic Hermitian matrices,
(2) 
The set of linear invertible transformations leaving the quadratic norm,
(3) 
invariant is the Uduality group . Using the dictionary (183) one finds,
(4) 
where,
(5) 
and is the metric,
(6) 
The black string entropy is proportional to the quadratic norm,
(7) 
There are two Uduality orbits, one 1/2BPS “small” orbit and one 1/4BPS “large” orbit [6, 48, 47]. Note, an orbit is referred as “small” in the sense that the associated Uduality invariant and, hence, BekensteinHawking entropy are vanishing. Correspondingly, for “large” orbits the Uduality invariant and BekensteinHawking entropy are nonvanishing. These orbits may distinguished by the Jordan rank of as detailed in C.1,

(8) 
which is precisely the condition originally presented in [48]. The orbits, their rank conditions, dimensions and representative states are summarized in Table 2.
Rank  Rank/orbit conditions  Representative state  Orbit  dim  SUSY  

2.2 Uduality orbits of
For quantized charges the continuous Uduality is broken to an infinite discrete subgroup, which for is given by [9]. The integral Jordan algebra, , of integral splitoctonionic Hermitian matrices provides a natural model for , which may used to analyse the discrete Uduality orbits. The quantized black string charge vector is given by,
(9) 
The discrete group is defined by the set of normpreserving invertible linear transformations,
(10) 
It is with this framework that we shall study the discrete Uduality orbits.
The first important observation is that the charge conditions defining the orbits in the continuous theory are manifestly invariant under the discrete subgroup and, hence, those states unrelated by Uduality in the classical theory remain unrelated in the quantum theory. There are two disjoint classes of orbits, one 1/2BPS and one 1/4BPS, corresponding to the two orbits of the continuous case. However, each of these classes is broken up into a countably infinite set of discrete orbits. To classify these orbits we use to bring an arbitrary charge vector into a diagonal reduced canonical form, which is uniquely defined by the following set of two discrete invariants,
(11) 
See C.3 for details.
diagonal reduced canonical form
Every element is equivalent to a diagonally reduced canonical form,
(12) 
The canonical form is uniquely determined by (11) since
(13) 
so that for arbitrary one obtains and .
Black string orbit classification

The complete set of distinct 1/2BPS charge vector orbits is given by,
(14) 
The complete set of distinct 1/4BPS charge vector orbits is given by,
(15)
3 Black holes in
3.1 Uduality orbits of
In the classical supergravity limit the 27 ( throughout) electric black hole charges transform as the fundamental of the continuous Uduality group . Under the breaks as
(16) 
where the singlet may be identified as the graviphoton charge descending from , the as the remaining NSNS sector charges and the as the RR sector charges. Further decomposing under one obtains
(17) 
In this basis the charges may be conveniently represented as an element of the cubic Jordan algebra of splitoctonionic Hermitian matrices,
(18) 
The cubic norm (207) is then given by the determinant like object,
(19) 
The set of invertible linear transformations leaving the cubic norm and trace bilinear form invariant is nothing but the Uduality group . Moreover,
(20) 
where,
(21) 
and is the invariant tensor.
The black hole entropy is simply given by the cubic norm,
(22) 
In this case there are three Uduality orbits, 1/2BPS and 1/4BPS “small” orbits and a single 1/8BPS “large” orbit [5]. These orbits may distinguished by the Jordan rank of ,

(23) 
where is the quadratic adjoint map (211). See D.1 for details. Note, transforms as a under . Similarly
(24) 
manifestly transforms as a under so that . Hence, these conditions are entirely equivalent to the conditions originally presented in [48],

(25) 
The orbits with their rank conditions, dimensions and representative states are summarized in Table 3. The 27 magnetic black string charges form the contragredient of . The orbit classification is identical to the black hole case.
Rank  Rank/orbit conditions  Representative state  Orbit  dim  SUSY  

3.2 relations for charge orbits and moduli spaces
Through the branching of the irrep. of Uduality with respect to duality [48, 46, 20] (, and throughout; cf. Eq. (16)) {subequations}
(26)  
(27) 
the unique electric and magnetic invariants (94) and (95) of the can be written in a manifestly invariant way respectively as follows [48, 50]: {subequations}
(28)  
(29) 
where ’s are the gamma matrices, and respectively are the electric and magnetic charge of the KaluzaKlein vector, and the unique quadratic invariant of the of is defined as follows (note the basis change compared to (5)(6) ): {subequations}
(30)  
(31) 
In order to study the relations among the various charge orbits of maximal supergravity in and (and the consequences for the related moduli spaces^{1}^{1}1In all the treatment , and respectively denote a scalar manifold, a charge orbit and a moduli space. is defined all along the scalar flow, from the near horizon geometry (if any, at the classical level) to asymptotically flat spatial infinity. Thus, if the corresponding is “large” , can be interpreted as the moduli space of attractor solutions (at the near horizon geometry) and the moduli space of the ADM mass (at spatial infinity). In the case of “small” , the attractor near horizon interpretation of corresponding breaks down.), let us briefly recall the Uinvariant classification of the charge orbits of black branes (black holes, respectively black membranes) in , supergravity [6, 48].
3.2.1 Résumé on black branes in ,
Without any loss of generality, let us consider a black hole () corresponding to the uplift of a black hole. In other words, we only consider the orbits of chiral spinor of determined by the branching (26) of of .

1/2BPS “small” orbit (charge solution) [6, 48]
(35) defined by
(36) In other words, the invariant constraint defining such an orbit is the fact that is a pure chiral spinor of . However, it is worth recalling that the nontriviality of the background implies that at least some exist such that
(37) As for case , the resulting BekensteinHawking entropy is vanishing (see (34)).
3.2.2 The 1/4BPS black string orbit under
A representative of the 1/4BPS “large” charge orbit
(38) 
of a dyonic black string in , supergravity is provided by {subequations}
(39) 
By plugging (39) into (28), one obtains
(40) 
The treatment splits in two separate cases, depending on the vanishing or not of the magnetic charge of KaluzaKlein vector:
Thus, at the level of charge orbits, Eqs. (3.2.2) correspond to the following picture:
(45) 
Indeed, at the level of the semisimple part of the orbit stabilizers, the embedding^{2}^{2}2“max” and “symm” respectively denote the maximality and symmetricity of the group embedding under consideration.
(46) 
holds. At the level of corresponding moduli spaces, (45) implies that (see also (118))
(47) 
satisfies
(48) 
which is nothing but a part of the embedding (120).
Thus, it follows that a “small” 1/4BPS black hole, as well as a “large” 1/8BPS black hole, of , supergravity can be uplifted to a “large” 1/4BPS dyonic black string of , supergravity.
It should be pointed out that the nonvanishing or not of charges of KaluzaKlein vector is crucial in order to discriminate among the various possible uplifts. A general result holding throughout the present treatment in can be stated as follows: if the charges of the KaluzaKlein vector are not switched on, the supersymmetrypreserving features of the solution are unaffected by the dimensional reduction.
3.2.3 The 1/2BPS black string orbit under
A representative of the 1/2BPS “small” charge orbit
(49) 
of a dyonic black string in , supergravity is provided by {subequations}
(50) 
By plugging (50) into (28), one obtains
(51) 
where is the unique possibly nonvanishing component of .
As above, depending on the vanishing or not of the magnetic charge of KaluzaKlein vector, the treatment splits as follows:
Thus, at the level of charge orbits, Eqs. (3.2.3) correspond to the following picture:
(56) 
Indeed, at the level of the semisimple part of the orbit stabilizers, the following embeddings trivially hold:
(57) 
At the level of corresponding moduli spaces, (56) implies that
(58) 
satisfies (the embedding in the second line being trivial; see also (118)) [51]
(59) 
Thus, it follows that a “small” 1/2BPS black hole, as well as a “small” 1/4BPS black hole, of , supergravity can be uplifted to a “small” 1/2BPS dyonic black string of , supergravity.
3.2.4 The 1/4BPS black hole orbit under
A representative of the 1/4BPS “small” charge orbit (32) of a black hole in , supergravity is provided by {subequations}
(60) 
and
(61) 
for at least some . By plugging (60), (61) into (28), one obtains
(62) 
where is the unique nonvanishing component of .
Thus, independently on the vanishing or not of , this case corresponds to a 1/4BPS “small” black hole of , supergravity.
At the level of charge orbits, Eqs. (3.2.4) depict the following situation:
(63) 
Indeed, at the level of the semisimple part of the orbit stabilizers, the following embedding trivially holds:
(64) 
At the level of corresponding moduli spaces, (63) implies that
(65) 
satisfies (see also (118)) [51]
(66) 
Thus, it follows that a “small” 1/4BPS black hole of , supergravity can be uplifted to a “small” 1/4BPS black hole of , supergravity.
3.2.5 The 1/2BPS black hole orbit under
A representative of the 1/2BPS “small” charge orbit (35) of a black hole in , supergravity is provided by {subequations}
(67) 
and
(68) 
By plugging (67), (68) into (28), one obtains
(69) 
Thus, independently on the vanishing or not of , this case corresponds to a 1/2BPS “small” black hole of , supergravity.
At the level of charge orbits, Eqs. (3.2.5) depict the following situation:
(70) 
Indeed, at the level of semisimple part of the stabilizers of orbits, the following embedding trivially holds:
(71) 
At the level of corresponding moduli spaces, (70) implies that
(72) 
satisfies [51]
(73) 
which is a trivial consequence of the embedding between the scalar manifolds of maximal supergravity in and .
Thus, it follows that a “small” 1/2BPS black hole of , supergravity can be uplifted to a “small” 1/2BPS black hole of , supergravity.
Summarizing the embeddings of moduli spaces (48), (59), (66) and (73), related to the various dimensional reductions considered in section 3.2.2section 3.2.5, the following result is achieved achieved [51]:
(74) 
3.3 Uduality orbits of
For quantized charges the continuous Uduality is broken to an infinite discrete subgroup, which for is given by [9]. The integral Jordan algebra of integral splitoctonionic Hermitian matrices provides a natural model for , which may used to analyse the discrete Uduality orbits. See [23, 24, 25] and D for further details. The quantized black hole charge vector is given by,