CC BY
58
UDC 530.1; 539.1
BLACK HOLE ORBITS IN SUPERGRAVITY1
K. S. Stelle
The Blackett Laboratory, imperial College London Prince Consort Road, London SW7 2AZ.
E-mail: k.stelle@imperial.ac. uk
Duality symmetries are used to organise symmetry orbits of supergravity black-hole solutions and to display their relation to extremal (i.e. BPS) solutions at the limits of such orbits. An important technique for this analysis uses a timelike dimensional reduction and exchanges the stationary black-hole problem for a nonlinear sigma-model problem. Families of BPS solutions are characterized by nilpotent orbits under the duality symmetries, based upon a tri-graded or penta-graded decomposition of the corresponding duality group algebra.
Keywords: supergravity, black-holes, duality symmetries, sigma-model.
The study of families of black-hole solutions is of considerable current interest because it touches upon many important issues in theoretical physics. For example, the classification of BPS and non-BPS black holes forms part of a more general study of branes in supergravity and superstring theory. Branes and their intersections, as well as their worldvolume modes and attached string modes, are also key elements in phenomenological approaches to the marriage of string theory with particle physics phenomenology. The related study of nonsingular and horizon-free BPS gravitational solitons is also central to the "fuzzball" proposal of BPS solutions as candidate black-hole quantum microstates. Brane solutions are also the basis for a number of early-universe cosmology candidates.
The search for supergravity solutions with assumed Killing symmetries can be recast as a. Kaluza-Klein problem [1-3]. To see this, consider a 4D theory with a nonlinear bosonic symmetry G4 (e.g. the "duality" symmetry E7 for maximal N = 8 supergravity). Scalar fields take their values in a target space $4 = G4/H4, where H4 is the corresponding linearly realized subgroup, generally the maximal compact subgroup of G4 (e.g. SU(8) C E7 for N = 8 SG). The search will be constrained by the following considerations:
• We assume that a solution spacetime is asymptotically flat or asymptotically Taub-NUT and that there is a 'radial' function r which is divergent in
the asymptotic region, dMrdvr ~ 1 + O(r-1).
•
assuming that a solution possesses a timelike Killing vector field km(x). Lie derivatives with respect to km are assumed to vanish on all fields. The Killing vector km
will be assumed to have W := —kmk' orthogonality, i.e. kv(dM«v — dvkm) '
any vielbein frame, the curvature will then fall off as Rabcd ~ O(r-3).
The 3D theory obtained after dimensional reduction with respect to a timelike Killing vector will have an Abelian principal bundle structure, with a metric
ds2 = -W (dt + Bidxi)2 + W Yij dxldxj
(1)
where t is a coordinate adapted to the timelike Killing vector k^ wid Yij is the metric on the 3-dimensional hypersurface M3 at const ant t. If the 4D theory also has Abelian vector fields they similarly reduce to 3D as
4 V4nGAMdxM = U (dt + Bidxi) + Aidxi.
(2)
~ 1 + O(r-1). H [2] hypersurface O(r-2). In
The timelike reduced 3D theory will have a G/H* coset space structure similar to the G/H coset space structure of a 3D theory reduced with a spacelike Killing vector. Thus, for the spacelike reduction of maximal supergravity down to 3D, one obtains an Eg/SO(16) theory from the sequence of dimensional reductions descending from D = 11 [4]. The resulting 3D theory has this exceptional symmetry because 3D Abelian vector fields can be dualized to scalars; this also happens for the analogous theory subjected to a timelike reduction to 3D. The resulting 3D theory
G/H*
model.
G
reduction is the same as that obtained in a spacelike
H*
is a noncompact form of the spacelike divisor group H [2]. A consequence of this H ^ H* change and the dualization of vectors is the appearance of negative-sign kinetic terms for some 3D scalars.
1An expanded version of this note will appear in the Festschrift in honor of the 75th birthday of Professor Andrei Alekseevich
Slavnov.
Consequently, maximal supergravity. after a timelike reduction to 3D and the subsequent dualization of 29 vectors to sc9.l9.rs 5 hclS a bosonic sector containing 3D gravity coupled to a Eg/SO*(16) nonlinear signia model with 128 scalar fields. As a consequence of the timelike dimensional reduction and vector dualizations. however, the scalars do not all have the same signs for their ''kinetic" terms:
• There are 72 positive-sign scalars: 70 descending directly from the 4D theory, one emerging from the 4D metric and one more coming from the D = 4 ^ D = 3
Kaluza-Klein vector, subsequently dualized to a scalar.
•
directly from the time components of the 28 4D vectors, and another 28 emerging from the 3D vectors obtained from spatial components of the 28 4D vectors, becoming then negative-sign scalars after dualization.
The sigma-model structure of this timelike reduced maximal theory is Eg/SO*(16). The SO*(16) divisor group is not an SO(p, q) group defined via preservation of an indefinite metric. Instead it is constructed starting from the SO(16) Clifford algebra {r1, rJ} = 2SIJ and then % forming the complex U(8)-covariant oscillators ai := 1 (r2i-i + ir2i) and a1 = (a,i) = 2(r2j—i — ir2j). These satisfy the standard fermi oscillator annihilation/creation anticomnmtation relations
[ai, a,j} = [a1,, aj} = 0
[a,i, aj} = 5i :
—= Vi(V77ij Gab (*)dj *B) = 0 V7
Rij (Y) = 2 Gab (m4>Adj *B
where Vi is a doubly covariant derivative (for the 3D space M3 and for the G/H* target space).
Now one can make the simplifying assumption that $A(x) = $A(a(x)), with a single intermediate map a(x). Subject to this assumption, the field equations become
V2^ + Yij diadj a[
da
d2*A ~dä2
+ rBc (G) ^ = 0,
da da
Rij
2 Gab <*) g o^j *B
(0)
Now one uses the gravitational Bianchi identity Vi(Rij — i Yij R) = 0 to obtain
0. Requiring
separation of the a(x) properties d
properties leads to the conditions
4 £ (Gab (*) ^)(Viadia) =
from the ^
d2*A da2
+ rBc (G)
V2a
d*B d*C
da da
— ( Gab (*) d*A d*B
da V da da
(7)
(8) (9)
(3)
*(16)
the 64 hermitian U(8) generaters a^' plus the 2 x 28 = 56 antihermitian combinations of aij ± aij. Under *(16)
spinor are pscndo-rcal, while the 128-dimensional chiral spinor representation is real. This is the representation under which the 72^56 scalar fields transform in the Eg/SO*(16) sigma model.
The 3D classification of extended supergravity stationary solutions via timelike reduction generalizes the 3D supergravity systems obtained from spacelike reduction [5]. This also connects with N = 2 models with coupled vectors [6] and N = 4 models with vectors, where solutions have also been generated using duality symmetries [7.8].
The process of timelike dimensional reduction down to 3 dimensions together with dualization of all form-fields to scalars produces an Euclidean gravity theory coupled to a G/H * nonlinear sigma model, Ia = f d3x^Y(R(y) — iGab ($)di$Adj$BYij) where GAB ($) is th e G/H * sigma-model target-space metric and Yij is the 3D metric. Varying this action produces the 3D field equations
a(x)
a harmonic map from the 3D space M3 into a curve 4>A(a) in th e G/H * target space. The second equation
(8) implies that 4>A(a) is a geodesic in G/H*. The third
a
decomposition of $ : M3 ^ G/H * into a harmonic map a : M3 ^ R and a geodesic $ : R ^ G/H * is in accordance with a general theorem on harmonic maps
[9] according to which the composition of a harmonic map with a totally geodesic one is again harmonic. Such factorization into geodesic and harmonic maps is also characteristic of general higher-dimensional p-brane supergravity solutions [1,3].
Here IS 9. sketch of the map composition:
(4)
(5)
D=3 Space M
Now define the Komar two-form K = dxM A dxv. This is invariant under the action of the timelike isometry and, by the asymptotic hypersnrface orthogonality assumption, is asymptotically horizontal. This condition is equivalent to the requirement that the scalar field B dual to the Kaluza-Klein vector arising out of the 4D metric must vanish like O(r-i) as r ^ <x>. In this case, one can define the Komar mass and NUT
0
charge by (where s* indicates a pull-back to a section) [101
m = — K
8n
s* *K,
dMs
n= — K
8n
s*K. (10)
dMa
The Maxwell field also defines charges. Using the Maxwell field equation d* F = 0, where F = 5C/5F is a linear combination of the two-form field strengths F
dF = 0 and magnetic charges:
¿K
s* * F,
dMa
s*F .
dMa
Now consider these charges from the three-dimensional point of view in order to clarify their transformation properties under the 3D duality group G
of a coset representative V G G/H*. The Maurer-Cartan form V-1dV for g decomposes as
V dV = Q + P, Q = Q^dx^ e h* P = P^dx^ e s © h* .
C 4n
*VPV
'dMa
P = C % + O(r-2) .
noncompact duality group E6,6, with the 42 D = 5 scalar fields taking their values in the coset space E6,6/USp(8), while the 1-form (i.e. vector) fields transform in the 27 of E6,6.
D=5
produce new scs.ls.rs upon dimensional reduction, and one also gets a new Kaluza-Klein scalar emerging from D=5
the 4D theory. These take their values in E7,7/SU(8), while the 4D vector field strengths transform in the 56 of E7,7. The new KK scalar corresponds to a gl1 grading genera,tor of E7,7, leading to a tri-graded decomposition of the E7,7 algebra as follows:
e7,7 ~ 27( 2) © (fl[1 © ee,6)(0) 0 27'
(H)
(2)
(15)
where the superscripts indicate the gl1 grading.
Continuing on down to 3D via a timelike reduction, one encounters a new phenomenon: 3D vectors can now be dualized to scalars. This is already clear in the timelike reduction of pure 4D GR to 3D, where one obtains a two-scalar system taking values in SL(2,R)/SO(2), where SL(2,R) is the Ehlers group [11]. Its generators can be written
(12)
Then the three-dimensional scalar-field equation of motion can be rewritten as d * VPV-1 = 0, so the g-valued "Noether current" is *VPV-1. Since the three-dimensional theory is Euclidean, one cannot properly speak of a conserved charge. Nevertheless, since *VPV-1 is d-closed, the integral of this 2-form over a given homology cycle does not depend on the particular representative of that cycle.
As a result, for stationary solutions, the integral of this three-dimensional 2-form current, taken over any spacelike closed surface dM3 containing in its interior all the singularities and topologically nontrivial subspaces of a solution, defines a g © lf)*-valued Noether-charge matrix C:
Y h © ee © <pf =
(16)
and its Lie algebra is
[h, e] = 2e , [h, f ] = —2f , [e, f ] = h.
Accordingly, in reducing from 4D to 3D a supergravity theory with 4D symmetry group G4, with corresponding Lie algebra g4 and with vectors transforming in the representation of g4, one obtains a penta-graded structure for the 3D Lie algebra g, with the Ehlers h now acting as the grading generator 1(0):
s ~ 1(-2) © û(-1) © (1 © S4)(0) © [4+1) © 1(2).
(17)
(13)
For example, in 3D maximal supergravity one obtains in this way e8,8:
es,s ^ 1(-2) © 56(-1) © (1 © e7,7)(0) © 56(+1) © 1(2)
This transforms in the adjoint representation of the G
linear action of G on V G G/H*. For asymptotically-V
at infinity to the identity matrix; the charge matrix C in that case is simply given by the asymptotic value of P
.
(18)
(14)
G
down a couple of steps in dimensional reduction. In D=5
Now apply this to the decomposition of the coset-space structure for the 3D scalar fields and the charge
C
g4 © l 4 l 4
H4
the 4D electric and magnetic charges q mid p is l4.
C
decomposed into three irreducible representations with respect to so(2) © )4 according to
g © )* = (sl(2,R) © so(2)) © [4 © (Kg4 © )4). (19)
The metric induced by the g algebra's Cartan-Killing metric on this coset space is positive definite for the first and last terms, but is negative definite for l4. One associates the sl(2,R) © so(2) components with the Komar mass and the Komar NUT charge, while the [4 components are associated with the electromagnetic charges. The remaining g4 © )4 charges belong to the Noether current of the 4D theory.
Breitenlohner, Gibbons and Maison [2] proved that if G is simple, all the non-extremal single-black-hole solutions of a given theory lie on the H* orbit of a Kerr solution. Moreover, all static solutions regular outside the horizon with a charge matrix satisfying Tr C2 > 0 lie on t he H*-orbit of a Schwarzschild solution. (Turning on and off angular momentum requires consideration of the D = 2 duality group generalizing the Geroch A11 group.)
Using Weyi coordinates, where the 4D metric takes the form
ds2 = f (x, p)-1[e2k(x, p)(dx2 + dp2)
+ p2d02] + f (x,p)(dt + A(x, p)d<)2 , (20)
the coset representative V associated to the Schwarzschild solution with mass m can be written in terms of the non-compact generator h of the Ehlers sl(2,R) only, i.e.
/1 r_mm \
V = exp - ln-h ^ C = mh. (21)
\2 r + m ) x '
For the maximal N = 8 theory with symmetry E8(8) (and also for the exceptional 'magic' N = 2 supergravity [12] with symmetry E8(_24)), one has h = diag[2, -, 0, -- -2], so
h5 = 5h3 - 4h (22)
C
the characteristic equation
C5 = 5c2C3 - 4c4C, (23)
where c2 = x/h2 C2 is the extremality parameter
c2 = 0 c2 = m2
for Schwarzschild). Moreover, for all but the two
8
C
C3 = c2C. (24)
The characteristic equation selects acceptable orbits of solutions, i.e. orbits not exclusively containing
C
in terms of the mass and NUT charge and the 4D
electromagnetic charges.
c2
discussion: c2 = v2. The Maxwell-Einstein theory-is the simplest example with an indefinite-signature sigma-model metric, for the scalar-field target space G/H* = SU(2, -)/S(U(-,-) x U(-)). The MaxwellEinstein charge matrix is
(m n — z/%/2\
n -m iz/V2 I € su(2, 2) © u(-,-)
z/V2 iz/v2 0 J
(25)
where z = q + ip is the complex electromagnetic charge. The Maxwell-Einstein extremality parameter is c2 = m2 + n2 - zz. Solutions fall into three
c2 > 0 c2 = 0
c2 < 0 hyperextremal. The hyperextremal solutions have naked singularities, while the nonextremal and extremal solutions have their singularities cloaked by-horizons.
c2 = 0
C C5 = 0 8
C3 = 0
For N extended supergravity theories, one finds H* = Spin*(2N) x Ho and the charge matrix C transforms as a Weyl spinor of Spin*(2N) also valued in a representation of lf)0 (where ho acts on the matter content of reducible N = 4 theories). As in the SO* (-6) case considered earlier, one defines the Spin*(2N) fermionic oscillators
ai := -(r2i_i +ir2j),
a' = (a*)* = -(r2i_i - ir2i) (26)
for i,j, ■ ■ ■ = -,..., N. These obey standard fermionic annihilation & creation anticommutation relations. Using this annihilation/creation oscillator basis, the
C
a* 0 = 0)
|C) = (Kw + Z'j a* aj
+ SijW aj akal + •••) K |0) . (27)
From the requirement that the dilatino fields be left invariant under the unbroken supersymmetry of a BPS solution, one derives a 'Dirac equation' for the charge state vector,
(eiai + ttai)e?a^ |C) =0 (28)
where (ela, ef) ^s ^^e asymptotic (for r ^ to) value of the Killing spinor and is a symplectic form on C2n
n/N
Note that c2 =0 ^^ (C|C) = 0 is a weaker condition than the supersymmetry Dirac equation. Extremal and BPS are not always synonymous conditions, although they coincide for N < 5 pure supergravities. They are not synonymous for N = 6 & 8 or for theories with vector matter coupling.
Earlier analysis of the orbits of the 4D symmetry
G4
9 = u(g,z) exp (ln A(s,z) ^ b(g,z) (29)
with U(gZ) G H4 mid b(g,Z) G Bz, where Bz C G4 is the parabolic subgroup that leaves the charges Z invariant up to a multiplicative factor A(g,z). This multiplicative factor can be compensated for by 'trombone' transformations combining Weyl scalings with compensating dilational coordinate transformations, leading to a formulation of active symmetry transformations that map solutions into other solutions with unchanged asymptotic values of the spacetime metric and scalar fields.
The 4D 'trombone' transformation finds a natural home in the parabolic subgroup of the 3D duality G
fact that the Iwasawa decomposition breaks down for noncompact divisor groups H*.
The Iwasawa decomposition does, however work "almost everywhere" in the 3D solution space. The places where it fails are precisely the extremal suborbits of the duality group. This has the G G
send c2 ^ 0, thus landing on an extremal (generally BPS) suborbit. However, one cannot then invert the map and return to a generic non-extremal solution from
G
The above framework applies equally to multi-centered as to single-centered solutions [14,15]. One may start from a general ansatz
V(x) = Vo exp(- YHn(x)Cn)
(30)
Defining as above V-1dV = Q + P and restricting P to depend linearly on the functions Hn(x), one finds the requirement [KCm, [Cn, Cp]] =0. The Einstein and scalar equations of motion then reduce to
R^v - 2R = Y d»HmdvHn Tr (31)
2
d*dUn = 0.
(32)
with Lie algebra elements Cn G g © )* and functions Hn(x) to be determined by the equations of motion.
Restricting attention to solutions where the 3-space is fiat then requires Tr CmCn = 0. The resulting system generalizes that found in [3]. Solving [KCm, [Cn, Cp]] = 0 = Tr CmCn is now reduced to an algebraic problem amenable to the above nilpotent-orbit analysis: non-extremal and extremal stationary solutions can be formed from extremal single-hole constituents.
In summary, what has been developed here is a quite general framework for the analysis of stationary supergravity solutions using duality orbits.
C
equation C5 = 5c2 C3 — 4c4C in the maximal E8 cases and C3 = c2C in the non-maximal cases, where c2 = -¡jTh? Tr C2 is the extremality parameter. Extremal
c2 = 0 C
C5 = 0 C3 = 0 extremal suborbits. BPS solutions have a charge
C
equation' which encodes the general properties of such solutions. This is a stronger condition than the
c2 = 0
G
G
decomposition for noncompact divisor groups H*. The Iwasawa failure set corresponds to the extremal suborbits.
References
[1] G. Neugebaur and D. Kramer, Ann. der Physik (Leipzig) 24 (1969) 62.
[2] P. Breitenlohner, D. Maison and G. W. Gibbons, Commun. Math. Phys. 120 (1988) 295.
[3] G. Clement and D. Gal'tsov, Phys. Rev. D 54 (1996) 6136, [hep-th/9607043]; D. V. Gal'tsov and O. A. Rytchkov, Phys. Rev. D 58 (1998) 122001, [hep-th/9801160],
[4] E. Cremmer and B. Julia, Nucl. Phys. B 159 (1979) 141.
[5] B. de Wit, A. K. Tollsten and H. Nicolai, Nucl. Phys. B 392 (1993) 3 [hep-th/9208074],
[6] P. Meessen and T. Ortin, Nucl. Phys. B 749 (2006) 291 [hep-th/0603099],
[7] M. Cvetic and D. Youm, Phys. Rev. D 53 (1996) 584 [hep-th/9507090],
[8] M. Cvetic and A. A. Tseytlin, Phys. Lett. B 366 (1996) 95 [hep-th/9510097],
[9] J. Eells, Jr. and J. H. Sampson, J. H„ Am. J. Math. 86 (1964) 109.
[10] G. Bossard, H. Nicolai and K. S. Stelle, JHEP 0907 (2009) 003 [arXiv:0902.4438 [hep-th]].
[11] J. Ehlers, "Konstruktion und Charakterisierungen von Lösungen der Einsteinschen Gravitationsgleichungen", Dissertation, Hamburg (1957).
[12] M. Gunaydin, G. Sierra and P. K. Townsend, Phys. Lett. B 133 (1983) 72.
[13] E. Cremmer, H. Lu, C. N. Pope and K. S. Stelle, Nucl. Phys. B 520 (1998) 132 [hep-th/9707207],
[14] G. Bossard and H. Nicolai, Gen. Rel. Grav. 42 (2010) 509 [arXiv:0906.1987 [hep-th]].
[15] G. Bossard and C. Ruef, Gen. Rel. Grav. 44 (2012) 21 [arXiv:1106.5806 [hep-th]].
Received, Ц.11.20Ц
К. С. Стелле
ОРБИТЫ ЧЕРНЫХ ДЫР В СУПЕРГРАВИТАЦИИ
Дуальные симметрии используются для построения симметричных орбит в решениях для черных дыр в супергравитации и для выявления их связи с внешними решениями (т.е. BPS) в пределе таких орбит. Используемый для данного анализа подход сводит задачу стационарной черной дыры к задаче нелинейной сигма-модели. Семейство BPS решений характеризуется нильпотентными орбитами дуальной симметрии.
Ключевые слова: супергравитация, черные дыры, симметрии дуальности, сигма-модель.
Стелле К. С. доктор, профессор. Имперский колледж Лондона.
Prince Consort Road, London SW7 2AZ, Великобритания. E-mail: k.stelle@imperial.ac.uk
