Duality, Entropy and ADM Mass in Supergravity

46 Pages·2009·0.53 MB·English

UCB-PTH-09/05
LBNL-1936E
CERN-PH-TH/2009-001
SU-ITP-09/04
Duality, Entropy and ADM Mass
in Supergravity
Bianca L. Cerchiai♠,♦, Sergio Ferrara♣,(cid:91),
Alessio Marrani♥,(cid:91) and Bruno Zumino♠,♦
9
0
0 ♠ Lawrence Berkeley National Laboratory,
2
Theory Group, Bldg 50A5104

n
1 Cyclotron Rd, Berkeley, CA 94720-8162, USA
u
J [email protected], [email protected]

4

♦ Department of Physics, University of California,
]
h Berkeley, CA 94720-8162, USA
t
-
p
♣ Theory division, CERN, Geneva, Switzerland
e
h CH 1211, Geneva 23, Switzerland
[
[email protected]

2
v (cid:91) INFN - LNF,
3
7 Via Enrico Fermi 40, I-00044 Frascati, Italy
9 [email protected]
3
.
2 ♥ Stanford Institute for Theoretical Physics
0
Department of Physics, 382 Via Pueblo Mall, Varian Lab,
9
0 Stanford University, Stanford, CA 94305-4060, USA
:
v
i
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r
a ABSTRACT
We consider the Bekenstein-Hawking entropy-area formula in four dimensional extended
ungauged supergravity and its electric-magnetic duality property.
Symmetries of both “large” and “small” extremal black holes are considered, as well as
the ADM mass formula for N = 4 and N = 8 supergravity, preserving different fraction of
supersymmetry.
The interplay between BPS conditions and duality properties is an important aspect of
this investigation.
Contents
1 Introduction 1
2 Electric-Magnetic Duality in Supergravity : Basic Facts 3
3 N = 8 8
4 N = 4 15
5 N = 2 26
6 ADM Mass for BPS Extremal Black Hole States 35
6.1 N = 8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
6.2 N = 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
1 Introduction
In d = 4 extended ungauged supergravity theories based on scalar manifolds which are (at
least locally) symmetric spaces
G
M = , (1.1)
H
it is known that the classification of static, spherically symmetric and asymptotically flat
extremal black hole (BH) solutions is made in terms of charge orbits of the corresponding
classical electric-magnetic duality group group G [1, 2, 3, 4, 5, 6] (later called U-duality1 in
string theory) .
These orbits correspond to certain values taken by a duality invariant2 combination of
the “dressed” central charges and matter charges. Denoting such an invariant by I, the set
of scalars parametrizing the symmetric manifold M by φ, and the set of “bare” magnetic
and electric charges of the (dyonic) BH configuration by the 2n×1 symplectic vector
 
pΛ
P ≡  , Λ = 1,...,n, (1.2)
q
Λ
then it holds that
∂ I(φ,P) = 0 ⇔ I = I(P). (1.3)
φ
1Here U-duality is referred to as the “continuous” version, valid for large values of the charges, of the
U-duality groups introduced by Hull and Townsend [7].
2By duality invariant, throughout our treatment we mean that such a combination is G-invariant. Thus,
it is actually independent on the scalar fields, and it depends only on “bare” electric and magnetic (asymp-
totical) charges (defined in Eq. (1.2)).
1
In some cases, the relevant invariant I is not enough to characterize the orbit, and additional
constraints are needed. This is especially the case for the so-called3 “small” BHs, in which
case I = 0 on the corresponding orbit [3, 4, 9].
An explicit expression for the E -invariant [10] was firstly introduced in supergravity
7(7)
in [11], and then adopted in the study of BH entropy in [12]. The additional U-invariant
constraints which specify charge orbits with higher supersymmetry were given in [3]. The
corresponding (large and small) charge orbits for N = 8 and exceptional N = 2 supergravity
were determined in [4], whereas the large orbits for all other symmetric N = 2 supergravities
were obtained in [6], and then in [13] for all N > 2-extended theories. Furthermore, the
invariant for N = 4 supergravity was earlier discussed in [14, 15].
The invariants play an important role in the attractor mechanism [16, 17, 18, 19, 20],
becausetheBekenstein-HawkingBHentropy[8], determinedbyevaluatingtheeffective black
hole potential ([18, 19, 20])
1
V (φ,P) ≡ − PTM(φ)P (1.4)
BH
2
at its critical points, actually coincides with the relevant invariant:
S
BH = V | = V (φ (P),P) = |I(P)|1/2 (or |I(P)|). (1.5)
π BH ∂φVBH=0 BH H
In Eq. (1.4) M stands for the 2n×2n real (negative definite) symmetric scalar-dependent
symplectic matrix
 ImN +ReN (ImN)−1|Ξ∆ReN −ReN (ImN)−1|ΞΣ 
ΛΣ ΛΞ ∆Σ ΛΞ
M(φ) ≡  , (1.6)
−(ImN)−1|Λ∆ReN (ImN)−1|ΛΣ
ΞΣ
defined in terms of the normalization of the Maxwell and topological terms4
ImN (φ)FΛFΣ, ReN (φ)FΛF˜Σ (1.7)
ΛΣ ΛΣ
of the corresponding supergravity theory (see e.g. [21, 22] and Refs. therein). Furthermore,
in Eq. (1.5) φ (P) denotes the set of charge-dependent, stabilized horizon values of the
H
scalars, solutions of the criticality conditions for V :
BH
(cid:12)
∂VBH (φ,P)(cid:12)
(cid:12) ≡ 0. (1.8)
∂φ (cid:12)
φ=φH(P)
For the case of charge orbits corresponding to small BHs, in the case of a single-center
solution I(P) = 0, and thus the event horizon area vanishes, and the solution is singular
3Throughout the present treatment, we will respectively call small or large (extremal) BHs those BHs
with vanishing or non-vanishing area of the event horizon (and therefore with vanishing or non-vanishing
Bekenstein-Hawking entropy [8]). Forsymmetricgeometries,theycanbeG-invariantlycharacterizedrespec-
tively by I =0 or by I =(cid:54) 0.
4AttentionshouldbepaidinordertodistinguishbetweenthenotationsofthenumberN ofsupercharges
of a supergravity theory and the kinetic vector matrix N introduced in Eqs. (1.6) and (1.7).
ΛΣ
2
(i.e. with vanishing Bekenstein-Hawking entropy). However, the charge orbits with van-
ishing duality invariant play a role for multi-center solutions as well as for elementary BH
constituents through which large (i.e. with non-vanishing Bekenstein-Hawking entropy) BHs
are made [23, 24, 25].
In the present investigation, we re-examine the duality invariant and the U-invariant
classification of charge orbits of N = 8, d = 4 supergravity, we give a complete analysis
of the N = 4 large and small charge orbits, and we also derive a diffeomorphism-invariant
expression of the N = 2 duality invariant, which is common to all symmetric spaces and
which is completely independent on the choice of a symplectic basis.
The paper is organized as follows.
In Sect. 2 we recall some basic facts about electric-magnetic duality in N-extended
supergravity theories, firstly treated in [2]. The treatment follows from the general analysis
of [1], and the dictionary between that paper and the present work is given.
In Sect. 3 we re-examine N = 8, d = 4 supergravity and the E -invariant charac-
7(7)
terization of its charge orbits. This refines, re-organizes and extends the various results of
[3, 4, 5, 9].
In Sect. 4 we reconsider matter coupled N = 4, d = 4 supergravity. The SL(2,R) ×
SO(6,M)-invariant characterization of all its BPS and non-BPS charge orbits, firstly ob-
tained in [3, 9], is the starting point of the novel results presented in this Section.
Sect. 5 is devoted to the analysis of the N = 2, d = 4 case [3]. Beside the generalities
on the special Ka¨hler geometry of Abelian vector multiplets’ scalar manifold, the results of
this Section are novel. In particular, a formula for the duality invariant is determined, which
is diffeomorphism-invariant and holds true for all symmetric special Ka¨hler manifolds (see
e.g. [26] and Refs. therein), regardless of the considered symplectic basis.
Sect. 6, starting from the analysis of [3, 9], deals with the issue of the ADM mass [27] in
N = 8 (Subsect. 6.1) and N = 4 (Subsect. 6.2), ungauged d = 4 supergravities. In general,
for all supersymmetric orbits the ADM mass has a known explicit expression, depending on
the number of supersymmetries preserved by the state which is supported by the considered
orbit (saturating the BPS [28] bound).
2 Electric-Magnetic Duality in Supergravity : Basic Facts
The basic requirement for consistent coupling of a non-linear sigma model based on a sym-
metric manifold (1.1) to N-extended, d = 4 supergravity (see e.g. [21] and Refs. therein)
is that the vector field strengths and their duals (through Legendre transform with respect
the Lagrangian density L)
δL
FΛ, G ≡ , (2.1)
Λ δFΛ
3
belong to a symplectic representation R of the global (classical, see Footnote 1) U-duality
s
group G, given by 2n×2n matrices with block structure
 
A B
  ∈ Sp(2n,R), (2.2)
C D
where A, B, C and D are n×n real matrices. By defining the 2n×2n symplectic metric
(each block being n×n)
 
0 −1
Ω ≡  , (2.3)
1 0
the finite symplecticity condition for a 2n×2n real matrix P
PTΩP = Ω (2.4)
yields the following relations to hold for the block components of the matrix defined in Eq.
(2.2):
ATC −CTA = 0; (2.5)
BTD−DTB = 0; (2.6)
ATD−CTB = 1. (2.7)
An analogous, equivalent definition of the representation R is the following one: R is real
s s
and it contains the singlet in its 2-fold antisymmetric tensor product
(R ×R ) (cid:51) 1. (2.8)
s s a
If the basic requirements (2.5)-(2.7) or (2.8) are met, the coset representative of M in
the symplectic representation R is given by the (scalar-dependent) 2n×2n matrix
s
 
A(φ) B(φ)
S(φ) ≡   ∈ Sp(2n,R). (2.9)
C(φ) D(φ)
A particular role is played by the two (scalar-dependent) complex n×n matrices f and h,
which do satisfy the properties
−f†h+h†f = i1, (2.10)
−fTh+hTf = 0. (2.11)
The constraining relations (2.10) and (2.11) are equivalent to require that
 
Ref −Imf
√
S(φ) = 2 , (2.12)
Reh −Imh
4
or equivalently:
1
f = √ (A−iB); (2.13)
2
1
h = √ (C −iD). (2.14)
2
In order to make contact with the formalism introduced by Gaillard and Zumino in [1],
it is convenient to use another (complex) basis, namely the one which maps an element
S ∈ Sp(2n,R) into an element U ∈ U (n,n)∩ Sp(2n,C). The change of basis is exploited
through the matrix
 
1 1
1
A ≡ √  , A−1 = A†. (2.15)
2
−i1 i1
The (scalar-dependent) matrix U is thus defined as follows:
 
f +ih f +ih
1
U (φ) ≡ A−1SA = √   ∈ U (n,n)∩Sp(2n,C). (2.16)
2
f −ih f −ih
This is the matrix named S in Eq. (5.1) of [1]. Correspondingly, the Sp(2n,R)-covariant
vector (cid:0)FΛ,G (cid:1)T is mapped into the vector
Λ
      
FΛ 1 i1 FΛ FΛ +iG
1 1 Λ
A−1  = √    = √  . (2.17)
2 2
G 1 −i1 G FΛ −iG
Λ Λ Λ
The kinetic vector matrix N appearing in Eqs. (1.6) and (1.7) is given by (in matrix
ΛΣ
notation)
N (φ) = hf−1 = (cid:0)f−1(cid:1)T hT, (2.18)
and it is named −iK in [1].
Thus, by introducing the 2n×1 (n×n matrix-valued) complex vector
 
f
Ξ ≡   (2.19)
h
and recalling the definition (1.6), the matrix M can be written as
M(φ) = −iΩ+2ΩΞ(ΩΞ)† = −iΩ−2ΩΞΞ†Ω =
 
−h
(cid:0) (cid:1)
= −iΩ−2  h†, −f† =
f
   
0 −1 hh† −hf†
= −i +2 . (2.20)
1 0 −fh† ff†
5
Eqs. (1.4), (1.6) and (2.20) imply that
1
(cid:0) (cid:1) (cid:0) (cid:1)
V (φ,P) ≡ − PTM(φ)P = Tr ZZ† = Tr Z†Z =
BH
2
N
(cid:88) AB I 1 AB I
= Z Z +Z Z = Z Z +Z Z =
AB I AB I
2
A>B=1
1 1
= Tr(cid:0)ZZ†(cid:1)+Z ZI = Tr(cid:0)Z†Z(cid:1)+Z ZI, (2.21)
I I
2 2
where (A, B = 1,...,N and I = 1,...,m throughout; recall Λ = 1,...,n)
Z ≡ PTΩΞ = qf −ph = (Z (φ,P), Z (φ,P)); (2.22)
AB I
(cid:109)
 
AB
Z (φ,P)
Z† ≡ −Ξ†ΩP = f†q −h†p =  ; (2.23)
 
I
Z (φ,P)
Z (φ,P) ≡ fΛ q −h pΛ; (2.24)
AB AB Λ AB|Λ
Z (φ,P) ≡ fΛq −h pΛ. (2.25)
I I Λ I|Λ
Thus, Eq. (2.21) yields the “BH potential” V (φ,P) to be nothing but the sum of the
BH
(cid:16) (cid:17)
Λ (cid:0) (cid:1)
squares of the “dressed” charges. It is here worth noticing that fΛ ,f and h ,h
AB I AB|Λ I|Λ
are n × n complex matrices, because it holds that5 fΛ = fΛ , h = h (thus
AB [AB] AB|Λ [AB]|Λ
implying Z = Z ), and
AB [AB]
N (N −1)
n = +m, (2.26)
2
where N stands for the number of spinorial supercharges (see Footnote 4), and m denotes
the number of matter multiplets coupled to the supergravity multiplet, except for N = 6,
d = 4 pure supergravity, for which m = 1.
Eqs. (2.24) and (2.25) are the basic relation between the (scalar-dependent) “dressed”
charges Z and Z and the (scalar-independent) “bare” charges P. It is worth remark-
AB I
ing that Z is the “central charge matrix function”, whose asymptotical value appears in
AB
the right-hand side of the N-extended (d = 4) supersymmetry algebra, pertaining to the
asymptotical Minkowski space-time background:
(cid:8) (cid:9)
QA,QB = (cid:15) ZAB(φ ,P), (2.27)
α β αβ ∞
whereφ denotesthesetofvaluestakenbythescalarfieldsatradial infinity (r → ∞)within
∞
the considered static, spherically symmetric and asymptotically flat dyonic extremal BH
background. Notice that the indices A, B of the central charge matrix are raised and lowered
5Unlessotherwisenoted, squarebracketsdenoteantisymmetrizationwithrespecttotheenclosedindices.
6
with the metric of the relevant R-symmetry group of the corresponding supersymmetry
algebra.
By denoting the ADM mass [27] of the considered BH background by M (φ ,P),
ADM ∞
the BPS bound [28] implies that
M (φ ,P) (cid:62) |Z (φ ,P)| (cid:62) ... (cid:62) (cid:12)(cid:12)Z (φ ,P)(cid:12)(cid:12), (2.28)
ADM ∞ 1 ∞ [N/2] ∞
where Z (φ,P),...,Z (φ,P) denote the set of skew-eigenvalues of Z (φ,P), and here
1 [N/2] AB
square brackets denote the integer part of the enclosed number. If 1 (cid:54) k (cid:54) [N/2] of
the bounds expressed by Eq. (2.28) are saturated, the corresponding extremal BH state is
named to be k-BPS. Thus, the minimal fraction of total supersymmetries (pertaining to the
N
asymptotically flat space-time metric) preserved by the extremal BH background within the
considered assumptions is 1 (for k = 1), while the maximal one is 1 (for k = N). See Sect.
N 2 2
6 for further details.
We end the present Section with some considerations on the issue of duality invariants.
A duality invariant I is a suitable linear combination (in general with complex coeffi-
cients) of (φ-dependent) H-invariant combinations of Z (φ,P) and Z (φ,P) such that Eq.
AB I
(1.3) holds, i.e. such that I is invariant under G, and thus φ-independent:
I = I(Z (φ,P),Z (φ,P)) = I(P). (2.29)
AB I
In presence of matter coupling, a charge configuration P (and thus a certain orbit of the
symplectic representation of the U-duality group G, to which P belongs) is called supersym-
metric iff, by suitably specifying φ = φ(P), it holds that
Z (φ(P),P) = 0, ∀I = 1,...,m. (2.30)
I
Notice that the conditions (2.30) cannot hold identically in φ, otherwise such conditions
would be G-invariant, which generally are not. Indeed, in order for the supersymmetry
constraints (2.30) to be invariant (or covariant) under G, the following conditions must hold
identically in φ:
∂ Z (φ,P) = 0, ∀φ ∈ M. (2.31)
φ I
Therefore, supersymmetry conditions are not generally G-invariant (i.e. U-invariant),
otherwise extremal BH attractors (which are large) supported by supersymmetric charge
configurations would not exist.
Nevertheless, insome supergravities itis possible to giveU-invariant supersymmetry con-
ditions. In light of previous reasoning, such U-invariant supersymmetric conditions cannot
stabilize the scalar fields in terms of charges (by implementing the attractor mechanism in
the considered framework), because such U-invariant conditions are actually identities, and
not equations, for the set of scalar fields φ. Actually, U-invariant supersymmetry conditions
can be given for all supersymmetric charge orbits supporting small BHs (for which the clas-
sical attractor mechanism does not hold). This can be seen e.g. in N = 8 (pure) and N = 4
(matter coupled) d = 4 supergravities, respectively treated in Sects. 3 and 4.
7
3 N = 8
The scalar manifold of the maximal, namely N = 8, supergravity in d = 4 is the symmetric
real coset
(cid:18) (cid:19)
G E
7(7)
= , dim = 70, (3.1)
R
H SU (8)
N=8,d=4
where the usual notation for non-compact forms of exceptional Lie groups is used, with
subscripts denoting the difference “# non-compact generators − # compact generators”.
This theory is pure, i.e. matter coupling is not allowed. The classical (see Footnote 1) U-
duality group is E . Moreover, the R-symmetry group is SU (8) and, due to the absence
7(7)
of matter multiplets, it is nothing but the stabilizer of the scalar manifold (3.1) itself.
The Abelian vector field strengths and their duals, as well the corresponding fluxes
(charges), sit in the fundamental representation 56 of the global, classical U-duality group
E . Such a representation determines the embedding of E into the symplectic group
7(7) 7(7)
Sp(56,R), which is the largest symmetry acting linearly on charges. The 56 of E ad-
7(7)
mits an unique invariant, which will be denoted by I throughout. I is quartic in
4,N=8 4,N=8
charges, and it was firstly determined in [11].
More precisely, I is the unique combination of Z (φ,P) satisfying
4,N=8 AB
E
7(7)
∂ I (Z (φ,P)) = 0, ∀φ ∈ . (3.2)
φ 4,N=8 AB
SU (8)
E
Eq. (3.2) can be computed by using the Maurer-Cartan Eqs. of the coset 7(7) (see e.g. [29]
SU(8)
and Refs. therein):
1
CD
∇Z = P Z , (3.3)
AB ABCD
2
E
or equivalently by performing an infinitesimal 7(7) -transformation of the central charge
SU(8)
matrix (see e.g. [29] and Refs. therein):
1
CD
δ Z = ξ Z , (3.4)
ξABCD AB 2 ABCD
where ∇ and P respectively denote the covariant differential operator and the Vielbein
ABCD
E E
1-form in 7(7) , and the infinitesimal 7(7) -parameters ξ satisfy the reality constraint
SU(8) SU(8) ABCD
1
EFGH
ξ = (cid:15) ξ . (3.5)
ABCD ABCDEFGH
4!
As firstly found in [11] and rigorously re-obtained in [29], the unique solution of Eq. (3.2)
reads:
(cid:20) (cid:18) (cid:19) (cid:21)
1 (cid:16) (cid:17)2 (cid:16) (cid:16) (cid:17)(cid:17)2
I = 22Tr Z ZBC − Tr Z ZBC +25Re(Pf (Z )) , (3.6)
4,N=8 22 AC AC AB
where the Pfaffian of Z is defined as [11]
AB
1
Pf (Z ) ≡ (cid:15)ABCDEFGHZ Z Z Z , (3.7)
AB 244! AB CD EF GH
8
and it holds that (see e.g. [29])
|Pf (Z )| = |det(Z )|1/2. (3.8)
AB AB
In [29] it was indeed shown that, although each of the three terms of the expression (3.6) is
SU (8)-invariant but scalar-dependent, only the combination given by the expression (3.6) is
actually E -independent and thus scalar-independent, satisfying
7(7)
δ I = 0, (3.9)
ξABCD 4,N=8
with Eqs. (3.4) and (3.5) holding true.
It is here worth commenting a bit further about formula (3.6). The first two terms in its
right-hand side are actually U (8)-invariant, while the third one, namely 25Re(Pf (Z )), is
AB
only SU (8)-invariant. Such a third term introduces an SU (8)-invariant phase ϕ , defined
Z
as (one fourth of) the overall phase of the central charge matrix, when this latter is reduced
to a skew-diagonal form in the so-called normal frame through an SU (8)-transformation:
 
e
1
Z S−U→(8) Z ≡ eiϕZ/4 e2 ⊗(cid:15), e ∈ R+, ∀i = 1,...,4, (3.10)
AB AB,skew−diag.  e3  i
e
4
where the ordering e (cid:62) e (cid:62) e (cid:62) e can be performed without any loss of generality, and
1 2 3 4
the 2×2 symplectic metric
(cid:18) (cid:19)
0 −1
(cid:15) ≡ (3.11)
1 0
has been introduced (notice (cid:15) = Ω for n = 1, as defined in Eq. (2.3)). For non-vanishing (in
generalalldifferent)skew-eigenvalues e ,thesymmetrygroupofZ is(USp(2))4 ∼
i AB,skew−diag.
(SU (2))4. Thus, beside the 4 skew-eigenvalues e and the phase ϕ , the generic Z is de-
i Z AB
(cid:16) (cid:17)
scribed by 51 = dim SU(8) “generalized angles”. Consistently, the total number of
R (SU(2))4
parameters is 4+1+51 = 56, which is the real dimension of the fundamental representation
56, defining the embedding of E into Sp(56,R).
7(7)
Equivalently, ϕ can be defined through the Pfaffian of Z as follows:
Z AB
Pf (Z )
e2iϕZ ≡ AB , (3.12)
(cid:0) (cid:1)
Pf Z
AB
(cid:0) (cid:1)
where clearly Pf Z = Pf (Z ), as yielded by the definition (3.7). It is then immediate
AB AB
to compute ϕ from Eq. (3.6):
Z
(cid:20) (cid:18) (cid:19) (cid:21)
(cid:16) (cid:17)2 (cid:16) (cid:16) (cid:17)(cid:17)2
BC AC
22I (P)−22Tr Z Z + Tr Z Z
4,N=8 AC AB
cosϕ (φ,P) = . (3.13)
Z (cid:16) (cid:16) (cid:17)(cid:17)1/4
BC
25 det Z Z
AC
9