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) +c.c.) + - . . . (41) 71=0
^
'
47
This formula is indeed quite general. By changing the factor 2 in the exponents of the partial derivatives and as the upper limit of the sum to N/4 the same action works for any N divisible by 8. For any such JV there is a self-conjugate field as in (40). We do not expect though, that such a theory will be consistent when higher orders of interactions is constructed. Also for N/4 being an odd integer the three-point action is correct if the field 4> {y) transforms in the adjoint representation of a non-abelian symmetry. The three-point coupling then needs a structure constant to soak up the indices of the fields. For N/4 = 1 the three-point coupling is the same as in (22), which can be shown by using partial integrations. Hence the three-point coupling given in (40) is general and can be used both for N = 4 super Yang-Mills as well as for N — 8 supergravity. When closing the superPoincare algebra the number of space derivatives in the three-point interaction term is fixed and related to the helicity of the superfield (and hence to the highest helicity of the multiplet). This leads to a unique dimension of the coupling constant. The N = 4 and the N = 8 cases then have coupling constants which differ by a mass dimension, which, of course, is known from the covariant formulations. This fact makes it relatively simple to construct the four-point coupling in the N — 4 theory since it does not involve any space derivatives. For the N = 8 case it is more difficult and we have so far not managed to do it. With modern computational techniques it should be straightforward though to construct it. In the supergravity case the algebra does not close at the four-point coupling but continues to all orders. Even so with the action up to the four-point term there are several issues that can be studied. One can for example study the one-loop graphs and one can investigate if there are symmetries other than the superPoincare and the non-abelian ones. Note that all symmetries that are found in the action are real symmetries since all gauge invariance has been used up.
4. Ten Dimensions The very compact formalism of the previous section was constructed for the N = 4 and the N = 8 theories in four dimensions. We now generalize this formalism to restore the theory in ten dimensions, 10 without changing the superfield, simply by introducing generalized derivative operators. Let us first treat the N = 4 case in some detail. First of all, the transverse light-cone variables need to be generalized to eight. We stick to the previous notation, and introduce the six extra
48
coordinates and their derivatives as antisymmetric bi-spinors 1
/ • -J= { Xm + 3 + I Xm + 6 ) ,
„ m \
gmi
=
V2
/ S ]•m +
3 + idm + e) , (42)
for m ^ 4, and their complex conjugates PI — 9 zpqmn
re„
V
PQ
(43)
u
Their derivatives satisfy ft I X PX q _ TPQ _ u Umnu' — V % n
x 1 X P) . u % n Ji
amn u
^ _ I xm xn _ ^pq — \ u pu q
xm u
qu
xn p
(44)
and Qmn
xpq
_
£ £pqrs
gmn
^
_
emnpq
(45)
There are no modifications to be made to the chiral superfield, except for the dependence on the extra coordinates A(y) = A(x,x,xmn,xmn,y
) , etc... .
(46)
These extra variables will be acted on by new operators that generate the higher-dimensional symmetries. 4.1. The SuperPoincare
Algebra in 10
Dimensions
The SuperPoincare algebra needs to be generalized from the form in Ref. [6]. One starts with the construction of the 50(8) little group using the decomposition 50(8) D 50(2) x 50(6). The 50(2) generator is the same; the 50(6) ~ 5f7(4) generators are given by
r
( CE
0pn
Xpn
emBn + endm +
0
m
v + 2^2 - d+-{d
dn-dndm) " "
;
8V2 9+
-(epBp-epdp)5mn
[dpdp~dpdp)5m
n
.
(47)
The extra terms with the d and d operators are not necessary for closure of the algebra. However they insure that the generators commute with the chiral derivatives. They satisfy the commutation relations J
Jm
= 0
jm
j
P
smq q j° p nn - 6<Jpn u r
q •
V^o)
The remaining 50(8) generators lie in the coset 5 0 ( 8 ) / ( 5 0 ( 2 ) x 50(6)) JPI
=
x
dpq - xpq d + - ^ 9 + ep 9" \/2
iV2— dp8q o+
V2d+
dpdq.
49 —
—
—
i
/— 1 — —
— —
Jmn = X dmn ~ Xmn 0 + ~^9+
6m6n-i
7
—
V2 — dm Bn + -j=—dm
—
dn . (49)
All 50(8) transformations are specially constructed so as not to mix chiral and antichiral superfields, [Jmn,
[Jmn,dp]
dp\ = 0 ;
= 0,
(50)
and satisfy the 50(8) commutation relations
J,Jmn jm
__ jmn
jpq
jmn f J , Jpq
•A Jmn
=
= 8qn Jmp - 6pn Jmq = SmpJ\ + 5\Jmp
-J„
6maQJnp - 6mVJ,nq '
jm j ° nt "ipq
u
- 5npJmq - 8mqJnp - (Smp5nq -
5npSmq)J.
Rotations between the 1 or 2 and 4 through 9 directions induce on the chiral fields the changes 1 (51) mn J ' n ^mn ) 4 > ,
-(h
where complex conjugation is like duality -
W
_1
P = i V2 ZJmn d+ em6n<j> 5J4> =
,
(74)
wjpq
j - ^ d + epeq -i V2~BpBq + -^i_a p s g J • (75)
We list the conjugate relations for completeness Sj$
= wpqJpq$
= iy/2 wpqd+6p§q$,
(76)
(77) The variations of the generalized derivative and its conjugate are given by Sj V = uJmn [ Jmn , V ] = - ujmn V m " , 5jV
= w™ [ Jpq , V] = j~= w*> dpdq A .
(78) (79)
Invariance under 50(8) is checked by doing a Sj variation on the cubic vertex, including its complex conjugate. The cubic vertex is 50(8) invariant if a = 1, and the generalized derivative is totally determined V = B+T-^dpdqd^
.
(80)
To obtain this result, we used the antisymmetry of the structure functions, the chiral constraints, the "inside-out" constraints, and performed integrations by parts on the coordinates and Grassmann variables. In this light-cone form, the Lorentz invariance in ten dimensions is automatic once
53
the little group invariance has been established. We have therefore shown ten-dimensional invariance, since the quartic term does not need to be changed. 5. Eleven Dimensions We can now redo the scheme above to construct the eleven-dimensional supergravity. 11 We extend the Lorentz symmetry to SO (9) by adding in the 50(7) which is missing in the four-dimensional case and construct also the the generators for 50(9)/SO(7).The translation generators are also extended. We use the same superfield as in four dimensions and and the only difference in the action is that we have to find corresponding generalized derivatives V and V. They transform together with d and 5 as 9-vectors. We do not give the details here but refer to the forthcoming paper. The final action is then
5=
Jdnxjd80d8e(^^4>
+ f {£* * B - 1 ) " Q ) ( V 2 - 8*n4> V" 3+ 2 ->) + c.c.) + . • • .(81) Again we find that this action has exactly the same structure as the tendimensional one apart from having infinitely many terms. However, those properties like the results of simple one-loop calculations can be taken over from one model to the other. We also know that those properties are very similar.12 It is also natural to use this formalism to study possible symmetries over and above the superPoincare symmetry. This work is in progress. 6. Further Extensions In a series of papers we have studied an extension of the 11-dimensional supergravity theory. 13 It is based on the observation that the Dynkin indices of supergravity multiplet has very interesting properties. 14 They match up bosons vs fermions for all but the largest of the Dynkin indices, I8. This property is also respected by an infinity of triplets of the little group 50(9). These triplets then correspond to higher spins. This result can be understood in the following way. SO(9) is the maximal subalgebra of FA. They have the same rank. For two algebras like this one can associate to every representation in the larger algebra an multiplet in the subalgebra with properties as above. 15 Kostant has constructed a Dirac-like equation
54
the solutions of which are exactly the multiplets above. In the case of 5 0 ( 9 ) we can write the Kostant equation 1 6 as 7
»Ti^
=
0
,
( 82 )
T* are the generators of F 0. It is then useful to order this expansion according to the height of the positive root a — Sj nl an, defined as ht(a) = £ j n \ The correspondence discussed above between the leading asymptotic evolution near a cosmological singularity (described by a billiard) and Weyl chambers of KM algebras involves only the terms in the field equation whose height is ht(a) < 1. By contrast, the authors of Ref. [6] managed to show, by explicit calculation, that there exists a way to define, at each spatial point x, a correspondence between the field variables g^(t,x), A^x(t,x) (and their gradients), and a (finite) subset
59
of the parameters defining an element of the (infinite-dimensional) coset space Eio/K(Eio) where K(Ei0) denotes the maximal compact subgroup of £10, such that the (PDE) field equations of supergravity get mapped onto the (ODE) equations describing a null geodesic in EIOJK(EIQ) up to terms of height 30. This tantalizing result suggests that the infinitedimensional hyperbolic Kac-Moody group Eio may be a "hidden symmetry" of supergravity in the sense of mapping solutions onto solutions (the idea that E\o might be a symmetry of supergravity was first suggested by Julia long ago 24 ' 25 ). Note that the conjecture here is that the continuous group Eio(M.) be a hidden symmetry group of classical supergravity. At the quantum level, i.e. for M theory, one expects only a discrete version of £10, say i?io (Z), to be a quantum symmetry. See [26] for recent work on extending the identification of [6] between roots of £ 1 0 and symmetries of supergravity/M-theory beyond height 30, and for references about previous suggestions of a possible role for EIQ. For earlier appearances of the Weyl groups of the E series in the context of [/-duality see [27,28,29]. A series of recent papers 30 " 34 has also explored the possible role of -En (a nonhyperbolic extension of £10) as a hidden symmetry of M theory. It is also tempting to assume that the KM groups underlying the other (special) Einstein-matter systems discussed above might be hidden (solution-generating) symmetries. For instance, in the case of pure Einstein gravity in D = 4 space-time, the conjecture is that AE3 be such a symmetry of Einstein gravity. This case, and the correspondence between the field variables and the coset ones is further discussed in [7]. Rigorous mathematical proofs 17 ' 35 ' 16 are however only available for 'non chaotic' billiards. In the remainder of this paper, we will outline various arguments explaining the above results; a more complete derivation can be found in [7]. 2. G e n e r a l M o d e l s The general systems considered here are of the following form
S[gMN,d>,A^]= J 1 2
dDx^i
V^ 2^(„
l +
l)|
R(g) - d.'Mp(P) e r
5M,
M1-Mp+1r
p(p)M1-Mp+1
(4)
Units are chosen such that IQTTGN = 1, GM is Newton's constant and the space-time dimension D = d + 1 is left unspecified. Besides the standard
60
Einstein-Hilbert term the above Lagrangian contains a dilaton b field <j> and a number of p-form fields A^ ... M (for p > 0). The p-form field strengths F^ = dA^ are normalized as F
= (P +
^^MI^MI-M^!]
=
9
± P permutations . (5) As a convenient common formulation we adopt the Einstein conformal frame and normalize the kinetic term of the dilaton 4> with weight one with respect to the Ricci scalar. The Einstein metric gMN has Lorentz signature (—h • —h) and is used to lower or raise the indices; its determinant is denoted by g. The dots in the action (4) above indicate possible modifications of the field strength by additional Yang-Mills or ChaplineManton-type couplings. 36,37 The real parameter Ap measures the strength of the coupling of A^ to the dilaton. When p — 0, we assume that Ao ^ 0 so that there is only one dilaton. M\-MP+1
M.4...MP+1
3. Dynamics in the Vicinity of a Spacelike Singularity The main technical points that will be reviewed here are the following • near the singularity, t —> 0, due to the decoupling of space points, the Einstein's (PDE) equations become ODE's with respect to time. • The study of these ODE's near t —> 0, shows that the d diagonal spatial metric components ugu" and the dilaton <j> move on a billiard in an auxiliary d + 1 dimensional Lorentz space. • All the other field variables {gij,i ^ j,-Ai1...jp,7rtl-,,lp) freeze as t-t0. • In many interesting cases, the billiard tables can be identified with the fundamental Weyl chamber of an hyperbolic KM algebra. • For SUGRAn, the KM algebra is _Ei0. Moreover, the PDE's are equivalent to the equations of a null geodesic on the coset space E10/K(EW), up to height 30. 3.1. Arnowitt-Deser-Misner
Hamiltonian
formalism
To focus on the features relevant to the billiard picture, we assume here that there are no Chern-Simons and no Chapline-Manton terms and that the The generalization to any number of dilatons is straightforward.
61
curvatures F ( p ) are abelian, F^ = cL4(p). That such additional terms do not alter the analysis has been proven in [7]. In any pseudo-Gaussian gauge and in the temporal gauge (g0i = 0 and A0i2...ip = 0, Vp), the ArnowittDeser-Misner Hamiltonian action 8 reads 9ij,^A^A^..Jp,ni^
J dx* J dfiz L^ + Wpi + ^ £ + J2 h ^'J°F£-iP
-
p
where the subscript \j stands for a spatially covariant derivative.
(14)
62
3.2. Iwasawa
decomposition
of the spatial
metric
We systematically use the Iwasawa decomposition of the spatial metric g^j and write d
0=1
where Af is an upper triangular matrix with l's on the diagonal. We will also need the Iwasawa coframe {0a}, 9a=Maldxi,
(17)
as well as the vectorial frame {ea} dual to the coframe {0a},
e 0 =AP a A
(18)
where the matrix A / \ is the inverse of Afai, i.e., NaiNlb = 8%. It is again an upper triangular matrix with l's on the diagonal. Let us now examine how the Hamiltonian action gets transformed when one performs, at each spatial point, the Iwasawa decomposition (16) of the spatial metric. The kinetic terms of the metric and of the dilaton in the Lagrangian (4) are given by the quadratic form
G^d^d/T
d I d \2 a 2 a = J > / 3 ) - £ d/3 \ + dcj>2, p = (/3a, 0). 0=1
\<J=I
(19)
/
The change of variables (#,j —> j3a,Nai) corresponds to a point transformation and can be extended to the momenta as a canonical transformation in the standard way via
Tr^y = Y, *«F + E Pt^ai a
•
(20)
a
Note that the momenta
P\ = J £ - =Ye^b-^^jNhAfib oNai
(21)
^
conjugate to the nonconstant off-diagonal Iwasawa components Hai are only defined for a < i; hence the second sum in (20) receives only contributions from a < i.
63
3.3. Splitting
of the
Hamiltonian
We next split the Hamiltonian density H (7) in two parts: Ho, which is the kinetic term for the local scale factors /3M = (/3a, (j>), and V, a "potential density" of weight 2, which contains everything else. Our analysis below will show why it makes sense to group the kinetic terms of both the offdiagonal metric components and the p-forms with the usual potential terms, i.e. the term M in (8). Thus, we write H = Ho + V,
(22)
with the kinetic term of the /3 variables H0 = \ G^n^
,
(23)
where GM" denotes the inverse of the metric Gav of Eq. (19). In other words, the right hand side of Eq. (23) is defined by
G^x^^E^-^ifE^j +*h where 7rM = {na^^) i.e.
( 24 )
are the momenta conjugate to (3a and , respectively, Tru=2N-lG^f3'/
= 2G^-^.
(25)
The total (weight 2) potential density, v is naturally split into a "centrifugal" part Vs linked to the kinetic energy of the off-diagonal components (the index S referring to "symmetry,"), a "gravitational" part VG, a term from the p-forms, £ ) p Vp, which is a s u m of an "electric" and a "magnetic" contribution and also a contribution to the potential coming from the spatial gradients of the dilaton V$. • "centrifugal" potential Vs = l52e-Wb-^(pibAf"J)\
(27)
a 0 which corresponds to /?M tending to future time-like infinity, we decompose /3M into hyperbolic polar coordinates (p,7^), i.e. P» = Pr, (40) where 7M are coordinates on the future sheet of the unit hyperboloid which are constrained by a^-f-f = r^
= -i,
(41)
and p is the time-like variable defined by p2 = -GMUp»p"
= - W
> 0,
(42)
which behaves like p ~ — lnt —» +oo at the BKL limit. In terms of these variables, the potential term looks like J2cA(dx/3»,P,Q)p2exp(-2pwA(1)). A
(43)
66
The essential point now is that, since p —> +00, each term p2 exp ( — 2PWA(I)) becomes a sharp wall potential, i.e. a function of WA{I) which is zero when WA{J) > 0, and +00 when WA(J) < 0. To formalize this behavior we define the sharp wall 9-function c as f0
if x < 0,
I +00 if x > 0 . A basic formal property of this 0-function is its invariance under multiplication by a positive quantity. Because all the relevant prefactors CA{dx(3^iP->Q) a r e generically positive near each leading wall, we can formally write lim
CA(dx^, Q, P)p2 exp ( - pwA{n)\ = cA(Q, P)Q( - 2 ^ ( 7 ) )
p—*oo
= 6(-21^(7)),
(45) 7
valid in spite of the increasing of the spatial gradients. Therefore, the limiting dynamics is equivalent to a free motion in the /3-space interrupted by reflections against hyperplanes in this /3-space given by WA(P) = 0 which correspond to a potential described by infinitely high step functions V(/3,P,Q) = ^ e ( - 2 ^ ( 7 ) ) -
(46)
A
The other dynamical variables (all variables but the /3M's) completely disappear from this limiting Hamiltonian and therefore they all get frozen as t->0. 4. Cosmological Singularities and Kac—Moody Algebras Two kinds of motion are possible according to the volume of the billiard table on which it takes place, i.e. the volume (after projection on hyperbolic space) of the region where V = 0 for t —> 0, also characterized by the conditions, wA((3)>0
VA
(47)
Depending on the fields present in the Lagrangian, on their dilatoncouplings and on the spacetime dimension, the (projected) billiard volume c
O n e should more properly write 0<x>(x), but since this is the only step function encountered here, we use the simpler notation Q(x).
67
Figure 3. Sketch of billiard tables describing the asymptotic cosmological behavior of Einstein-matter systems. is either finite or infinite. The finite volume case corresponds to neverending, chaotic oscillations for the /3's while in the infinite volume case, after a finite number of reflections off the walls, they tend to an asymptotically monotonic Kasner-like behavior, see Fig. 3. In Fig. 3 the upper panels are drawn in the Lorentzian space spanned by ((3^) = (j3a,<j>). The billiard tables are represented as "wedges" in (d + 1)dimensional (or oo as T —> 0). In this frame the complete evolution equations of D = 11
f
In this section, the proper time is denoted by T while the variable t denotes the parameter of the one-dimensional cr-model introduced above.
72
supergravity read d0(Gacd0Gcb) d0{Gf°abc)
= \GTa^bTbfil5
- ±GFa^5Fa/3l55ab
-
2GRab{T,C),
^eabca^a>b^b°b*F0aia2a3fblb2b3bi
= +
lGJrde[abcc}de
_
GCedefdabc
_
dd(pjrdabc^
f
doFabcd = §Foe[abGecd] + 49[ a ^ r 0 6cd] i
(57)
where a,b £ { 1 , . . . , 10} and a, (3 € { 0 , 1 , . . . , 10}, and Rab(T, C) denotes the spatial Ricci tensor; the (frame) connection components are given by ^GadT bc = Cabc + Cbca — Ccab + dbGca + dcGab — daGbc with Cabc = GadCdbc being the structure coefficients of the zehnbein d9a = \Cabc6b A 6C. (Note the change in sign convention here compared to above.) The frame derivative is da = Ela(x)di (with EaiElb = 5%). To determine the solution at any given spatial point x requires knowledge of an infinite tower of spatial gradients; one should thus augment (57) by evolution equations for daGbc,daJ:'obcd,daJ-'bcde, etc., which in turn would involve higher and higher spatial gradients. The main result of concern here is the following: there exists a map between geometrical quantities constructed at a given spatial point x from the supergravity fields G)lu(x0,x) and A^vp{x°,x) and the one-parameterdependent quantities gab(t),Aabc(t),... entering the coset Lagrangian (53), under which the supergravity equations of motion (57) become equivalent, up to 30th order in height, to the Euler-Lagrange equations of (53). In the gauge (56) this map is defined by t = x° = J dT/^/G and
9ab(t)
=Gab(t,x),
IM0l-a«(t) = DA»\ai...a8{t)
=
-iea-M*b*b*rblb2b3lH(t,x), 3 e a 1 ...a 8 6 l6a ( C! 6 6i62(a;)
+
f^C^]^)) .
(58)
Let us also mention in passing (from [39]) that the Eio coset action is not compatible with the addition of an eleven-dimensional cosmological constant in the supergravity action (an addition which has been proven to be incompatible with supersymmetry in [40]).
73
6. Conclusions We have reviewed the finding that the general solution of many physically relevant (bosonic) Einstein-matter systems, in the vicinity of a space-like singularity, exhibits a remarkable mixture of chaos and symmetry. Near the singularity, the behavior of the general solution is describable, at each (generic) spatial point, as a billiard motion in an auxiliary Lorentzian space or, after a suitable "radial" projection, as a billiard motion on hyperbolic space. This motion appears to be chaotic in many physically interesting cases including pure Einstein gravity in any space-time dimension D < 10 and the particular Einstein-matter systems arising in string theory. Also, for these cases, the billiard tables can be identified with the Weyl chambers of some Lorentzian Kac-Moody algebras. In the case of the bosonic sector of supergravity in 11-dimensional space-time the underlying Lorentzian algebra is that of the hyperbolic Kac-Moody group E\Q, and there exists some evidence of a correspondence between the general solution of the Einstein-three-form system and a null geodesic in the infinite-dimensional coset space E\Q/K(EIO), where K(Eio) is the maximal compact subgroup of Ew Acknowledgments It is a pleasure to thank Sophie de Buyl and Christiane Schomblond for their help in trimming the manuscript and in improving the figures. Appendix A. Kac-Moody Algebras A KM algebra Q(A) can be constructed out of a generalized Cartan matrix A, (i.e. an r x r matrix such that i) An = 2, a = 1, ...,r; ii) —Aij € N for i ^ j and Hi) A^ — 0 implies Aji — 0) according to the following rules for the Chevalley generators {hi, e%, ft}, i — 1,..., r: l e ii Jj\
=
\JXi,&j\ = [hiijj]
=
"ijili, J\.ijCj, ~-™-ijJj,
[hi,hj] — 0. The generators must also obey the Serre's relations, namely ( a d e i ) 1 - A y e j = 0,
(ad/O1^/;^,
74
and the Jacobi identity. Q(A) admits a triangular decomposition G(A) = n-@hen+,
(A.l)
where n_ is generated by the multicommutators of the form [fix, [/j2,...]], n + by the multicommutators of the form [e^, [ej2,...]] and h is the Cartan subalgebra. The algebras Q{A) build on a symmetrizable Cartan matrix A have been classified according to properties of their eigenvalues • if A is positive definite, Q{A) is a finite dimensional Lie algebra; • if A admits one null eigenvalue and the others are all strictly positive, Q(A) is an Affine KM algebra; • if A admits one negative eigenvalue and all the others are strictly positive, G(A) is a Lorentzian KM algebra. A KM algebra such that the deletion of one node from its Dynkin diagram gives a sum of finite or affine algebras is called an hyperbolic KM algebra. These algebras are all known; in particular, there exists no hyperbolic algebra with rang higher than 10. References 1. T. Damour and M. Henneaux, Phys. Rev. Lett. 85, 920 (2000) [hepth/0003139]; see also a short version in Gen. Rel. Gray. 32, 2339 (2000). 2. T. Damour and M. Henneaux, Phys. Lett. B488, 108 (2000) [hepth/0006171]. 3. T. Damour and M. Henneaux, Phys. Rev. Lett. 86, 4749 (2001) [hepth/0012172]. 4. T. Damour, M. Henneaux, B. Julia and H. Nicolai, Phys. Lett. B509, 323 (2001) [hep-th/0103094]. 5. T. Damour, S. de Buyl, M. Henneaux and C. Schomblond, JEEP 0208, 030 (2002) [hep-th/0206125]. 6. T. Damour, M. Henneaux and H. Nicolai, Phys. Rev. Lett. 89, 221601 (2002) [hep-th/0207267]. 7. T. Damour, M. Henneaux and H. Nicolai, Class. Quant. Grav. 20, R145 (2003) [hep-th/0212256]. 8. R. Arnowitt, S. Deser and C. W. Misner, The Dynamics Of General Relativity, gr-qc/0405109. 9. V.A. Belinskii, I.M. Khalatnikov and E.M. Lifshitz, Adv. Phys. 19, 525 (1970). 10. V.A. Belinskii, I.M. Khalatnikov and E.M. Lifshitz, Sov. Phys. JETP 35, 838 (1972).
75 11. V.A. Belinskii, I.M. Khalatnikov and E.M. Lifshitz, Adv. Phys. 31, 639 (1982). 12. E.M. Lifshitz, I.M. Lifshitz and I.M. Khalatnikov, Sov. Phys. JETP 32, 173 (1971). 13. D.F. Chernoff and J.D. Barrow, Phys. Rev. Lett. 50, 134 (1983). 14. D.M. Chitre, Ph. D. Thesis, University of Maryland, 1972. 15. C.W. Misner, in D. Hobill et al., eds., Deterministic chaos in general relativity, p. 317 (Plenum, 1994) [gr-qc/9405068]. 16. T. Damour, M. Henneaux, A.D. Rendall and M. Weaver, Annales Henri Poincare3, 1049 (2002) [gr-qc/0202069]; 17. L. Andersson and A.D. Rendall, Commun. Math. Phys. 218, 479 (2001) [grqc/0001047]. 18. A.A. Kirillov, Sov. Phys. JETP 76, 355 (1993). 19. A.A. Kirillov and V.N. Melnikov, Phys. Rev. D52, 723 (1995) [grqc/9408004], 20. V.D. Ivashchuk, A.A. Kirillov and V.N. Melnikov, JETP Lett. 60, 235 (1994) [Pisma Zh. Eksp. Teor. Fiz. 60, 225 (1994)]. 21. V.D. Ivashchuk and V.N. Melnikov, Class. Quant. Grav. 12, 809 (1995). 22. V.G. Kac, Infinite Dimensional Lie Algebras, Third Edition (Cambridge University Press, 1990). 23. S. de Buyl and C. Schomblond, Hyperbolic Kac Moody algebras and Einstein billiards, hep-th/0403285. 24. B. Julia, Report LPTENS 80/16, Invited Paper Presented at the Nuffield Gravity Workshop, Cambridge, England, June 22 - July 12, 1980. 25. B. Julia, in Lectures in Applied Mathematics, AMS-SIAM, vol. 21 (1985), p. 355. 26. J. Brown, O.J. Ganor and C. Helfgott, M-theory and E\Q: Billiards, Branes, and Imaginary Roots, hep-th/0401053. 27. H. Lu, C.N. Pope and K.S. Stelle, Nucl. Phys. B476, 89 (1996) [hepth/9602140], 28. N.A. Obers, B. Pioline and E. Rabinovici, Nucl. Phys. B525, 163 (1998) [hep-th/9712084]. 29. T. Banks, W. Fischler and L. Motl, JHEP 9901, 019 (1999) [hep-th/9811194]. 30. P.C. West, Class. Quant. Grav. 18, 4443 (2001) [hep-th/0104081]. 31. I. Schnakenburg and P.C. West, Phys. Lett. B517, 421 (2001) [hepth/0107181]. 32. I. Schnakenburg and P.C. West, Phys. Lett. B540, 137 (2002) [hepth/0204207]. 33. F. Englert, L. Houart, A. Taormina and P. West, JHEP 0309, 020 (2003) [hep-th/0304206]. 34. F. Englert and L. Houart, JHEP 0405, 059 (2004) [hep-th/0405082]. 35. A.D. Rendall and M. Weaver, Class. Quant. Grav. 18, 2959 (2001) [grqc/0103102]. 36. E. Bergshoeff, M. de Roo, B. de Wit and P. van Nieuwenhuizen, Nucl. Phys. B195, 97 (1982). 37. G.F. Chapline and N.S. Manton, Phys. Lett. B120, 105 (1983).
76
38. E. Cremmer, B. Julia and J. Scherk, Phys. Lett. B76, 409 (1978). 39. T. Damour and H. Nicolai, Eleven dimensional supergravity and the E\o/K(Eio) a-model at low Ag levels, invited contribution to the XXV International Colloquium on Group Theoretical Methods in Physics, 2-6 August 2004, Cocoyoc, Mexico; to appear in the proceedings, hep-th/0410245. 40. K. Bautier, S. Deser, M. Henneaux and D. Seminara, Phys. Lett. B406, 49 (1997) [hep-th/9704131].
GENERALIZED HOLONOMY IN M-THEORY
A. BATRACHENKO, M.J. DUFF, JAMES T. LIU and W.Y. WEN Michigan Center for Theoretical Physics Randall Laboratory, Department of Physics, University of Michigan Ann Arbor, MI 48109-1040, USA E-mail: {abat,mduff, jimliu,wenw}@umich. edu
In M-theory vacua with vanishing 4-form F^, one can invoke ordinary Riemannian holonomy H C Spin(10,1) to account for unbroken supersymmetries n = 1, 2, 3, 4, 6, 8, 16, 32. However, in the presence of non-zero f(4), Riemannian holonomy must be extended to generalized holonomy H C SL(32,R) to account for more exotic fractions of supersymmetry. The resulting number of M-theory vacuum supersymmetries, 0 < n < 32, is then given by the number of singlets appearing in the decomposition of the 32 of SL(32,R) under H C SL(32,R).
1. I n t r o d u c t i o n M-theory not only provides a non-perturbative unification of t h e five consistent superstring theories, b u t also embraces earlier work on supermembranes and eleven-dimensional supergravity. 1 It is regarded by m a n y as the dreamed-of final theory and has accordingly received an enormous amount of attention. It is curious, therefore, t h a t two of the most basic questions of M-theory have until now remained unanswered: i) What are the symmetries of M-theory? ii) How many supersymmetries can vacua of M-theory preserve? While t h e issue of hidden symmetries of M-theory is an i m p o r t a n t one in its own right, here we focus on the more direct task of counting supersymmetries preserved by any given M-theory vacuum state. In the supergravity limit, the equations of M-theory display the maximum number of supersymmetries N = 32, and so n, the number of supersymmetries preserved by a particular vacuum, must be some integer 0 < n < 32. Furthermore, physical arguments restricting maximum spin and spacetime dimension give us confidence t h a t this upper limit of 32 supersymmetries extends t o the full M-theory as well. To proceed, however, we content ourselves with working in the super77
78
gravity limit. In this case, in vacua with vanishing 4-form F( 4 ), it is well known that n is given by the number of singlets appearing in the decomposition of the 32 of Spin(l,10) under H C Spin(l,10) where H is the holonomy group of the usual Riemannian connection DM
=
9M
+
(1)
^AB-
I^M
This connection can account for vacua with n = 0, 1, 2, 3, 4, 6, 8, 16, 32. Vacua with non-vanishing F^ allow more exotic fractions of supersymmetry, including 16 < n < 32. Here, however, it is necessary to generalize the notion of holonomy to accommodate the generalized connection that results from a non-vanishing F^ VM = DM-
^(TMNpQR
- 86MTpQR)FNPQR.
(2)
As discussed in Ref. [2], the number of M-theory vacuum supersymmetries is now given by the number of singlets appearing in the decomposition of the 32 of Q under H c Q where Q is the generalized structure group and 7i is the generalized holonomy group. Discussions of generalized holonomy may also be found in Refs. [3,4]. In subsequent papers by Hull 5 and Papadopoulos and Tsimpis 6 it was shown that Q may be as large as SL(32,R) and that an M-theory vacuum admits precisely n Killing spinors if and only if SL(31 - n,R) K (n + l ) R ( 3 1 - n ) 2 U C SL(32 - n,R) K nR (32 -™\
(3)
i.e. the generalized holonomy is contained in SL(32 — n,R)ixnR ( 3 2 - ? l ) but is not contained in SL(31 - n,R) ix (n + l ) R ( 3 1 - n ) . Here we expand on the use of generalized holonomy as a means of classifying M-theory vacua, and provide some examples of n = 16 and n = 8 generalized holonomy groups. 2. Holonomy and supersymmetry The number of supersymmetries preserved by an M-theory background depends on the number of covariantly constant spinors, VMt
= 0,
(4)
called Killing spinors. It is the presence of the terms involving the 4-form F(4) in (2) that makes this counting difficult. Hence we first consider vacua for which F(4) vanishes. In this case, Killing spinors satisfy the first order integrability condition [DM, DN]Z = \RMNABTABe
= 0,
(5)
79
where RMN is the Riemann tensor. The subgroup of Spin(10,1) generated by this linear combination of Spin(10,1) generators TAB corresponds to the holonomy group H of the connection % . We note that the same information is contained in the first order Killing spinor equation (4) and second-order integrability condition (5). One implies the other, at least locally. The number of supersymmetries, n, is then given by the number of singlets appearing in the decomposition of the 32 of Spin(10,1) under H. In Euclidean signature, connections satisfying (5) are automatically Ricci-flat and hence solve the field equations when Fu\ = 0. In Lorentzian signature, however, they need only be Ricci-null so Ricci-flatness has to be imposed as an extra condition. In Euclidean signature, the holonomy groups have been classified.7 In Lorentzian signature, much less is known but the question of which subgroups H of Spin(10,1) leave a spinor invariant has been answered.8 There are two sequences according as the Killing vector VA = eT^e is timelike or null. Since v2 < 0, the spacelike VA case does not arise. The timelike VA case corresponds to static vacua, where H C Spin(10) C Spin(10,1) while the null case to non-static vacua where H C ISO(9) C Spin(10,1). It is then possible to determine the possible n-values and one finds n = 2, 4, 6, 8, 16, 32 for static vacua, and n = 1 2, 3, 4, 8, 16, 32 for non-static vacua. 9 ' 10 ' 11
2.1. Generalized
holonomy
In general we want to include vacua with F^ ^ 0. Such vacua are physically interesting for a variety of reasons. In particular, they typically have fewer moduli than their zero F^ counterparts. 12 Now, however, we face the problem that the connection in (2) is no longer the spin connection to which the bulk of the mathematical literature on holonomy groups is devoted. In addition to the Spin(10,1) generators TAB, it is apparent from (2) that there are terms involving TABC and TABCDE- In fact, the generalized connection takes its values in the Clifford algebra SL(32,R). Note, however, that some generators are missing from the covariant derivative. Denoting the antisymmetric product of k Dirac matrices by T^k\ the complete set of SL(32,R) generators involve { r ^ . r ^ . T ^ . r ^ . r ^ } whereas only { r ( 2 ) , r ( 3 ) , r ( 5 ' } appear in the covariant derivative. Another way in which generalized holonomy differs from the Riemannian case is that, although the vanishing of the covariant derivative of the spinor implies the vanishing of the commutator, the converse is not true, as discussed below
80
in section 2.2. This generalized connection can preserve exotic fractions of supersymmetry forbidden by the Riemannian connection. For example, M-branes at angles 13 include n = 5, 11-dimensional pp-waves 14,15 ' 16 ' 17 include n = 18, 20, 22, 24, 26, squashed N(l,l) spaces 18 and M5-branes in a pp-wave background 19 include n — 12 and Godel universes 20 ' 21 include n — 14, 18, 20, 22, 24. These various fractions of supersymmetry may be quantified in terms of generalized holonomy groups. Here generalized holonomy means that one can assign a holonomy Ti C Q to the generalized connection appearing in the supercovariant derivative V where Q is the generalized structure group. The number of unbroken supersymmetries is then given by the number of H singlets appearing in the decomposition of the 32 dimensional representation of Q under H C Q. For generic backgrounds we require that Q be the full SL(32, R) while for special backgrounds smaller Q are sufficient.5 To see this, we write the supercovariant derivative as VM = DM + XM
(6)
for some other connection DM and some covariant 32 x 32 matrix XM- If we now specialize to backgrounds satisfying XMe = 0,
(7)
then the relevant structure group is G C Q. Consider, for example, for the connection D arising in dimensional reduction of D = 11 supergravity. One can show2 that the lower dimensional gravitino transformation may be written
—m.
83
We now ask how the 8 of SO(8) decomposes under H. In the case of the left squashed S7, H = SO(7)~, whereupon 8 —*• 1 + 7 and TV = 1, but for the right squashed S7, H = SO(7) + , so that 8 -> 8 and N = 0. Prom the first order integrability condition alone, however, we would have incorrectly concluded that H = G2 C SO (7), for which 8 —> 1 + 7 and hence that both orientations give N = l.22 2.3. Higher order
corrections
Another context in which generalized holonomy may prove important is that of higher loop corrections to the M-theory Killing spinor equations. As shown in Ref. [30], higher loops yield non-Riemannian corrections to the supercovariant derivative, even for vacua with F^ = 0, thus rendering the Berger classification inapplicable. Although the Killing spinor equation receives higher order corrections, so too does the metric, ensuring, for example, that H = G2 Riemannian holonomy 7-manifolds still yield N = 1 in D — 4 when the non-Riemannian corrections are taken into account. This would require a generalized holonomy H for which the decomposition 8 —> 1 + 7 continues to hold. 3. Generalized holonomy for n = 16 We now turn to a generalized holonomy analysis of some basic supergravity solutions. These results were presented in Ref. [31], where we refer the reader to for additional details. We first note that the maximally supersymmetric backgrounds (n = 32), namely E 1 ' 1 0 , AdS 7 x S 4 , AdS4 x S7 and Hpp, all have trivial generalized holonomy, in accord with (3). However, only flat space may be described by (trivial) Riemannian holonomy. Somewhat more interesting to consider are the four basic objects of Mtheory preserving half of the supersymmetries (corresponding to n = 16). These are the M5-brane, M2-brane, M-wave (MW) and the Kaluza-KIein monopole (MK). The latter two have F(4) = 0 and may be categorized using ordinary Riemannian holonomy, with H C Spin(10,1). 3.1. The
M5-brane
The familiar supergravity M5-brane solution 32 may be written in isotropic coordinates as ds2 = Hs1/3dxl
+
Fijki = eijkimdmHr,,
H25/3dy2, (18)
84
where H$(y) is harmonic in the six-dimensional transverse space spanned by {y l L and e^/m = ±1- While the transverse space only needs to be Ricci flat, we take it to be E 5 , so as not to further break the supersymmetry. A simple computation of the generalized covariant derivative on this background yields \Tf}P+H-3'2diH,
ViL = dil-
Vi = di + § i y P+d,- In H - \T{5)di In H,
(19)
where P^ = ±(1 ± r< 5) ) is the standard 1/2-BPS projection for the M5brane and T^ = ^€ijkimX'i:'klm. All quantities with bars indicate tangent space indices. To obtain the generalized holonomy of the M5-brane, we examine the commutator of covariant derivatives. Denning MMN
= [VM,VN],
(20)
we find that MM„ = 0, so that the holonomy is trivial in the longitudinal directions along the brane. On the other hand, the transverse commutator Mij yields a set of Hermitian generators T^ = — ^TjjP^. It is easily seen that they generate the SO(5) algebra [Tij,Tki] = i(SikTji - duTjk - SjkTu + SjiTik).
(21)
As a result, the transverse holonomy is simply SO(5) + , where the + refers to the sign of the M5-projection. Turning next to the mixed commutator, MM», we see that it introduces an additional set of generators, K^i = T^P^. Since r^P^" = PgTp, it is clear that the K^ generators commute among themselves. On the other hand, commuting K^ with the SO(5)+ generators Tij yields the additional combinations K^ = T^P^ and K^j = r ^ j P 5 + . Picking a set of Cartan generators T 12 and T34 for SO(5)+, we may see that the complete set {K^, K^i, K^ij} has weights ±1/2. As a result, they transform as a set of 4-dimensional spinor representations of S0(5)+. We conclude that the generalized holonomy of the M5-brane is HM5 = SO(5)+K6M4(4).
(22)
Note that the additional commutations required to close the algebra arise naturally from the higher order integrability conditions.
85
3.2. The
M2-brane
Turning next to the M2-brane, its supergravity solution may be written as 3
ds^H^dxl
+
H^dy2,
H2
A similar examination of the commutator of generalized covariant derivatives, (20), for this solution indicates the presence of both compact generators Tij = — | r y P 2 + and non-compact ones K^ — T^jP^. Here, P£ = £(1 ± ri)+(i,i6)_
(36)
The generalized holonomy analysis for the M5/MW solution 36 ds2 = H~1/3(2dx+dx-
+Kdx+2+
dz42) +
H2/3dy2,
Fijkl = CijlclmdmHs,
(37)
is similar. Here the functions H$ and K are harmonic on the fivedimensional overall transverse space. This corresponds to a superposition of a M5-brane with a delocalized wave, where the latter has R 5 holonomy. Closing the holonomy algebra over the M5 and MW generators yields the generalized holonomy H M 5 /MW = [SO(5) X SU*(8) X 4R [SO(8) x SU(2) x 3R2(8] X 8R( 8 - 1 )+( 1 - 16 ) [SO(5) x SU*(8) x 4R t = u, l+t2
= ^ - .
(13)
Hence for N = 2 vacua the D7 and D3-brane positions are still moduli while the axion-dilaton and T 2 complex structure are stabilised. 2.3. N = 1,0 critical
points
The N = 1 critical points in flat space studied in [4] were first obtained by setting go, gi 7^ 0 and g2 = g3 = 0, with k% = g0 along the direction qu = Cm=l
a n d ku = gi
a l o n g t h e d i r e c t i o n qu =
Cm=2
Constant Killing spinors. By imposing Se2 f = 0 for the variations of the fermionic fields / we get the following: From the hyperino variations: 5e2CAa = 0 => e^ = 0 m = l , 2 ; a = l , . . . , 1 9 , • vanishing of the gravitino variation.
(14)
The gravitino variation vanishes if: + ig1X1=0.
S22 = -goX°
(15)
From the gaugino variations we obtain: 5t2 {\l)A = 0 =» e% VXA (diXA + (diK)XA)axA2
= 0,
(16)
the second term (with diK) gives a contribution proportional to the gravitino variation while the first term, for i = u, t, xk respectively gives: ~g0daX0 ~g0dtX°
+ ig1duX1 +
= Q,
ig1dtX1=0,
-godxkX°
= 0, (17)
r
for i = y the equation is identically satisfied. From the last equation we get xk = 0 and the other two, together with S22 = 0 give u = t = - i , 9o = 0 i .
101
So we see that for N = 1 vacua the D7-brane coordinates are frozen while the D3-brane coordinates remain moduli. This agrees with the analysis of [3]. If go jt gx the above solutions give critical points with vanishing cosmological constant but with no supersymmetry left. More general N = 1,0 vacua can be obtained also in this case by setting 92, 53 7^ 0. The only extra conditions coming from the gaugino variations for N = 1 vacua is that e ^ 1 , 2 — 0. This eliminates from the spectrum two extra metric scalars eg =1 ' 2 and the Ca=i,2 axions. These critical points preserve N = 1 or not depending on whether |go| = |pi| or not. We can describe the N = 1 —* N = 0 transition with an N = 1 noscale supergravity 34,35 based on a constant superpotential and a non-linear sigma-model which is U ( l , l + n3) SO(2,18) l U ( l ) x U ( l + n3) SO(2) x SO(18) ' ' where the two factors come from vector multiplets and hypermultiplets, respectively. This model has vanishing scalar potential, reflecting the fact that there are not further scalars becoming massive in this transition. 4 We further note that any superpotential W(y) for the D3 brane coordinates would generate a potential 36 term eK Kyy dyWdvW,
(19)
which then would require the extra condition dyW = 0 for a critical point with vanishing vacuum energy. The residual moduli space of K3 metrics at fixed volume is locally given by ( j SO(17) • We again remark that we have considered vacua with vanishing vacuum energy. We do not consider here the possibility of other vacua with nonzero vacuum energy, as i.e. in [20].
2.4. More general
vacua
There are more general critical points defined by values of t, u different form — i and depending on ratios of fluxes. Let us give an instance of this for the N — 2 preserving vacua. Consider the situation with generic flux / P A , p = (m, a), A = 0 , . . . , 3, which corresponds to the charge-couplings: V „ C = 0/iC*, + / * A ^ .
(21)
102
For a N = 2 vacuum, for the vanishing of the gravitino and gaugino variations, we need V% = 0, where Vl=u>lkl=u>$fl.
(22)
From the hyperino variations we have k\XK
= fAXK
= 0.
(23)
We take A = 2, 3 with / | ) 3 ^ 0 for p = a, (a = 1 . . . , 19) and / £ = 0 otherwise. The hyperino variation then is: fa2X2 a
a
Setting f 2 = af
+ fa3X3=0.
(24)
we obtain
3
f%(aX2+X3)
= 0,
(25)
that is Xl=l+tu-^=-a
=
( 2 6 )
-J^-
The condition V\ = 0 on the other hand implies eI„/°2,3=0, a
(27)
a
but since / 2 = a / 3 then the above equation is equivalent to the following single condition exafa2
= 0,
(28)
namely it fixes only one triplet of metric moduli. This vacuum preserves N = 2 supersymmetry with one massive vector multiplet corresponding to a combination of A^ and Az. Moreover condition (26) fixes the T2 complex structure modulus in terms of the axiondilaton and the xk moduli of the D7 brane coordinates. Note that in the previous solution 22 X2 = X3 = 0, u = t, t2 = - 1 + ^ y - and xk were still unfixed. For a = 0 or oo we get X3 or X2 vanishing which corresponds to the example given in [3]. 3. The potential The general form of the N = 2 scalar potential is: V = 4 eKhuvkuAkvE XA ~XE + eKgij k\k3s XA +eK(UAE
-3eKXAXE)VZT>Z,
I
r
(29)
103
where the second term is vanishing for abelian gaugings. Here huv is the quaternionic metric and k^ the quaternionic Killing vector of the hypermultiplet cr-model. The scalar potential, at the extremum of the e^ scalars, has the following formc: 3
14>0K
V = 4e^e
1 2
/
A
Y,(9A) \X f + Ugl+g\){t - t ) ( ( u - « ) AA=0
Z
V
l(xk- -xk)2\ 2 (t--i) ) (30)
From the above expression we see that in the N = 2 case, namely for go = 9i — 0, the potential depends on yr only through the factor eK and vanishes identically in yr for the values of the t, u scalars given in (13), for which X2 = X3 = 0. If g0 or g\ are non-vanishing (N = 1, 0 cases) the extremisation of the potential with respect to xk, namely dxkV — 0 fixes xk = 0. For xk = 0 the potential depends on yT only through the factor eK and vanishes identically in yr for t = u — —i.
4. Special coordinates, solvable coordinates and B.I. action The prepotential for the spatial geometry of the £>3 — Dl system, given in (3), was obtained in [22], by using arguments based on duality symmetry, four dimensional Chern-Simons terms coming from the p-brane couplings as well as couplings of vector multiplets in D = 4 and D = 8. A similar result was advocated in [37,38] by performing first a K3 reduction to D = 6 and then further compactifying the theory to D = 4 on T2. The subtlety of this derivation is that the naive Born-Infeld action derived for D5 and D9 branes in D = 6 gives kinetic terms for the scalar fields which, at the classical level, are inconsistent with TV = 2 supersymmetry. This is a consequence of the fact that anomalies are present in the theory, as in the D — 10 case. The mixed anomaly local counterterms are advocated to make the Lagrangian N = 2 supersymmetric in D = 4. Therefore the corrected Lagrangian, in the original brane coordinates is highly non-polynomial. In fact the original Born-Infeld, Chern-Simons c
Note that there is a misprint in Eq. (5.1) of Ref. [4]. The term e 2 * eK g0gi(X0Xi X I X Q ) is actually absent.
+
104
naive (additive) classical scalar action \ds' + crddr\2 (s' - s')2
s = s
Idu' +
tfdtfl2
(u' - u')2
\dryr
xi = al+tbl
\t' ddr + dd
+
(s' - s') {f - V) WdV + daH2 («' - u') (*' - V)
; u' =u-1-bixi ; yr = 6'
\dt ' | 2 (f - t')2 '
; t' = t,
tdr ,
(31)
has a metric which was shown 38 to be Kahler with Kahler potential K
-log
r
:-.r\2
{s-s)(t-t)--(y
- l o g (u-u)(t-F)--(xi-xi)2 = -logYSK-log(l
+
+ log(t - 1 )
XA —^),
(32)
ISG
where X« =
(x*-x*)2(yr-yr) 4(t-t)
r,r\2
YSK = (a - s)(t - t)(u -u)-l-{u-
u)(yr - f)2
- l ( s - 5) (a* - x')2 , (33)
where here and in the following summation over repeated indices is understood. Therefore the correction to the scalar metric in the brane coordinates is:
dpd?AK = d p 0 ? log(l + •£*-)
(34)
ISG
It is clear that the classical brane coordinates are not good "supersymmetric" coordinates, in that the corrected action is not polynomial in them. Prom the fact that the combined system is a homogeneous space, we indeed expect that suitable coordinates exist such that the quantum corrected (N = 2 supersymmetric) action has a simple polynomial dependence on them, including the interference term. Such coordinates do indeed exist and allow to write the combined Born-Infeld action and supersymmetric Y$K differs by a factor —t from the special geometry formula obtained from the prepotential in (3).
105
counterterms, in a manifest supersymmetric way. Modulo field redefinitions, these coordinates reduce to the standard brane coordinates when either the D3 or the £>7-branes are absent, in which cases the homogeneous space becomes a symmetric space. This parametrization in terms of "supersymmetric" coordinates, corresponds to the solvable Lie algebra description of the manifold first introduced by Alekseevski, 39,40 which we shall discuss in what follows. In Alekseevski's notation the manifold under consideration is of type K(n3,n?) which can be written as: K{n3,n7)
Y±,Z±),
= W(ga,ha,
d i m ( y ± ) = n 3 ; dim(Z ± ) = n7 ,
(35)
where 123 and n7 denote the number of D3 and D7-branes respectively. Our identification of the scalar fields with solvable parameters is described by the following expression for a generic solvable Lie algebra element: Vaha + 6tgt + 0ugu + 6sg3 + yr±Y±
Solv = { Y,
+ zl±Zf}
,
a—t,u,s
6t = et+yr+yr-+zi+zi-,
(36)
where (yr+ ,yr~) and (zl+, zl~) are related to the real and imaginary parts of the D3 and D7-branes complex coordinates along T 2 . The non trivial commutation relations between the above solvable generators are: [huY±}=1-Y±
; [/ l t ) Z ± ] = i z
[h.,Y±]=±±Y± [g.,Y-]
[gu,Z~] = Z+ ,
[Yr , Yf] = 5rs gt ; [ha,ga\=
,
[hu,Z±]=±±Z±,
;
=y+ ;
+
±
[Zf, Zj] = 8ij gt ; r, s = 1 , . . . , n 3 i, j = 1 , . . . , n7 ,
9a ; a = t,u,s.
We exponentiate representative: L = es9s
the solvable
(37) algebra
using the following
eV'-Y- e„-+y+ eeu9u eJ-z;
e^zf
e§t 9t eV"ha
_
coset-
(38)
The order of the exponentials in the coset representative and the particular parameter 6t used for gt, have been chosen in such a way that the axions 6S, 8t, 6U, yr+, zl+ appear in the resulting metric only covered by
106
derivatives. The metric reads: ds2 = (d<pa)2 + e- 2 «" Uet + ^deu(z~)2 +e-2^d92u
+ e-2,fi°d62 + e'^-^
+e-Vt-'p°{dyr+
+ d6syr-)2
+
+ \d6s{y-f
{dzi+ + dduZ^f
+
dyrA
e~^+^{dzi-)2
e-f'+^idy1-)2,
n-j
(z+) 2 = ^ > < + ) 2 ;
+ z*"dz i+ + yr~
«3
{y+)2^(yr+)2.
i=l
(39)
r=l
Identifying the axionic coordinates 6S, 9t, 6U, yr+, zl+ with the real part of the special coordinates s, t, u, yr, xl, and comparing the corresponding components of the metric one easily obtains the following relations between the solvable coordinates and the special coordinates:
t = 6t~l- (e*» +\e^ x^z^+'-e^z'-
{z-f + \e*" (y~)2
; yr=yr++l-e*°yr-.
(40)
Note that the classical B-I+C-S action (31), with no interference term in the D3 (c, d) and Dl (a, b) brane coordinates is still described by a homogeneous manifold spanned by the following 2 713 + 2 717 + 6 isometries: u —> e u u ; Su = UQ + alQb%, s —> eXs s ; 5s = so + cr0dr , t -> eXt t ; St = t0, cr —» e "2 c r ; 5cr =todr , a* -» e _!k 2 _t a* ; e ^ * 6* ; 56* = bQ .
(41)
The underlying homogeneous space is generated by the following rank 3 solvable Lie algebra {T£, Tg, T£, T^, hs, ht, hu, gs, 9t, 9u} whose non triv-
107
ial commutation relations are: [TZ,TI}=5rsgs,
ra,n]=5^9u-
[n, gt] = Ta ; [Tdr, gt] = Tl, [ha, ga]= 9a,
a =
lh„TS) = ±TS; [K,Ti]
=
l
-Tl-
s,t,u, [h3,Tr]=l-T;,
[hu,Ti]
l
=
\ht,Trd} = -\Trd;
\huTl\
[huTl]=-l-Ti;
[ht,12\=±7i,
=
-Ti,
\Tl, (42)
where the nilpotent generators have been labeled by the corresponding axionic scalar fields. This space is not a subspace of the original quaternionic space, but it becomes so if we set either a, b = 0 and exchange the role of s and t or if we set c, d = 0 and exchange the role of u and t. The amazing story is that the coordinates in D = 4 corresponding to the supersymmetric theory, deform this space into an other homogeneous space generated by the isometries in (37) which corresponds to an ./V = 2 special geometry. The relation between the solvable Lie algebra generators {T^, T6*, T£, TJ, hSl ht, hu , gs, gt, gu} corresponding to the classical coordinates and the solvable generators {Y±, Z±, ha, ga} corresponding to the "supersymmetric" coordinates is the following:
Tl = Zi+ ; Tl = fr- , where Y and Z are the generators with opposite grading with respect to Y and Z respectively. It can be shown that in the manifold K(n3,ri7), Y or Z are isometries only if n-j = 0 or 713 = 0 respectively. Indeed in these two cases the manifold is symmetric and each solvable nilpotent isometry has a "hidden" counterpart with opposite grading. Otherwise the manifold spanned by the classical coordinates and the manifold parametrized by the "supersymmetric" ones are in general different. 5. Conclusions The present investigation allows us to study in a fairly general way the potential for the 3-form flux compactification, in presence of both bulk
108
and open string moduli. In absence of fluxes the D3, D7 dependence of the Kahler potential is rather different since this moduli couple in different ways to the bulk moduli. Moreover, in the presence of 3-form fluxes which break N — 2 —> TV = 1,0 the D7 moduli are stabilised while the D3 moduli are not. For small values of the coordinates xk, yr the dependence of their kinetic term is (for u = t = — i), —(dflyrd!*yr)/lm(s) for the D3-brane moduli, and — {dllxkd^xk) for the D7-brane moduli. This is in accordance with the suggestion of [16]. Note that the above formulae, at x = 0, u = t = — i are true up to corrections 0(Im(y) 2 /Im(s)), since y and s are moduli even in presence of fluxes. The actual dependence of these terms on the compactification volume is important in order to further consider models for inflatons where the terms in the scalar potential allow to stabilise the remaining moduli. Finally, we have not considered here the gauging of compact gauge groups which exist on the brane world-volumes. This is, for instance, required 41 ' 42 ' 20 in models with hybrid inflation.43 This issue will be considered elsewhere. Acknowledgments Work supported in part by the European Community's Human Potential Program under contract HPRN-CT-2000-00131 Quantum Space-Time, in which R. D'A. is associated to Torino University. The work of S.F. has been supported in part by European Community's Human Potential Program under contract HPRN-CT-2000-00131 Quantum Space-Time, in association with INFN Frascati National Laboratories and by D.O.E. grant DE-FG0391ER40662, Task C. Appendix A. Some relevant formulae We are interested in gauging the 22 translations in the coset SO(4,20)/(SO(3,19) x 0(1,1)). Let us denote by L the coset representative of SO(3,19)/SO(3) x SO(19). It will be written in the form:
L=(il+ef)hn
-eT
I m
T
a
"" , V (l + e T e)2 /
(A.1) v
'
where e = {e a}, e = {e m} , m = 1,2,3 and a = 1,...,19, are the coordinates of the manifold. The 22 nilpotent Peccei-Quinn generators are
109
denoted by {Zm, Za} and the gauge generators are: tA = fmAZm
+ haAZa ,
(A.2)
the corresponding Killing vectors have non vanishing components: k™ = fmA and k\ = haA- The moment maps are: VXA = V2 (e+ {lTx)*m
fmA + e* ( L " 1 ) * , h%) ,
(A.3)
where is the T 2 volume modulus: 4 e" 2 * = Vol(T2) and x = 1,2,3. The metric along the Peccei-Quinn directions I = (m, a) is: hIj = e2HSu+2eaIeaj).
(A.4)
The potential has the following form: V = 4 e 2 * (fmA
fmE
+ 2 eame\
fmA fns
+2 e 2 * {UAE - 3 LA LE) (fmA +2 [(1 + e e T ) 5 ] V « fmiA
+ haA haE) LA LE
fmE
+ eamean
fmA
haE) + enaenb haA hb^j
fns .
(A.5)
In all the models we consider, at the extremum point of the potential in the special Kahler manifold the following condition holds: (UAS — 3 LA LE) |Q fm(A ha£) = 0 . As a consequence of this, as it is clear from (A.5), the potential in this point depends on the metric scalars e™ only through quadratic terms in the combinations emahaA and eamfmA. Therefore V is extremised with respect to the e™ scalars once we restrict ourselves to the moduli defined as follows: moduli:
emahaA
= eamfmA=0.
(A.6)
The vanishing of the potential implies {UAE - LA LE) ,0 fm{A
fmE) + 2 {LA LE)
|Q
h\A haE) = 0 -
(A.7)
Furthermore, one may notice that, as in [4], the following relations hold in all the models under consideration: (UAE - LA LE)
|0
r
i A
fmE) = (LA LE)
|0
h\A haE) = 0.
(A.8)
Our analysis is limited to the case in which the only non-vanishing / and h constants are: f \ = 9o I / 2 i = 9i ; hl2 = 92 ; h23 = g3 ; h2+k3+k h
2+n +r
?
3+n7+r
= gl.
= g\ , (A.9)
110 A p p e n d i x B . T h e m a t r i x A/" Using the special geometry formula (9) it is possible to compute the matrix TVAE f ° r any choice of the symplectic section, including those cases for which no prepotential exists. For the sake of simplicity we will suppress the indices k and r in xk and yr by considering the case 713 = nj = 1. Moreover we will express the complex coordinates in terms of their real and imaginary parts: s = si+is2;
t = ti+it2;
u = ui+iu2;
yi+iy2 (B.l) Let the Dl and D 3 brane vectors correspond to the values A = 4 , 5 respectively. We shall list below the independent components of the real and imaginary p a r t s of M: 1
Re(A0o,o = si-
-uiyi 2
+
(— 1 + t\ui
x = xi+\x2;
u2 ( - 2 + 2 t i u i -xi2) — 5 2*2^2 — 2:2 — ^xi2)
(2tiu22
y =
2/12/2
+ x2 (—2u2X\ + u\x2))
V22
(-2t2u2+x22)2 2 + 4 (h + M-,I ) W22/2) 2/1 (. - 2 i 2- U2- „2/1 . +x2 yi ---. L Re(Af)o,l = * -;—g + 2 8*2 "2 — 4X2 8*2 "2 - 4X2^ 2 2/2 2 (2u2 (2-2ti (ti+2ui)+xi2^ + 4 (i 2 +ui)
u2x1x2^
2 2
A{-2t2u2+x2 ) (2-2ui
( 2 t i + « i ) + x i 2 ) z 2 2 2/2 2 4(-2t2W2 + K 2 2 ) 2
1 «2 (2*1 MI - x i 2 J yi2/2 2 Re(A0o,2 = ^"12/1 + 2 lyl -2t2u2 + x22 m u i - 5 xi2)
(2t\u22
+x2
(-2u2xi
+ u\
x2)j
2/2
2
(-2i2U2+ai22)2 2/1 ( - 2 t 2 « 2 2 / i + a ; 2 2 2 / l + 4 () 2
Re(A^)3,3 = - s i +
2
— 2
(h-Ui) (2U2 -X2 ) —± - ^ (-2t2"2+X22)
V2
( 2«2 2 xi + 2 (-£1 + ui) «2 »2 - x\ X22) J/22
M^hA
=
-p-
^2
^ 2 ( - 2 t 2 U 2 + Z2 2 ) \[2 ( 2 t 2 u 2 2 / i - £ 2 2 2 / l + 2 {-t\ + w i ) «2 2/2) Re(AT)3,5 = — ^ 5—5 *3:1X2 3/22 —2 (-2i2U2+z22) _2U2JC12/2 . — o
-.A „ Re(7V)4,4 = - s i R
Re(AT)4,5-
4M2
- 2 £2 1*2+£2
Re(AT)5,5 = - U l
(B.2)
As far as Im(A/") is concerned, its independent entries are:
, ,»A S2(4 + 4 ( t l 2 + t 2 2 ) ( U l 2 + U 2 2 ) + X 1 4 + X 2 4 + 2 X 1 2 ( 2 + X22)) , lm(A/) 0 ,o = — —5 V 8t 2 U 2 - 4X2^ S2t2 ( —2mxiX2 + M2 (xi — X2) ( l l + X 2 )) 2 t 2 « 2 — X2 2 $2*1 (2ll2XlX 2 + Ul (2 + Xi 2 - X 2 2 ) ) 2 are taken to vanish in the background. We remark that
121
• No general ansatz is known for the scalar fields deformations 9mn{)2-\e&**G3*G3,
(15)
where the 3-form G3 is closed G 3 = dB2,
(16)
and
Now any group G admits both a left and a right action by itself and thus it may be regarded as a coset of G x G with respect to G. It turns out that for the very special field content of the bosonic string Cvetic, Hu, Pope and myself were able to show1 that • There is a Pauli Reduction on 5 3 thought of as 5 0 ( 4 ) / 5 0 ( 3 ) giving the full complement of the 50(4) gauge bosons. • There is strong evidence that this works in general for the general case G x G/G. 4.2.
Salam-Sezgin
This is a particular case of a general class of six-dimensional supergravities found by Sezgin and Nishino. They have received some attention recently by phenomenologist working on the large extra dimensions scenario. The Lagrangian is
R - i(9^) 2 - ^ * f f 3 A f f s - ^ A A - 8g2e-l* . Note that the theory has • A 3-form H3 = dB2 + \F2 A Au • an abelian fields strength F2 = dA\, • and a scalar <j> with a positive Liouville type potential.
(18)
124
It was discovered long ago by Salam and Sezgin that this model admits a supersymmetric background of the form S2 x E 3 ' 1
+ monopole
(19)
with the scalar cf> = constant. Chris Pope and I have recently shown2 that there is a completely consistent Pauli reduction yielding the gauge group SU(2) and no cosmological term in 3 + 1 dimensions. If /x* are three Cartesian coordinates defining the unit sphere ^u1 = 1 in E 3 , then the ansatz is ds62 = ei+dsl + e-^9mn{dm
+ 2gAiKD(dm
F2 = 2ge^eabea H3 = H3-
+ 2gAiK™),
(20)
/\ebu.iF\
(21)
2gFi A K?{ea + 2gA^K^),
(22)
$ = -.
(23)
The four dimensional Lagrangian form which the equations can be derived is
R - \(d(t>)2 - \^{^f
- \e~**pi
A
^ + \pi
Api
>
(24)
where we have dualized the three-form H3 = e2^ * da .
(25)
What we have in four dimensions is • SU(2) Yang-Mills Fl coupled to an axion a and a dilaton <j>, but with no potential for the scalar's. • Remarkably, it turns out that there is a completely consistent reduction for the fermion sector, in other words: there is a completely consistent reduction to a supersymmetric theory in 3+1 dimensions. This theory, which is the first example known to us of a realization of Pauli's original idea, has many remarkable properties and is certainly the simplest and most completely constructed example of any known reduction, other than the original example of Kaluza.
125
4.3. Uniqueness
of the Ground
State
The currently favoured Calabi-Yau reductions (which are inconsistent in the sense being used here) are highly non-unique, with many moduli. The Salam-Sezgin model has the significant advantage: if one assumes that the six-dimensional spacetime is a product M6 = M2 x M4 ,
(26)
with M4 being maximally symmetric, i.e., de-Sitter ds4, Anti-de-Sitter AdSi or Minkowski spacetime E 3 ' 1 , and that that two-dimensional 'internal'space M2 is a smooth and closed, then Rahmi Giiven, Chris Pope and I have proved3 that the only solution is that of Salam and Sezgin. In other words Local Lorentz-invariance =» SUSY
(27)
One may also construct solutions with conical defects, representing 3branes which break SUSY. 4.4. Super symmetric
Solutions
Marco Cariglia, Rahmi Giiven, Chris Pope and I 5 have found all solutions of the four-dimensional reduced theory which admit Killing spinor, i.e. we have found all supersymmetric solutions. Interestingly, in the Yang-Mills sector, they contain the non-abelian plane waves discovered by Coleman long ago and whose significance has been somewhat obscure. The solutions take the form ds2 = 2dudv + H{u, x, y)du2 + dx2 + dy2 , Ai = Ai(u,x,y)du,
(28) (29)
with {&l + efyA* = 0.
(30)
Thus one may take Ai = ^(Xi(u,z)+xi(u,z)),
(31)
with x*(u> z) holomorphic in z = x + iy but arbitrary in u and the profile function H is given by H(u, z, z) = K(u, z) + K(u, z) - \e-*{\X\u,
z)\2 + (
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