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The holonomy of IIB supercovariant connection

[.2cm]

Department of Mathematics

King’s College London

Strand

London WC2R 2LS

We show that the holonomy of the supercovariant connection of IIB supergravity is contained in . We also find that the holonomy reduces to a subgroup of for IIB supergravity backgrounds with Killing spinors. We give the necessary and sufficient conditions for a IIB background to admit Killing spinors. A IIB supersymmetric probe configuration can involve up to 31 linearly independent planar branes and preserves one supersymmetry.

Supersymmetric backgrounds in various supergravity theories can be characterized by the holonomy of their supercovariant connection. For example, the holonomy of the supercovariant connection of maximally supersymmetric solutions is such that the curvature of the supercovariant connection vanishes. The latter property has been used in  to classify all maximally supersymmetric solutions in eleven and ten dimensions. The holonomy of the supercovariant connection of a generic eleven-dimensional supergravity background is a subgroup of [3, 4]. For backgrounds with Killing spinors it reduces to . This has been used in  to give the necessary and sufficient conditions for an eleven-dimensional background to admit Killing spinors. The holonomy of the supercovariant connection is known for many eleven-dimensional backgrounds, see [3, 5] and references therein. For another approach to investigating supersymmetric solutions in eleven-dimensions see .

In this note, we shall show that the (restricted) holonomy, , of the supercovariant connection, , of IIB supergravity is , i.e. it is the same as that of the supercovariant connection of eleven-dimensional supergravity. Consequently, the holonomy of the supercovariant connection of a background with Killing spinors is a subgroup of . A IIB background admits exactly Killing spinors iff

 SL(31−N,\mybb R)⋉(⊕N+1\mybb R31−N)⊉hol(D)⊆SL(32−N,\mybb R)⋉(⊕N% \mybb R32−N) . (0.1)

Adapting the results of  to ten dimensions, we conclude that there is no topological obstruction for the existence of Killing spinors. The first obstruction can occur for provided that the top cohomology class of spacetime does not vanish. If spacetime has topology , an obstruction can occur for . The holonomy of the supercovariant connections of standard and massive IIA supergravities takes values in as well.

The Clifford algebra as a vector space is isomorphic to , , and so it has dimension . (For a summary see ). The Clifford algebra as an algebra is isomorphic to

 Cliff(\mybb R9,1)=M32(\mybb R) , (0.2)

where is the space of matrices with real entries. We have

 Cliff(\mybb R9,1)=Cliffeven(\mybb R9,1)⊕Cliffodd(\mybb R9,1) ,

corresponding to the decomposition of in terms of even- and odd-degree forms. It is known that

 Cliffeven(\mybb R9,1)=M16(\mybb R)⊕M16(\mybb R) (0.3)

and so . has one irreducible (pinor) representation of dimension 32, given by the standard action of on . The even part of the Clifford algebra has two irreducible representations given by the standard action of on , one for each factor in (0.3). These are the well-known Majorana-Weyl spinor representations of . Therefore,

 Spin(9,1)⊂SL(16,\mybb R)⊂GL(16,\mybb R) . (0.4)

The product of two spinor representations can be decomposed as

 Δ±⊗Δ±=Λ1(\mybb R9,1)⊕Λ3(\mybb R9,1)⊕Λ5±(\mybb R9,1) ,

and

 Δ+⊗Δ−=Λ0(\mybb R9,1)⊕Λ2(\mybb R9,1)⊕Λ4(\mybb R9,1) .

In particular, we have

 Δ±⊗Δ±=M16(\mybb R)

for all choices of representations. In what follows, we denote by the spinor bundles associated with the representation , respectively.

IIB supergravity [8, 9] has two Killing spinor equations. One is associated with the supersymmetry variation of the gravitino and the other with the supersymmetry variation of the axionino-dilatino. The former is differential and it is the parallel transport equation of the supercovariant connection. The latter is algebraic. We shall be concerned mostly with the gravitino Killing spinor equation. There are many ways to write the Killing spinor equations of IIB supergravity. For our purpose, it is convenient to use the form given in  after adjusting for conventions111We use a mostly plus metric, , and the gamma matrices are real.. The gravitino Killing spinor equation in the string frame is

 DMϵ=0 , (0.5)

where is a chiral spinor doublet, i.e. a section of the bundle , is the supercovariant connection,

 DM=∇M+18HMABΓAB⊗σ3+116eϕ5∑n=1(−1)n−1(2n−1)!GA1…A2n−1ΓA1…A2n−1ΓM⊗λn , (0.6)

is the Levi-Civita connection and is the dilaton. We have used the notation , for even, and , for odd; are the Pauli matrices. In addition, we have222Our conventions for a k-degree form are .

 H = dB (0.7) G2n+1 = dC2n−H∧C2n−2 , (0.8)

where () is the NS-NS three-(two-)form field strength (gauge potential) and () are the R-R - (-)form field strengths (gauge potentials). The independent RR field strengths are , and , where , , and is self-dual.

The supercovariant connection of IIB supergravity is a connection on the bundle . This bundle can be viewed as an associated vector bundle of a principal bundle with fibre group for or . The representation is the standard representation of on restricted to the subgroup . The bundle is also an associated bundle of and . This can been seen by observing that and can be diagonally embedded in and in . In order to investigate the properties of the supercovariant connection, it is best to view as an associated bundle of .

The holonomy of the superconnection can be any subgroup of . We shall find that it is a subgroup of . To show this, we have to compute the curvature

 RMN=[DM,DN] (0.9)

of the supercovariant derivative. The supercovariant curvature can be expanded in a basis involving gamma matrices as follows

 RMN = 3∑μ=1ϕMNμ 116⊗τμ (0.10) + 3∑μ=02∑n=1ϕMNA1…A2nμΓA1…A2n⊗τμ , (0.11)

where , , and . Observe that is expanded in even powers of gamma matrices. The coefficients are functions of the bosonic fields of IIB supergravity. As we have seen the basis involving even powers of gamma matrices spans . These are tensored with the space of matrices. Thus the curvature of the supercovariant derivative takes values in333In general .

 M16(\mybb R)⊗M2(\mybb R)=M32(\mybb R) .

However the component of along the generator vanishes. To see this observe that the only terms that can contribute to the component of along the generator , are proportional to

 HMA1A2HNB1B2[ΓA1A2,ΓB1B2]⊗12 (0.12)

and

 GA1…A2n−1GB1…B2n−1[ΓA1…A2n−1ΓM,ΓB1…B2n−1ΓN]⊗12 . (0.13)

The remaining terms are always of the type for and , where denotes the skew-symmetric product of gamma matrices. It is easy to see that (0.12) vanishes along . The only terms in the decomposition of (0.13) which are along are proportional to either

 GMA1…A2n−2GNA1…A2n−2 (0.14)

or

 gMNGA1…A2n−1GA1…A2n−1 . (0.15)

In both cases they vanish because of the anti-symmetry in the indices. Thus the non-vanishing components of are along the generators , and ; , . These span the subspace of of matrices that have zero trace, and . Thus we conclude that

 hol(D)⊆SL(32,\mybb R) . (0.16)

The existence of parallel (Killing) spinors with respect to the supercovariant derivative , i.e. spinors such that

 DMϵ=0 ,

implies

 RMNϵ=0

and the reduction of the holonomy group to a subgroup of . If the background admits Killing spinors, then

 hol(D)⊆SL(32−N,\mybb R)⋉(⊕N% \mybb R32−N) .

The proof is similar to the one given in the case of eleven-dimensional supergravity. Similarly, a background admits precisely Killing spinors iff

 SL(31−N,\mybb R)⋉(⊕N+1\mybb R31−N)⊉hol(D)⊆SL(32−N,\mybb R)⋉(⊕N% \mybb R32−N) . (0.17)

These conditions together with the field equations of IIB supergravity are the necessary and sufficient conditions for a IIB supergravity background to admit Killing spinors.

Conditions (0.17) can be expressed in terms of conditions on the supercovariant curvature. For this we adopt a moving frame along the parallel spinors and introduce the basis of given for example in . Since is another basis of , there is an invertible transformation such that

 ΓA1…A2n⊗τμ=YA1…A2n,μabmab , (0.18)

where and have the appropriate ranges. Therefore we can expand

 RMN=RMNabmab . (0.19)

The conditions for the existence of Killing spinors are

 RMNab=0 ,a=1,…,32     b=1,…,N . (0.20)

These are similar to those found in eleven-dimensional supergravity.

For completeness, we give the curvature of the supercovariant derivative of IIB supergravity in the notation of  to which the reader is referred for further details. is given by

 DMϵ:=DMϵ+UMϵ+VMϵ⋆ , (0.21)

where

 DMϵ := (∂M+14ωMabΓab−i2QM)ϵ UM := iκ48ΓL1…L4FML1…L4 VM := κ96(ΓML1L2L3GL1L2L3−9ΓL1L2GML1L2) . (0.22)

A computation yields

 D[NDM]ϵ=12RNMϵ=Sϵ+Tϵ⋆ , (0.23)

where the 16 by 16 complex matrices , are given by

 S = 18RNML1L2ΓL1L2−12P[NP⋆M]+iκ48ΓL1…L4D[NFM]L1…L4 (0.24) +κ224(−ΓL1L2F[N|L1Q1Q2Q3F|M]L2Q1Q2Q3+12ΓL1…L4FNML1Q1Q2FL2L3L4Q1Q2 +12Γ[NL1L2L3FM]L1Q1Q2Q3FL2L3Q1Q2Q3) +κ232(−12G[NL1L2G⋆M]L1L2+148ΓNMGL1L2L3G⋆L1L2L3 −14Γ[NL1GM]L2L3G⋆L1L2L3+18Γ[N|QGQL1L2G⋆|M]L1L2 +316ΓL1L2GNML3G⋆L1L2L3−ΓL1L2G[N|L1QG⋆|M]L2Q −316ΓL1L2GL1L2QG⋆NMQ+116ΓNML1L2GL1Q1Q2G⋆L2Q1Q2 −116ΓL1…L4GL1L2L3G⋆NML4+18Γ[N|L1L2L3GL1L2QG⋆|M]L3Q +14ΓL1…L4G[N|L1L2G⋆|M]L3L4+116ΓL1…L4GNML1G⋆L2L3L4 +14Γ[N|L1L2L3G|M]L1QG⋆L2L3Q+124Γ[N|L1…L5G|M]L1L2G⋆L3L4L5 −148Γ[N|L1…L5GL1L2L3G⋆|M]L4L5−132ΓNML1…L4GL1L2QG⋆L3L4Q −1288ΓNML1…L6GL1L2L3G⋆L4L5L6)

and

 T = −κ96(Γ[NL1L2L3DM]GL1L2L3+9ΓL1L2D[NGM]L1L2) (0.25) +iκ232(13FNML1L2L3GL1L2L3+ΓL1L2F[N|L1L2Q1Q2G|M]Q1Q2 +13Γ[NQFM]QL1L2L3GL1L2L3−12ΓL1…L4FNML1L2QGL3L4Q +12Γ[NL1L2L3FM]L1L2Q1Q2GL3Q1Q2+14ΓL1…L4FL1…L4QGNMQ −12Γ[N|L1L2L3FL1L2L3Q1Q2G|M]Q1Q2) .

Using the doubling trick

 i→(01−10),     ϵ=ϵ1+iϵ2→(ϵ1ϵ2) , (0.26)

we can write (0.23) in terms of 32 by 32 real matrices,

 12RNM=Re(S)⊗τ0+Im(T)⊗τ1−Im(S)⊗τ2+Re(T)⊗τ3 . (0.27)

It can be seen again that takes values in .

As in the case of M-theory , the maximal number of linearly independent IIB branes that can take part in a supersymmetric probe configuration in is 31. This number is determined by the dimension of the Cartan subalgebra of spanned by symmetric traceless matrices. Such brane configurations can be constructed by reducing a supersymmetric M-brane configuration of M-theory with 31 linearly independent branes to IIA and then T-dualizing to IIB. The subgroup of rotates one IIB supersymmetric planar brane configuration to another. Generic rotations by do not preserve the hermiticity of projectors.

The holonomy of the supercovariant connections of standard and massive IIA supergravities also takes values in . In the former case this follows either from the similar result in eleven-dimensions or by directly computing the supercovariant curvature. In the latter case it follows from the computation of the supercovariant curvature of massive IIA supergravity . Therefore all our results regarding the reduction of the holonomy for backgrounds with N Killing spinors and the above discussion regarding planar brane configurations apply to this case as well. It is rather striking that the holonomy of the supercovariant connections of D=11 and type II D=10 supergravities is . However, the spaces on which acts in each case are different. One explanation for this may be the underlying duality relations between these theories. This common property of ten- and eleven-dimensional supergravity theories merits further investigation.

Acknowledgements

DT thanks the European Union, grant number HPRN-2000-00122, which includes the Department of Physics, Queen Mary College University of London, as a sub-contractor.
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