The complete D-brane spectrum in ZZ[squared] orientifolds is computed. Stable non-BPS D-branes with both integral and torsion charges are found. The relation to K-theory is discussed and a new K-theory relevant to orientifolds is suggested

Bertolini, M.

Campos V.L

Ferretti, G.

Fre, P.

Salomonson, P.

Trigiante, Mario

English

arXiv

Matthias R. Gaberdiel

Crossref

D-branes from conformal field theory

Nicolas Boulanger

MUCC (Crossref)

Multi-graviton theories: yes-go and no-go results.

Bergshoeff, Eric

University of Groningen Research Database

Supersymmetry in singular spaces

Laura Andrianopoli

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N = 2 → N = 1 supergravity reduction in four dimensions
Laura Andrianopoli 1
1 Dip. Fisica, Politecnico di Torino, Corso Duca degli Abruzzi 24, I-10129 Torino
e-mail: andrianopoli@athena.polito.it
Abstract: We discuss the reduction of N = 2 supergravity to N = 1, by a consistent
truncation of the second gravitino multiplet.
It is well known that for globally supersymmetric theories with particle content of
spin 0, 1
2
, 1 any theory with N supersymmetries can be regarded as a particular case of
a theory with a number N ′ < N of supersymmetries [1]. To prove this it is sufficient
to decompose the N supersymmetry–extended multiplets into N ′-multiplets. Of course
N -extended supersymmetry is more restrictive than N ′ < N supersymmetry implying
that the former will only allow some restricted couplings of the latter. As we are going
to show, the same argument does not apply to supergravity theories [2]. Indeed, let us
consider a standard N -extended supergravity with N gravitini and a given number of
matter multiplets: then the N ′-extended supergravity obtained by reduction from the
mother theory will no longer be standard because a certain number N − N ′ of spin 3
2
multiplets appear in the decomposition. To obtain a standard N ′-extended supergravity
one must truncate out at least the N−N ′ spin 3
2
multiplets and all the non-linear couplings
that they generate in the supergravity action.
Here I will report on the truncation of N = 2 matter-coupled supergravity theory to
N = 1, showing that the reduction does indeed imply the truncation of part of the matter
content, besides of the second gravitino multiplet. The present discussion is based on
[2]. A more detailed analysis, with proofs of consistency, and including the truncation of
N = 8 supergravity down to general N theories can be found in [2] to which I refer also
for the notations and conventions and for a complete list of references.
The supersymmetry reduction N = 2 → N = 1 is obtained by truncating the N = 1
spin 3/2 multiplet containing the second gravitino ψµ2 and the graviphoton.
Let us write down the supersymmetry transformation laws of the fermions for the
N = 2 theory, up to 3-fermions terms [3]:
δ ψAµ = ∇̂µ ǫA +
(
i g SABηµν + ǫABT
−
µν
)
γνǫB (1)
δ λIA = i∇µ zIγµǫA +G−Iµν γµνǫBǫAB + gW IABǫB (2)
δ ζα = iUBβu ∇µ qu γµǫAǫAB Cαβ + gNAα ǫA (3)
where: ∇̂µ ǫA = DµǫA + ω̂ Bµ|A ǫB + Q̂µǫA, with the SU(2) and U(1) 1-form “gauged”
connections respectively given by: ω̂ BA = ω
B
A + g(Λ)A
Λ P x
Λ
(σx) BA ; Q̂ = Q+ g(Λ)AΛ P 0Λ,
1
ω BA and Q = − i2(∂IKdzI − ∂ĪKdz̄Ī) being the SU(2) and U(1) connections of the
ungauged theory. T−µν appearing in the supersymmetry transformation law of the N = 2
left-handed gravitini is the “dressed” graviphoton defined as:
T−µν ≡ 2iImNΛΣLΣ
[
FΛ−µν + 3 fermions terms
]
(4)
while
GI−µν ≡ −gIJ̄ ImNΛΣ f̄ΣJ̄
[
FΛ−µν + 3 fermions terms
]
(5)
are the “dressed” field strengths of the vectors inside the vector multiplets (the “minus”
apex means taking the self-dual part.). Moreover the fermionic shifts SAB,W
I AB and NAα
are given in terms of the prepotentials and Killing vectors of the quaternionic geometry
as follows:
SAB = i
1
2
PABΛ L
Λ ≡ i1
2
P x
Λ
σxABL
Λ (6)
W I AB = iPAB
Λ
gIJ̄ fΛJ̄ + ǫ
ABkI
Λ
L
Λ
(7)
NAα = 2UAαu kuΛL
Λ
(8)
Since we are going to compare the N = 2 reduced theory with the standard N = 1
supergravity, I also quote the supersymmetry transformation laws of fermions in the
latter theory [4],[5]. We have, up to 3-fermions terms:
N = 1 transformation laws
δψ•µ = Dµǫ• +
i
2
Q̂µǫ• + iL(z, z̄)γµε
• (9)
δχi = i
(
∂µz
i + g(Λ)A
Λ
µk
i
Λ
)
γµε• +N
iε• (10)
δλΛ• = F (−)Λµν γµνε• + iDΛε• (11)
where Q̂ is defined in a way analogous to the N = 2 definition and:
L(z, z̄) = W (z) e
1
2
K(1)(z,z̄) , ∇ı̄L = 0 (12)
N i = 2 gi̄∇̄ L̄ (13)
DΛ = −2(ImfΛΣ)−1PΣ(z, z̄) (14)
andW (z),K(1)(z, z̄), PΣ(z, z̄), fΛΣ(z) are the superpotential, Kähler potential, Killing pre-
potential and vector kinetic matrix respectively [5], [4], [6]. Note that for the gravitino
and gaugino fields we have denoted by a lower (upper) dot left-handed (right-handed)
chirality. For the spinors of the chiral multiplets χ, instead, left-handed (right-handed)
chirality is encoded via an upper holomorphic (antiholomorphic) world index (χi, χı̄).
To perform the truncation we set A=1 and 2 successively, putting ψ2µ = ǫ2 = 0, and
for the gravitino we get, from equation (1):
δ ψ1µ = Dµǫ1 − Q̂µǫ1 − ω̂ 1µ|1 ǫ1 + i g S11ηµνγνǫ1 (15)
while, for consistency:
δ ψ2µ ≡ 0 = −ω̂ 1µ|2 ǫ1 +
(
i g S21ηµν − T−µν
)
γνǫ1 (16)
Comparing (9) with (15), we learn that we must identify ψ1µ ≡ ψ•µ ; ǫ1 ≡ ǫ•. Fur-
thermore, g S11 must be identified with the superpotential of the N = 1 theory, that is to
2
the covariantly holomorphic section L of the N = 1 Kähler-Hodge manifold. Therefore
we have [7] L(q, z, z̄) = i
2
g(Λ)P
x
Λ
(σx)11L
Λ = i
2
g(Λ) (P
1
Λ
− iP 2
Λ
)LΛ. As it has been shown in
[2], after consistent reduction of the special-Kähler manifold MSK and of the quaternionic
σ-model MQ, L is in fact a covariantly holomorphic function of the Kähler coordinates
ws of the reduced manifold MKH ⊂ MQ and of some subset zi ∈ MR of the scalars zI
of the N = 2 special-Kähler manifold MSK .
Furthermore, for a consistent truncation we must set to zero all the following bosonic
structures:
T−µν = 0 (17)
ω̂
1
µ|2 = 0 (18)
S21 = 0 (19)
As it has been proven in [2], equations (17) and (18) are “orthogonality” conditions
on the scalar sectors of N = 2 supergravity which imply a reduction of both special
Kähler manifold (MSK(nV )) [8],[9], [10], [3] and quaternionic manifold (MQ(nH)), where
nV and nH are the number of vector multiplets and hypermultiplets respectively, and
imply therefore a truncation of part of the matter multiplets. We note that similar
orthogonality conditions, leading to the same reduction of the matter content of the
theory, can be found through a complementary analysis, by looking at the 3-fermions
terms in the transformation laws, instead of at the bosonic ones [11].
Equation (17) is satisfied by imposing a suitable constraint on the set of vectors and of
scalar sections which can be retained in the reduction. Indeed, if we decompose the index
Λ labelling the vectors into two disjoint sets Λ ⇒ (Λ, X),Λ = 1, · · · , n′V = nV − nC ;X =
0, 1, · · · , nC , we may satisfy the relation (17) as an “orthogonality relation” between the
subset Λ running on the retained vectors and the subset X running on the retained scalar
sections. That is we set:
FXµν = 0; (20)
ImNΛΣLΣ = TΛ = 0 (21)
We note that if we delete the electric field strengths F−X we must also delete their
magnetic counterpart G−X = N̄XY F−Y + N̄XΣF−Σ = 0 so that we must also impose
NXΣ = 0. Then, the constraint (21) reduces to ImNΛΣLΣ = 0 which implies
LΣ = 0 (22)
since the vector-kinetic matrix ImNΛΣ has to be invertible. Note that condition (22)
implies a reduction of the N = 2 scalar manifold MSK → MR, since it says that some
coordinate dependent sections on MSK have to be zero in the reduced theory.
Let us now quote the reduction of the gauginos transformation law. When ǫ2 = 0 we get,
up to three fermions terms:
δ λI1 = i∇µ zIγµǫ1 + W I11ǫ1 (23)
δ λI2 = −G−Iµν γµνǫ1 + gW I21ǫ1. (24)
From eqs. (23) and (24) we immediately see that the spinors λI1 transform into the
scalars zI (and should therefore belong to N = 1 chiral multiplets) while the spinors
λI2 transform into the matter vectors field strengths G−Iµν (and should then be identified
with the gauginos of the N = 1 vector multiplets). Let us decompose the world indices
3
I of the N = 2 special-Kähler σ-model as follows: I ⇒ (i, α), with i = 1, · · · , nC ,
α = 1, · · · , n′V = nV − nC , where nC and n′V are respectively the number of chiral and
vector multiplets of the reduced N = 1 theory while nV is the number of N = 2 vector
multiplets. By an appropriate choice of coordinates, we call zi the coordinates on MR,
zα the coordinates on the orthogonal complement. Then it is easy to see that the metric
gIJ̄ has only components gi̄, gαβ̄, while giᾱ = 0. Then, if we decompose the gauginos
λI2 ⇒ (λi2, λα2), the above truncation implies, by supersymmetry, λi2 = 0 and, for
consistency,
δ λi2 = −G−iµνγµνǫ1 + gW i21ǫ1 = 0. (25)
Setting G−iµν = 0 gives:
G−iµν = −giJ̄∇J̄ L̄ΛImNΛΣF−Σµν = −gi̄∇̄L̄ΛImNΛΣF−Σµν = 0 (26)
implying
∇̄L̄Λ = f̄Λ̄ = 0. (27)
Moreover, W i21 = 0 implies P 3
Λ̇
= 0, ki
Λ̇
= 0. Note that the integrability condition of
equation (27) is:
∇i∇jLΛ = iCijKgKK̄∇K̄L̄Λ = iCijkgkk̄∇k̄L̄Λ + iCijαgαᾱ∇āL̄Λ = 0. (28)
where Cijk is the 3-index symmetric tensor appearing in the equations defining the special
geometry (see e.g. ref. [12],[3]). Since the first term on the r.h.s. of equation (28) is zero
on MR (equation (27)), equation (28) is satisfied by imposing:
Cijα = 0 (29)
so that only the N = 2 special-Kähler manifolds satisfying the constraint (29) are suitable
for reduction.
Let us now consider the other condition coming from the reduction of the gravitino
transformation law, eq. (18). It implies, on the curvature Ω 12 of the SU(2) connection
ω 12 :
Ω 12 = −i ICαβUα1 ∧ Uβ2 = 0 (30)
Let us now note that the scalars in N = 1 supergravity must lie in chiral multiplets, and
span in general a Kähler-Hodge manifold. It is therefore required that the holonomy of
the quaternionic manifold be reduced: Hol
(
MQ
)
⊂ SU(2)× Sp(2nH) → Hol
(
MKH
)
⊂
U(1) × SU(n). Therefore the SU(2) and the Sp(2nH) indices have to be decomposed
accordingly. We set α → (I, İ) ∈ U(1) × SU(nH) ⊂ Sp(2nH) and the symplectic metric
ICαβ reduces to ICIJ̇ = − ICJ̇I = δIJ̇ . From equation (30) we see that the constraint (18)
for involution is satisfied iff U1I ∧ U1İ = 0 that is if, say, the subset U2I =
(
U1İ
)∗
of the
quaternionic vielbein is set to zero on a submanifold MKH ⊂ MQ.
When this condition is imposed, the submanifold MKH has dimension at most half the
dimension of the quaternionic manifold and the SU(2) connection is reduced to a U(1)
connection, whose curvature on MKH is:
Ω3|MKH = iλU1I ∧ U1I = iλU1I ∧ U1I (31)
so that the SU(2)-bundle of the quaternionic manifold is reduced to a U(1)-Hodge bundle
for the nH dimensional complex submanifold spanned by the nH complex vielbein U1I .
The truncation corresponds therefore to select a nH-complex dimensional submanifold
4
MKH ⊂ MQ spanned by the vielbein U1I and to ask that, on the submanifold, the 2nH
extra degrees of freedom are frozen, that is: U2I |MKH = (U1İ)∗ |MKH = 0.
In order to consistently impose the constraint U2I = 0, it is necessary to check its
involution:
dU2I = i
2
(ω1 + iω2) ∧ U1I + i
2
ω3 ∧ U2I −∆IJ ∧ U2J −∆IJ̇ ∧ U2J̇ = 0. (32)
Eq. (32) implies that we must set to zero, for a consistent reduction, also the “off-
diagonal” part of the Sp(2nH) connection, ∆
I
J̇
. This in turn implies a constraint on the
four-fold symmetric tensor Ωαβγδ appearing in the symplectic part of the quaternionic
curvature [2]:
ΩIJKL̇|MKH = 0. (33)
Finally, let us look at the reduction of the hypermultiplets transformation laws, after
imposing (18). They become, after putting ǫ2 = 0:
U1Iu δ qu = ζ̄Iǫ1 ; U2Iu δ qu = − ICIJ̇ ζ̄J̇ǫ1 (34)
δ ζI = iU2J̇u ICIJ̇ ∇µ qu γµǫ1 + gN1I ǫ1 =
(
δ ζI
)∗
(35)
δ ζJ̇ = iU2Iu ICJ̇I ∇µ qu γµǫ1 + gN1J̇ǫ1 =
(
δ ζ J̇
)∗
(36)
Equations (34) show that the fermionic partners of the retained scalars U1I are the spinors
ζI , while the partners of the scalars U2I which have to be dropped out are the spinors ζİ .
Consistency of the truncation then imposes that the gauging contribution gN1
J̇
has to be
zero on the reduced N = 1 theory.
In the sequel, we just quote the main results on the gauged theory, which are extensively
discussed in [2]. The truncation N = 2 → N = 1 implies, on the gauged theory:
- The D-term of the N = 1-reduced gaugino λΛ = −2fΛi λi2 is:
DΛ = −2fΛαW α21 = −2g(Λ)(Imf)−1ΛΣ
(
P 3Σ(w
s) + P 0Σ(z
i)
)
(37)
- The N = 1-reduced superpotential, that is the gravitino mass, is
L(z, w) =
i
2
g(X)L
X
(
P 1X − iP 2X
)
(38)
and is a holomorphic function of its coordinates zi and ws.
- The fermion shifts of the N = 1 chiral spinors χi = λi1 coming from the N = 2 gaugini
are N i ≡ gW i11 = 2gi̄∇̄L̄.
- The fermion shifts of the N = 1 chiral spinors ζs =
√
2P I,sζI coming from N = 2
hypermultiplets (P I,s are the scalar vielbein on MKH) are N s = −4g(X)ktXL̄XU1İt Us2İ =
2gss̄∇s̄L̄. Furthermore, the consistency of the truncation of the second gravitino multiplet
δψµ2 = 0 and of the spinors ζİ in the hypermultiplets sector for the gauged theory gives
the gauging constraints:
ω̂ 21 = 0 =⇒ g(Λ)AΛ
(
P 1
Λ
− iP 2
Λ
)
= 0 ⇒ g(Λ)AΛ
(
P 1Λ − iP 2Λ
)
= 0 (39)
S12 = 0 =⇒ g(Λ)LΛP 3Λ = 0 ⇒ g(X)LXP 3X = 0 (40)
δζİ = 0 =⇒ g(Λ)ksΛL̄Λ = 0 ⇒ g(X)ksXL̄X = 0 (41)
If we call G(2) the gauge group of the N = 2 theory and G(1) ⊆ G(2) the gauge group
of the corresponding N = 1 theory, then we have that the adjoint representation of G(2)
5
decomposes as Adj(G(2)) ⇒ Adj(G(1))+R(G(1)), where R(G(1)) denotes some representa-
tion of (G(1)). The gauged vectors of the N = 1 theory are restricted to the subset {AΛ}
generating Adj(G(1)) (that is, with the same decomposition of the ungauged theory, we set
Λ → (Λ, X), with Λ ∈ Adj(G(2) andX ∈ R(G(1))). The quaternionic Killing vectors of the
N = 2 theory then decompose as ku
Λ
⇒ {ksΛ, ks̄Λ, ktΛ, ksX , ks̄X, ktX}. Obviously, we must have
that ksX = 0 since the Killing vectors of the reduced submanifold have to span the adjoint
representation of G(1). Viceversa, the Killing vectors with world index in the orthogonal
complement, kt
Λ
, must obey ktΛ = 0, while k
t
X are in general different from zero. Indeed,
the isometries generated by kt
Λ
would not leave invariant the hypersurface describing the
submanifold MKH ⊂ MQ. Since we have found that ksX = 0, equation (41) is identically
satisfied. Eq.s (39) and (40) are satisfied by requiring: P 1Λ = P
2
Λ = 0 ; P
3
X = 0.
We are left with an N = 1 theory coupled to n′V vector multiplets (Λ = 1, · · · , n′V ) and
nC + nH chiral multiplets (X = 0, 1, · · · , nC) with superpotential (38). All the isometries
of the scalar manifolds are in principle gauged since the D-term of the reduced N = 1
theory depends on P 0Λ(z, z̄) + P
3
Λ(w, w̄).
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6


N = 2 → N = 1 supergravity reduction in four dimensions

The complete D-brane spectrum in Z(2) orientifolds is computed. Stable non-BPS D-branes with both integral and torsion charges are found. The relation to K-theory is discussed and a new K-theory relevant to orientifolds is suggested

D-branes, Orientifolds and K-theory

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