A Kahler-Einstein inspired anzatz for Spin(7) holonomy metrics and its solution

0.15 MB·English

by  O. P. Santillan

A K¨ahler-Einstein inspired anzatz for Spin(7) holonomy
metrics and its solution
O.P.Santillan
∗
7
0
0
2

n Abstract
a
J
We construct propose an anzatz for Spin(7) metrics as an R bundle over closed G
7 2
1 structures. These G2 structures are R3 bundles over 4-dimensional compact quaternion
Ka¨hler spaces. The inspiration for the anzatz metric comes from the Bryant-Salamon

4
construction of G holonomy metrics and from the fact that the twistor space of any
v 2
8 compact quaternion Ka¨hler space is Ka¨hler-Einstein. The reduction of the holonomy to
8 a subgroup of Spin(7) gives non linear system relating three unknown functions of one
0
variable. We obtain aparticular solution andwefindthattheresultingmetric isaCalabi-
9
0 Yau cone over an Einstein-Sassaki manifold which means that the holonomy is reduced to
6 SU(4) Spin(7). Another coordinate change show us that our metrics are hyperkahler
0 ⊂
conesknownasSwannbundles,thustheholonomyisreducedtoSp(2) SU(4) Spin(7)
/
h ⊂ ⊂
and the cone is tri-Sassakian. We revert our argument and state that the Swann bundle
t
- define a closed G structure by reduction along an isometry. We calculate the torsion
p 2
e classes for such structure explicitly.
h
:
v
i 1. Introduction
X
r
a
There exist a growing interest in the construction of Spin(7) holonomy metrics due to their
application in supergravity compactification preserving certain amount of supersymmetry [13]-
[30]. The present work is concerned with this task and from the analysis performed here it is
obtained the following proposition.
Proposition Let us consider a compact quaternion K¨ahler space M in d = 4 with metric
g and with cosmological constant Λ normalized to 3. For any of such metrics there always exist
q
a basis ea such that g = δ ea eb for which the Sp(1) part of the spin connection ωa and the
q ab −
⊗
negative oriented K¨ahler triplet J defined by
i
ωa = ωa ǫ ωb, J = e1 e2 e3 e4,
− 0 − abc c 1 ∧ − ∧
J = e1 e3 e4 e2 J = e1 e4 e2 e3,
2 3
∧ − ∧ ∧ − ∧
satisfy the relations
dωi +ǫ ωj ωk = J , dJi = ǫ Jj ωk. (1.1)
− ijk − − i ijk −
∧ ∧
∗
fi[email protected]
Let τ and u be four new coordinates, u = √uiui, α = dui +ǫijkωj uk and H a τ-independent
i i −
one form. Then the 8-dimensional metric
(dt+H)2
1 1
g = +e2f+2hα α +e2g+2hg (1.2)
8 3 i i q
e2h
together with the four form
ǫ
Φ = (dt+H) e3fα α α +ef+2gα J +e2(f+g)+h ijkα α J +e4g+he e e e ,
4 1 2 3 i i i j k 1 2 3 4
∧(cid:18) ∧ ∧ ∧ (cid:19) 2 ∧ ∧ ∧ ∧ ∧
constitute an Spin(7) structure preserved by the Killing vector ∂ . Moreover if f, g and h are
τ
functions of u related by
ef+2g
ue3f = (ef+2g)′, λ(e3f ) = (e2(f+g)+h)′,
− u2
4ue2(f+g)+h 2λef+2g = (e4g+h)′,
−
and the 1-form H satisfy
ǫ
ijk
dH = u J + u θ θ , (1.3)
i i i j k
− 2 ∧
being θ = d(ui)+ǫijkωj uk and ui = ui/ue, then Φ ewill be closed and therefore, the holonomy
i − 4
of the metric (2.29) will be included in Spin(7).
e e e
We will show below that the integrability condition for (1.3), namely
ǫ
ijk
d(u J u θ θ ) = 0
i i i j k
− 2 ∧
e e
is always satisfied due to the fact that the twistor space Z of any compact quaternion K¨ahler
space M carries a K¨ahler-Einstein metric of positive scalar curvature [1] and the left hand
side of (1.3) is the K¨ahler form for such metric. Also, the closure of Φ follows directly from
4
the formulas (2.13) given below. By construction, the vector field ∂ is Killing and if the
τ
quaternion K¨ahler basis possess an isometry group G which preserves the forms ωi , then G
−
will be an isometry of g . The construction of Spin(7) holonomy metrics that follows from the
8
proposition can also be applied to quaternion K¨ahler orbifolds.
2. Closed G and Spin(7) structures and K¨ahler-Einstein
2
metrics
Quaternion K¨ahler spaces in brief
A key ingredient in order to construct the metrics (2.29) are quaternion K¨ahler manifolds
andso, it isconvenient togive abrief description of their properties. Bydefinition, a quaternion
K¨ahler space M is an euclidean 4n dimensional space with holonomy group Γ included into
the Lie group Sp(n) Sp(1) SO(4n) [4]-[6]. This affirmation is non trivial if D > 4, but
× ⊂
in D = 4 we have the well known isomorphism Sp(1) Sp(1) SU(2) SU(2) SO(4)
L R
× ≃ × ≃
and so to state that Γ Sp(1) Sp(1) is the same that to state that Γ SO(4). The last
⊆ × ⊆
affirmation is trivially satisfied for any oriented space and gives almost no restrictions about
the space, therefore the definition of quaternion K¨ahler spaces should be modified in d = 4.
Their main properties are the following.
- There exists three automorphism Ji (i = 1 ,2, 3) of the tangent space TM at a given
x
point x with multiplication rule Ji Jj = δ +ǫ Jk, and for which the metric g is quaternion
ij ijk q
· −
hermitian, that is
g (X,Y) = g(JiX,JiY), (2.4)
q
being X and Y arbitrary vector fields.
- The automorphisms Ji satisfy the fundamental relation
Ji = ǫ Jjωk, (2.5)
X ijk −
∇
with theLevi-CivitaconnectionofM andωi itsSp(1)part. Asaconsequence ofhermiticity
X −
∇
of g, the tensor Ji = (Ji)cg is antisymmetric, and the associated 2-form
ab a cb
Ji = Ji ea eb
ab ∧
satisfies
dJi = ǫ Jj ωk, (2.6)
ijk −
∧
being d the usual exterior derivative.
- Corresponding to the Sp(1) connection we can define the 2-form
Fi = dωi +ǫ ωj ωk.
− ijk − −
∧
Then for a quaternion K¨ahler manifold
Ri = 2nκJi, (2.7)
−
Fi = κJi, (2.8)
being Λ certain constant and κ the scalar curvature. The tensor Ra is the Sp(1) part of the
−
curvature. The last two conditions implies that g is Einstein with non zero cosmological con-
stant, i.e, R = 3κ(g ) being R the Ricci tensor constructed from g . Notice that (2.8) is
ij q ij ij q
equivalent to (1.1) if we choose the normalization κ = 1.
- For any quaternion K¨ahler space the (0,4) and (2,2) tensors
1 1 2 2 3 3
Θ = J J +J J +J J ,
∧ ∧ ∧
Ξ = J1 J1 +J2 J2 +J3 J3
⊗ ⊗ ⊗
are globally defined and covariantly constant with respect to the usual Levi Civita connection.
- Any quaternion K¨ahler space is orientable.
- In four dimensions the K¨ahler triplet J and the one forms ωa are
2 −
ωa = ωa ǫ ωb, J = e1 e2 e3 e4,
− 0 − abc c 1 ∧ − ∧
J = e1 e3 e4 e2 J = e1 e4 e2 e3.
2 3
∧ − ∧ ∧ − ∧
In this dimension quaternion K¨ahler spaces are defined by the conditions (2.8) and (2.7). This
definition is equivalent to state that quaternion K¨ahler spaces are Einstein and with self-dual
Weyl tensor.
The twistor space of a quaternion K¨ahler space
Anothervery importantpropertyaboutcompact quaternionK¨ahlerspaces isthatitstwistor
space is K¨ahler-Einstein. In order to define the twistor space let us note that any linear
combination of the form J = u J is an almost complex structure on M, and the metric g is
i i q
hermitic with respect to it. Here we have defined the scalar fields ui = ui/u and it is evident
e
that they are constrained by the condition uiui = 1. This means that the bundle of almost
e
complex structures over M is parameterized by points on the two sphere S2. This bundle is
e e
known as the twistor space Z of M. The space Z is endowed with the metric
g = θ θ +g , (2.9)
6 i i q
where we have defined
θ = d(ui)+ǫijkωj uk.
i −
The metric (2.9) is six dimensional due to thee constrainteuiui = 1. Corresponding to this metric
we have the K¨ahler two form
ǫ e e
ijk
J = u J u θ θ . (2.10)
i i i j k
− 2 ∧
e e
It has been proved in [1] that J is integrable and J is closed (see also [3]), therefore J is truly
a complex structure and g is K¨ahler. The calculation of the Ricci tensor of g shows that it is
6 6
also Einstein, therefore the space Z is K¨ahler-Einstein. We are using the normalization κ = 1
here, for other normalization certain coefficients must be included in (2.10). Let us introduce
the covariant derivative
α = dui +ǫijkωj uk, (2.11)
i −
which is related to θ by
i
α u du
θi = i i .
u − u2
With the help of this relation and the definition (2.10) it follows that
u ǫ α α α du
i ijk j k k
J = J u ∧ ǫ u u ∧ (2.12)
u i − 2 i u3 − ijk i j u4
u ǫ α α
i ijk j k
= J u ∧ .
u i − 2 i u3
The last expression will be needed in the following, although it is completely equivalent to
(2.10). Also, the following formulae
J J = 2δ e e e e ,
i j ij 1 2 3 4
∧ − ∧ ∧ ∧
−
dα = ǫ (u J +α ω ),
i ijk j k k ∧ j
d(uiui) = d(u2) = 2udu = 2uiα ,
i
d(ǫ α α α ) = udu α J , (2.13)
ijk i j k i i
∧ ∧ − ∧ ∧
d(α J ) = 0,
i i
∧
d(e3f) α α α = (e3f)′du α α α = 0,
1 2 3 1 2 3
∧ ∧ ∧ ∧ ∧ ∧
relating J and α will be useful for our purposes. 1 For instance, the closure of (2.12) is a
i i
direct consequence of the second (2.13) together with (2.6).
A proof of the proposition
Let us go back to our task of constructing the metric (2.29). Our starting point is an
eight-dimensional metric anzatz of the form
(dt+H)2 1
g = +e2hg , (2.14)
8 3 7
e2h
being g a metric over a 7-manifold Y and h a function over Y. Neither the one form H nor the
7
function h depends on t, therefore the vector field ∂ is, by construction, Killing. Associated to
t
the metric (2.14) we can construct the octonionic 4-form
Φ = (dt+H) Φ+eh Φ, (2.15)
4
∧ ∗
being Φ a G invariant three form corresponding to the metric g and Φ its dual. The precise
2 7
∗
form for Φ will be found below. If we impose the condition dΦ = 0 then the metric g will
4 8
have Spin(7) holonomy. 2 We will suppose that the seven dimensional metric g is of the form
7
g = e2fα α +e2gg , (2.16)
7 i i q
being g a quaternion K¨ahler metric in d = 4 and α defined in (2.11). The functions f, g will
q i
depend only on the ”radius” u = √uiui. The form (2.16) for the 7-metric is well known and
is inspired in the Bryant-Salamon construction for G holonomy metrics [2]. For a metric with
2
G holonomy we have dΦ = d Φ = 0 but we will not suppose that the holonomy of (2.16)
2
∗
is G , as in the Bryant-Salamon case. Instead we will consider 7-spaces for which the form Φ
2
is closed but not co-closed. These are known as closed G structures [8]-[12]. In this case the
2
Spin(7) holonomy condition dΦ = 0 for (2.15) will reduce to
4
dH Φ = d(eh Φ). (2.17)
∧ − ∗
We will find below a suitable H and g for which (2.17) gets simplified even more. It seems
7
reasonable for us to choose H in (2.15) such that
dH = λJ. (2.18)
−
The reason for this election is that the integrability condition dJ = 0 will be automatically
satisfied because, as we have seen above, the two form J is the K¨ahler formof a K¨ahler-Einstein
metric. Here λ is a parameter, and the minus sign was introduced by convenience. Also, by
selecting the basis
e = efα , i = 1,2,3 e = ege α = 1,2,3,4,
i i α α
1A more complete aeccount of formulae can be found,efor instance, in [31].
2The converse of this affirmation is not true, that is, the holonomy group of g8 could be Spin(7) and (2.15)
could be not closed. The form (2.15) is preserved by the Killing vector, but there could exist cases for which
this simplifying condition do not hold and the holonomy is still Spin(7). In such cases there will exist another
closed 4-form Φ4 which is not preserved by ∂t.
for (2.16), we can construct the G invariant three form
2
Φ = c ea eb ec = e3fα α α +ef+2gα J (2.19)
abc 1 2 3 i i
∧ ∧ ∧ ∧ ∧
e e e
and its dual
ǫ
Φ = e2(f+g) ijkα α J +e4ge e e e . (2.20)
i j k 1 2 3 4
∗ 2 ∧ ∧ ∧ ∧ ∧
Here eα is a basis for the quaternion K¨ahler metric g . As we stated above, we will consider
q
7-spaces for which the form Φ is closed but not co-closed. In other words we will have that
dΦ = 0 but d Φ = 0. We also suppose that f, g and h are functions of the radius u only. By
∗ 6
using (2.13) it follows that the closure condition dΦ = 0 for (2.19) leads to the equation
ue3f = (ef+2g)′, (2.21)
where the tilde implies the derivation with respect to u. This is one of the equations that we
need.
By another side, with the election (2.18) for H and using that dΦ = 0 it follows from (2.15)
that
dΦ = dH Φ+d(eh Φ) = λJ Φ+d(eh Φ) (2.22)
4
∧ ∗ − ∧ ∗
and therefore the Spin(7) holonomy condition dΦ = 0 is equivalent to
4
λJ Φ = d(eh Φ). (2.23)
∧ ∗
From (2.19) and (2.12) we see that the left side of (2.23) is
e3f ef+2g ef+2g
λJ Φ = λ( )uiJi α α α 2λ uiα e e e e . (2.24)
∧ u − u3 ∧ 1 ∧ 2 ∧ 3 − u i ∧ 1 ∧ 2 ∧ 3 ∧ 4
By using the formula uiα = udu and that
i
ǫ
uiJi α α α = udu ijkα α Jk,
1 2 3 i j
∧ ∧ ∧ ∧ 2 ∧ ∧
we can reexpress (2.24) as
ef+2g ǫ
λJ Φ = λ(e3f )du ijkα α Jk 2λef+2gdu e e e e . (2.25)
∧ − u2 ∧ 2 i ∧ j ∧ − ∧ 1 ∧ 2 ∧ 3 ∧ 4
By another side, the right side of the equation (2.23) is found directly from (3.41) and (2.13),
the result is
ǫ
d(eh Φ) = (e2(f+g)+h)′du ijkα α Jk+ (e4g+h)′ 4ue2(f+g)+h du e e e e . (2.26)
i j 1 2 3 4
∗ ∧ 2 ∧ ∧ (cid:18) − (cid:19) ∧ ∧ ∧ ∧
By equating (2.26) with (2.25) and taking into account the closure condition (2.21) we obtain
the following differential system
ef+2g
ue3f = (ef+2g)′, λ(e3f ) = (e2(f+g)+h)′,
− u2
4ue2(f+g)+h 2λef+2g = (e4g+h)′. (2.27)
−
From this system of equations we obtain the proposition stated above.
2.1 A particular solution: the Swann bundle
If we were able to solve the system (2.27) then we will obtain a family of Spin(7) holonomy
metrics based on arbitrary quaternion K¨ahler spaces. We do not know its general solution, but
we have found a particular one. It is not difficult to check that indeed
ef = u−1/3, eg = u2/3, eh = λu−2/3, (2.28)
is a solution of (2.27). By introducing it into the expression (2.14) and by defining the variable
τ = t/λ and rescaling by g λ−1g we obtain the following metric
8 8
→
α2 +α2 +α2
g = u(dτ +H)2 + 1 2 3 +ug , (2.29)
8 q
u
and the corresponding closed four form is
α α α ǫ
Φ = (dτ +H) 1 ∧ 2 ∧ 3 +uα J + ijkα α J +u2e e e e .
4 i i i j k 1 2 3 4
∧(cid:18) u ∧ (cid:19) 2 ∧ ∧ ∧ ∧ ∧
The closure of this form follows directly from formulas (2.13). An inspection of this metric
shows that it is a cone over an Einstein-Sassaki metric, thus the holonomy is in SU(4) G . In
2
∈
order to see this we need to show the orthogonality condition u θ = 0, which is a consequence
i i
of the following calculation
e
u θ = u du +ǫijkuiωj uk = u du = d(u u ) = 0,
i i i i − i i i i
e e e e e e e e e
we have used u u = 1 in the last line. We also have that
i i
e e u du
uθi + i = α ,
i
u
and by inserting this expression in (2.29), applying after the orthogonality condition and by
defining the new radius r = 2u1/2 gives the following conical form of the metric
g = dr2 +r2g , (2.30)
8 7
being g given by
7
g = (dτ +H)2 +g = (dτ +H)2 +θ θ +g . (2.31)
7 6 i i q
We have seen that the six dimensional metric g is K¨ahler-Einstein and therefore g is Einstein-
6 7
Sassaki (see the lectures [33] and references therein). Any cone over an Einstein-Sassaki space
is Calabi-Yau and therefore its holonomy is in SU(4) Spin(7).
⊂
But more information about these metrics can be found by finding explicitly the one form
H, which is defined by dH = J. In order to solve dH = J we need to simplify the expression
(2.10). Let us remind that
ǫ
ijk
J = u J u θ θ
i i i j k
− 2 ∧
and that θ = d(ui)+ǫijkωj uk. The ortheogonality econdition u θ = 0 is equivalent to
i − i i
e e e
(u θ +u θ )
1 1 2 2
θ = .
3
− u
e 3e
e
From the last relation it follows that
ǫ θ θ
ijk 1 2
u θ θ = ∧ .
i j k
2 ∧ u
3
e
After certain calculation we obtain e
θ θ du du ǫijk
1 ∧ 2 = 1 ∧ 2 dui ω−i + uiω−j ω−k.
u u − ∧ 2 ∧
3 e 3 e
e e
Therefore e e
ǫ du du ǫijk
ijkuiθj θk = 1∧ 2 dui ω−i + uiω−j ω−k. (2.32)
2 ∧ u − ∧ 2 ∧
e 3 e
e e e
By another side we have the fundamental relation for quaternion Kahler manifolds, which is
e
ǫijk
J = dωi + ωj ωk. (2.33)
i − − −
2 ∧
e
Inserting expressions (2.32) and (2.33) into (2.10) give us a remarkably simple expression for
J, namely
du du
J = d(u ωi ) 1 ∧ 2 (2.34)
i −
− u
e 3 e
e
By parameterizing the coordinates u in the spherical form
i e
e
u = cosθ, u = sinθcosϕ, u = sinθsinϕ,
1 2 3
e e e
we find out that
du du
1 2
∧ = dϕ dcosθ.
u − ∧
e 3 e
With the help of the last expression can reexpress (2.34) as
e
J = d(u ωi ) dϕ dcosθ,
i −
− ∧
e
from where it follows directly that the form H such that dH = J is given by
H = u ωi +cosθdϕ, (2.35)
i −
up to a total differential term. By introducing the expression (2.35) into (2.31) we find directly
the following expression for the Einstein-Sasaki metric
g = (dτ +cosθdϕ+cosθω1 +sinθcosϕω2 +sinθsinϕω3)2 +(dθ sinϕω2 +cosϕω3)2
7 − − − − −
−
+(sinθdϕ+sinθω1 cosθcosϕω2 cosθsinϕω3)2 +g . (2.36)
− − − q
− −
Let us introduce the coordinates u written in spherical form
i
u = u sinθcosϕcosφ,
1
| |
u = u sinθcosϕsinφ,
2
| |
u = u sinθsinϕ,
3
| |
u = u cosθ,
4
| |
Then it is not difficult to check that the cone g = dr2 +r2g being g given at (2.36), can be
8 7 7
expressed as
g = g u 2g +f[(du u ωi )2 +(du +u ωi +ǫ u ωk)2]. (2.37)
s 0 i − i 0 − ijk k −
| | −
The coordinates u can be extended to a single quaternion valued coordinate
i
u = u +u I +u J +u K, u = u u I u J u K.
0 1 2 3 0 1 2 3
− − −
Here I,J,K denote the unit quaternions, and it follows that du 2 = (du )2+(du )2+(du )2+
0 1 2
| |
(du )2. The Sp(1) SU(2) triplet ωi can be used to define a quaternion valued one form
3 −
∼
ω− = ω−1I +ω−2J +ω−3K,
a 1 2 3
andtheKahlertripletJ canbeextendedtoaquaternionvaluedtwoformJ = J I+J J+J K.
The metric (2.37) can be expressed in this notation as
g8 = u 2gq + du+uω− 2, (2.38)
| | | |
Under the transformation u Gu with G : M SU(2) the SU(2) instanton ω− is gauge
transformed as ω− Gω−G−→1 +GdG−1. Therefor→e the form du+ω−u is transformed as
→
du+uω− d(Gu)+(Gω−G−1 +GdG−1)Gu = Gdu+(dG+Gω− dG)u = G(du+uω−),
→ −
and it is seen that du+ω−u is a well defined quaternion-valued one form over the chiral bundle.
Associated to the metric (2.38) we have the quaternion valued two form
J = uJu+(du+uω−) (du+uω−), (2.39)
∧
e
and it can be checked that the metric (2.37) is hermitic with respect to any of the components
of (2.39). We have that
dJ = du (J +dω− ω− ω−)u+u (J +dω− ω− ω−)du
∧ − ∧ ∧ − ∧
e
+u(dJ +ω− dω− dω− ω−)u.
∧ − ∧
The first two terms of the last expression are zero due to (2.8). Also by introducing (2.8) into
the relation (2.6) we obtain that
dJ +ω− dω− dω− ω− = 0
∧ − ∧
and therefore the third term is also zero. This means that the metric (2.38) is hyperkahler with
respect to the triplet J and the holonomy is reduced to Sp(2) SU(4) Spin(7).
⊂ ⊂
The hyperkahler meetrics (2.38) are indeed well known. They are the Swann principal CO(3)
bundle of co-frames over a quaternion K¨ahler spaces [32]. Hyperkahler quotients ofsuch metrics
bytri-holomorphicisometriesarerelatedtoquaternionKahlerquotientsofthebasespaces. The
hyperkahler condition for the Swann metric implies that the seven dimensional cone of (2.31) is
not only Einstein-Sassaki, but tri-Sasaki. The vector field ∂ is the Reeb vector of the tri-Sasaki
φ
metric.
The self duality of the spin connection
Although we have found that our example is hyperkahler, it is instructive to check that the
spin connection ω of the metric (2.29) is self-dual. We choose the basis
ab
αi
eα = u1/2eα, ei = , e8 = u1/2(dτ +H)
u1/2
being eα a basis for g with α = 1,2,3,4 and i = 1,2,3. With the help of the first Cartan
q
equation
dem +ωˆ en = 0,
mn
∧
where m an index that can be latin or greek, we obtain the following components of the spin
connection
u[iαj] u
ωˆ = ǫ ωk +ǫ k(dτ +H),
ij 2u2 − ijk − ijk u
u u
ωˆ = ω ǫ k(J ) αi i(J ) (dτ +H),
αβ αβ − ijku2 j αβ − u i αβ
ui u
ωˆ = eα ǫ k(J ) eβ,
αi ijk j αβ
2 − u
u u
ωˆ = i(dτ +H)+ǫ jαk,
8i u ijku2
u
ωˆ = i(J ) eβ.
8α u2 j αβ
By using that 2ωi = ω (J ) and the representation
− αβ j αβ
0 1 0 0 0 0 1 0
− −
 1 0 0 0   0 0 0 1 
J1 = , J2 =
0 0 0 1 1 0 0 0
   
 −   
 0 0 1 0   0 1 0 0 
   − 
0 0 0 1
−
 0 0 1 0 
J3 = J1J2 = − , (2.40)
0 1 0 0
 
 
 1 0 0 0 
 
for the matrix (J ) , it can be checked that
j αβ
ωˆ = (ωˆ +ωˆ +ωˆ ), ωˆ = (ωˆ +ωˆ +ωˆ ),
81 23 65 47 82 31 46 57
− −
ωˆ = (ωˆ +ωˆ +ωˆ ), ωˆ = (ωˆ +ωˆ +ωˆ ),
83 12 54 67 84 62 35 71
− −
ωˆ = (ωˆ +ωˆ +ωˆ ), ωˆ = (ωˆ +ωˆ +ωˆ ),
85 16 43 72 86 15 24 73
− −
ωˆ = (ωˆ +ωˆ +ωˆ ).
87 14 36 25
−
These conditions for ωˆ can be written more concisely as
mn
ωˆ = c ωˆ ,
8i imn mn
−
and this is equivalent to say that, for the basis em, the spin connection ωˆ is self-dual.
mn