3 Affine Connections with Torsion in (Para-)complexified Structures
45
case, let T
+
p M and T
−
p M be the eigenbundles of L corresponding to the eigenvalues
±α, i.e., at each point p ∈ M, the fiber is defined by
T
+
p M : = {Z ∈ T
L
p M : L p (Z ) = α Z } ,
T
−
p M : = {Z ∈ T
L
p M : L p (Z ) = −α Z } .
with
T
L
M ≡ T M ⊗ L =
p∈M
T
+
p M ⊕ T
−
p M = T
+
M ⊕ T
−
M.
We will refer to vectors to be of “type +” or (1, 0), or “type −” or (0, 1), if they
take values in T
+
p M or T
−
p M, respectively. Moreover, define π
+ and π
− to be the
projections of a (para-)complexified vector field to T
+
M =
p T
+
p M and T
−
M =
p T
−
p M, respectively:
π
+
(Z ) =
1
2
(Z + α
−1 L Z),
π
−
(Z ) =
1
2
(Z − α
−1 L Z).
As subbundles of the (para-)complexified tangent bundle T
L
M, T
+
M and T
−
M
are distributions. A distribution is called a foliation if it is closed under the Lie bracket
[·, ·] .
Let T
(n,m)
p
M denotes the tensor product of n-copies of T
+
p M ≡ T
(1,0)
p
M and
m-copies of T
−
p M ≡ T
(0,1)
p
M:
T
(n,m)
p
M = (T
+
p M × · · · × T
+
p M
n times
) × (T
−
p M × · · · × T
−
p M
m times
).
In particular, T
(2,0)
p
M will supply two vectors fields both drawn from T
+
p M, T
(0,2)
p
M
will supply two vectors fields both drawn from T
−
p M, and T
(1,1)
p
M will supply two
vectors fields drawn respectively from T
+
p M and T
−
p M. That is, the duplet vector
fields (X, Y ) take in value, respectively, from (+, +), from (+, −) or (−, +), or
from (−, −), which are shorthand notations for T
(2,0)
p
M, T
(1,1)
p
M, or T
(0,2)
p
M.
It is important to understand what (para-)complexification accomplishes. Take
L = J , for example. Because J
2
= −id, the J operator on a dim(M) = 2n dimensional vector space (i.e., J maps a real-valued 2n-vector to a real-valued 2n-vector)
does not have any eigenvector over the real field R: that is, there is no X ∈ T p M such
that J X = α X where α = ±i are the two eigenvalues of the 2n × 2n real-valued
operator J .
Given a vector X ∈ T p M on the real even-dimensional manifold, (para-)
complexification amounts to supplying it with an “imaginary” component, to yield
an element Z ∈ T
L
p M ≡ T p M ⊗ L. Even better, we can turn X into either X
+ or
X
− , where
45
case, let T
+
p M and T
−
p M be the eigenbundles of L corresponding to the eigenvalues
±α, i.e., at each point p ∈ M, the fiber is defined by
T
+
p M : = {Z ∈ T
L
p M : L p (Z ) = α Z } ,
T
−
p M : = {Z ∈ T
L
p M : L p (Z ) = −α Z } .
with
T
L
M ≡ T M ⊗ L =
p∈M
T
+
p M ⊕ T
−
p M = T
+
M ⊕ T
−
M.
We will refer to vectors to be of “type +” or (1, 0), or “type −” or (0, 1), if they
take values in T
+
p M or T
−
p M, respectively. Moreover, define π
+ and π
− to be the
projections of a (para-)complexified vector field to T
+
M =
p T
+
p M and T
−
M =
p T
−
p M, respectively:
π
+
(Z ) =
1
2
(Z + α
−1 L Z),
π
−
(Z ) =
1
2
(Z − α
−1 L Z).
As subbundles of the (para-)complexified tangent bundle T
L
M, T
+
M and T
−
M
are distributions. A distribution is called a foliation if it is closed under the Lie bracket
[·, ·] .
Let T
(n,m)
p
M denotes the tensor product of n-copies of T
+
p M ≡ T
(1,0)
p
M and
m-copies of T
−
p M ≡ T
(0,1)
p
M:
T
(n,m)
p
M = (T
+
p M × · · · × T
+
p M
n times
) × (T
−
p M × · · · × T
−
p M
m times
).
In particular, T
(2,0)
p
M will supply two vectors fields both drawn from T
+
p M, T
(0,2)
p
M
will supply two vectors fields both drawn from T
−
p M, and T
(1,1)
p
M will supply two
vectors fields drawn respectively from T
+
p M and T
−
p M. That is, the duplet vector
fields (X, Y ) take in value, respectively, from (+, +), from (+, −) or (−, +), or
from (−, −), which are shorthand notations for T
(2,0)
p
M, T
(1,1)
p
M, or T
(0,2)
p
M.
It is important to understand what (para-)complexification accomplishes. Take
L = J , for example. Because J
2
= −id, the J operator on a dim(M) = 2n dimensional vector space (i.e., J maps a real-valued 2n-vector to a real-valued 2n-vector)
does not have any eigenvector over the real field R: that is, there is no X ∈ T p M such
that J X = α X where α = ±i are the two eigenvalues of the 2n × 2n real-valued
operator J .
Given a vector X ∈ T p M on the real even-dimensional manifold, (para-)
complexification amounts to supplying it with an “imaginary” component, to yield
an element Z ∈ T
L
p M ≡ T p M ⊗ L. Even better, we can turn X into either X
+ or
X
− , where