50
J. Zhang and G. Khan
Therefore, to show that the (para-)complexified Nijenhuis tensor vanishes, it is sufficient to show that its (2, 0)-part does. The nature of N
(2,0) will be investigated in
Sect. 3.3.5.1.
3.3.3.2 (Para-)complexifying MC1 and MC2
The torsion tensor T
∇ of a connection ∇ on M is a 1,2-tensor, that is, it takes in two
(real-valued) vector fields X, Y and outputs one (real-valued) vector-field denoted
by T
∇
(X, Y ). When the vector fields are (para-)complexified, we can treat T
∇ as
mapping two (para-)complexified vector fields to one (para-)complexified vector
field as an output. Naturally, we can investigate its various components, taking inputs
in, T
+
p M × T
+
p M or (+, +) called “(2,0)-part”, T
+
p M × T
−
p M or (+, −) called
“(1,1)-part”, T
−
p M × T
−
p M or (−, −) called “(0,2)-part”.
Because torsion can be decomposed into (2,0)-, (1,1)-, or (0,2)-part, we can
also investigate the 1st Matching Condition (MC1) and 2nd Matching Condition (MC2) between T
∇ and L when torsion T
∇ breaks down in terms of these
separate components. We refer to this as “(para-)complexifying” torsion-coupling
and torsion-compatibility conditions. In doing so, we hope to obtain a deeper
appreciation of whether enforcing these conditions separately on each (para-)
holomorphic/anti-(para-)holomorphic components affects integrability of L and/or
(para-)holomorphicity of ∇.
With respect to the “torsion-coupling” of (∇, L) or MC2, we will show that
the (2, 0) + (0, 2)-part is equivalent to the vanishing of the Nijenhuis tensor, while
the (1, 1)-part is equivalent to the holomorphicity of ∇. Since torsion-coupling of ∇
implies torsion-coupling of ∇
L , enforcing (1, 1)-part of torsion-coupling also implies
that ∇
L is holomorphic (just as ∇ is). Furthermore, the (1, 1)-part of torsion-coupling
implies that the (1, 1)-component of the (para-)complexified torsion is 0, both for ∇
and ∇
L .
With respect to the “torsion-compatibility” of (∇, L) or MC1, we will show
that the (para-)complexified version of this condition Eq. (3.11) implies that the
(2, 0) + (0, 2)-part of the torsion must vanish; the torsion carried by ∇, if any, is
entirely of (1, 1) type. Because the (2,0)- and (0,2)-part are equivalent, which is
meant by (2,0) + (0,2)-part, we will use the phrase (2,0)-part or (2,0)-component
from now on.
The next two subsections give the details of these assertions.
3.3.4 Torsion-Compatibility (MC1)
3.3.4.1 (1,1)-Part of Torsion-Compatibility
It is easy to verify that
J. Zhang and G. Khan
Therefore, to show that the (para-)complexified Nijenhuis tensor vanishes, it is sufficient to show that its (2, 0)-part does. The nature of N
(2,0) will be investigated in
Sect. 3.3.5.1.
3.3.3.2 (Para-)complexifying MC1 and MC2
The torsion tensor T
∇ of a connection ∇ on M is a 1,2-tensor, that is, it takes in two
(real-valued) vector fields X, Y and outputs one (real-valued) vector-field denoted
by T
∇
(X, Y ). When the vector fields are (para-)complexified, we can treat T
∇ as
mapping two (para-)complexified vector fields to one (para-)complexified vector
field as an output. Naturally, we can investigate its various components, taking inputs
in, T
+
p M × T
+
p M or (+, +) called “(2,0)-part”, T
+
p M × T
−
p M or (+, −) called
“(1,1)-part”, T
−
p M × T
−
p M or (−, −) called “(0,2)-part”.
Because torsion can be decomposed into (2,0)-, (1,1)-, or (0,2)-part, we can
also investigate the 1st Matching Condition (MC1) and 2nd Matching Condition (MC2) between T
∇ and L when torsion T
∇ breaks down in terms of these
separate components. We refer to this as “(para-)complexifying” torsion-coupling
and torsion-compatibility conditions. In doing so, we hope to obtain a deeper
appreciation of whether enforcing these conditions separately on each (para-)
holomorphic/anti-(para-)holomorphic components affects integrability of L and/or
(para-)holomorphicity of ∇.
With respect to the “torsion-coupling” of (∇, L) or MC2, we will show that
the (2, 0) + (0, 2)-part is equivalent to the vanishing of the Nijenhuis tensor, while
the (1, 1)-part is equivalent to the holomorphicity of ∇. Since torsion-coupling of ∇
implies torsion-coupling of ∇
L , enforcing (1, 1)-part of torsion-coupling also implies
that ∇
L is holomorphic (just as ∇ is). Furthermore, the (1, 1)-part of torsion-coupling
implies that the (1, 1)-component of the (para-)complexified torsion is 0, both for ∇
and ∇
L .
With respect to the “torsion-compatibility” of (∇, L) or MC1, we will show
that the (para-)complexified version of this condition Eq. (3.11) implies that the
(2, 0) + (0, 2)-part of the torsion must vanish; the torsion carried by ∇, if any, is
entirely of (1, 1) type. Because the (2,0)- and (0,2)-part are equivalent, which is
meant by (2,0) + (0,2)-part, we will use the phrase (2,0)-part or (2,0)-component
from now on.
The next two subsections give the details of these assertions.
3.3.4 Torsion-Compatibility (MC1)
3.3.4.1 (1,1)-Part of Torsion-Compatibility
It is easy to verify that