In mathematics, the Bismut connection
∇
is the unique connection on a complex Hermitian manifold that satisfies the following conditions,
- It preserves the metric
∇
g
=
0
- It preserves the complex structure
∇
J
=
0
- The torsion
T
(
X
,
Y
)
contracted with the metric, i.e.
T
(
X
,
Y
,
Z
)
=
g
(
T
(
X
,
Y
)
,
Z
)
, is totally skew-symmetric.
Bismut has used this connection when proving a local index formula for the Dolbeault operator on non-Kähler manifolds. Bismut connection has applications in type II and heterotic string theory.
The explicit construction goes as follows. Let
⟨
−
,
−
⟩
denote the pairng of two vectors using the metric that is Hermitian w.r.t the complex structure, i.e.
⟨
X
,
J
Y
⟩
=
−
⟨
J
X
,
Y
⟩
. Further let
∇
be the Levi-Civita connection. Define first a tensor
T
such that
T
(
Z
,
X
,
Y
)
=
−
1
2
⟨
Z
,
(
∇
X
J
)
Y
⟩
. It is easy to see that this tensor is anti-symmetric in the first and last entry, i.e. the new connection
∇
+
T
still preserves the metric. In concrete terms, the new connection is given by
Γ
β
γ
α
−
1
2
J
δ
α
∇
β
J
γ
δ
with
Γ
β
γ
α
being the Levi-Civita connection. It is also easy to see that the new connection preserves the complex structure. However, the tensor
T
is not yet totall anti-symmetric, in fact the anti-symmetrization will lead to the Nijenhuis tensor. Denote the anti-symmetrization as
T
(
Z
,
X
,
Y
)
+
cyc~in~
X
,
Y
,
Z
=
T
(
Z
,
X
,
Y
)
+
S
(
Z
,
X
,
Y
)
, with
S
given explicitly as
S
(
Z
,
X
,
Y
)
=
−
1
2
⟨
X
,
J
(
∇
Y
J
)
Z
⟩
−
1
2
⟨
Y
,
J
(
∇
Z
J
)
X
⟩
.
We show that
S
still preserves the complex structure (that it preserves the metric is easy to see), i.e.
S
(
Z
,
X
,
J
Y
)
=
−
S
(
J
Z
,
X
,
Y
)
.
S
(
Z
,
X
,
J
Y
)
+
S
(
J
Z
,
X
,
Y
)
=
−
1
2
⟨
J
X
,
(
−
(
∇
J
Y
J
)
Z
−
(
J
∇
Z
J
)
Y
+
(
J
∇
Y
J
)
Z
+
(
∇
J
Z
J
)
Y
)
⟩
=
−
1
2
⟨
J
X
,
R
e
(
(
1
−
i
J
)
[
(
1
+
i
J
)
Y
,
(
1
+
i
J
)
Z
]
)
⟩
.
So if
J
is integrable, then above term vanishes, and the connection
Γ
β
γ
α
+
T
β
γ
α
+
S
β
γ
α
.
gives the Bismut connection.
