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In mathematics, the Bismut connection 
is the unique connection on a complex manifold that satisfies the following conditions,
- It preserves the metric
- It preserves the complex structure
- The torsion
contracted with the metric, i.e. 
, 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. 
. Further let be the Levi-Civita connection. Define first a tensor 
such that 
. It is easy to see that this tensor is anti-symmetric in the first and last entry, i.e. the new connection 
still preserves the metric. In concrete terms, the new connection is given by 
with 
being the Levi-Civita connection. It is also easy to see that the new connection preserves the complex structure. However, the tensor is not yet totall anti-symmetric, in fact the anti-symmetrization will lead to the Nijenhuis tensor. Denote the anti-symmetrization as 
, with 
given explicitly as
We show that still preserves the complex structure (that it preserves the metric is easy to see), i.e. 
.
![\begin{align}
S(Z,X,JY)+S(JZ,X,Y)&=-\frac12\langle JX, \big(-(\nabla_{JY}J)Z-(J\nabla_ZJ)Y+(J\nabla_YJ)Z+(\nabla_{JZ}J)Y\big)\rangle\\
&=-\frac12\langle JX, Re\big((1-iJ)[(1+iJ)Y,(1+iJ)Z]\big)\rangle.\end{align}](http://upload.wikimedia.org/math/c/0/1/c01da5390d23c19897cbf3f9787a143a.png)
So if 
is integrable, then above term vanishes, and the connection
gives the Bismut connection.
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