Bismut connection

In mathematics, the Bismut connection $\nabla$ is the unique connection on a complex Hermitian manifold that satisfies the following conditions,

It preserves the metric $\nabla g =0$
It preserves the complex structure $\nabla J=0$
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 $\langle-,-\rangle$ denote the pairng of two vectors using the metric that is Hermitian w.r.t the complex structure, i.e. $\langle X,JY\rangle=-\langle JX,Y\rangle$ . Further let $\nabla$ be the Levi-Civita connection. Define first a tensor $T$ such that $T(Z,X,Y)=-\frac12\langle Z,(\nabla_{X}J)Y\rangle$ . It is easy to see that this tensor is anti-symmetric in the first and last entry, i.e. the new connection $\nabla+T$ still preserves the metric. In concrete terms, the new connection is given by $\Gamma^{\alpha}_{\beta\gamma}-\frac12 J^{\alpha}_{~\delta}\nabla_{\beta}J^{\delta}_{~\gamma}$ with $\Gamma^{\alpha}_{\beta\gamma}$ 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 $T(Z,X,Y)+\textrm{cyc~in~}X,Y,Z=T(Z,X,Y)+S(Z,X,Y)$ , with $S$ given explicitly as