Bismut connection

In mathematics, the Bismut connection $$\nabla$$ is the unique connection on a complex Hermitian manifold that satisfies the following conditions, 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.
 * 1) It preserves the metric $$\nabla g =0$$
 * 2) It preserves the complex structure $$\nabla J=0$$
 * 3) The torsion $$T(X,Y)$$ contracted with the metric, i.e. $$T(X,Y,Z)=g(T(X,Y),Z)$$, is totally skew-symmetric.

The explicit construction goes as follows. Let $$\langle-,-\rangle$$ denote the pairing 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 $$T(Z,X,Y)=-\frac12\langle Z,J(\nabla_{X}J)Y\rangle $$. 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. The new connection also preserves the complex structure. However, the tensor $$T$$ is not yet totally anti-symmetric; 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


 * $$S(Z,X,Y)=-\frac12\langle X,J(\nabla_{Y}J)Z\rangle-\frac12\langle Y,J(\nabla_{Z}J)X\rangle.$$

$$S$$ still preserves the complex structure, i.e. $$S(Z,X,JY)=-S(JZ,X,Y)$$.


 * $$\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}$$ So if $$J$$ is integrable, then above term vanishes, and the connection


 * $$\Gamma^{\alpha}_{\beta\gamma}+T^{\alpha}_{~\beta\gamma}+S^{\alpha}_{~\beta\gamma}.$$

gives the Bismut connection.