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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

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

We show that S still preserves the complex structure (that it preserves the metric is easy to see), 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.

Mathematics Encyclopedia