Bismut connection

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

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.

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 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 $T(Z,X,Y)=-{\frac {1}{2}}\langle Z,J(\nabla _{X}J)Y\rangle$. 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 $\Gamma _{\beta \gamma }^{\alpha }-{\frac {1}{2}}J_{~\delta }^{\alpha }\nabla _{\beta }J_{~\gamma }^{\delta }$ with $\Gamma _{\beta \gamma }^{\alpha }$ 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

$S(Z,X,Y)=-{\frac {1}{2}}\langle X,J(\nabla _{Y}J)Z\rangle -{\frac {1}{2}}\langle Y,J(\nabla _{Z}J)X\rangle .$

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

${\begin{aligned}S(Z,X,JY)+S(JZ,X,Y)&=-{\frac {1}{2}}\langle JX,{\big (}-(\nabla _{JY}J)Z-(J\nabla _{Z}J)Y+(J\nabla _{Y}J)Z+(\nabla _{JZ}J)Y{\big )}\rangle \\&=-{\frac {1}{2}}\langle JX,Re{\big (}(1-iJ)[(1+iJ)Y,(1+iJ)Z]{\big )}\rangle .\end{aligned}}$

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

$\Gamma _{\beta \gamma }^{\alpha }+T_{~\beta \gamma }^{\alpha }+S_{~\beta \gamma }^{\alpha }.$

gives the Bismut connection.