Bismut connection

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

1. It preserves the metric $\mathrm{\nabla }g=0$
2. It preserves the complex structure $\mathrm{\nabla }J=0$
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 $⟨-,-⟩$ denote the pairing of two vectors using the metric that is Hermitian w.r.t the complex structure, i.e. $⟨X,JY⟩=-⟨JX,Y⟩$. Further let $\mathrm{\nabla }$ be the Levi-Civita connection. Define first a tensor $T$ such that $T(Z,X,Y)=-\frac{1}{2}⟨Z,J({\mathrm{\nabla }}_{X}J)Y⟩$. This tensor is anti-symmetric in the first and last entry, i.e. the new connection $\mathrm{\nabla }+T$ still preserves the metric. In concrete terms, the new connection is given by ${\mathrm{\Gamma }}_{\beta \gamma }^{\alpha }-\frac{1}{2}{J}_{\delta }^{\alpha }{\mathrm{\nabla }}_{\beta }{J}_{\gamma }^{\delta }$ with ${\mathrm{\Gamma }}_{\beta \gamma }^{\alpha }$ 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 $T(Z,X,Y)+\text{<mrow data-mjx-texclass="ORD">cyc<mspace width="0.5em">in<mspace width="0.5em"></mspace></mspace></mrow>}X,Y,Z=T(Z,X,Y)+S(Z,X,Y)$, with $S$ given explicitly as

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

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

$\begin{array}{rl}S(Z,X,JY)+S(JZ,X,Y)& =-\frac{1}{2}⟨JX,(-({\mathrm{\nabla }}_{JY}J)Z-(J{\mathrm{\nabla }}_{Z}J)Y+(J{\mathrm{\nabla }}_{Y}J)Z+({\mathrm{\nabla }}_{JZ}J)Y)⟩\\ & =-\frac{1}{2}⟨JX,Re((1-iJ)[(1+iJ)Y,(1+iJ)Z])⟩.\end{array}$

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

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

gives the Bismut connection.