Bismut connection

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

1. It preserves the metric ${\displaystyle \nabla g=0}$
2. It preserves the complex structure ${\displaystyle \nabla J=0}$
3. The torsion ${\displaystyle T(X,Y)}$ contracted with the metric, i.e. ${\displaystyle T(X,Y,Z)=g(T(X,Y),Z)}$, 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 ${\displaystyle \langle -,-\rangle }$ denote the pairing of two vectors using the metric that is Hermitian w.r.t the complex structure, i.e. ${\displaystyle \langle X,JY\rangle =-\langle JX,Y\rangle }$. Further let ${\displaystyle \nabla }$ be the Levi-Civita connection. Define first a tensor ${\displaystyle T}$ such that ${\displaystyle T(Z,X,Y)=-{\frac {1}{2}}\langle Z,J(\nabla _{X}J)Y\rangle }$. This tensor is anti-symmetric in the first and last entry, i.e. the new connection ${\displaystyle \nabla +T}$ still preserves the metric. In concrete terms, the new connection is given by ${\displaystyle \Gamma _{\beta \gamma }^{\alpha }-{\frac {1}{2}}J_{~\delta }^{\alpha }\nabla _{\beta }J_{~\gamma }^{\delta }}$ with ${\displaystyle \Gamma _{\beta \gamma }^{\alpha }}$ being the Levi-Civita connection. The new connection also preserves the complex structure. However, the tensor ${\displaystyle T}$ is not yet totally anti-symmetric; the anti-symmetrization will lead to the Nijenhuis tensor. Denote the anti-symmetrization as ${\displaystyle T(Z,X,Y)+{\textrm {cyc~in~}}X,Y,Z=T(Z,X,Y)+S(Z,X,Y)}$, with ${\displaystyle S}$ given explicitly as

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

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

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

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

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

gives the Bismut connection.