Bismut connection

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

1. It preserves the metric [math]\displaystyle{ \nabla g =0 }[/math]
2. It preserves the complex structure [math]\displaystyle{ \nabla J=0 }[/math]
3. The torsion [math]\displaystyle{ T(X,Y) }[/math] contracted with the metric, i.e. [math]\displaystyle{ T(X,Y,Z)=g(T(X,Y),Z) }[/math], 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 [math]\displaystyle{ \langle-,-\rangle }[/math] denote the pairing of two vectors using the metric that is Hermitian w.r.t the complex structure, i.e. [math]\displaystyle{ \langle X,JY\rangle=-\langle JX,Y\rangle }[/math]. Further let [math]\displaystyle{ \nabla }[/math] be the Levi-Civita connection. Define first a tensor [math]\displaystyle{ T }[/math] such that [math]\displaystyle{ T(Z,X,Y)=-\frac12\langle Z,J(\nabla_{X}J)Y\rangle }[/math]. This tensor is anti-symmetric in the first and last entry, i.e. the new connection [math]\displaystyle{ \nabla+T }[/math] still preserves the metric. In concrete terms, the new connection is given by [math]\displaystyle{ \Gamma^{\alpha}_{\beta\gamma}-\frac12 J^{\alpha}_{~\delta}\nabla_{\beta}J^{\delta}_{~\gamma} }[/math] with [math]\displaystyle{ \Gamma^{\alpha}_{\beta\gamma} }[/math] being the Levi-Civita connection. The new connection also preserves the complex structure. However, the tensor [math]\displaystyle{ T }[/math] is not yet totally anti-symmetric; the anti-symmetrization will lead to the Nijenhuis tensor. Denote the anti-symmetrization as [math]\displaystyle{ T(Z,X,Y)+\textrm{cyc~in~}X,Y,Z=T(Z,X,Y)+S(Z,X,Y) }[/math], with [math]\displaystyle{ S }[/math] given explicitly as

[math]\displaystyle{ S(Z,X,Y)=-\frac12\langle X,J(\nabla_{Y}J)Z\rangle-\frac12\langle Y,J(\nabla_{Z}J)X\rangle. }[/math]

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

[math]\displaystyle{ \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} }[/math]

So if [math]\displaystyle{ J }[/math] is integrable, then above term vanishes, and the connection

[math]\displaystyle{ \Gamma^{\alpha}_{\beta\gamma}+T^{\alpha}_{~\beta\gamma}+S^{\alpha}_{~\beta\gamma}. }[/math]

gives the Bismut connection.