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Bismut connection
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.
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