Bismut connection

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

1. It preserves the metric ∇ g = 0
2. It preserves the complex structure ∇ 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 pairng of two vectors using the metric that is Hermitian w.r.t the complex structure, i.e. ⟨ X , J Y ⟩ = − ⟨ J X , Y ⟩ . Further let ∇ be the Levi-Civita connection. Define first a tensor T such that T ( Z , X , Y ) = − 1 2 ⟨ Z , ( ∇ X J ) Y ⟩ . It is easy to see that this tensor is anti-symmetric in the first and last entry, i.e. the new connection ∇ + T still preserves the metric. In concrete terms, the new connection is given by Γ β γ α − 1 2 J   δ α ∇ β J   γ δ with Γ β γ α 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 ) + cyc~in~ X , Y , Z = T ( Z , X , Y ) + S ( Z , X , Y ) , with S given explicitly as

We show that S still preserves the complex structure (that it preserves the metric is easy to see), i.e. S ( Z , X , J Y ) = − S ( J Z , X , Y ) .

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

gives the Bismut connection.