In mathematics, the Bismut connection 
  
    
      
        ∇
      
    
    {\displaystyle \nabla }
  
 is the unique connection on a complex Hermitian manifold that satisfies the following conditions,
 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. 
  
    
      
        ⟨
        X
        ,
        J
        Y
        ⟩
        =
        −
        ⟨
        J
        X
        ,
        Y
        ⟩
      
    
    {\displaystyle \langle X,JY\rangle =-\langle JX,Y\rangle }
  
. Further let 
  
    
      
        ∇
      
    
    {\displaystyle \nabla }
  
 be the Levi-Civita connection. Define first a tensor 
  
    
      
        T
      
    
    {\displaystyle T}
  
 such that 
  
    
      
        T
        (
        Z
        ,
        X
        ,
        Y
        )
        =
        −
        
          
            1
            2
          
        
        ⟨
        Z
        ,
        J
        (
        
          ∇
          
            X
          
        
        J
        )
        Y
        ⟩
      
    
    {\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 
  
    
      
        ∇
        +
        T
      
    
    {\displaystyle \nabla +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
          
        
      
    
    {\displaystyle \Gamma _{\beta \gamma }^{\alpha }-{\frac {1}{2}}J_{~\delta }^{\alpha }\nabla _{\beta }J_{~\gamma }^{\delta }}
  
 with 
  
    
      
        
          Γ
          
            β
            γ
          
          
            α
          
        
      
    
    {\displaystyle \Gamma _{\beta \gamma }^{\alpha }}
  
 being the Levi-Civita connection. The new connection also preserves the complex structure. However, the tensor 
  
    
      
        T
      
    
    {\displaystyle 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
        )
        +
        
          
            cyc~in~
          
        
        X
        ,
        Y
        ,
        Z
        =
        T
        (
        Z
        ,
        X
        ,
        Y
        )
        +
        S
        (
        Z
        ,
        X
        ,
        Y
        )
      
    
    {\displaystyle T(Z,X,Y)+{\textrm {cyc~in~}}X,Y,Z=T(Z,X,Y)+S(Z,X,Y)}
  
, with 
  
    
      
        S
      
    
    {\displaystyle S}
  
 given explicitly as
 
  
    
      
        S
      
    
    {\displaystyle S}
  
 still preserves the complex structure, i.e. 
  
    
      
        S
        (
        Z
        ,
        X
        ,
        J
        Y
        )
        =
        −
        S
        (
        J
        Z
        ,
        X
        ,
        Y
        )
      
    
    {\displaystyle S(Z,X,JY)=-S(JZ,X,Y)}
  
.
 So if 
  
    
      
        J
      
    
    {\displaystyle J}
  
 is integrable, then above term vanishes, and the connection
 gives the Bismut connection.
 
 This differential geometry-related article is a stub. You can help Wikipedia by expanding it.