Spinor bundle

<p>In <a href="/facts/Differential_geometry/c04XmyLu">differential geometry</a>, given a <a href="/facts/Spin_structure/BF78kFz8">spin structure</a> on an 
  
    
      
        n
      
    
    {\displaystyle n}
  
-dimensional orientable <a href="/facts/Riemannian_manifold/5KYnmEgF">Riemannian manifold</a> 
  
    
      
        (
        M
        ,
        g
        )
        ,
        
      
    
    {\displaystyle (M,g),\,}
  
 one defines the spinor bundle to be the <a href="/facts/Complex_vector_bundle/0nE8TEIn">complex vector bundle</a> 
  
    
      
        
          π
          
            
              S
            
          
        
        :
        
          
            S
          
        
        →
        M
        
      
    
    {\displaystyle \pi _{\mathbf {S} }\colon {\mathbf {S} }\to M\,}
  
 associated to the corresponding <a href="/facts/Principal_bundle/bbB2D94B">principal bundle</a> 
  
    
      
        
          π
          
            
              P
            
          
        
        :
        
          
            P
          
        
        →
        M
        
      
    
    {\displaystyle \pi _{\mathbf {P} }\colon {\mathbf {P} }\to M\,}
  
 of spin frames over 
  
    
      
        M
      
    
    {\displaystyle M}
  
 and the <a href="/facts/Spin_representation/upLgk4zS">spin representation</a> of its <a href="/facts/Spin_group/tfvR9cqM">structure group</a> 
  
    
      
        
          
            S
            p
            i
            n
          
        
        (
        n
        )
        
      
    
    {\displaystyle {\mathrm {Spin} }(n)\,}
  
 on the space of <a href="/facts/Spinor/sKFNnEwP">spinors</a> 
  
    
      
        
          Δ
          
            n
          
        
      
    
    {\displaystyle \Delta _{n}}
  
.
</p><p>A section of the spinor bundle 
  
    
      
        
          
            S
          
        
        
      
    
    {\displaystyle {\mathbf {S} }\,}
  
 is called a spinor field.
</p>

Spinor bundle open-in-new

Spinor bundle