Spinor bundle

<h2 id="formal-definition">Formal definition</h2>
<p>Let 
  
    
      
        (
        
          
            P
          
        
        ,
        
          F
          
            
              P
            
          
        
        )
      
    
    {\displaystyle ({\mathbf {P} },F_{\mathbf {P} })}
  
 be a <a href="/facts/Spin_structure/BF78kFz8">spin structure</a> on a <a href="/facts/Riemannian_manifold/5KYnmEgF">Riemannian manifold</a> 
  
    
      
        (
        M
        ,
        g
        )
        ,
        
      
    
    {\displaystyle (M,g),\,}
  
that is, an <a href="/facts/Equivariant/KkIkDwIH">equivariant</a> lift of the oriented <a href="/facts/Orthonormal_frame_bundle/Gugt3qyL">orthonormal frame bundle</a> 
  
    
      
        
          
            F
          
          
            S
            O
          
        
        (
        M
        )
        →
        M
      
    
    {\displaystyle \mathrm {F} _{SO}(M)\to M}
  
 with respect to the double covering 
  
    
      
        ρ
        :
        
          
            S
            p
            i
            n
          
        
        (
        n
        )
        →
        
          
            S
            O
          
        
        (
        n
        )
      
    
    {\displaystyle \rho \colon {\mathrm {Spin} }(n)\to {\mathrm {SO} }(n)}
  
 of the <a href="/facts/Special_orthogonal_group/jKWM1KDM">special orthogonal group</a> by the <a href="/facts/Spin_group/tfvR9cqM">spin group</a>.
</p><p>The spinor bundle 
  
    
      
        
          
            S
          
        
        
      
    
    {\displaystyle {\mathbf {S} }\,}
  
 is defined <a class="footnote-ref" id="fnref:1" href="#fn:1"><sup>1</sup></a> to be the <a href="/facts/Complex_vector_bundle/0nE8TEIn">complex vector bundle</a>

S
          
        
        =
        
          
            P
          
        
        
          ×
          
            κ
          
        
        
          Δ
          
            n
          
        
        
      
    
    {\displaystyle {\mathbf {S} }={\mathbf {P} }\times _{\kappa }\Delta _{n}\,}

associated to the <a href="/facts/Spin_structure/BF78kFz8">spin structure</a> 
  
    
      
        
          
            P
          
        
      
    
    {\displaystyle {\mathbf {P} }}
  
 via the <a href="/facts/Spin_representation/upLgk4zS">spin representation</a> 
  
    
      
        κ
        :
        
          
            S
            p
            i
            n
          
        
        (
        n
        )
        →
        
          
            U
          
        
        (
        
          Δ
          
            n
          
        
        )
        ,
        
      
    
    {\displaystyle \kappa \colon {\mathrm {Spin} }(n)\to {\mathrm {U} }(\Delta _{n}),\,}
  
 where 
  
    
      
        
          
            U
          
        
        (
        
          
            W
          
        
        )
        
      
    
    {\displaystyle {\mathrm {U} }({\mathbf {W} })\,}
  
 denotes the <a href="/facts/Group_(mathematics)/IQTYqINt">group</a> of <a href="/facts/Unitary_operator/rkypfQQ9">unitary operators</a> acting on a <a href="/facts/Hilbert_space/aDFMh4lV">Hilbert space</a> 
  
    
      
        
          
            W
          
        
        .
        
      
    
    {\displaystyle {\mathbf {W} }.\,}
  
 The spin representation 
  
    
      
        κ
      
    
    {\displaystyle \kappa }
  
 is a faithful and <a href="/facts/Unitary_representation/W1XRatG8">unitary representation</a> of the group 
  
    
      
        
          
            S
            p
            i
            n
          
        
        (
        n
        )
        .
      
    
    {\displaystyle {\mathrm {Spin} }(n).}
  
<a class="footnote-ref" id="fnref:2" href="#fn:2"><sup>2</sup></a>
</p>
<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Clifford_bundle/FawjG7Ia">Clifford bundle</a></li>
<li><a href="/facts/Clifford_module_bundle/ESqf4ECI">Clifford module bundle</a></li>
<li><a href="/facts/Orthonormal_frame_bundle/Gugt3qyL">Orthonormal frame bundle</a></li>
<li><a href="/facts/Spin_geometry/1utupJnD">Spin geometry</a></li>
<li><a href="/facts/Spinor/sKFNnEwP">Spinor</a></li>
<li><a href="/facts/Spinor_representation/upLgk4zS">Spinor representation</a></li></ul>
<h2 id="notes">Notes</h2>

<h2 id="further-reading">Further reading</h2>
<ul><li><a href="/facts/H._Blaine_Lawson/mcTSg513">Lawson, H. Blaine</a>; <a href="/facts/Marie-Louise_Michelsohn/SCDjRoTw">Michelsohn, Marie-Louise</a> (1989). <i>Spin Geometry</i>. <a href="/facts/Princeton_University_Press/kwvFlKEj">Princeton University Press</a>. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-691-08542-5.</li>
<li>Friedrich, Thomas (2000), <i>Dirac Operators in Riemannian Geometry</i>, <a href="/facts/American_Mathematical_Society/fr5gotCt">American Mathematical Society</a>, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-8218-2055-1</li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1"><p>Friedrich, Thomas (2000), Dirac Operators in Riemannian Geometry, American Mathematical Society, ISBN 978-0-8218-2055-1 page 53 <a href="978-0-8218-2055-1" target="_blank">978-0-8218-2055-1</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li>
<li id="fn:2"><p>Friedrich, Thomas (2000), Dirac Operators in Riemannian Geometry, American Mathematical Society, ISBN 978-0-8218-2055-1 pages 20 and 24 <a href="978-0-8218-2055-1" target="_blank">978-0-8218-2055-1</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li>
</ol>

Spinor bundle open-in-new

Spinor bundle