Modular group representation

<p>In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, the modular group representation (or simply modular representation) of a modular tensor category 
  
    
      
        
          
            C
          
        
      
    
    {\displaystyle {\mathcal {C}}}
  
 is a representation of the <a href="/facts/Modular_group/wL6l14Hk">modular group</a> 
  
    
      
        
          
            SL
          
          
            2
          
        
        (
        
          Z
        
        )
      
    
    {\displaystyle {\text{SL}}_{2}(\mathbb {Z} )}
  
 associated to 
  
    
      
        
          
            C
          
        
      
    
    {\displaystyle {\mathcal {C}}}
  
. It is from the existence of the modular representation that modular tensor categories get their name.
</p><p>From the perspective of <a href="/facts/Topological_quantum_field_theory/zHjrp3b8">topological quantum field theory</a>, the modular representation of 
  
    
      
        
          
            C
          
        
      
    
    {\displaystyle {\mathcal {C}}}
  
 arrises naturally as the representation of the <a href="/facts/Mapping_class_group/buQm001q">mapping class group</a> of the <a href="/facts/Torus/J9SkaV1W">torus</a> associated to the <a href="/facts/Reshetikhin%E2%80%93Turaev_invariant/xQhD411L">Reshetikhin–Turaev topological quantum field theory</a> associated to 
  
    
      
        
          
            C
          
        
      
    
    {\displaystyle {\mathcal {C}}}
  
. As such, modular tensor categories can be used to define projective representations of the mapping class groups of all closed surfaces.
</p>

Modular group representation open-in-new

Modular group representation