Neha Patil

Superconformal algebra

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In theoretical physics, the superconformal algebra is a graded Lie algebra or superalgebra that combines the conformal algebra and supersymmetry. In two dimensions, the superconformal algebra is infinite-dimensional. In higher dimensions, superconformal algebras are finite-dimensional and generate the superconformal group (in two Euclidean dimensions, the Lie superalgebra does not generate any Lie supergroup).

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Superconformal algebra in dimension greater than 2

The conformal group of the ( p + q ) -dimensional space R p , q is S O ( p + 1 , q + 1 ) and its Lie algebra is s o ( p + 1 , q + 1 ) . The superconformal algebra is a Lie superalgebra containing the bosonic factor s o ( p + 1 , q + 1 ) and whose odd generators transform in spinor representations of s o ( p + 1 , q + 1 ) . Given Kač's classification of finite-dimensional simple Lie superalgebras, this can only happen for small values of p and q . A (possibly incomplete) list is

  • o s p ( 2 N | 2 , 2 ) in 3+0D thanks to u s p ( 2 , 2 ) s o ( 4 , 1 ) ;
  • o s p ( N | 4 ) in 2+1D thanks to s p ( 4 , R ) s o ( 3 , 2 ) ;
  • s u ( 2 N | 4 ) in 4+0D thanks to s u ( 4 ) s o ( 5 , 1 ) ;
  • s u ( 2 , 2 | N ) in 3+1D thanks to s u ( 2 , 2 ) s o ( 4 , 2 ) ;
  • s l ( 4 | N ) in 2+2D thanks to s l ( 4 , R ) s o ( 3 , 3 ) ;
  • real forms of F ( 4 ) in five dimensions
  • o s p ( 8 | 2 N ) in 5+1D, thanks to the fact that spinor and fundamental representations of s o ( 8 , C ) are mapped to each other by outer automorphisms.
  • Superconformal algebra in 3+1D

    According to the superconformal algebra with N supersymmetries in 3+1 dimensions is given by the bosonic generators P μ , D , M μ ν , K μ , the U(1) R-symmetry A , the SU(N) R-symmetry T j i and the fermionic generators Q α i , Q ¯ i α ˙ , S i α and S ¯ α ˙ i . Here, μ , ν , ρ , denote spacetime indices; α , β , left-handed Weyl spinor indices; α ˙ , β ˙ , right-handed Weyl spinor indices; and i , j , the internal R-symmetry indices.

    The Lie superbrackets of the bosonic conformal algebra are given by

    [ M μ ν , M ρ σ ] = η ν ρ M μ σ η μ ρ M ν σ + η ν σ M ρ μ η μ σ M ρ ν [ M μ ν , P ρ ] = η ν ρ P μ η μ ρ P ν [ M μ ν , K ρ ] = η ν ρ K μ η μ ρ K ν [ M μ ν , D ] = 0 [ D , P ρ ] = P ρ [ D , K ρ ] = + K ρ [ P μ , K ν ] = 2 M μ ν + 2 η μ ν D [ K n , K m ] = 0 [ P n , P m ] = 0

    where η is the Minkowski metric; while the ones for the fermionic generators are:

    { Q α i , Q ¯ β ˙ j } = 2 δ i j σ α β ˙ μ P μ { Q , Q } = { Q ¯ , Q ¯ } = 0 { S α i , S ¯ β ˙ j } = 2 δ j i σ α β ˙ μ K μ { S , S } = { S ¯ , S ¯ } = 0 { Q , S } = { Q , S ¯ } = { Q ¯ , S } = 0

    The bosonic conformal generators do not carry any R-charges, as they commute with the R-symmetry generators:

    [ A , M ] = [ A , D ] = [ A , P ] = [ A , K ] = 0 [ T , M ] = [ T , D ] = [ T , P ] = [ T , K ] = 0

    But the fermionic generators do carry R-charge:

    [ A , Q ] = 1 2 Q [ A , Q ¯ ] = 1 2 Q ¯ [ A , S ] = 1 2 S [ A , S ¯ ] = 1 2 S ¯ [ T j i , Q k ] = δ k i Q j [ T j i , Q ¯ k ] = δ j k Q ¯ i [ T j i , S k ] = δ j k S i [ T j i , S ¯ k ] = δ k i S ¯ j

    Under bosonic conformal transformations, the fermionic generators transform as:

    [ D , Q ] = 1 2 Q [ D , Q ¯ ] = 1 2 Q ¯ [ D , S ] = 1 2 S [ D , S ¯ ] = 1 2 S ¯ [ P , Q ] = [ P , Q ¯ ] = 0 [ K , S ] = [ K , S ¯ ] = 0

    Superconformal algebra in 2D

    There are two possible algebras with minimal supersymmetry in two dimensions; a Neveu–Schwarz algebra and a Ramond algebra. Additional supersymmetry is possible, for instance the N = 2 superconformal algebra.

    References

    Superconformal algebra Wikipedia


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