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).
Superconformal algebra in dimension greater than 2 The conformal group of the ( p + q ) {\displaystyle (p+q)} -dimensional space R p , q {\displaystyle \mathbb {R} ^{p,q}} is S O ( p + 1 , q + 1 ) {\displaystyle SO(p+1,q+1)} and its Lie algebra is s o ( p + 1 , q + 1 ) {\displaystyle {\mathfrak {so}}(p+1,q+1)} . The superconformal algebra is a Lie superalgebra containing the bosonic factor s o ( p + 1 , q + 1 ) {\displaystyle {\mathfrak {so}}(p+1,q+1)} and whose odd generators transform in spinor representations of s o ( p + 1 , q + 1 ) {\displaystyle {\mathfrak {so}}(p+1,q+1)} . Given Kac's classification of finite-dimensional simple Lie superalgebras, this can only happen for small values of p {\displaystyle p} and q {\displaystyle q} . A (possibly incomplete) list is
o s p ∗ ( 2 N | 2 , 2 ) {\displaystyle {\mathfrak {osp}}^{*}(2N|2,2)} in 3+0D thanks to u s p ( 2 , 2 ) ≃ s o ( 4 , 1 ) {\displaystyle {\mathfrak {usp}}(2,2)\simeq {\mathfrak {so}}(4,1)} ;
o s p ( N | 4 ) {\displaystyle {\mathfrak {osp}}(N|4)} in 2+1D thanks to s p ( 4 , R ) ≃ s o ( 3 , 2 ) {\displaystyle {\mathfrak {sp}}(4,\mathbb {R} )\simeq {\mathfrak {so}}(3,2)} ;
s u ∗ ( 2 N | 4 ) {\displaystyle {\mathfrak {su}}^{*}(2N|4)} in 4+0D thanks to s u ∗ ( 4 ) ≃ s o ( 5 , 1 ) {\displaystyle {\mathfrak {su}}^{*}(4)\simeq {\mathfrak {so}}(5,1)} ;
s u ( 2 , 2 | N ) {\displaystyle {\mathfrak {su}}(2,2|N)} in 3+1D thanks to s u ( 2 , 2 ) ≃ s o ( 4 , 2 ) {\displaystyle {\mathfrak {su}}(2,2)\simeq {\mathfrak {so}}(4,2)} ;
s l ( 4 | N ) {\displaystyle {\mathfrak {sl}}(4|N)} in 2+2D thanks to s l ( 4 , R ) ≃ s o ( 3 , 3 ) {\displaystyle {\mathfrak {sl}}(4,\mathbb {R} )\simeq {\mathfrak {so}}(3,3)} ; real forms of F ( 4 ) {\displaystyle F(4)} in five dimensions
o s p ( 8 ∗ | 2 N ) {\displaystyle {\mathfrak {osp}}(8^{*}|2N)} in 5+1D, thanks to the fact that spinor and fundamental representations of s o ( 8 , C ) {\displaystyle {\mathfrak {so}}(8,\mathbb {C} )} are mapped to each other by outer automorphisms.
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