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).
The superconformal algebra is a Lie superalgebra containing the bosonic factor
and whose odd generators transform in spinor representations of
Given Kac's classification of finite-dimensional simple Lie superalgebras, this can only happen for small values of
A (possibly incomplete) list is According to [1][2] the superconformal algebra with
supersymmetries in 3+1 dimensions is given by the bosonic generators
μ ν
μ , ν , ρ , …
denote spacetime indices;
α , β , …
left-handed Weyl spinor indices;
right-handed Weyl spinor indices; and
the internal R-symmetry indices.
The Lie superbrackets of the bosonic conformal algebra are given by where η is the Minkowski metric; while the ones for the fermionic generators are: The bosonic conformal generators do not carry any R-charges, as they commute with the R-symmetry generators: But the fermionic generators do carry R-charge: Under bosonic conformal transformations, the fermionic generators transform as: 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.
This quantum mechanics-related article is a stub.