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).

Superconformal algebra in dimension greater than 2[edit]

The conformal group of the -dimensional space is and its Lie algebra is . 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 and . A (possibly incomplete) list is

  • in 3+0D thanks to ;
  • in 2+1D thanks to ;
  • in 4+0D thanks to ;
  • in 3+1D thanks to ;
  • in 2+2D thanks to ;
  • real forms of in five dimensions
  • in 5+1D, thanks to the fact that spinor and fundamental representations of are mapped to each other by outer automorphisms.

Superconformal algebra in 3+1D[edit]

According to [1][2] the superconformal algebra with supersymmetries in 3+1 dimensions is given by the bosonic generators , , , , the U(1) R-symmetry , the SU(N) R-symmetry and the fermionic generators , , and . Here, denote spacetime indices; 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:

Superconformal algebra in 2D[edit]

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.

See also[edit]


  1. ^ West, P. C. (2002). "Introduction to Rigid Supersymmetric Theories". Confinement, Duality, and Non-Perturbative Aspects of QCD. NATO Science Series: B. Vol. 368. pp. 453–476. arXiv:hep-th/9805055. doi:10.1007/0-306-47056-X_17. ISBN 0-306-45826-8. S2CID 119413468.
  2. ^ Gates, S. J.; Grisaru, Marcus T.; Rocek, M.; Siegel, W. (1983). "Superspace, or one thousand and one lessons in supersymmetry". Frontiers in Physics. 58: 1–548. arXiv:hep-th/0108200.