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

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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

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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

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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

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References

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  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. Bibcode:2001hep.th....8200G.