Algebra combining both supersymmetry and conformal symmetry
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
-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
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
![{\displaystyle [M_{\mu \nu },M_{\rho \sigma }]=\eta _{\nu \rho }M_{\mu \sigma }-\eta _{\mu \rho }M_{\nu \sigma }+\eta _{\nu \sigma }M_{\rho \mu }-\eta _{\mu \sigma }M_{\rho \nu }}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
![{\displaystyle [M_{\mu \nu },P_{\rho }]=\eta _{\nu \rho }P_{\mu }-\eta _{\mu \rho }P_{\nu }}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
![{\displaystyle [M_{\mu \nu },K_{\rho }]=\eta _{\nu \rho }K_{\mu }-\eta _{\mu \rho }K_{\nu }}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
![{\displaystyle [M_{\mu \nu },D]=0}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
![{\displaystyle [D,P_{\rho }]=-P_{\rho }}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
![{\displaystyle [D,K_{\rho }]=+K_{\rho }}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
![{\displaystyle [P_{\mu },K_{\nu }]=-2M_{\mu \nu }+2\eta _{\mu \nu }D}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
![{\displaystyle [K_{n},K_{m}]=0}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
![{\displaystyle [P_{n},P_{m}]=0}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
where η is the Minkowski metric; while the ones for the fermionic generators are:
![{\displaystyle \left\{Q_{\alpha i},{\overline {Q}}_{\dot {\beta }}^{j}\right\}=2\delta _{i}^{j}\sigma _{\alpha {\dot {\beta }}}^{\mu }P_{\mu }}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
![{\displaystyle \left\{Q,Q\right\}=\left\{{\overline {Q}},{\overline {Q}}\right\}=0}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
![{\displaystyle \left\{S_{\alpha }^{i},{\overline {S}}_{{\dot {\beta }}j}\right\}=2\delta _{j}^{i}\sigma _{\alpha {\dot {\beta }}}^{\mu }K_{\mu }}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
![{\displaystyle \left\{S,S\right\}=\left\{{\overline {S}},{\overline {S}}\right\}=0}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
![{\displaystyle \left\{Q,S\right\}=}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
![{\displaystyle \left\{Q,{\overline {S}}\right\}=\left\{{\overline {Q}},S\right\}=0}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
The bosonic conformal generators do not carry any R-charges, as they commute with the R-symmetry generators:
![{\displaystyle [A,M]=[A,D]=[A,P]=[A,K]=0}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
![{\displaystyle [T,M]=[T,D]=[T,P]=[T,K]=0}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
But the fermionic generators do carry R-charge:
![{\displaystyle [A,Q]=-{\frac {1}{2}}Q}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
![{\displaystyle [A,{\overline {Q}}]={\frac {1}{2}}{\overline {Q}}}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
![{\displaystyle [A,S]={\frac {1}{2}}S}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
![{\displaystyle [A,{\overline {S}}]=-{\frac {1}{2}}{\overline {S}}}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
![{\displaystyle [T_{j}^{i},Q_{k}]=-\delta _{k}^{i}Q_{j}}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
![{\displaystyle [T_{j}^{i},{\overline {Q}}^{k}]=\delta _{j}^{k}{\overline {Q}}^{i}}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
![{\displaystyle [T_{j}^{i},S^{k}]=\delta _{j}^{k}S^{i}}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
![{\displaystyle [T_{j}^{i},{\overline {S}}_{k}]=-\delta _{k}^{i}{\overline {S}}_{j}}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
Under bosonic conformal transformations, the fermionic generators transform as:
![{\displaystyle [D,Q]=-{\frac {1}{2}}Q}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
![{\displaystyle [D,{\overline {Q}}]=-{\frac {1}{2}}{\overline {Q}}}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
![{\displaystyle [D,S]={\frac {1}{2}}S}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
![{\displaystyle [D,{\overline {S}}]={\frac {1}{2}}{\overline {S}}}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
![{\displaystyle [P,Q]=[P,{\overline {Q}}]=0}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
![{\displaystyle [K,S]=[K,{\overline {S}}]=0}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
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.
See also
References
- ^ 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.
- ^ 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.