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).
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.
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 }}](https://wikimedia.org/api/rest_v1/media/math/render/svg/687b4acc2b5ab48f39823e263cdc8c750f3d6c7a)
![{\displaystyle [M_{\mu \nu },P_{\rho }]=\eta _{\nu \rho }P_{\mu }-\eta _{\mu \rho }P_{\nu }}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4625856a3989de9a59b85e66f47d9895e31b8a6c)
![{\displaystyle [M_{\mu \nu },K_{\rho }]=\eta _{\nu \rho }K_{\mu }-\eta _{\mu \rho }K_{\nu }}](https://wikimedia.org/api/rest_v1/media/math/render/svg/49e9685cee498faaa78ac46a70a1841c34fcf079)
![{\displaystyle [M_{\mu \nu },D]=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/11b11341d586db686770ca7851f2c5a676ca2867)
![{\displaystyle [D,P_{\rho }]=-P_{\rho }}](https://wikimedia.org/api/rest_v1/media/math/render/svg/942dfb47712324e239aef315cc24a2df9b192a7e)
![{\displaystyle [D,K_{\rho }]=+K_{\rho }}](https://wikimedia.org/api/rest_v1/media/math/render/svg/74fa88fcd77c04dc3671e519a93abcf7f70f333a)
![{\displaystyle [P_{\mu },K_{\nu }]=-2M_{\mu \nu }+2\eta _{\mu \nu }D}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b17a14d13b529b2a5745332adc5b5be70202b7de)
![{\displaystyle [K_{n},K_{m}]=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d293742c0c53a24590b3021ac4c80b432e667416)
![{\displaystyle [P_{n},P_{m}]=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8c1b4ac4978865c990d9547753314dfa09e0a281)
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 }}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4a448cf5e94614f0c31ead321c0949dfcda6609b)
![{\displaystyle \left\{Q,Q\right\}=\left\{{\overline {Q}},{\overline {Q}}\right\}=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/89a88c11ccb291bb40f5c787a50d7e4349872eff)
![{\displaystyle \left\{S_{\alpha }^{i},{\overline {S}}_{{\dot {\beta }}j}\right\}=2\delta _{j}^{i}\sigma _{\alpha {\dot {\beta }}}^{\mu }K_{\mu }}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5332acbf4672ba22d7ef8f2177a8da76f4df7f5a)
![{\displaystyle \left\{S,S\right\}=\left\{{\overline {S}},{\overline {S}}\right\}=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0f177cb85b108ef79be48fd3315d447ccdc96487)
![{\displaystyle \left\{Q,S\right\}=}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c5f2989b3f5f90ccd25d76223989cd033658d791)
![{\displaystyle \left\{Q,{\overline {S}}\right\}=\left\{{\overline {Q}},S\right\}=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3363f51937b79eaa8235f300c2eede962f4c686f)
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}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4833fa63196ae09bc9dbf8019994892ec55e0b21)
![{\displaystyle [T,M]=[T,D]=[T,P]=[T,K]=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/56cd8446ecf271f7cfec74ad82cdad929595fdee)
But the fermionic generators do carry R-charge:
![{\displaystyle [A,Q]=-{\frac {1}{2}}Q}](https://wikimedia.org/api/rest_v1/media/math/render/svg/69afc1315b121c8e39f9bba64a64275714d83abe)
![{\displaystyle [A,{\overline {Q}}]={\frac {1}{2}}{\overline {Q}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d9133710212899ec561f70e03ff3eb04953c13ea)
![{\displaystyle [A,S]={\frac {1}{2}}S}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0f6a8645f68a80dbbb500462f5dad836d3d435ec)
![{\displaystyle [A,{\overline {S}}]=-{\frac {1}{2}}{\overline {S}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/579c65649ca2c9220f0dd4793569401ed7f4f326)
![{\displaystyle [T_{j}^{i},Q_{k}]=-\delta _{k}^{i}Q_{j}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a8a6ebf8c08f368281b0f2a1f1ca1a26a9c70773)
![{\displaystyle [T_{j}^{i},{\overline {Q}}^{k}]=\delta _{j}^{k}{\overline {Q}}^{i}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/17f5defc6a7813508071aaffb7d43ef150691aa7)
![{\displaystyle [T_{j}^{i},S^{k}]=\delta _{j}^{k}S^{i}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fecaa6b3c925c62e2ac9b16ff7aba7b4b8a65132)
![{\displaystyle [T_{j}^{i},{\overline {S}}_{k}]=-\delta _{k}^{i}{\overline {S}}_{j}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ec5f645adf1cd9d3cfd22db64bf8e0c2c98fa265)
Under bosonic conformal transformations, the fermionic generators transform as:
![{\displaystyle [D,Q]=-{\frac {1}{2}}Q}](https://wikimedia.org/api/rest_v1/media/math/render/svg/56ca193e6b9f56ff00a50dad8823f3f2393adb0e)
![{\displaystyle [D,{\overline {Q}}]=-{\frac {1}{2}}{\overline {Q}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1ab3f77f7221402080b35b77134456c7c0fb135f)
![{\displaystyle [D,S]={\frac {1}{2}}S}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d4268043eef612d2898de14e4068ed9d982c82c3)
![{\displaystyle [D,{\overline {S}}]={\frac {1}{2}}{\overline {S}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7093e155ae8b5aa0e8761b473e053f79038c7fd4)
![{\displaystyle [P,Q]=[P,{\overline {Q}}]=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2a910fbf5011bd0b61125eb3cc1ffbdefd913430)
![{\displaystyle [K,S]=[K,{\overline {S}}]=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3c7941dbb196b7aaeb96f7b6be3caa362583e8a4)
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
- Conformal symmetry
- Super Virasoro algebra
- Supersymmetry algebra
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.
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