Superconformal algebra

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 $(p+q)$-dimensional space $\mathbb{R}^{p,q}$ is ${\displaystyle SO(p+1,q+1)}$ and its Lie algebra is ${\displaystyle {\mathfrak {so}}(p+1,q+1)}$. The superconformal algebra is a Lie superalgebra containing the bosonic factor ${\displaystyle {\mathfrak {so}}(p+1,q+1)}$ and whose odd generators transform in spinor representations of ${\displaystyle {\mathfrak {so}}(p+1,q+1)}$. Given Kač's classification of finite-dimensional simple Lie superalgebras, this can only happen for small values of $p$ and $q$. A (possibly incomplete) list is

• ${\displaystyle {\mathfrak {osp}}^{*}(2N|2,2)}$ in 3+0D thanks to ${\displaystyle {\mathfrak {usp}}(2,2)\simeq {\mathfrak {so}}(4,1)}$;
• ${\displaystyle {\mathfrak {osp}}(N|4)}$ in 2+1D thanks to ${\displaystyle {\mathfrak {sp}}(4,\mathbb {R} )\simeq {\mathfrak {so}}(3,2)}$;
• ${\displaystyle {\mathfrak {su}}^{*}(2N|4)}$ in 4+0D thanks to ${\displaystyle {\mathfrak {su}}^{*}(4)\simeq {\mathfrak {so}}(5,1)}$;
• ${\displaystyle {\mathfrak {su}}(2,2|N)}$ in 3+1D thanks to ${\displaystyle {\mathfrak {su}}(2,2)\simeq {\mathfrak {so}}(4,2)}$;
• ${\displaystyle {\mathfrak {sl}}(4|N)}$ in 2+2D thanks to ${\displaystyle {\mathfrak {sl}}(4,\mathbb {R} )\simeq {\mathfrak {so}}(3,3)}$;
• real forms of $F(4)$ in five dimensions
• ${\displaystyle {\mathfrak {osp}}(8^{*}|2N)}$ in 5+1D, thanks to the fact that spinor and fundamental representations of ${\displaystyle {\mathfrak {so}}(8,\mathbb {C} )}$ are mapped to each other by outer automorphisms.

Superconformal algebra in 3+1D

According to ^[1][2] the superconformal algebra with ${\mathcal {N}}$ supersymmetries in 3+1 dimensions is given by the bosonic generators $P_{\mu }$, $D$, $M_{\mu\nu}$, $K_\mu$, the U(1) R-symmetry $A$, the SU(N) R-symmetry $T_{j}^{i}$ and the fermionic generators $Q^{{\alpha i}}$, $\overline {Q}_{i}^{{{\dot \alpha }}}$, $S_{i}^{\alpha }$ and ${\overline {S}}^{{{\dot \alpha }i}}$. Here, $\mu ,\nu ,\rho ,\dots$ denote spacetime indices; $\alpha ,\beta ,\dots$ left-handed Weyl spinor indices; ${\dot \alpha },{\dot \beta },\dots$ right-handed Weyl spinor indices; and $i,j,\dots$ the internal R-symmetry indices.

The Lie superbrackets of the bosonic conformal algebra are given by

$[M_{{\mu \nu }},M_{{\rho \sigma }}]=\eta _{{\nu \rho }}M_{{\mu \sigma }}-\eta _{{\mu \rho }}M_{{\nu \sigma }}+\eta _{{\nu \sigma }}M_{{\rho \mu }}-\eta _{{\mu \sigma }}M_{{\rho \nu }}$

$[M_{{\mu \nu }},P_{\rho }]=\eta _{{\nu \rho }}P_{\mu }-\eta _{{\mu \rho }}P_{\nu }$

$[M_{{\mu \nu }},K_{\rho }]=\eta _{{\nu \rho }}K_{\mu }-\eta _{{\mu \rho }}K_{\nu }$

$[M_{{\mu \nu }},D]=0$

$[D,P_{\rho }]=-P_{\rho }$

$[D,K_{\rho }]=+K_{\rho }$

$[P_{\mu },K_{\nu }]=-2M_{{\mu \nu }}+2\eta _{{\mu \nu }}D$

$[K_{n},K_{m}]=0$

$[P_{n},P_{m}]=0$

where η is the Minkowski metric; while the ones for the fermionic generators are:

$\left\{Q_{{\alpha i}},\overline {Q}_{{{\dot {\beta }}}}^{j}\right\}=2\delta _{i}^{j}\sigma _{{\alpha {\dot {\beta }}}}^{{\mu }}P_{\mu }$

$\left\{Q,Q\right\}=\left\{\overline {Q},\overline {Q}\right\}=0$

$\left\{S_{{\alpha }}^{i},\overline {S}_{{{\dot {\beta }}j}}\right\}=2\delta _{j}^{i}\sigma _{{\alpha {\dot {\beta }}}}^{{\mu }}K_{\mu }$

$\left\{S,S\right\}=\left\{\overline {S},\overline {S}\right\}=0$

$\left\{Q,S\right\}=$

$\left\{Q,\overline {S}\right\}=\left\{\overline {Q},S\right\}=0$

The bosonic conformal generators do not carry any R-charges, as they commute with the R-symmetry generators:

$[A,M]=[A,D]=[A,P]=[A,K]=0$

$[T,M]=[T,D]=[T,P]=[T,K]=0$

But the fermionic generators do carry R-charge:

$[A,Q]=-{\frac {1}{2}}Q$

$[A,\overline {Q}]={\frac {1}{2}}\overline {Q}$

$[A,S]={\frac {1}{2}}S$

$[A,\overline {S}]=-{\frac {1}{2}}\overline {S}$

$[T_{j}^{i},Q_{k}]=-\delta _{k}^{i}Q_{j}$

$[T_{j}^{i},{\overline {Q}}^{k}]=\delta _{j}^{k}{\overline {Q}}^{i}$

$[T_{j}^{i},S^{k}]=\delta _{j}^{k}S^{i}$

$[T_{j}^{i},\overline {S}_{k}]=-\delta _{k}^{i}\overline {S}_{j}$

Under bosonic conformal transformations, the fermionic generators transform as:

$[D,Q]=-{\frac {1}{2}}Q$

$[D,\overline {Q}]=-{\frac {1}{2}}\overline {Q}$

$[D,S]={\frac {1}{2}}S$

$[D,\overline {S}]={\frac {1}{2}}\overline {S}$

$[P,Q]=[P,\overline {Q}]=0$

$[K,S]=[K,\overline {S}]=0$

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

• Conformal symmetry
• Super Virasoro algebra
• Supersymmetry algebra

References

1. West, Peter C. (1997). "Introduction to rigid supersymmetric theories". arXiv:hep-th/9805055.
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.