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 $SO(p+1,q+1)$ and its Lie algebra is ${\mathfrak {so}}(p+1,q+1)$. The superconformal algebra is a Lie superalgebra containing the bosonic factor ${\mathfrak {so}}(p+1,q+1)$ and whose odd generators transform in spinor representations of ${\mathfrak {so}}(p+1,q+1)$. 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

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

Superconformal algebra in 3+1D

According to 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

Main article: super Virasoro algebra

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