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 ${\displaystyle (p+q)}$-dimensional space ${\displaystyle \mathbb {R} ^{p,q))$ 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 Kac's classification of finite-dimensional simple Lie superalgebras, this can only happen for small values of ${\displaystyle p}$ and ${\displaystyle q}$. 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 ${\displaystyle F(4)}$ 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 ${\displaystyle {\mathcal {N))}$ supersymmetries in 3+1 dimensions is given by the bosonic generators ${\displaystyle P_{\mu ))$, ${\displaystyle D}$, ${\displaystyle M_{\mu \nu ))$, ${\displaystyle K_{\mu ))$, the U(1) R-symmetry ${\displaystyle A}$, the SU(N) R-symmetry ${\displaystyle T_{j}^{i))$ and the fermionic generators ${\displaystyle Q^{\alpha i))$, ${\displaystyle {\overline {Q))_{i}^{\dot {\alpha ))}$, ${\displaystyle S_{i}^{\alpha ))$ and ${\displaystyle {\overline {S))^((\dot {\alpha ))i))$. Here, ${\displaystyle \mu ,\nu ,\rho ,\dots }$ denote spacetime indices; ${\displaystyle \alpha ,\beta ,\dots }$ left-handed Weyl spinor indices; ${\displaystyle {\dot {\alpha )),{\dot {\beta )),\dots }$ right-handed Weyl spinor indices; and ${\displaystyle i,j,\dots }$ the internal R-symmetry indices.

The Lie superbrackets of the bosonic conformal algebra are given by

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

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

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

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

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

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

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

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

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

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

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

${\displaystyle \left\{Q,Q\right\}=\left\((\overline {Q)),{\overline {Q))\right\}=0}$

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

${\displaystyle \left\{S,S\right\}=\left\((\overline {S)),{\overline {S))\right\}=0}$

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

${\displaystyle \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:

${\displaystyle [A,M]=[A,D]=[A,P]=[A,K]=0}$

${\displaystyle [T,M]=[T,D]=[T,P]=[T,K]=0}$

But the fermionic generators do carry R-charge:

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

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

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

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

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

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

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

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

Under bosonic conformal transformations, the fermionic generators transform as:

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

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

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

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

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

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

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.