Superconformal algebra

Template:Short description Template:Cleanup 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 $\mathfrak{𝔰}\mathfrak{𝔬}(p+1,q+1)$. The superconformal algebra is a Lie superalgebra containing the bosonic factor $\mathfrak{𝔰}\mathfrak{𝔬}(p+1,q+1)$ and whose odd generators transform in spinor representations of $\mathfrak{𝔰}\mathfrak{𝔬}(p+1,q+1)$. Given Kac's classification of finite-dimensional simple Lie superalgebras, this can only happen for small values of $p$ and $q$. A (possibly incomplete) list is

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

Superconformal algebra in 3+1D

According to the superconformal algebra with ${𝒩}$ 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 $Q^{\alpha i}$, ${\bar{Q}}_i^{\dot{\alpha }}$, $S_i^{\alpha }$ and ${\bar{S}}^{\dot{\alpha }i}$. 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:

$\{Q_{\alpha i},{\bar{Q}}_{\dot{\beta }}^j\}=2{\delta }_i^j{\sigma }_{\alpha \dot{\beta }}^{\mu }{P}_{\mu }$

$\{Q,Q\}=\{\bar{Q},\bar{Q}\}=0$

$\{S_{\alpha }^i,{\bar{S}}_{\dot{\beta }j}\}=2{\delta }_j^i{\sigma }_{\alpha \dot{\beta }}^{\mu }{K}_{\mu }$

$\{S,S\}=\{\bar{S},\bar{S}\}=0$

$\{Q,S\}=$

$\{Q,\bar{S}\}=\{\bar{Q},S\}=0$

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:

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

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

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

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

$[T_j^i,Q_k]=-{\delta }_k^i{Q}_j$

$[T_j^i,{\bar{Q}}^k]={\delta }_j^k{\bar{Q}}^i$

$[T_j^i,S^k]={\delta }_j^k{S}^i$

$[T_j^i,{\bar{S}}_k]=-{\delta }_k^i{\bar{S}}_j$

Under bosonic conformal transformations, the fermionic generators transform as:

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

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

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

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

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

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

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

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