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 ^[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

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.

References

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

Superconformal algebra in dimension greater than 2

Superconformal algebra in 3+1D

Superconformal algebra in 2D

See also

References