Superconformal algebra

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

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

According to {{cite book |doi=10.1007/0-306-47056-X_17|chapter=Introduction to Rigid Supersymmetric Theories |title=Confinement, Duality, and Non-Perturbative Aspects of QCD |series=NATO Science Series: B |year=2002 |last1=West |first1=P. C. |volume=368 |pages=453–476 |arxiv=hep-th/9805055 |isbn=0-306-45826-8 |s2cid=119413468 }}

{{cite journal

| last = Gates | first = S. J.

| last2 = Grisaru | first2 = Marcus T.

| last3 = Rocek | first3 = M.

| author3-link = Martin Rocek

| last4 = Siegel | first4 = W.

| author4-link = Warren Siegel

| year = 1983

| title = Superspace, or one thousand and one lessons in supersymmetry

| journal = Frontiers in Physics

| volume = 58 | pages = 1–548

| arxiv = hep-th/0108200

| bibcode = 2001hep.th....8200G

| doi =

}} the superconformal algebra with $\backslash mathcal\{N\}$ supersymmetries in 3+1 dimensions is given by the bosonic generators $P\_\backslash mu$, $D$, $M\_\{\backslash mu\backslash nu\}$, $K\_\backslash mu$, the U(1) R-symmetry $A$, the SU(N) R-symmetry $T^i\_j$ and the fermionic generators $Q^\{\backslash alpha\; i\}$, $\backslash overline\{Q\}^\{\backslash dot\backslash alpha\}\_i$, $S^\backslash alpha\_i$ and $\{\backslash overline\{S\}\}^\{\backslash dot\backslash alpha\; i\}$. Here, $\backslash mu,\backslash nu,\backslash rho,\backslash dots$ denote spacetime indices; $\backslash alpha,\backslash beta,\backslash dots$ left-handed Weyl spinor indices; $\backslash dot\backslash alpha,\backslash dot\backslash beta,\backslash dots$ right-handed Weyl spinor indices; and $i,j,\backslash dots$ the internal R-symmetry indices.

The Lie superbrackets of the bosonic conformal algebra are given by

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

:$[M\_\{\backslash mu\backslash nu\},P\_\backslash rho]=\backslash eta\_\{\backslash nu\backslash rho\}P\_\backslash mu-\backslash eta\_\{\backslash mu\backslash rho\}P\_\backslash nu$

:$[M\_\{\backslash mu\backslash nu\},K\_\backslash rho]=\backslash eta\_\{\backslash nu\backslash rho\}K\_\backslash mu-\backslash eta\_\{\backslash mu\backslash rho\}K\_\backslash nu$

:$[M\_\{\backslash mu\backslash nu\},D]=0$

:$[D,P\_\backslash rho]=-P\_\backslash rho$

:$[D,K\_\backslash rho]=+K\_\backslash rho$

:$[P\_\backslash mu,K\_\backslash nu]=-2M\_\{\backslash mu\backslash nu\}+2\backslash eta\_\{\backslash mu\backslash nu\}D$

:$[K\_n,K\_m]=0$

:$[P\_n,P\_m]=0$

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

:$\backslash left\backslash \{\; Q\_\{\backslash alpha\; i\},\; \backslash overline\{Q\}\_\{\backslash dot\{\backslash beta\}\}^j\; \backslash right\backslash \}\; =\; 2\; \backslash delta^j\_i\; \backslash sigma^\{\backslash mu\}\_\{\backslash alpha\; \backslash dot\{\backslash beta\}\}P\_\backslash mu$

:$\backslash left\backslash \{\; Q,\; Q\; \backslash right\backslash \}\; =\; \backslash left\backslash \{\; \backslash overline\{Q\},\; \backslash overline\{Q\}\; \backslash right\backslash \}\; =\; 0$

:$\backslash left\backslash \{\; S\_\{\backslash alpha\}^i,\; \backslash overline\{S\}\_\{\backslash dot\{\backslash beta\}j\}\; \backslash right\backslash \}\; =\; 2\; \backslash delta^i\_j\; \backslash sigma^\{\backslash mu\}\_\{\backslash alpha\; \backslash dot\{\backslash beta\}\}K\_\backslash mu$

:$\backslash left\backslash \{\; S,\; S\; \backslash right\backslash \}\; =\; \backslash left\backslash \{\; \backslash overline\{S\},\; \backslash overline\{S\}\; \backslash right\backslash \}\; =\; 0$

:$\backslash left\backslash \{\; Q,\; S\; \backslash right\backslash \}\; =$

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

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:

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

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

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

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

:$[T^i\_j,Q\_k]=\; -\; \backslash delta^i\_k\; Q\_j$

:$[T^i\_j,\{\backslash overline\{Q\}\}^k]=\; \backslash delta^k\_j\; \{\backslash overline\{Q\}\}^i$

:$[T^i\_j,S^k]=\backslash delta^k\_j\; S^i$

:$[T^i\_j,\backslash overline\{S\}\_k]=\; -\; \backslash delta^i\_k\; \backslash overline\{S\}\_j$

Under bosonic conformal transformations, the fermionic generators transform as:

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

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

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

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

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

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

{{main|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.

• Conformal symmetry
• Super Virasoro algebra
• Supersymmetry algebra

{{reflist}}

{{Supersymmetry topics}}

{{String theory topics |state=collapsed}}

{{DEFAULTSORT:Superconformal Algebra}}

Category:Conformal field theory

Category:Supersymmetry

Category:Lie algebras

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