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

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

Superconformal algebra in 3+1D

According to
the superconformal algebra with $\mathcal$ supersymmetries in 3+1 dimensions is given by the bosonic generators $P_\mu$, $D$, $M_$, $K_\mu$, the U(1) R-symmetry $A$, the SU(N) R-symmetry $T^i_j$ and the fermionic generators $Q^$, $\overline^_i$, $S^\alpha_i$ and $^$. 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_\eta_M_-\eta_M_+\eta_M_-\eta_M_
: _,P_\rho\eta_P_\mu-\eta_P_\nu
: _,K_\rho\eta_K_\mu-\eta_K_\nu
: _,D0
: ,P_\rho-P_\rho
: ,K_\rho+K_\rho
: _\mu,K_\nu-2M_+2\eta_D
: _n,K_m0
: _n,P_m0
where η is the Minkowski metric; while the ones for the fermionic generators are:
:$\left\ = 2 \delta^j_i \sigma^_P_\mu$
:$\left\ = \left\ = 0$
:$\left\ = 2 \delta^i_j \sigma^_K_\mu$
:$\left\ = \left\ = 0$
:$\left\ =$
:$\left\ = \left\ = 0$

The bosonic conformal generators do not carry any R-charges, as they commute with the R-symmetry generators:
: ,M ,D ,P ,K0
: ,M ,D ,P ,K0

But the fermionic generators do carry R-charge:
: ,Q-\fracQ
: ,\overline\frac\overline
: ,S\fracS
: ,\overline-\frac\overline

: ^i_j,Q_k - \delta^i_k Q_j
: ^i_j,^k \delta^k_j ^i
: ^i_j,S^k\delta^k_j S^i
: ^i_j,\overline_k - \delta^i_k \overline_j

Under bosonic conformal transformations, the fermionic generators transform as:
: ,Q-\fracQ
: ,\overline-\frac\overline
: ,S\fracS
: ,\overline\frac\overline
: ,Q ,\overline0
: ,S ,\overline0

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

Conformal field theory
Supersymmetry
Lie algebras

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