Superconformal algebra
In theoretical physics, the superconformal algebra is a graded Lie algebra or superalgebra that combines the conformal algebra and supersymmetry. It generates the superconformal group in some cases (In two Euclidean dimensions, the Lie superalgebra doesn't generate any Lie supergroup.).
In two dimensions, the superconformal algebra is infinite-dimensional. In higher dimensions, there is a finite number of known examples of superconformal algebras.
Superconformal algebra in 3+1D
According to,^{[1]}^{[2]} the \mathcal{N}=1 superconformal algebra in 3+1D 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^i_j and the fermionic generators Q^{\alpha i}, \overline{Q}^{\dot\alpha}_i, S^\alpha_i and \overline{S}^{\dot\alpha i}. \mu,\nu,\rho,\dots denote spacetime indices, \alpha,\beta,\dots left-handed Weyl spinor indices and \dot\alpha,\dot\beta,\dots right-handed Weyl spinor indices, and i,j,\dots the internal R-symmetry indices.
The Lie superbrackets 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
This is the bosonic conformal algebra. Here, η is the Minkowski metric.
- [A,M]=[A,D]=[A,P]=[A,K]=0
- [T,M]=[T,D]=[T,P]=[T,K]=0
The bosonic conformal generators do not carry any R-charges.
- [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^i_j,Q_k]= - \delta^i_k Q_j
- [T^i_j,\overline{Q}^k]= \delta^k_j \overline{Q}^i
- [T^i_j,S^k]=\delta^k_j S^i
- [T^i_j,\overline{S}_k]= - \delta^i_k \overline{S}_j
But the fermionic generators do.
- [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
Tells us how the fermionic generators transform under bosonic conformal transformations.
- \left\{ Q_{\alpha i}, \overline{Q}_{\dot{\beta}}^j \right\} = 2 \delta^j_i \sigma^{\mu}_{\alpha \dot{\beta}}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^i_j \sigma^{\mu}_{\alpha \dot{\beta}}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
Superconformal algebra in 2D
See super Virasoro algebra. There are two possible algebras; a Neveu-Schwarz algebra and a Ramond 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. Bibcode:2001hep.th....8200G. arXiv:hep-th/0108200.