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| 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]].).
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| In two dimensions, the superconformal algebra is infinite-dimensional. In higher dimensions, there is a finite number of known examples of superconformal algebras.
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| == Superconformal algebra in 3+1D ==
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| According to,<ref>{{cite arxiv
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| | last = West
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| | first = Peter C.
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| | title = Introduction to rigid supersymmetric theories
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| | year = 1997
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| | eprint = hep-th/9805055
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| }}</ref><ref>
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| {{cite journal
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| | last = Gates | first = S. J.
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| | author-link = Sylvester Gates
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| | last2 = Grisaru | first2 = Marcus T.
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| | author2-link = Marcus Grisaru
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| | last3 = Rocek | first3 = M.
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| | author3-link = Martin Rocek
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| | last4 = Siegel | first4 = W.
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| | author4-link = Warren Siegel
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| | year = 1983
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| | title = Superspace, or one thousand and one lessons in supersymmetry
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| | journal = [[Frontiers in Physics]]
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| | volume = 58 | pages = 1–548
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| | arxiv = hep-th/0108200
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| | bibcode = 2001hep.th....8200G
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| | doi =
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| }}</ref> the <math>\mathcal{N}=1</math> superconformal algebra in 3+1D is given by the bosonic generators <math>P_\mu</math>, <math>D</math>, <math>M_{\mu\nu}</math>, <math>K_\mu</math>, the U(1) [[R-symmetry]] <math>A</math>, the SU(N) R-symmetry <math>T^i_j</math> and the fermionic generators <math>Q^{\alpha i}</math>, <math>\overline{Q}^{\dot\alpha}_i</math>, <math>S^\alpha_i</math> and <math>\overline{S}^{\dot\alpha i}</math>. <math>\mu,\nu,\rho,\dots</math> denote spacetime indices, <math>\alpha,\beta,\dots</math> left-handed Weyl spinor indices and <math>\dot\alpha,\dot\beta,\dots</math> right-handed Weyl spinor indices, and <math>i,j,\dots</math> the internal R-symmetry indices.
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| The Lie superbrackets are given by
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| :<math>[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}</math>
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| :<math>[M_{\mu\nu},P_\rho]=\eta_{\nu\rho}P_\mu-\eta_{\mu\rho}P_\nu</math>
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| :<math>[M_{\mu\nu},K_\rho]=\eta_{\nu\rho}K_\mu-\eta_{\mu\rho}K_\nu</math>
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| :<math>[M_{\mu\nu},D]=0</math>
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| :<math>[D,P_\rho]=-P_\rho</math>
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| :<math>[D,K_\rho]=+K_\rho</math>
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| :<math>[P_\mu,K_\nu]=-2M_{\mu\nu}+2\eta_{\mu\nu}D</math>
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| :<math>[K_n,K_m]=0</math> | |
| :<math>[P_n,P_m]=0</math>
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| This is the bosonic [[conformal symmetry|conformal algebra]]. Here, η is the [[Minkowski metric]].
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| :<math>[A,M]=[A,D]=[A,P]=[A,K]=0</math>
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| :<math>[T,M]=[T,D]=[T,P]=[T,K]=0</math>
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| The bosonic conformal generators do not carry any R-charges.
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| :<math>[A,Q]=-\frac{1}{2}Q</math>
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| :<math>[A,\overline{Q}]=\frac{1}{2}\overline{Q}</math>
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| :<math>[A,S]=\frac{1}{2}S</math>
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| :<math>[A,\overline{S}]=-\frac{1}{2}\overline{S}</math>
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| :<math>[T^i_j,Q_k]= - \delta^i_k Q_j</math>
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| :<math>[T^i_j,\overline{Q}^k]= \delta^k_j \overline{Q}^i</math>
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| :<math>[T^i_j,S^k]=\delta^k_j S^i</math>
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| :<math>[T^i_j,\overline{S}_k]= - \delta^i_k \overline{S}_j</math>
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| But the fermionic generators do.
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| :<math>[D,Q]=-\frac{1}{2}Q</math>
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| :<math>[D,\overline{Q}]=-\frac{1}{2}\overline{Q}</math>
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| :<math>[D,S]=\frac{1}{2}S</math>
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| :<math>[D,\overline{S}]=\frac{1}{2}\overline{S}</math>
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| :<math>[P,Q]=[P,\overline{Q}]=0</math>
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| :<math>[K,S]=[K,\overline{S}]=0</math>
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| <!--
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| :<math>[P_\mu,S_{\alpha}^i]=\sigma_\mu^{\alpha\dot{\beta}} \overline{Q}_{\dot{\beta}}</math>
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| :<math>[P_\mu,\overline{S}_{\dot{\alpha}}_i]= Q</math>
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| :<math>[K_\mu,Q_{\alpha i}]=\overline{S}</math>
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| :<math>[K_\mu,\overline{Q}_{\dot{\alpha}}^i]=S</math>
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| -->
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| Tells us how the fermionic generators transform under bosonic conformal transformations.
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| <!--
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| :<math>[M_{\mu\nu}, Q^\alpha] = ??? (\overline{\sigma}_\mu\sigma_\nu - \overline{\sigma}_\nu \sigma_\mu) Q</math>
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| :<math>[M_{\mu\nu}, S^\alpha] = ??? (\overline{\sigma}_\mu\sigma_\nu - \overline{\sigma}_\nu \sigma_\mu) S</math>
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| -->
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| :<math>\left\{ Q_{\alpha i}, \overline{Q}_{\dot{\beta}}^j \right\} = 2 \delta^j_i \sigma^{\mu}_{\alpha \dot{\beta}}P_\mu</math>
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| :<math>\left\{ Q, Q \right\} = \left\{ \overline{Q}, \overline{Q} \right\} = 0</math>
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| :<math>\left\{ S_{\alpha}^i, \overline{S}_{\dot{\beta}j} \right\} = 2 \delta^i_j \sigma^{\mu}_{\alpha \dot{\beta}}K_\mu</math>
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| :<math>\left\{ S, S \right\} = \left\{ \overline{S}, \overline{S} \right\} = 0</math>
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| :<math>\left\{ Q, S \right\} = </math>
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| :<math>\left\{ Q, \overline{S} \right\} = \left\{ \overline{Q}, S \right\} = 0</math>
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| <!--
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| :<math>[Q_\alpha,D]={1\over2}Q_\alpha</math>
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| :<math>[S_\alpha,D]=-{1\over2}S_\alpha</math>
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| :<math>[Q_\alpha,K_\nu]=-(\gamma_\nu)_\alpha^\beta
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| S_\beta</math>
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| :<math>[S_\alpha,P_n]=(\gamma
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| _n)_\alpha^\beta Q_\beta</math>
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| -->
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| <!--
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| :<math>[Q_\alpha^i,T_r]=\big( \delta_\alpha^\beta(\tau_{r_1})^i_j
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| +(\gamma_5)_\alpha^
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| \beta(\tau_{r_2})^i_j\big) Q_\beta^j</math>
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| :<math>[S_\alpha^i,T_r]=\big(\delta_\alpha^\beta(\tau_{r_1})^i_j-(\gamma_5)_\alpha^
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| \beta(\tau_{r_2})^i_j\big)Q_\beta^j</math>
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| :<math>[Q_\alpha^i,A]=-i\frac{3}{4}(\gamma_5)_\alpha^\beta
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| Q_\beta^i</math>
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| :<math>[S_\alpha^i,A]={4-N\over4N}i(\gamma_5)_\alpha^\beta S_\beta^i\cr
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| \{Q_\alpha^i,S_\beta^j\}=-2(C^{-1}_{\alpha\beta})
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| D\delta^{ij}+(\gamma^{mn}C^{-
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| 1})_{\alpha\beta}J_{mn}\delta^{ij}+4i(\gamma_5C^{-1}_{\alpha\beta})
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| A\delta^{ij}</math>
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| :<math>
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| &\quad-2(\tau_{r_1})^{ij}(C^{-1})_{\alpha\beta}+\big((\tau_{r_2})^{ij}(\gamma_
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| 5C^{-1})_{\alpha\beta}\big)T_r&(2.41)}</math>
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| -->
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| == Superconformal algebra in 2D ==
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| See [[super Virasoro algebra]]. There are two possible algebras; a Neveu-Schwarz algebra and a Ramond algebra.
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| == References ==
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| {{reflist}}
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| == See also ==
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| * [[Conformal symmetry]]
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| * [[Super Virasoro algebra]]
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| * [[Supersymmetry algebra]]
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| {{DEFAULTSORT:Superconformal Algebra}}
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| [[Category:Conformal field theory]]
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| [[Category:Supersymmetry]]
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| [[Category:Lie algebras]]
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