In der theoretischen Physik ist die superkonforme Algebra eine abgestufte Lie-Algebra oder Superalgebra, die konforme Algebra und Supersymmetrie kombiniert. In zwei Dimensionen ist die Superkonformalgebra unendlich groß. In höheren Dimensionen sind Superkonformalgebren endlichdimensional und erzeugen die Superkonformal-Gruppe (in zwei euklidischen Dimensionen erzeugt die Lie-Superalgebra keine Lie-Supergruppe).
Superkonformalgebra in Dimensionen größer als 2 [ edit Die konforme Gruppe von ( p q q ] { displaystyle (p + q)} R p q {19659017] { displaystyle mathbb {R} ^ {p, q}} S O ( p + ] 1 q + 1 ) {Anzeigestil SO (p + 1, q + 1)} s o ( p + 1 + + 1 ) { displaystyle { mathfrak {so}} (p + 1, q + 1)} s o p + 1 q + . 1 ) { displaystyle { mathfrak {so}} (p + 1, q + 1)} s o p + 1 + + transformieren. 19659025] 1 ) { displaystyle { mathfrak {so}} (p + 1, q + 1)} p { displaystyle p} q {19659077] { displaystyle q geschehen }
o s p 19 ( 2 N | 2 2 ) ] { displaystyle { mathfrak {osp}} ^ {*} (2N | 2,2)} u s p ( 2 2 ) 19 s o ( 4 1 ) { displaystyle { mathfrak {usp}} (2,2) simeq { mathfrak {so}} (4 , 1)} o s s [19659035] p ( N | 4 4] { displaystyle { mathfrak {osp}} (N | 4)} s p ( 4 R ) s o ( 3 2 ) [19659] 010] { displaystyle { mathfrak {sp}} (4, mathbb {R}) simeq { mathfrak {so}} (3,2)} s u ( 2 N | 4 ) { displaystyle { mathfrak {su}} ^ {*} (2N | 4)} s u ∗ (19659025) 4 ] s o ( 5 1 ) { displaystyle { mathfrak {su}} ^ {*} (4) simeq { mathfrak { so}} (5,1)} u ( 2 2 | N ) {19659010] { displaystyle { mathfrak {su}} (2, 2 | N)} s u ( ) 25] 2 2 ) 19 s o (19659025) 4 2 2 ) { displaystyle { mathfrak {su}} (2,2) simeq { mathfrak {so}} (4,2)} s L ( 4 | N ] {19659010] { displaystyle { mathfrak {sl}} (4 | N)} s l ( 4 R ) 19659035] 19659036] 3 3 3 3 3 ]) { displaystyle { mathfrak {sl}} (4, mathbb {R}) simeq { mathfrak {so}} (3,3)} reale Formen von F ( 4 4 ]) { displaystyle F (4)} o 19659035] 8 ] 2 N ) { displaystyle { mathfrak {osp}} (8 ^ {*} | 2N)} in 5 + 1D, dank der Tatsache, dass Spinor und grundlegende Darstellungen von s o (19659025) 8 C ]) { displaystyle { mathfrak {so}} (8, mathbb {C})} Superkonformale Algebra in 3 + 1D [ edit Gemäß [1] [2] der N [1965979]. Supersymmetrien in 3 + 1 Dimensionen werden durch die bosonischen Generatoren P μ { displaystyle gegeben. [displaystyle{mathcal{N}}} D { displaystyle D} M µ ν {19659286] {19659286] { Displaystyle M _ { mu nu }} R-Symmetrie A { Displaystyle A} T j i { displaystyle T_ {j} ^ {i}} Q α i { displaystyle Q ^ { alpha i}} Q i α 19 ] { displaystyle { overline {Q}} _ {i} ^ { dot { alpha}}} S i α { displaystyle S_ {i} ^ { alpha}} α ˙ [19659328] i { displaystyle { overline {S}} ^ {{ dot { alpha}} i}} μ ν ρ (19659010]) nu, rho, dots} α β ,… {193.963] α ˙ β 19 … {19st_ {alpha} }, { dot { beta}}, dots} i j … { displaystyle i, j, dots}
Die Lie-Superbalken der bosonischen -formalen Algebra sind von gegeben
[ M μ [1965921] ν M ρ = = 19659107] ν ρ M µ σ - η ρ M ν σ + η ν M M M μ - η μ M ρ [19456592] displaystyle [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 μ P P P P P P P P P ] = η ν ρ P μ - η μ ρ P ν { displaystyle [M_{mu nu },P_{rho }] = eta _ { nu rho} P _ { mu} - eta _ { mu rho} P _ { nu}} M μ K K K ] = η ν ρ K μ - η μ ρ K ν { displaystyle [M_{mu nu },K_{rho }] = eta _ { nu rho} K _ { mu} - eta _ { mu rho} K _ { nu}} M μ D D D ]] = 0 { displaystyle [M_{mu nu },D] = 0}
[ D P "/> ρ ]. 19659107] - P ρ { displaystyle [D,P_{rho }] = - P _ { rho}} D K "/> ρ ]. 19659107] + K ρ { displaystyle [D,K_{rho }] = + K _ { rho}} [ P μ K ] 19659107] = - 2 M μ [196590373] + 2 ] ν D { displaystyle [P_{mu },K_{nu }] = - 2M _ { mu nu} +2 eta _ { mu nu} D} K 19659511] K m m 19659107] = 0 { displaystyle [K_{n},K_{m}] = 0} P 19659511] P m 19659107] = 0 { displaystyle [P_{n},P_{m}] = 0} wobei η die Minkowski-Metrik ist; Die für die Fermionikgeneratoren sind:
{ Q α i Q β j j j j = 2 δ i j σ & agr; & bgr; & mgr; 19659552] P μ { displaystyle left {Q _ { alpha i}, { overline {Q}} _ { dot { beta}} ^ {j} right } = 2 delta _ {i} ^ {j} sigma _ { alpha { dot { beta}}} ^ { mu} P _ { mu}} { Q Q Q Q ] = 0 {[displaystyle left {Q, Q right ] } = left {{ overline {Q}}, { overline {Q}} right } = 0} { S α i und S β 19 j = 2 δ j. α β 19 µ K µ {[displaystyleleft{S_{alpha}{{}}{overline{S}}_{{dot{beta}}j}right}=2delta_{j}^{i}sigma_{alpha{dot{beta}}}^{mu}K_{mu}} {19659557] S S S S S ] = 0 {[Displaystyle left {S, S right ] } = left {{ overline {S}}, { overline {S}} right } = 0} { Q S ] ] { displaystyle left {Q, S right } =} { Q S ¯ [196596568] ¯ ¯ ¯ ] ] = { Q S } = 0 {[displaystyleleft{Q{overline{S}}right}=left{{overline{Q}}Sright}=0} Die bosonischen Konformgeneratoren tragen keine R-Ladungen, da sie mit den R-Symmetriegeneratoren pendeln:
A M = 19659647] A D D D A P = A K { displaystyle [A,M] = [A,D] = [A,P] = [A,K] = 0} T M ] = = 19659006] T D ] = [ T P T K ] = 0 { displaystyle [T,M] = [T,D] = [T,P] = [T,P] = [T,K] = 0} Aber die Fermionikgeneratoren tragen R-Ladung:
A Q - 1 2 Q Q - { frac {1} {2}} Q} [ A Q ]. 19659715] 1 2 Q { displaystyle [A,{overline {Q}}] = { frac {1} {2}} { overline {Q}}] A S ] = = ]. 19659704] 2 S { displaystyle [A,S] = { frac {1} {2}} S} [ A S ]. 19659107] - 1 2 S { displaystyle [A,{overline {S}}] = - { frac {1} {2}} { overline {S}}} [19659745] = - { frac {1} {2}} overline {S} "/> ] [ T j i Q k = - δ δ ] k i Q j { displaystyle [T_{j}^{i},Q_{k}] = - delta _ {k} ^ {i} Q_ {j}} T j i Q Q 19659312] k ] = δ j k Q i i { Displaystyle [19456534] = _ {j} ^ {k} { overline {Q}} ^ {i}} ] T j i S S S. 19659771]] = δ j k S i { displaystyle [T_{j}^{i},S^{k}] = delta _ {j} ^ {k} S ^ {i}} T j i S S S. 19659312] k ] = - δ k S. j j j j j j ¯ displaystyle [T_{j}^{i},{overline {S}}_{k}] = - delta _ {k} ^ {i} { overline {S}} _ {j}} Unter bosonischen konformen Transformationen transformieren die Fermionikgeneratoren wie folgt:
D Q = 19659107] 1 2 Q - { frac {1} {2}} Q} D Q ]. 19659107] - 1 2 Q { displaystyle [D,{overline {Q}}] = - { frac {1} {2}} { overline {Q}}] [19659842] = - { frac {1} {2}} overline {Q} "/> D S ] = = ]. 19659704] 2 S { displaystyle [D,S] = { frac {1} {2}} S} [ D S ]. 19659715] 1 2 S { displaystyle [D,{overline {S}}] = { frac {1} {2}} { overline {S}}] P Q ] 19659006] P Q ] = 0 {Displaystyle [P,Q] = [P,Q] = 0} K S 19659006] K S ] 0 {Displaystyle [K,S] = [K,{overline {S}}] = 0} = [K,overline {S}] = 0 "/> Superkonformale Algebra in 2D edit Es gibt zwei mögliche Algebren mit minimaler Supersymmetrie in zwei Dimensionen. eine Neveu-Schwarz-Algebra und eine Ramond-Algebra. Eine zusätzliche Supersymmetrie ist möglich, beispielsweise die N = 2-Superkonformalgebra .
Siehe auch [ edit Referenzen [ edit 
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in 3 + 1D dank 
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. Hier 
bezeichnen Raumzeitindizes; 
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![[M_{{mu nu }},P_{rho }] = eta _ {{ nu rho}} P _ { mu} - eta _ {{ mu rho}} P _ { nu}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4625856a3989de9a59b85e66f47d9895e31b8a6c)
![[M_{{mu nu }},K_{rho }] = eta _ {{ nu rho}} K _ { mu} - eta _ {{ mu rho}} K _ { nu}](https://wikimedia.org/api/rest_v1/media/math/render/svg/49e9685cee498faaa78ac46a70a1841c34fcf079)
![[M_{{mu nu }},D] = 0](https://wikimedia.org/api/rest_v1/media/math/render/svg/11b11341d586db686770ca7851f2c5a676ca2867)
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![[A,M] = [A,D] = [A,P] = [A,K] = 0](https://wikimedia.org/api/rest_v1/media/math/render/svg/4833fa63196ae09bc9dbf8019994892ec55e0b21)
![[T,M] = [T,D] = [T,P] = [T,K] = 0](https://wikimedia.org/api/rest_v1/media/math/render/svg/56cd8446ecf271f7cfec74ad82cdad929595fdee)
![[A,Q] = - { frac {1} {2}} Q](https://wikimedia.org/api/rest_v1/media/math/render/svg/69afc1315b121c8e39f9bba64a64275714d83abe)
![[A,overline {Q}] = { frac {1} {2}} overline {Q}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d9133710212899ec561f70e03ff3eb04953c13ea)
![[A,S] = { frac {1} {2}} S](https://wikimedia.org/api/rest_v1/media/math/render/svg/0f6a8645f68a80dbbb500462f5dad836d3d435ec)
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![[T_{j}^{i},{overline {Q}}^{k}] = delta _ {j} ^ {k} { overline {Q}} ^ {i}](https://wikimedia.org/api/rest_v1/media/math/render/svg/17f5defc6a7813508071aaffb7d43ef150691aa7)
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![[D,Q] = - { frac {1} {2}} Q](https://wikimedia.org/api/rest_v1/media/math/render/svg/56ca193e6b9f56ff00a50dad8823f3f2393adb0e)
![[D,S] = { frac {1} {2}} S](https://wikimedia.org/api/rest_v1/media/math/render/svg/d4268043eef612d2898de14e4068ed9d982c82c3)
![[D,overline {S}] = { frac {1} {2}} overline {S}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7093e155ae8b5aa0e8761b473e053f79038c7fd4)
![[P,Q] = [P,overline {Q}] = 0](https://wikimedia.org/api/rest_v1/media/math/render/svg/2a910fbf5011bd0b61125eb3cc1ffbdefd913430)
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