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In theoretical physics, the supersymmetry algebra is a mathematical formalism for describing the relation between bosons and fermions. In a supersymmetric world, every boson would have a partner fermion of equal rest mass. To explore the consequences of this assertion—and to attempt to explain why the present-day world does not appear supersymmetric—physicists and mathematicians have developed an algebraic method for describing the symmetries involved.
SUSY is the acronym preferred for whichever grammatical variation of supersymmetry occurs in a sentence.Consequently, we can speak of a SUSY algebra.
Introduction
Traditional symmetries in physics are generated by objects that transform under the various tensor representations of the Poincaré group. Supersymmetries, on the other hand, are generated by objects that transform under the spinor representations. According to the spin-statistics theorem bosonic fields commute while fermionic fields anticommute. In order to combine the two kinds of fields into a single algebra requires the introduction of a Z2-grading under which the bosons are the even elements and the fermions are the odd elements. Such an algebra is called a Lie superalgebra.
The simplest supersymmetric extension of the Poincaré algebra contains two Weyl spinors with the following anti-commutation relation:
- <math>\{Q_{\alpha}, \bar Q_{\dot{\beta}}\} = 2{\sigma^\mu}_{\alpha\dot{\beta}}P_\mu <math>
and all other anti-commutation relations between the Qs and Ps vanish. In the above expression <math>P_\mu<math> are the generators of translation and <math>\sigma^\mu<math> are the Pauli matrices.
Just as one can have representations of a Lie algebra, one can also have representations of a Lie superalgebra. For each Lie algebra, there exists an associated Lie group which is connected and simply connected. Unique up to isomorphism, this Lie group is canonically associated with the Lie algebra, and the algebra's representations can be extended to create group representations. In the same way, representations of a Lie superalgebra can sometimes be extended into representations of a Lie supergroup.
SUSY in 3 + 1 Minkowski spacetime
In 3+1 Minkowski spacetime, because of the Coleman-Mandula restriction, the SUSY algebra with N spinor generators is as follows.
The even part of the star Lie superalgebra is the direct sum of the Poincaré algebra and a reductive Lie algebra B (such that its self-adjoint part is the tangent space of a real compact Lie group). The odd part of the algebra would be
- <math>(\frac{1}{2},0)\otimes V\oplus(0,\frac{1}{2})\otimes V^*<math>
where <math>(1/2,0)<math> and <math>(0,1/2)<math> are specific representations of the Poincaré algebra. Both components are conjugate to each other under the * conjugation. V is an N-dimensional complex representation of B and V* is its dual representation. The Lie bracket for the odd part is given by a symmetric intertwiner {.,.} from the odd part "squared" to the even part. In particular, its reduced intertwiner from <math>[(\frac{1}{2},0)\otimes V]\otimes[(0,\frac{1}{2})\otimes V^*]<math> to the ideal of the Poincaré algebra generated by translations is given as the product of a nonzero intertwiner from <math>(\frac{1}{2},0)\otimes(0,\frac{1}{2})<math> to (1/2,1/2). The "contraction intertwiner" from <math>V\otimes V^*<math> to the trivial representation and the reduced intertwiner from <math>[(\frac{1}{2},0)\otimes V]\otimes [(\frac{1}{2},0)\otimes V]<math> is the product of a (antisymmetric) intertwiner from (1/2,0) squared to (0,0) and an antisymmetric intertwiner A from <math>N^2<math> to B. * conjugate it to get the corresponding case for the other half.
N = 1
B is now <math>u(1)<math> (called R-symmetry) and V is the 1D representation of <math>u(1)<math> with "charge" 1. A (the intertwiner defined above) would have to be zero since it is antisymmetric.
Actually, there are two versions of N=1 SUSY, one without the <math>u(1)<math> (i.e. B is zero dimensional) and the other with <math>u(1)<math>.
N = 2
B is now <math>su(2)\oplus u(1)<math> and V is the 2D doublet representation of <math>su(2)<math> with a zero <math>u(1)<math> "charge". Now, A is a nonzero intertwiner to the <math>u(1)<math> part of B.
Alternatively, V could be a 2D doublet with a nonzero <math>u(1)<math> "charge". In this case, A would have to be zero.
Yet another possibility would be to let B be <math>u(1)_A\oplus u(1)_B \oplus u(1)_C<math>. V is invariant under <math>u(1)_B<math> and <math>u(1)_C<math> and decomposes into a 1D rep with <math>u(1)_A<math> charge 1 and another 1D rep with charge -1. The intertwiner A would be complex with the real part mapping to <math>u(1)_B<math> and the imaginary part mapping to <math>u(1)_C<math>.
Or we could have B being <math>su(2)\oplus u(1)_A\oplus u(1)_B<math> with V being the doublet rep of <math>su(2)<math> with zero <math>u(1)<math> charges and A being a complex intertwiner with the real part mapping to <math>u(1)_A<math> and the imaginary part to <math>u(1)_B<math>.
This doesn't even exhaust all the possibilities. We see that there is more than one <math>N=2<math> supersymmetry; likewise, the SUSYs for <math>N > 2<math> are also not unique (in fact, it only gets worse).
See also representation of the superPoincaré algebra.
SUSY in various dimensions
In 0+1, 2+1, 3+1, 4+1, 6+1, 7+1, 8+1, 10+1 dimensions, etc., a SUSY algebra is classified by a positive integer N.
In 1+1, 5+1, 9+1 dimensions, etc., a SUSY algebra is classified by two nonnegative integers (M,N), at least one of which is nonzero. M represents the number of left handed SUSYs and N represents the number of right handed SUSYs.
The reason of this has to do with the reality conditions of the spinors.
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