Super-Poincaré algebra

The Poincaré algebra describes the isometries of Minkowski spacetime. From the representation theory of the Lorentz group, it is known that the Lorentz group admits two inequivalent complex spinor representations, dubbed ${\displaystyle 2}$ and ${\displaystyle {\overline {2}}}$.[nb 1] Taking their tensor product, one obtains ${\displaystyle 2\otimes {\overline {2}}=3\oplus 1}$; such decompositions of tensor products of representations into direct sums is given by the Littlewood–Richardson rule.

Normally, one treats such a decomposition as relating to specific particles: so, for example, the pion, which is a chiral vector particle, is composed of a quark-anti-quark pair. However, one could also identify ${\displaystyle 3\oplus 1}$ with Minkowski spacetime itself. This leads to a natural question: if Minkowski space-time belongs to the adjoint representation, then can Poincaré symmetry be extended to the fundamental representation? Well, it can: this is exactly the super-Poincaré algebra. There is a corresponding experimental question: if we live in the adjoint representation, then where is the fundamental representation hiding? This is the program of supersymmetry, which has not been found experimentally.

The super-Poincaré algebra was first proposed in the context of the Haag–Łopuszański–Sohnius theorem, as a means of avoiding the conclusions of the Coleman–Mandula theorem. That is, the Coleman–Mandula theorem is a no-go theorem that states that the Poincaré algebra cannot be extended with additional symmetries that might describe the internal symmetries of the observed physical particle spectrum. However, the Coleman–Mandula theorem assumed that the algebra extension would be by means of a commutator; this assumption, and thus the theorem, can be avoided by considering the anti-commutator, that is, by employing anti-commuting Grassmann numbers. The proposal was to consider a supersymmetry algebra, defined as the semidirect product of a central extension of the super-Poincaré algebra by a compact Lie algebra of internal symmetries.

The simplest supersymmetric extension of the Poincaré algebra contains two Weyl spinors with the following anti-commutation relation:

${\displaystyle \{Q_{\alpha },{\bar {Q}}_{\dot {\beta }}\}=2{\sigma ^{\mu }}_{\alpha {\dot {\beta }}}P_{\mu }}$

and all other anti-commutation relations between the Qs and Ps vanish.[1] The operators ${\displaystyle Q_{\alpha },{\bar {Q}}_{\dot {\alpha }}}$ are known as supercharges. In the above expression ${\displaystyle P_{\mu }}$ are the generators of translation and ${\displaystyle \sigma ^{\mu }}$ are the Pauli matrices. The index ${\displaystyle \alpha }$ runs over the values ${\displaystyle \alpha =1,2.}$ A dot is used over the index ${\displaystyle {\dot {\beta }}}$ to remind that this index transforms according to the inequivalent conjugate spinor representation; one must never accidentally contract these two types of indexes. The Pauli matrices can be considered to be a direct manifestation of the Littlewood–Richardson rule mentioned before: they indicate how the tensor product ${\displaystyle 2\otimes {\overline {2}}}$ of the two spinors can be re-expressed as a vector. The index ${\displaystyle \mu }$ of course ranges over the space-time dimensions ${\displaystyle \mu =0,1,2,3.}$

It is convenient to work with Dirac spinors instead of Weyl spinors; a Dirac spinor can be thought of as an element of ${\displaystyle 2\oplus {\overline {2}}}$; it has four components. The Dirac matrices are thus also four-dimensional, and can be expressed as direct sums of the Pauli matrices. The tensor product then gives an algebraic relation to the Minkowski metric ${\displaystyle g^{\mu \nu }}$ which is expressed as:

${\displaystyle \{\gamma ^{\mu },\gamma ^{\nu }\}=2g^{\mu \nu }}$

and

${\displaystyle \sigma ^{\mu \nu }={\frac {i}{2}}\left[\gamma ^{\mu },\gamma ^{\nu }\right]}$

This then gives the full algebra[2]

${\displaystyle {\begin{aligned}\left[M^{\mu \nu },Q_{\alpha }\right]&={\frac {1}{2}}(\sigma ^{\mu \nu })_{\alpha }^{\;\;\beta }Q_{\beta }\\\left[Q_{\alpha },P^{\mu }\right]&=0\\\{Q_{\alpha },{\bar {Q}}_{\dot {\beta }}\}&=2(\sigma ^{\mu })_{\alpha {\dot {\beta }}}P_{\mu }\\\end{aligned}}}$

which are to be combined with the normal Poincaré algebra. It is a closed algebra, since all Jacobi identities are satisfied and can have since explicit matrix representations. Following this line of reasoning will lead to supergravity.

Just as the Poincaré algebra generates the Poincaré group of isometries of Minkowski space, the super-Poincaré algebra, an example of a Lie super-algebra, generates what is known as a supergroup. This can be used to define superspace with ${\displaystyle {\mathcal {N}}}$ supercharges: these are the right cosets of the Lorentz group within the ${\displaystyle {\mathcal {N}}}$ super-Poincaré group.

Just as ${\displaystyle P_{\mu }}$ has the interpretation as being the generator of spacetime translations, the charges ${\displaystyle Q_{\alpha }^{I},{\bar {Q}}_{\dot {\alpha }}^{I}}$, with ${\displaystyle I=1,\cdots ,{\mathcal {N}}}$, have the interpretation as generators of superspace translations in the 'spin coordinates' of superspace. That is, we can view superspace as the direct sum of Minkowski space with 'spin dimensions' labelled by coordinates ${\displaystyle \theta _{\alpha }^{I},{\bar {\theta }}^{I{\dot {\alpha }}}}$. The supercharge ${\displaystyle Q_{\alpha }^{I}}$ generates translations in the direction labelled by the coordinate ${\displaystyle \theta _{\alpha }^{I}.}$ By counting, there are ${\displaystyle 4{\mathcal {N}}}$ spin dimensions.

In (3 + 1) Minkowski spacetime, the Haag–Łopuszański–Sohnius theorem states that 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

${\displaystyle \left({\frac {1}{2}},0\right)\otimes V\oplus \left(0,{\frac {1}{2}}\right)\otimes V^{*}}$

where ${\displaystyle (1/2,0)}$ and ${\displaystyle (0,1/2)}$ are specific representations of the Poincaré algebra. (Compared to the notation used earlier in the article, these correspond ${\displaystyle {\overline {2}}\oplus 1}$ and ${\displaystyle 1\oplus 2}$, respectively, also see the footnote where the previous notation was introduced). 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 equivariant pairing {.,.} on the odd part with values in the even part. In particular, its reduced intertwiner from ${\displaystyle \left[\left({\frac {1}{2}},0\right)\otimes V\right]\otimes \left[\left(0,{\frac {1}{2}}\right)\otimes V^{*}\right]}$ to the ideal of the Poincaré algebra generated by translations is given as the product of a nonzero intertwiner from ${\displaystyle \left({\frac {1}{2}},0\right)\otimes \left(0,{\frac {1}{2}}\right)}$ to (1/2,1/2) by the "contraction intertwiner" from ${\displaystyle V\otimes V^{*}}$ to the trivial representation. On the other hand, its reduced intertwiner from ${\displaystyle \left[\left({\frac {1}{2}},0\right)\otimes V\right]\otimes \left[\left({\frac {1}{2}},0\right)\otimes V\right]}$ is the product of a (antisymmetric) intertwiner from ${\displaystyle \left({\frac {1}{2}},0\right)\otimes \left({\frac {1}{2}},0\right)}$ to (0,0) and an antisymmetric intertwiner A from ${\displaystyle N^{2}}$ to B. Conjugate it to get the corresponding case for the other half.

In 0 + 1, 2 + 1, 3 + 1, 4 + 1, 6 + 1, 7 + 1, 8 + 1, and 10 + 1 dimensions, a SUSY algebra is classified by a positive integer N.

In 1 + 1, 5 + 1 and 9 + 1 dimensions, 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.

Hereafter d = 9 means d = 8 + 1 in Minkowski signature, etc. The structure of supersymmetry algebra is mainly determined by the number of the fermionic generators, that is the number N times the real dimension of the spinor in d dimensions. It is because one can obtain a supersymmetry algebra of lower dimension easily from that of higher dimensionality by the use of dimensional reduction.

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