theoretical physics Theoretical physics is a branch of physics that employs mathematical models and abstractions of physical objects and systems to rationalize, explain and predict natural phenomena. This is in contrast to experimental physics, which uses experi ...

, a supersymmetry algebra (or SUSY algebra) is a mathematical formalism for describing the relation between

boson In particle physics, a boson ( ) is a subatomic particle whose spin quantum number has an integer value (0,1,2 ...). Bosons form one of the two fundamental classes of subatomic particle, the other being fermions, which have odd half-integer s ...

s and

fermion In particle physics, a fermion is a particle that follows Fermi–Dirac statistics. Generally, it has a half-odd-integer spin: spin , spin , etc. In addition, these particles obey the Pauli exclusion principle. Fermions include all quarks and ...

s. The supersymmetry algebra contains not only the

Poincaré algebra Poincaré is a French surname. Notable people with the surname include: * Henri Poincaré (1854–1912), French physicist, mathematician and philosopher of science * Henriette Poincaré (1858-1943), wife of Prime Minister Raymond Poincaré * Luci ...

and a compact subalgebra of internal symmetries, but also contains some fermionic supercharges, transforming as a sum of ''N'' real

spinor representation In mathematics, the spin representations are particular projective representations of the orthogonal or special orthogonal groups in arbitrary dimension and signature (i.e., including indefinite orthogonal groups). More precisely, they are two equi ...

s of the

Poincaré group The Poincaré group, named after Henri Poincaré (1906), was first defined by Hermann Minkowski (1908) as the group of Minkowski spacetime isometries. It is a ten-dimensional non-abelian Lie group that is of importance as a model in our und ...

. Such symmetries are allowed by the Haag–Łopuszański–Sohnius theorem. When ''N''>1 the algebra is said to have

extended supersymmetry In theoretical physics, extended supersymmetry is supersymmetry whose infinitesimal generators Q_i^\alpha carry not only a spinor index \alpha, but also an additional index i=1,2 \dots \mathcal where \mathcal is integer (such as 2 or 4). Extend ...

. The supersymmetry algebra is a semidirect sum of a central extension of the

super-Poincaré algebra In theoretical physics, a super-Poincaré algebra is an extension of the Poincaré algebra to incorporate supersymmetry, a relation between bosons and fermions. They are examples of supersymmetry algebras (without central charges or internal symme ...

by a compact

Lie algebra In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an operation called the Lie bracket, an alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow \mathfrak g, that satisfies the Jacobi iden ...

''B'' of internal symmetries.

Bosonic field In quantum field theory, a bosonic field is a quantum field whose quanta are bosons; that is, they obey Bose–Einstein statistics. Bosonic fields obey canonical commutation relations, as distinct from the canonical anticommutation relations obeyed ...

s commute while

fermionic field In quantum field theory, a fermionic field is a quantum field whose quanta are fermions; that is, they obey Fermi–Dirac statistics. Fermionic fields obey canonical anticommutation relations rather than the canonical commutation relations of bo ...

s anticommute. In order to have a transformation that relates the two kinds of fields, the introduction of a Z₂-grading under which the even elements are bosonic and the odd elements are fermionic is required. Such an algebra is called a

Lie superalgebra In mathematics, a Lie superalgebra is a generalisation of a Lie algebra to include a Z2 grading. Lie superalgebras are important in theoretical physics where they are used to describe the mathematics of supersymmetry. In most of these theories, th ...

. Just as one can have representations of a

, one can also have representations of a Lie superalgebra, called supermultiplets. For each Lie algebra, there exists an associated Lie group which is connected and

simply connected In topology, a topological space is called simply connected (or 1-connected, or 1-simply connected) if it is path-connected and every path between two points can be continuously transformed (intuitively for embedded spaces, staying within the spa ...

, unique up to

isomorphism In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word i ...

, and the representations of the algebra can be extended to create

group representation In the mathematical field of representation theory, group representations describe abstract groups in terms of bijective linear transformations of a vector space to itself (i.e. vector space automorphisms); in particular, they can be used t ...

s. In the same way, representations of a Lie superalgebra can sometimes be extended into representations of a

Lie supergroup The concept of supergroup is a generalization of that of group. In other words, every supergroup carries a natural group structure, but there may be more than one way to structure a given group as a supergroup. A supergroup is like a Lie group in ...

Structure of a supersymmetry algebra

The general supersymmetry algebra for spacetime dimension ''d'', and with the fermionic piece consisting of a sum of ''N'' irreducible real spinor representations, has a structure of the form :(''P''×''Z'').''Q''.(''L''×''B'') where *''P'' is a bosonic abelian vector normal subalgebra of dimension ''d'', normally identified with translations of spacetime. It is a vector representation of ''L''. *''Z'' is a scalar bosonic algebra in the center whose elements are called central charges. *''Q'' is an abelian fermionic spinor subquotient algebra, and is a sum of ''N'' real spinor representations of ''L''. (When the signature of spacetime is divisible by 4 there are two different spinor representations of ''L'', so there is some ambiguity about the structure of ''Q'' as a representation of ''L''.) The elements of ''Q'', or rather their inverse images in the supersymmetry algebra, are called supercharges. The subalgebra (''P''×''Z'').''Q'' is sometimes also called the supersymmetry algebra and is nilpotent of length at most 2, with the Lie bracket of two supercharges lying in ''P''×''Z''. *''L'' is a bosonic subalgebra, isomorphic to the Lorentz algebra in ''d'' dimensions, of dimension ''d''(''d''–1)/2 *''B'' is a scalar bosonic subalgebra, given by the Lie algebra of some compact group, called the group of internal symmetries. It commutes with ''P'',''Z'', and ''L'', but may act non-trivially on the supercharges ''Q''. The terms "bosonic" and "fermionic" refer to even and odd subspaces of the superalgebra. The terms "scalar", "spinor", "vector", refer to the behavior of subalgebras under the action of the Lorentz algebra ''L''. The number ''N'' is the number of irreducible real spin representations. When the signature of spacetime is divisible by 4 this is ambiguous as in this case there are two different irreducible real spinor representations, and the number ''N'' is sometimes replaced by a pair of integers (''N''₁, ''N''₂). The supersymmetry algebra is sometimes regarded as a real super algebra, and sometimes as a complex algebra with a hermitian conjugation. These two views are essentially equivalent, as the real algebra can be constructed from the complex algebra by taking the skew-Hermitian elements, and the complex algebra can be constructed from the real one by taking tensor product with the complex numbers. The bosonic part of the superalgebra is isomorphic to the product of the Poincaré algebra ''P''.''L'' with the algebra ''Z''×''B'' of internal symmetries. When ''N''>1 the algebra is said to have extended supersymmetry. When ''Z'' is trivial, the subalgebra ''P''.''Q''.''L'' is the

References

* * {{Industrial and applied mathematics Supersymmetry Lie algebras

Structure of a supersymmetry algebra

See also

References