Super-Poincaré Algebra: Is There a Case for Nontrivial Geometry in the Bosonic Sector?


In one dimension there is no Lorentz group and therefore all bosonic and fermionic fields have no space-time indices. The simplest free action for one bosonic field φ and one fermionic field ψ reads

S = γ∫dt [φ.2 – i/2ψψ.] —– (1)

We treat the scalar field as dimensionless and assign dimension cm−1/2 to fermions. Therefore, all our actions will contain the parameter γ with the dimension [γ] = cm. (1) provides the first example of a supersymmetric invariant action and is invariant w.r.t. the following transformations:

δφ = −iεψ, δψ = −εφ ̇ —– (2)

The infinitesimal parameter ε anticommutes with fermionic fields and with itself. What is really important about transformations (2) is their commutator

δ2δ1φ = δ2(−ε1ψ) = iε1ε2φ ̇

δ1δ2φ = iε2ε1φ ̇ ⇒ [δ2, δ1] φ = 2iε1ε2φ ̇ —– (3)

Thus, from (3) we may see the main property of supersymmetry transformations: they commute on translations. In our simplest one-dimensional framework this is the time translation. This property has the followin form in terms of the supersymmetry generator Q:

{Q, Q} = −2P —– (4)

The anticommutator (4), together with

[Q, P] = 0 —– (5)

describe N = 1 super-Poincaré algebra in d = 1. The structure of N-extended super-Poincaré algebra includes N real super-charges QA , A = 1, . . . , N with the following anti commutators:

{QA, QB} = −2δABP, [QA, P] = 0 —– (6)

Let us stress that the reality of the supercharges is very important, as well as having the same sign in the r.h.s. of QA, QB ∀ QA.

From (2) we see that the minimal N = 1 supermultiplet includes one bosonic and one fermionic field. A natural question arises: how many components do we need, in order to realize the N-extended superalgebra (6)? In order to mimic the transformations (2) for all N supertranslations

δφi = −iεA(LA)i ψ, δψ = −εA (RA)iφ ̇i —– (7)

Here the indices i = 1,…,db and iˆ = 1,…,df count the numbers of bosonic and fermionic components, while (LA)i and (RA)i are N arbitrary, for the time being, matrices. The additional conditions one should impose on the transformations (7) are

  • they should form the N-extended superalgebra (6)
  • they should leave invariant the free action constructed from the involved fields.

When N > 8 the minimal dimension of the supermultiplets rapidly increases and the analysis of the corresponding theories becomes very complicated. For many reasons, the most interesting case seems to be the N = 8 supersymmetric mechanics. Being the highest N case of minimal N-extended supersymmetric mechanics admitting realization on N bosons (physical and auxiliary) and N fermions, the systems with eight supercharges are the highest N ones, among the extended supersymmetric systems, which still possess a nontrivial geometry in the bosonic sector. When the number of supercharges exceeds 8, the target spaces are restricted to be symmetric spaces.

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