The Bethe/Gauge Correspondence

1.2. Gauge: supersymmetry 11Because of this, the θ a are called fermionic, or odd, coordinates; the (commuting) x µ are bosonicor even. The proper mathematical way to think about superspace, denoted by M 3,1|4 , is in termsof supermanifolds. Much more about supermanifolds can be found in [35]; see also §2 of [36].Supersymmetry algebra. The superspace symmetries are again made up of two parts: spacetimesymmetries and supersymmetries. The former is described by the Poincaré algebra (1.20)as before, and we have met the supersymmetries, or supercharges, Q too. They are the generatorsof translations in the ‘super directions’, very much like the momentum operators P µ generateordinary spacetime translations. On superspace, the supersymmetries have four components Q a .We’ll shortly see how these components are related to the θ a .Because of their fermionic nature, the Q a are characterized by their anticommutation relations.This can be easily seen as follows. As always, supersymmetry transformations δ ε arebosonic. They act on a field Φ asδ ε Φ = ε a Q a · Φ ,where ε a is a (spinorial) parameter that specifies in which (infinitesimal) ‘super direction’ wetransform the field. Thus, ε a and Q a anticommute, and we compute for the commutator of twosupersymmetry transformations, with parameters ε a and ζ b ,[ δ ε , δ ζ ] · Φ = [ ε a Q a , ζ b Q b ] · Φ= ε a Q a · (ζ b Q b · Φ) − ζ b Q b · (ε a Q b · Φ)= −ε a ζ b (Q a Q b + Q b Q a ) · Φ= −ε a ζ b {Q a , Q b } · Φ .The anticommutator of the Q a is given by the supersymmetry algebra 5{Q a , Q b } = 2 (γ µ ) ab P µ . (1.21)Thus, the difference between applying two supersymmetry transformations in one order and theother is proportional to an ordinary translation.The fact that Q a is a spinor is exhibited by the action of the generators of so(3, 1)[ M µν , Q a ] = (Σ µν ) a b Q b , (1.22)with Σ µν in the spinor representation (1.19). This relation and[ P µ , Q a ] = 0 (1.23)are the compatibility relations which allow us to combine the Poincaré algebra (1.20) and (1.21)into the Poincaré superalgebra iso(3, 1|4). (Unsurprisingly, the mathematical setting of thePoincaré superalgebra is that of Lie superalgebras; again, see [35].)Some of the nice features of supersymmetric theories directly follow from the supersymmetryalgebra. For example:• Any two states related by supersymmetry must have the same mass. Indeed, (1.23) immediatelyimplies [P 2 , Q] = 0. (This is how we know that supersymmetry must be brokenif it occurs in nature: otherwise we would have already found the superpartners of theparticles in the Standard Model.)• The vacuum energy in a supersymmetric theory vanishes. In terms of Feynman diagrams,this can be understood as follows. For each loop diagram in the non-supersymmetrictheory, there now is a corresponding loop diagram with the superpartner. Recall thatfermion loops always get a minus sign. It’s this sign that causes the two diagrams tocancel.5 The γ µ appears because we use the notation θ a for four-component spinors; see also the remark aboutnotation in the Preface. In the Weyl basis (0.2) the supersymmetry algebra (1.21) directly gives {Q α, ¯Q ˙α } =2 (σ µ ) α ˙α P µ.