2.3 BPS bound

The general 𝒩=2 supersymmetry algebra has the following form

{QαI,Qβ̇J̄} = δIJ̄Pμσαβ̇μ, (2.3.1) {QαI,QβJ} = 𝜖IJ𝜖αβZ. (2.3.2)

Here I = 1, 2 are the index distinguishing two supersymmetry generators, and Z is a complex quantity which commutes with everything. Let us take the coordinate system where

Pμ = (M, 0, 0, 0). (2.3.3)

This choice breaks the Lorentz symmetry SO(3, 1) to the spatial rotation SO(3), which allows us to identify the undotted and the dotted spinor indices. Let us then define

(φ)Qα = 1 2(Qα1 + eiφσ0αβ̇Qβ̇2) (2.3.4)

for which we have

{(φ)Qα,(φ)Qβ} = δαβ(M Re(eiφZ)). (2.3.5)

In general, if there is an operator a satisfying {a,a} = c with a constant c, c is necessarily non-negative. Indeed, take a ket vector |ψ then

a|ψ2 + a|ψ2 = ψ|aa|ψ + ψ|aa|ψ = cψ|ψ, (2.3.6)

meaning that c 0. From (2.3.5), then, we see

M Re(eiφZ) (2.3.7)

for all φ. Choosing φ = ArgZ, we find the inequality

M |Z|. (2.3.8)

In general, the multiplet of the supertranslations QαI and QJα generates 24 = 16 states in the supermultiplet. When the inequality (2.3.8) is saturated, c in the equation (2.3.6) for aα = (ArgZ)Qα is zero, forcing the operators (ArgZ)Qα themselves to vanish. Then the supertranslations only generate 22 = 4 states. Such multiplets are called BPS, and those multiplets with 16 states under the action of supertranslations are called non-BPS. A BPS state is rather robust: under a generic perturbation, the number of states in a multiplet can not jump. Therefore the BPS state will generically stay BPS.

What is this quantity Z, which commutes with everything? A quantity commuting with everything is by definition a conserved charge. When the low-energy theory is a weakly-coupled U(1) gauge theory, Z is a linear combination of the electric charge n, the magnetic charge m, and the flavor charges fi. We define the coefficients appearing in the linear combination to be a, aD and μi in the quantum theory:

Z = na + maD + iμifi. (2.3.9)

When the theory is weakly-coupled, we can identify a to be the diagonal entry of the field Φ, aD to be 2τa, and μi to be the coefficients of the mass terms in the Lagrangian, by comparing the quantum BPS mass formula (2.3.8) and its classical counterpart (2.2.16). In the strongly-coupled regime, there is no meaning in saying that a is the diagonal entry of a gauge-dependent field Φ. Rather, we should think of (2.3.9) as the definition of the quantity a.