BPS bound

2.3 BPS bound

The general $𝒩 = 2$ supersymmetry algebra has the following form

\begin{align} {Q_{α}^{I}, Q^{†}_{\dot{β}}^{\bar{J}}} & = δ^{I \bar{J}} P_{μ} σ_{α \dot{β}}^{μ}, & (2.3.1) \\ {Q_{α}^{I}, Q_{β}^{J}} & = 𝜖^{I J} 𝜖_{α β} Z . & (2.3.2) \end{align}

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

P_{μ} = (M, 0, 0, 0) .

(2.3.3)

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

^{(φ)} Q_{α} = \frac{1}{\sqrt{2}} (Q_{α}^{1} + e^{- i φ} σ^{0}_{α}^{\dot{β}} Q^{†}_{\dot{β}}^{2})

(2.3.4)

for which we have

{^{(φ)} Q_{α},^{(φ)} Q_{β}^{†}} = δ_{α β} (M - R e (e^{- i φ} Z)) .

(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^{†} | ψ ⟩|}^{2} + {|a | ψ ⟩|}^{2} = ⟨ ψ | a a^{†} | ψ ⟩ + ⟨ ψ | a^{†} a | ψ ⟩ = c ⟨ ψ | ψ ⟩,

(2.3.6)

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

M \geq R e (e^{- i φ} Z)

(2.3.7)

for all $φ$ . Choosing $φ = A r g Z$ , we ﬁnd the inequality

M \geq | Z | .

(2.3.8)

In general, the multiplet of the supertranslations $Q_{α}^{I}$ and $Q^{J}_{α}^{†}$ generates $2^{4} = 16$ states in the supermultiplet. When the inequality (2.3.8) is saturated, $c$ in the equation (2.3.6) for $a_{α} =^{(A r g Z)} Q_{α}$ is zero, forcing the operators $^{(A r g Z)} Q_{α}$ themselves to vanish. Then the supertranslations only generate $2^{2} = 4$ 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 $f_{i}$ . We deﬁne the coeﬃcients appearing in the linear combination to be $a$ , $a_{D}$ and $μ_{i}$ in the quantum theory:

Z = n a + m a_{D} + \sum_{i} μ_{i} f_{i} .

(2.3.9)

When the theory is weakly-coupled, we can identify $a$ to be the diagonal entry of the ﬁeld $Φ$ , $a_{D}$ to be $2 τ a$ , and $μ_{i}$ 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 $a$ .

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