$\mathcal{N} \ge 2$ Supersymmetry massive supermultiplets

Question

In Bertolinis SUSY notes [https://people.sissa.it/~bertmat/susycourse.pdf] we have defined: $$ \{Q^I_\alpha,\bar{Q}_\dot{\beta}^J\}=2m\delta_{\alpha\dot{\beta}}\delta^{IJ}\tag{3.24} $$ $$ \{Q^I_\alpha,Q^J_\beta\}=\epsilon_{\alpha\beta}Z^{IJ} $$ $$ \{\bar{Q}_{I\dot{\alpha}},\bar{Q}_{J\dot{\beta}}\}=\epsilon_{\dot{\alpha}\dot{\beta}}\bar{Z}_{IJ} $$

Also we define: $$ a_\alpha=\frac{1}{\sqrt{2m}}Q_\alpha\;\;\;\;\;a_\dot{\alpha}^\dagger=\frac{1}{\sqrt{2m}}\bar{Q}_\dot{\alpha} $$ Lastly, the central charges $Z^{IJ}$ can be written in the form $$Z^{IJ}= \left(\matrix{0 & Z_1 \\-Z_1 & 0\\&&0&Z_2\\&&-Z_2&0\\&&&&\ddots\\&&&&&0&Z_{\mathcal{N/2}}\\&&&&& -Z_{\mathcal{N}/2}&0\\} \right)\tag{3.28} $$

(Where the charges are non-zero only for even $\mathcal{N}$)

From these we define the following:$$ a^r_\alpha=\frac{1}{\sqrt{2}}\left(Q_\alpha^{2r-1}+\epsilon_{\alpha\beta}(Q_\beta^{2r})^\dagger\right) $$ $$ b^r_\alpha=\frac{1}{\sqrt{2}}\left(Q_\alpha^{2r-1}-\epsilon_{\alpha\beta}(Q_\beta^{2r})^\dagger\right) $$ where $r= 1,\dots,\mathcal{N}/2$

These equations satisfy the oscillator algebra: $$ \{a^r_\alpha,(a^s_\beta)^\dagger\}=(2m+Z_r)\delta_{rs}\delta_{\alpha\beta} $$ $$ \{b^r_\alpha,(b^s_\beta)^\dagger\}=(2m-Z_r)\delta_{rs}\delta_{\alpha\beta} $$

How does one "see" that we need to define those equations for $a^r_\alpha$,$b^r_\alpha$?

Answer 1

The point is to devise an algorithm that constructs supermultiplets given only the commutation relations.

Since you have a nice basis of harmonic oscillator esque creation/annihilation operators, you can define a supermultiplet by postulating a vacuum state $|s\rangle$ annihilated by $a,b$, so that every state in the supermultiplet is given by hitting $|s\rangle$ with creation operators $a^\dagger,b^\dagger$. Since the $a^\dagger$ and $b^\dagger$ are all fermionic the procedure will terminate at a finite number of steps, leading to a finite-dimensional supermultiplet.

You get different supermultiplets by postulating different $|s\rangle$s. You can label them by Poincare casimirs like helicity, mass squared etc. and central charge