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 $a^r_\alpha$,$b^r_\alpha$?


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


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.