Using the Shapiro’s lemma

One can show that:


In particular, the vertex operators (corresponding to ) are:


This formula may seem mysterious, but it has a transparent physical meaning. Indeed, it implies that the space of solutions of super-Maxwell equations is dual (as a linear space) to the space generated by and its derivatives .

This is what we expect: and its derivatives exhaust the gauge-invariant operators, i.e. gauge invariant linear functionals on the space of Maxwell solutions. Therefore:

This approach leads to the classification of gauge-invariant operators

Notice that automatically satisfies the Dirac equation: