Zero-dimensional supersymmetric field theory

A zero-dimensional model is a finite-dimensional integral with bosonic and Grassmann variables, rather than a field depending on spacetime. Odd graded derivations preserve its action and integration measure. Such models expose supersymmetric Ward identities and supersymmetric localization without spacetime dynamics.

For $P=W^{\prime }$ and $S=|P{|}^2-P^{\prime }{\theta }_1{\theta }_2-\overline{P^{\prime }}{\bar{\theta }}_1{\bar{\theta }}_2$, four odd graded derivations preserve $S$: ${\theta }_1{\partial }_z-\bar{P}{\partial }_{{\theta }_2}$, ${\theta }_2{\partial }_z+\bar{P}{\partial }_{{\theta }_1}$ and their barred counterparts. Left Grassmann derivatives fix the signs.