Supersymmetric localization

A symmetry-preserving deformation can leave protected observables unchanged while concentrating their integral on a smaller critical locus. Supersymmetric Ward identities justify the deformation; convergence and boundary terms must be controlled.

With $P=W^{\prime }$ a nonconstant polynomial, positive area measure and Berezin integration oriented positively, a holomorphic polynomial insertion in $S=|P{|}^2-P^{\prime }{\theta }_1{\theta }_2-\overline{P^{\prime }}{\bar{\theta }}_1{\bar{\theta }}_2$ equals $\pi {\sum }_a{m}_{a}f(z_a)$, where $P(z_a)=0$ and $m_a$ is the multiplicity of a root. Rescaling $P$ by $t>0$ leaves the insertion fixed and localizes it as $t\to \infty$.