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Rescale $W\mapsto tW$ with $t>0$. After Berezin integration, the unnormalized holomorphic function insertion isIts derivative is an ordinary total derivative:The last equality again follows from polynomial growth and exponential decay. Thus the insertion is independent of $t$, and supersymmetric localization allows evaluation as $t\to \infty$ at the zeros of $P$, as in localization of a zero-dimensional polynomial model.

For a simple zero $z_a$, $P(z)=P^{\prime }(z_a)(z-z_a)+O((z-z_a{)}^2)$. The local Gaussian integral cancels the factor $|P^{\prime }(z_a){|}^2$ and contributes $\pi f(z_a)$. For a zero with multiplicity of a root $m_a$, write $P(z)=a_a(z-z_a{)}^{m_a}+\cdots$. Its leading radial integral isThe insertion tends to $f(z_a)$ there; contributions away from the zeros vanish. Therefore the full answer, including degenerate critical points, isThis is an unnormalized expectation. For simple zeros all $m_a=1$; division by $Z=\pi r$ would give the normalized average, but is not part of the requested insertion. Taking $f=1$ recovers the partition function, and taking $f=Pg$ recovers its vanishing supersymmetric Ward identity.

The odd symmetries of a zero-dimensional polynomial model preserve both the action and the flat integration measure: their coefficients have zero superdivergence. The confining polynomial weight removes boundary terms at infinity. Thus the integral of an odd-symmetry derivative is zero, a supersymmetric Ward identity for this finite-dimensional integral in this zero-dimensional supersymmetric field theory.

Since $g(z)$ is a holomorphic function, the barred symmetry obeys ${\overline{V}}_1({\overline{\theta }}_{2}g(z))=-P(z)g(z)$. Applying the supersymmetric Ward identity gives the requested unnormalized insertion:One can also verify the result after Berezin integration, without invoking the symmetry terminology:The final integration by parts uses ${\partial }_{\bar{z}}(g{P}^{\prime })=0$ and rapid decay. The derivative on $W^{\prime }$ in the insertion is essential.

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.

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.