The trace potential x squared on the one-loop quiver

The trace definition in [F1] gives $W_1(z)=z^2$. Its differential is $d{W}_1=2zdz$; since the coefficient field has characteristic zero, this vanishes exactly at $z=0$, where $W_1(0)=0$. Thus the critical locus and critical value are as stated.

For any sufficiently small $ϵ\mathrm{\ne }0$, the local Milnor fiber $z^2=ϵ$ in a disk about zero consists of the two distinct points $\sqrt{ϵ}$ and $-\sqrt{ϵ}$. Its reduced zeroth cohomology is $\mathbb{Q}$, generated by the difference of the two point classes. When $ϵ$ goes once around zero, the two roots exchange, so this generator maps to its negative. The normalized vanishing-cycle stalk $\mathrm{\Phi }$ is therefore one-dimensional in degree one, with monodromy $-1$, by [F3]. The homological coefficient is its normalized equivariant Verdier dual as in [F1]; it must not be confused with an unshifted Milnor cohomology stalk.

This local calculation agrees with the degree-one normalization of [F3] and, under Thom–Sebastiani, with degree two for $x^2+y^2$. Trace additivity [F2] gives the extension value convention. The full operation package is proved by its separate supplier; this example verifies the specified local fibre, degree and monodromy.