Quiver potentials and realized monodromic coefficient data

Assume AC as inherited from the equivariant constructible category. Let $Q$ be a finite complex quiver. A potential is a finite $\mathbb{C}$-linear combination of directed cycles in $\mathbb{C}Q$, modulo cyclic rotation. For $h_1:i_0\to i_1,\dots ,h_{\mathrm{\ell }}:i_{\mathrm{\ell }-1}\to i_0$, its trace on a representation $X$ is $\mathrm{tr}(X_{h_{\mathrm{\ell }}}\cdots X_{h_1})$. Extending linearly defines the invariant regular function $W_{\mathbf{d}}$ on $X_{\mathbf{d}}={\mathrm{Rep}}_Q(\mathbf{d})$. Its critical locus is the zero scheme of $d{W}_{\mathbf{d}}$.

An on-shell monodromic homological coefficient datum consists of the following. For each dimension vector choose $S_{\mathbf{d}}\subset \mathbb{C}$ containing its critical values, with $S_{\mathbf{a}}+S_{\mathbf{b}}\subset S_{\mathbf{a}+\mathbf{b}}$. One may take every $S_{\mathbf{d}}=\mathbb{C}$. For every value supply a bounded equivariant constructible complex $K_{\mathbf{d},z}$ on the representation action nerve, supported on $\mathrm{Crit}(W_{\mathbf{d}})\cap W_{\mathbf{d}}^{-1}(z)$, with an invertible monodromy automorphism. These are PRIMARY HOMOLOGICAL coefficients in the intrinsic grading of Intrinsic Borel–Moore grading on quiver quotient stacks. They are actual Cartesian complexes on the indicated quotient presentations; no full six-functor category on arbitrary Artin stacks is presumed.

For the subobject-first correspondence $B={\mathfrak{M}}_{\mathbf{a}}\times {\mathfrak{M}}_{\mathbf{b}}\stackrel{q}{\leftarrow }E={\mathfrak{E}}_{\mathbf{a},\mathbf{b}}\stackrel{p}{\to }{\mathfrak{M}}_{\mathbf{a}+\mathbf{b}}$ and $z=z_1+z_2$, put $r=-{\chi }_Q(\mathbf{b},\mathbf{a})$. Supply a coefficient $K_{E,z_1,z_2}$, a smooth oriented identification and a proper coefficient morphism $q^{\ast }(K_{\mathbf{a},z_1}⊠K_{\mathbf{b},z_2})[2r]\simeq K_{E,z_1,z_2},R{p}_{\ast }{K}_{E,z_1,z_2}⟶K_{\mathbf{a}+\mathbf{b},z}\mathrm{.}$ Here the smooth operation, including the nonrepresentable map $q$, is interpreted through the affine-bundle/Levi and induced fixed-group frame models; the datum includes compatibility with those identifications. Require these maps to preserve monodromy and external products, to obey proper composition and proper/smooth Cartesian pasting, and to give the same identifications on every iterated block flag under the tensor associator. Require $K_{\mathbf{0},0}=\mathbb{Q}$ and the identity maps in both endpoint correspondences; dimension-zero coefficients at other values are zero. These are explicit coherence requirements for an arbitrary supplied datum, not an existence axiom for vanishing cycles. They retain arbitrary coherent coefficient complexes. In particular a choice $\omega \otimes \mathrm{\Phi }$ remains admissible whenever its required operations actually exist; it is not silently identified with $D\mathrm{\Phi }$ for arbitrary $\mathrm{\Phi }$.

The standard rational instance uses the normalized functor of Normalized rational vanishing cycles have coherent correspondence operations: ${\mathrm{\Phi }}_{\mathbf{d},z}=V_{W_{\mathbf{d}}-z}{\mathbb{Q}}_{X_{\mathbf{d}}},K_{\mathbf{d},z}=D_{X_{\mathbf{d}}}^{G_{\mathbf{d}}}{\mathrm{\Phi }}_{\mathbf{d},z}[-2\dim G_{\mathbf{d}}]\mathrm{.}$ The equivariant duality here is VARIETY normalized; the displayed extra shift is indispensable. The positive-loop automorphism is $T$ on $\mathrm{\Phi }$, and the homological coefficient has contragredient transport $D(T^{-1})$. For an isolated critical point, $H^k(\mathrm{\Phi }{)}_x={\tilde{H}}^{k-1}(F_x;\mathbb{Q})$; reduced $H^0$ is in degree one. This is consistent with $V_0\mathbb{Q}=\mathbb{Q}$ and the unit. For example $x^2$ on ${\mathbb{A}}^1$ has its one-dimensional stalk in degree one, with monodromy $-1$.

On the extension space, Thom–Sebastiani identifies the intrinsic vanishing cycles of the RESTRICTED trace function with the smooth pullback of the two diagonal coefficients. Proper functoriality supplies a morphism from the total-dimension vanishing cycles to the proper image of that extension coefficient; its dual gives the displayed homological coefficient morphism. There is no asserted isomorphism to the inverse image of the total-dimension coefficient. The full standard realization, its degree and coherent operations are proved in Standard rational quiver vanishing cycles realize the critical coefficient package ↗, recorded in justified_by. Mixed Hodge enrichment and off-shell rapid-decay cohomology are not part of this rational construction.