Motivic Hall scattering data for a quiver with potential

Fix quiver-with-potential data $(Q,W)$, with the finite quiver and Jacobian algebra conventions stated earlier on this page. Put $N={\mathbb{Z}}^{Q_0}$, $N^{\oplus }={\mathbb{N}}^{Q_0}$, $N^{+}=N^{\oplus }\setminus \{0\}$, $M={\mathrm{Hom}}_{\mathbb{Z}}(N,\mathbb{Z})$, and $M_{\mathbb{R}}=M{\otimes }_{\mathbb{Z}}\mathbb{R}$. For $d\in N^{\oplus }$, let ${\mathrm{Rep}}_d(J)$ be the finite-type scheme of collections of matrices for the arrows satisfying the Jacobian relations and having dimension vector $d$; its representation stack is ${\mathfrak{M}}_d=[{\mathrm{Rep}}_d(J)\mathrm{/}{\prod }_{i\in Q_0}{\mathrm{GL}}_{d_i}(\mathbb{C})]$. Let ${\mathfrak{M}}_d^{ss}(\theta )$ be the substack whose geometric points are the $\theta$-semistable representations under the King convention fixed earlier on this page.

Let $e_i$ be the standard basis of $N$. The source skew-symmetric form is the bilinear form determined by $⟨e_i,e_j⟩=\mathrm{\#}\{a:j\to i\}-\mathrm{\#}\{a:i\to j\}$, and the associated Poisson bracket on the monoid algebra is $\{x^{d_1},x^{d_2}\}=⟨d_1,d_2⟩x^{d_1+d_2}$.

For a stack $M$ locally of finite type over $\mathbb{C}$ with affine stabilizers, let $K(\mathrm{S}\mathrm{t}\mathrm{/}M)$ be the abelian group generated by isomorphism classes $[X\to M]$ of finite-type Artin stacks with affine stabilizers, modulo the closed-substack/open-complement relation and the relation identifying Zariski locally trivial fibrations with the same fibers. It is a module over $K(\mathrm{S}\mathrm{t}\mathrm{/}\mathbb{C})$. The source homomorphism $\mathrm{{\rm Y}}:K(\mathrm{S}\mathrm{t}\mathrm{/}\mathbb{C})\to \mathbb{C}(t)$ sends a smooth projective complex variety $Y$ to its Poincar'e polynomial ${\sum }_i{\dim }_{\mathbb{C}}{H}^i(Y^{\mathrm{a}\mathrm{n}},\mathbb{C})t^i$. Set $K_{\mathrm{{\rm Y}}}(\mathrm{S}\mathrm{t}\mathrm{/}M)=K(\mathrm{S}\mathrm{t}\mathrm{/}M){\otimes }_{K(\mathrm{S}\mathrm{t}\mathrm{/}\mathbb{C})}\mathbb{C}(t)$, with the additional relation $[Y\times X\to M]=\mathrm{{\rm Y}}([Y])[X\to M]$.

The source motivic Hall algebra $H(Q,W)$ is the graded $\mathbb{C}(t)$-algebra with degree-$d$ part $K_{\mathrm{{\rm Y}}}(\mathrm{S}\mathrm{t}\mathrm{/}{\mathfrak{M}}_d)$ and convolution product along the stack of short exact sequences $0\to E_1\to E\to E_2\to 0$. Its degree completion is the inverse limit over quotients retaining only vectors $d$ with $\delta (d):={\sum }_{i\in Q_0}{d}_i\le r$, as $r\to \mathrm{\infty }$. The source regular coefficient ring is

and $H_{\mathrm{r}\mathrm{e}\mathrm{g}}(Q,W)$ is the ${\mathbb{C}}_{\mathrm{r}\mathrm{e}\mathrm{g}}(t)$-subalgebra generated by classes $[X\to {\mathfrak{M}}_d]$ with $X$ a variety. Its source Poisson bracket is $\{a,b{\}}_{\mathrm{H}\mathrm{a}\mathrm{l}\mathrm{l}}=(t^2-1{)}^{-1}[a,b]$, where $[a,b]=a\ast b-b\ast a$ is the Hall commutator. The positive-degree completion ${\widehat{\mathfrak{g}}}_{\mathrm{H}\mathrm{a}\mathrm{l}\mathrm{l}}=\widehat{H}(Q,W{)}_{>0}$ is a pronilpotent Lie algebra under the commutator; its pro-unipotent group is ${\widehat{G}}_{\mathrm{H}\mathrm{a}\mathrm{l}\mathrm{l}}=1+\widehat{H}(Q,W{)}_{>0}$.

A scattering diagram with values in the completed Hall Lie group consists of rational codimension-one walls in $M_{\mathbb{R}}$, each contained in $n^{\perp }=\{\theta :\theta (n)=0\}$ for a primitive $n\in N^{+}$, and carrying an element of $\exp ({\prod }_{k\ge 1}\widehat{H}(Q,W{)}_{kn})$. The degree completion means that only finitely many walls of degree at most $r$ meet a compact set, for each finite truncation degree $r$. For a generic piecewise-smooth path transverse to walls and disjoint from joints, its path-ordered product is the ordered product of the wall labels in crossing order, using the inverse label when the path crosses a wall in the reverse orientation. The diagram is consistent when every generic closed path has identity path-ordered product modulo every finite degree truncation. Equivalence means equality of all such generic path products.

For a weight $\theta$, let ${\mathbf{1}}_d^{ss}(\theta )=[{\mathfrak{M}}_d^{ss}(\theta )\to {\mathfrak{M}}_d]$ and ${\mathbf{1}}^{ss}(\theta )={\sum }_{d\in N^{\oplus }}{\mathbf{1}}_d^{ss}(\theta )$ in the degree completion, with the zero-dimensional term equal to the Hall unit. It has the source logarithmic factorization

where each $ϵ(d,\theta )\in H_{\mathrm{r}\mathrm{e}\mathrm{g}}(Q,W{)}_d$ is the degree-$d$ term. Terms with no $\theta$-semistable representations are zero. For a finite $2$-acyclic quiver with finite potential, the source integration homomorphism is defined with the Euler specialization ${\mathrm{ev}}_{-1}:{\mathbb{C}}_{\mathrm{r}\mathrm{e}\mathrm{g}}(t)\to \mathbb{C}$, $t\mapsto -1$. This map is well-defined because every inverted polynomial $1+t^2+\cdots +t^{2k}$ takes the nonzero value $k+1$ at $-1$. Moreover, the stated positive-Betti Poincar'e polynomial satisfies $\mathrm{{\rm Y}}(Y){\mathrm{\mid }}_{t=-1}={\sum }_i(-1{)}^i\dim H^i(Y^{\mathrm{a}\mathrm{n}},\mathbb{C})=e(Y)$ for smooth projective $Y$, so this specialization respects the coefficient relation. After this specialization the integration map $I:H_{\mathrm{r}\mathrm{e}\mathrm{g}}(Q,W)\to \mathbb{C}[N^{\oplus }]$ sends $[X\to {\mathfrak{M}}_d]$ to $e(X)x^d$, where $e(X)$ is the topological Euler characteristic; for a finite-type complex variety it agrees with compactly supported Euler characteristic. Theorem 11.1 and §11.3 of the cited source establish that, under these potential and $2$-acyclic hypotheses, this is a Poisson homomorphism for the bracket above (the source uses a coefficient parameter sent to $1$, whereas the present positive-Betti Poincar'e parameter is sent to $-1$). Thus $J(d,\theta ):=e(ϵ(d,\theta ))$ is the source Joyce invariant and the integrated wall element at $\theta$ is $\exp ({\sum }_{d\in N^{+}:\theta (d)=0}J(d,\theta )x^d)$. This normalization is not an arbitrary numerical or vanishing-cycle DT invariant. This is a definition of data and consistency; it does not assert that a consistent diagram exists.

Quiver potential and King theta stability data