Finite-quiver Hall spectrum generators

Let $Q=(Q_0,Q_1,s,t)$ be a finite acyclic quiver, where $Q_0,Q_1$ are finite sets and $s,t:Q_1\to Q_0$ are the source and target maps. Let $k={\mathbb{F}}_q$ with $q>1$, and assign to each vertex $i\in Q_0$ a charge $z_i\in \mathbb{H}:=\{z\in \mathbb{C}:\mathrm{Im}z>0\}$. A finite-dimensional representation $M$ assigns a finite-dimensional $k$-vector space $M_i$ to each vertex and a linear map $M_a:M_{s(a)}\to M_{t(a)}$ to each arrow; its dimension vector is $\mathbf{d}(M)=({\dim }_k{M}_i{)}_{i\in Q_0}$. Put $Z(\mathbf{d})={\sum }_i{d}_i{z}_i$ and, for nonzero $\mathbf{d}\in {\mathbb{N}}^{Q_0}$, put ${\varphi }_Z(\mathbf{d})=\arg Z(\mathbf{d})\mathrm{/}\pi$ using the principal argument. For $M\mathrm{\ne }0$, set ${\varphi }_Z(M)={\varphi }_Z(\mathbf{d}(M))$. Call $M$ $Z$-semistable when every nonzero subrepresentation has phase at most ${\varphi }_Z(M)$. Include the zero representation as semistable with no assigned phase.

Use the small finitary abelian skeleton of these representations. Its Hall algebra has basis ${\delta }_M$ indexed by isomorphism classes, and the subobject-first convention is $(f\star g)(L)={\sum }_{U\subseteq L}f(U)g(L\mathrm{/}U)$, with distinct embedded subobjects counted separately. Its dimension completion is the product of the finite-dimensional homogeneous components. For integer dimension vectors define the bilinear Euler form and its skew-symmetrization by ${\chi }_Q(\mathbf{d},\mathbf{e})={\sum }_{i\in Q_0}{d}_i{e}_i-{\sum }_{a:i\to j}{d}_i{e}_j,⟨\mathbf{d},\mathbf{e}⟩={\chi }_Q(\mathbf{d},\mathbf{e})-{\chi }_Q(\mathbf{e},\mathbf{d})\mathrm{.}$

For a phase $\varphi$, let $S_Z(\varphi )={\delta }_0+{\sum }_{\begin{array}{c}[M]\mathrm{\ne }[0]\\ MZ\text{-semistable}\\ {\varphi }_Z(M)=\varphi \end{array}}{\delta }_M,$ where ${\delta }_M$ is the characteristic function of the isomorphism class $[M]$ and ${\delta }_0$ is the Hall unit. The Euler and skew integrations are the continuous maps $I(f)={\sum }_{[M]}\frac{f(M)}{\mathrm{\mid }\mathrm{Aut}(M)\mathrm{\mid }}{X}^{\mathbf{d}(M)},J(f)={\sum }_{[M]}{t}^{{\chi }_Q(\mathbf{d}(M),\mathbf{d}(M))}\frac{f(M)}{\mathrm{\mid }\mathrm{Aut}(M)\mathrm{\mid }}{x}^{\mathbf{d}(M)},t=\sqrt{q}>0,$ as proved for this finite-field category by the Euler and skew integration theorem. Their completed targets have bases $X^{\mathbf{d}}$ and $x^{\mathbf{d}}$ and products $X^{\mathbf{d}}{X}^{\mathbf{e}}=q^{-{\chi }_Q(\mathbf{e},\mathbf{d})}{X}^{\mathbf{d}+\mathbf{e}},x^{\mathbf{d}}\ast x^{\mathbf{e}}=t^{-⟨\mathbf{e},\mathbf{d}⟩}{x}^{\mathbf{d}+\mathbf{e}}\mathrm{.}$

Write ${\mathrm{\Phi }}_Z=\{{\varphi }_Z(\mathbf{e}):\mathbf{e}\in {\mathbb{N}}^{Q_0},\mathbf{e}\mathrm{\ne }0\}$ and order its elements decreasingly. Define the Euler and skew Hall spectrum generators by In each product the largest phase is the leftmost factor and multiplication is in the corresponding completed torus. In any fixed total-dimension degree $D$, only dimension vectors with ${\sum }_i{d}_i\le D$ and their finitely many phases can contribute. Thus these are degreewise formal products in the specified completion; no analytic convergence is asserted.

These generators describe the Euler or skew integration of finite-field semistable representation groupoids. Calling such data protected BPS indices, dimensions of physical state spaces, or spectra of a four-dimensional supersymmetric model is an additional dictionary, not part of this mathematical definition.

Finite-field stability data for an acyclic quiver

Subobject-first Hall completion and quantum tori for finite quivers

Euler and skew integration are multiplicative for finite acyclic quivers