Hall factorization, stability chambers and conditional BPS 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

Let $Q$ be any finite acyclic quiver over $k={\mathbb{F}}_q$, with no finite-representation-type restriction, and let $Z_{-}$ and $Z_{+}$ assign a charge in $\mathbb{H}=\{z\in \mathbb{C}:\mathrm{Im}z>0\}$ to every vertex. Use the subobject-first dimension completion and the Euler and skew integrations defined on this A page. Then $P_{Z_{-}}^E=P_{Z_{+}}^E,P_{Z_{-}}^{\mathrm{s}\mathrm{k}}=P_{Z_{+}}^{\mathrm{s}\mathrm{k}}\mathrm{.}$ Each is a unit with constant term $1$ in its own completed quantum torus. Consequently conjugation by the two generators defines the same continuous automorphism on each corresponding completed torus. The products and automorphisms are formal in the dimension completion. This is a finite-field Hall identity only; it makes no physical state-counting identification.

Finite-quiver Hall spectrum generators

Completed Hall wall crossing for finite acyclic quivers

The finite-field generator and wall-crossing equality defined earlier on this A page do not themselves define a Calabi–Yau-threefold Donaldson–Thomas invariant, a cohomological Hall invariant, or a physical BPS index. Define a CY3/CoHA protected-state bridge package to be the complete data below for one specified model:

1. A specified 3-Calabi–Yau category $\mathcal{C}$ over a specified field $k$: a $k$-linear triangulated or dg category with finite-dimensional Hom spaces in the stated subcategory and functorial perfect dualities ${\mathrm{Hom}}_{\mathcal{C}}(E,F[i])\cong {\mathrm{Hom}}_{\mathcal{C}}(F,E[3-i]{)}^{\ast }$ for all stated objects and degrees. Alternatively, for the quiver-with-potential branch using the potential and rational CoHA suppliers below, specify a finite quiver $Q=(Q_0,Q_1,s,t)$, with finite vertex and arrow sets and source and target maps, and its path algebra over $\mathbb{C}$. Give a potential $W$, a finite $\mathbb{C}$-linear combination of directed cycles modulo cyclic rotation. Give the exact Jacobian, derived, or critical-locus category and coefficient data constructed from $(Q,W)$. A different base field or coefficient theory requires suppliers and integration results with that same specified coefficient data.

2. A stability package on a specified abelian category or heart $\mathcal{A}$ of the category in (1); all subobjects, exact sequences, and filtrations below are taken in $\mathcal{A}$. Here $K_0(\mathcal{A})$ is generated by symbols $[E]$ for isomorphism classes of objects, with relation $[E]=[F]+[G]$ for each short exact sequence $0\to F\to E\to G\to 0$. Its numerical charge lattice means a specified finite-rank free abelian group $\mathrm{\Gamma }$ and a class homomorphism $\mathrm{cl}:K_0(\mathcal{A})\to \mathrm{\Gamma }$. If $\mathrm{\Gamma }$ is asserted to be a canonical numerical quotient, specify the pairing and the quotient relation that define it. Give a homomorphism $Z:\mathrm{\Gamma }\to \mathbb{C}$. For the phase convention used here, require $Z(\mathrm{cl}(E))\in \mathbb{H}:=\{z\in \mathbb{C}:\mathrm{Im}z>0\}$ for every nonzero $E\in \mathcal{A}$, and set ${\varphi }_Z(E)=\arg Z(\mathrm{cl}(E))\mathrm{/}\pi \in (0,1)$ using the principal argument. Call $E\mathrm{\ne }0$ semistable if every nonzero subobject $F\subseteq E$ has ${\varphi }_Z(F)\le {\varphi }_Z(E)$. The required Harder–Narasimhan property is that every nonzero $E$ has a finite filtration by subobjects $0=E_0⊊E_1⊊\cdots ⊊E_n=E$ whose nonzero factors $E_j\mathrm{/}{E}_{j-1}$ are semistable and satisfy ${\varphi }_Z(E_1\mathrm{/}{E}_0)>\cdots >{\varphi }_Z(E_n\mathrm{/}{E}_{n-1})$; this filtration is unique as a filtration by subobjects of $E$. State any different phase or semistability convention exactly and verify that it supplies the precise filtration property used. The support property used here is: for a specified norm on ${\mathrm{\Gamma }}_{\mathbb{R}}:=\mathrm{\Gamma }{\otimes }_{\mathbb{Z}}\mathbb{R}$, some constant $C>0$ satisfies $\mathrm{\mid }{Z}_{\mathbb{R}}(\mathrm{ȷ}(\gamma ))\mathrm{\mid }\ge C\mathrm{\parallel }\mathrm{ȷ}(\gamma )\mathrm{\parallel }$ for every nonzero class $\gamma =\mathrm{cl}(E)$ represented by a semistable object, where $\mathrm{ȷ}:\mathrm{\Gamma }\to {\mathrm{\Gamma }}_{\mathbb{R}}$ is $\gamma \mapsto \gamma \otimes 1$ and $Z_{\mathbb{R}}:{\mathrm{\Gamma }}_{\mathbb{R}}\to \mathbb{C}$ is the unique real-linear map satisfying $Z_{\mathbb{R}}(\mathrm{ȷ}(\gamma ))=Z(\gamma )$. Any alternative support-property convention must be stated exactly.

3. The orientation data required by the chosen motivic or cohomological integration theory, when that theory requires it. This means a specified square root of its determinant line, together with the compatibility maps under the exact-sequence operations and the conventions required by that theory; the name alone is not the data.

4. A specified completed motivic or cohomological Hall algebra, its coefficient theory and completion, and a proved continuous integration homomorphism to a named completed quantum torus. The motivic stack-function construction uses convolution along the exact-sequence correspondence: the first factor is the quotient and the second is the subrepresentation. On the quiver-with-potential branch of (1), an untwisted rational equivariant Borel–Moore CoHA choice uses the finite quiver over $\mathbb{C}$ specified there. Its graded groups are ${\mathcal{H}}_{\mathbf{d}}=H_{\ast +2\dim G_{\mathbf{d}}}^{\mathrm{B}\mathrm{M},G_{\mathbf{d}}}({\mathrm{Rep}}_Q(\mathbf{d});\mathbb{Q})$ and the product $\alpha \ast \beta =p_{\ast }{q}^{!}(\alpha ⊠\beta )$, where the first factor is the subrepresentation, the second is the quotient, and the homological shift is $-2{\chi }_Q(\mathbf{b},\mathbf{a})$ when the subrepresentation has dimension $\mathbf{a}$ and the quotient has dimension $\mathbf{b}$. State the exact coefficient map, Euler or skew form, sign, square-root and orientation conventions, and every hypothesis of the integration theorem. This displayed formula is the untwisted theory and has no $W$-dependent vanishing-cycle coefficients. If the quiver-with-potential branch uses $W$-dependent coefficients, select the critical CoHA whose groups and product use the specified vanishing-cycle coefficient datum and coherent exact-sequence operations. Its standard rational datum agrees with the untwisted theory when $W=0$; no identification for nonzero $W$, or integration map from the critical CoHA to a quantum torus, is implicit. That map must be supplied and proved separately as required above. These constructions remain distinct from finite-field groupoid integration.

5. For a physical interpretation, a named four-dimensional supersymmetric model and its protected sector: specify the state space or supertrace being counted, the charges and preserved supersymmetry, and a proved comparison identifying that protected index with the selected algebraic invariant. This comparison is an additional dictionary; it is not supplied by the finite-field factorization.

This definition is a list of required inputs, not an assertion that any bridge package exists or that arbitrary Hall modules are BPS Hilbert spaces. The displayed support property is the convention used here; any cited source using different hypotheses must be translated and checked before use. The rational equivariant Borel–Moore construction and standard quiver-potential coefficient data referenced below assume the Axiom of Choice; this contract records that supplier assumption and does not use it to establish a finite-field Hall result.

Finite-quiver Hall spectrum generators

Hall wall crossing preserves the completed spectrum generators

The Axiom of Choice

Quiver potentials and realized monodromic coefficient data

The motivic Hall algebra of a finite quiver

The Euler-twisted dimension-vector algebra of a quiver

The untwisted equivariant Borel–Moore CoHA of a finite quiver

Critical quiver CoHA and its standard rational realization