CY3, CoHA, and protected-state extension contract

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