Conditional physical BPS wall crossing from an exact Hall bridge

Assume ${\mathrm{A}\mathrm{C}}_{\omega }$ as required by the supplied trace-class BPS-index data. Use the selected pure four-dimensional rigid $\mathcal{N}=2$ Yang–Mills model with global gauge group $SU(2)$ and no microscopic matter hypermultiplets, in natural units $c=\mathrm{\hslash }=1$. On a simply connected regular patch $U$, the local periods $a,a_D:U\to \mathbb{C}$ are holomorphic and satisfy $\mathrm{Im}(\mathrm{\partial }{a}_D\mathrm{/}\mathrm{\partial }a)>0$. Its formal charge-label lattice is ${\mathrm{\Gamma }}_{\mathrm{S}\mathrm{W}}={\mathbb{Z}}^2$, with $z_u(n_e,n_m)=\sqrt{2}(n_{e}a(u)+n_m{a}_D(u)),⟨(n_e,n_m),(n_e^{\mathrm{\prime }},n_m^{\mathrm{\prime }}){⟩}_{\mathrm{D}\mathrm{S}\mathrm{Z}}=2(n_m{n}_e^{\mathrm{\prime }}-n_e{n}_m^{\mathrm{\prime }})\mathrm{.}$ For each supplied stable one-particle BPS rest sector, ${\mathrm{\Omega }}_u(\gamma )$ is the second-helicity trace $-\frac{1}{2}{\mathrm{tr}}_{E_{u,\gamma }}\text{\hspace{0.17em}⁣}((-1{)}^{2J_3}(2J_3{)}^2)$ from the physical charge-data definition; it is set to zero off the selected stable support. Let $u_{-},u_{+}\in U$ be two specified points.

The finite-field Hall input is a finite acyclic quiver $Q$ over ${\mathbb{F}}_q$, where $q>1$ is a prime power. For integer dimension vectors, define ${\chi }_Q(\mathbf{d},\mathbf{e})={\sum }_{i\in Q_0}{d}_i{e}_i-{\sum }_{a:i\to j}{d}_i{e}_j,⟨\mathbf{d},\mathbf{e}{⟩}_Q={\chi }_Q(\mathbf{d},\mathbf{e})-{\chi }_Q(\mathbf{e},\mathbf{d}),t=\sqrt{q}>0.$

Suppose an additive map $\rho :{\mathbb{Z}}^{Q_0}\to {\mathrm{\Gamma }}_{\mathrm{S}\mathrm{W}}$ and a physical charge cone ${\mathrm{\Gamma }}_{+}$ are supplied, where ${\mathrm{\Gamma }}_{+}$ contains $0$, is closed under addition and is pointed, such that $\rho ({\mathbb{N}}^{Q_0})\subseteq {\mathrm{\Gamma }}_{+}$. Supply a positive additive degree $\mathrm{\ell }:{\mathrm{\Gamma }}_{+}\to \mathbb{N}$ with finite sublevel sets and require $\mathrm{\ell }(\rho (\mathbf{d}))={\sum }_i{d}_i$ for every $\mathbf{d}\in {\mathbb{N}}^{Q_0}$. Assume also, for all $\mathbf{d},\mathbf{e}\in {\mathbb{Z}}^{Q_0}$, that

$⟨\rho (\mathbf{d}),\rho (\mathbf{e}){⟩}_{\mathrm{D}\mathrm{S}\mathrm{Z}}=⟨\mathbf{d},\mathbf{e}{⟩}_Q\mathrm{.}$

Let $\mathbb{H}=\{z\in \mathbb{C}:\mathrm{Im}z>0\}$. Choose one real phase rotation $\alpha$ such that for both $u\in \{u_{-},u_{+}\}$ the vertex charges $Z_u({\mathbf{e}}_i)=e^{-i\alpha }{z}_u(\rho ({\mathbf{e}}_i))$ lie in $\mathbb{H}$, and extend each assignment additively by $Z_u(\mathbf{d})={\sum }_i{d}_i{Z}_u({\mathbf{e}}_i)=e^{-i\alpha }{z}_u(\rho (\mathbf{d}))$. A representation $M$ assigns a finite-dimensional ${\mathbb{F}}_q$-vector space to each vertex and a linear map to each arrow; its dimension vector is $\mathbf{d}(M)\in {\mathbb{N}}^{Q_0}$. The additive central charge $Z_u(\mathbf{d})$ was specified above. For nonzero $\mathbf{d}$, set ${\varphi }_{Z_u}(\mathbf{d})=\arg Z_u(\mathbf{d})\mathrm{/}\pi \in (0,1)$; this range follows because a nonzero sum of upper-half-plane vertex charges lies in $\mathbb{H}$. For every nonzero dimension vector, the defining equality for $Z_u$ identifies ${\varphi }_{Z_u}(\mathbf{d})$ with $\arg (e^{-i\alpha }{z}_u(\rho (\mathbf{d})))\mathrm{/}\pi$. A nonzero $M$ is $Z_u$-semistable when every nonzero subrepresentation $U\subseteq M$ satisfies ${\varphi }_{Z_u}(\mathbf{d}(U))\le {\varphi }_{Z_u}(\mathbf{d}(M))$.

On the isomorphism-class basis ${\delta }_M$, the subobject-first Hall product is $(f\star g)(L)={\sum }_{U\subseteq L}f(U)g(L\mathrm{/}U)$, with distinct embedded subobjects counted separately. Complete by total dimension, that is, take the product over dimension vectors of the finite-dimensional homogeneous components. For a phase $\varphi$, put $S_{Z_u}(\varphi )={\delta }_0+{\sum }_{[M]\mathrm{\ne }[0],M{Z}_u\text{-semistable},{\varphi }_{Z_u}(\mathbf{d}(M))=\varphi }{\delta }_M$, where ${\delta }_0$ is the Hall unit. Its skew integration is $J(f)={\sum }_{[M]}{t}^{{\chi }_Q(\mathbf{d}(M),\mathbf{d}(M))}f(M)\mathrm{\mid }\mathrm{Aut}(M){\mathrm{\mid }}^{-1}{x}^{\mathbf{d}(M)}$, and the completed skew torus has product $x^{\mathbf{d}}\ast x^{\mathbf{e}}=t^{-⟨\mathbf{e},\mathbf{d}{⟩}_Q}{x}^{\mathbf{d}+\mathbf{e}}$. The resulting Hall spectrum generator is the decreasing-phase product , with the largest phase leftmost.

For each of the two specified points define the nonzero index support ${\mathrm{\Sigma }}_u:=\{\gamma \in {\mathrm{\Gamma }}_{+}\setminus \{0\}:{\mathrm{\Omega }}_u(\gamma )\mathrm{\ne }0\}\mathrm{.}$ Assume $\{\gamma \in {\mathrm{\Gamma }}_{\mathrm{S}\mathrm{W}}\setminus \{0\}:{\mathrm{\Omega }}_u(\gamma )\mathrm{\ne }0\}\subseteq {\mathrm{\Gamma }}_{+}$ for both specified points, so ${\mathrm{\Sigma }}_u$ is the full nonzero index support. Assume $\{\gamma \in {\mathrm{\Sigma }}_u:\mathrm{\ell }(\gamma )\le N\}$ is finite for every $N\ge 0$, ${\mathrm{\Omega }}_u(\gamma )\in \mathbb{Z}$ on ${\mathrm{\Sigma }}_u$, and $e^{-i\alpha }{z}_u(\gamma )\in \mathbb{H}$ for every $u\in \{u_{-},u_{+}\}$ and $\gamma \in {\mathrm{\Sigma }}_u$. Thus every phase ${\varphi }_u(\gamma )=\arg (e^{-i\alpha }{z}_u(\gamma ))\mathrm{/}\pi$ used below is defined. Zero charge is excluded from ${\mathrm{\Sigma }}_u$ and from all phase-indexed products.

Let $K_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}$ be a coefficient field with a field embedding $\sigma :\mathbb{Q}(t)\to K_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}$, and write $v:=\sigma (t)$ for the dimensionless Hall coefficient parameter. Here $\mathbb{Q}(t)=\mathbb{Q}(\sqrt{q})$ is the field generated by the specified algebraic number $t$, so $v^2=q$ and $v\mathrm{\ne }1$; $v$ is distinct from the physical constant $\mathrm{\hslash }=1$ fixed by natural units. Use the supplied additive degree $\mathrm{\ell }:{\mathrm{\Gamma }}_{+}\to \mathbb{N}$ with $\mathrm{\ell }(0)=0$, $\mathrm{\ell }(\gamma )>0$ for $\gamma \mathrm{\ne }0$, and finite sublevel sets. Define the commutative formal charge-series algebra and the quantum charge torus by

${\widehat{\mathbb{A}}}_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}:={\prod }_{\gamma \in {\mathrm{\Gamma }}_{+}}{K}_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}{y}^{\gamma },y^{\gamma }{y}^{\delta }=y^{\gamma +\delta },$

${\widehat{\mathbb{T}}}_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}:={\prod }_{\gamma \in {\mathrm{\Gamma }}_{+}}{K}_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}{Y}^{\gamma },Y^{\gamma }{\star }_v{Y}^{\delta }=v^{-⟨\delta ,\gamma {⟩}_{\mathrm{D}\mathrm{S}\mathrm{Z}}}{Y}^{\gamma +\delta }\mathrm{.}$

For either algebra and $N\ge 0$, let $F^N$ consist of series supported on charges $\gamma$ with $\mathrm{\ell }(\gamma )\ge N$. Give both algebras the linear topology whose neighborhoods of zero are the $F^N$. Finite sublevel sets make every truncation modulo $F^N$ finite-dimensional; the algebras are separated and complete, and their products are degreewise finite. Give ${\widehat{\mathbb{T}}}_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}^{\times }$ the subspace topology from ${\widehat{\mathbb{T}}}_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}$. For each $N\ge 1$, let $G_N$ be the group of $K_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}$-algebra automorphisms of ${\widehat{\mathbb{A}}}_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}\mathrm{/}{F}^N$ preserving every induced quotient filtration subspace $F^j\mathrm{/}{F}^N$ for $1\le j<N$, with reduction as the transition maps and the discrete topology on each $G_N$. Define with the inverse-limit topology. It is the group of continuous automorphisms and continuous inverses preserving the degree filtration.

The charge map and coefficient embedding define the degree-continuous torus map, extended from the total-dimension-completed Hall skew torus by continuity,

$\mathrm{\Phi }(x^{\mathbf{d}})=Y^{\rho (\mathbf{d})},\mathrm{\Phi }(c{x}^{\mathbf{d}})=\sigma (c)Y^{\rho (\mathbf{d})}\mathrm{.}$

The degree identity $\mathrm{\ell }(\rho (\mathbf{d}))={\sum }_i{d}_i$ for $\mathbf{d}\in {\mathbb{N}}^{Q_0}$ ensures this continuity. On the commutative completed charge series, supply a sign refinement $\epsilon :{\mathrm{\Gamma }}_{+}\to \{\pm 1\}$ with $\epsilon (0)=1$ and $\epsilon (\gamma +\delta )=(-1{)}^{⟨\gamma ,\delta {⟩}_{\mathrm{D}\mathrm{S}\mathrm{Z}}}\epsilon (\gamma )\epsilon (\delta )$. Define the physical KS factor for each $\gamma \in {\mathrm{\Sigma }}_u$ by

${\mathcal{K}}_{\gamma ,u}(y^{\delta })=y^{\delta }(1-\epsilon (\gamma )y^{\gamma }{)}^{{\mathrm{\Omega }}_u(\gamma )⟨\delta ,\gamma {⟩}_{\mathrm{D}\mathrm{S}\mathrm{Z}}}\mathrm{.}$

The integer power is interpreted by its formal binomial series in the degree topology. When charges have equal phase, order their labels lexicographically as integer pairs $(n_e,n_m)$. Let ${\mathcal{S}}_u$ be the composition of these factors over ${\mathrm{\Sigma }}_u$ in decreasing phase order, with the largest phase leftmost and the stated lexicographic order within each phase. Finally assume an exact model-specific bridge: there is a continuous group homomorphism $\mathrm{\Theta }:{\widehat{\mathbb{T}}}_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}^{\times }⟶{\mathrm{Aut}}_{\mathrm{f}\mathrm{i}\mathrm{l}}({\widehat{\mathbb{A}}}_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}),$ and for every phase $\varphi$ and either specified point it satisfies, using the same within-phase order, $\mathrm{\Theta }\text{\hspace{0.17em}⁣}(\mathrm{\Phi }(J(S_{Z_u}(\varphi ))))={\prod }_{\begin{array}{c}\gamma \in {\mathrm{\Sigma }}_u\\ {\varphi }_u(\gamma )=\varphi \end{array}}^{\searrow }{\mathcal{K}}_{\gamma ,u}\mathrm{.}$

Here $S_{Z_u}(\varphi )$ and $J$ are the semistable phase factor and skew integration from the finite-field Hall definition. Then

${\mathcal{S}}_{u_{-}}={\mathcal{S}}_{u_{+}}\mathrm{.}$

This is a conditional transport theorem. The specified per-phase bridge is an additional physical input; neither the finite-field Hall identity nor this theorem asserts that the pure-SU(2) model supplies it, that the resulting index counts exist, or that the model has a continuum quantum-field-theory construction.

Physical BPS charge and second-helicity index data: Pure SU(2) physical BPS charge and index data.

Finite-field Hall spectrum generators: Finite-quiver Hall spectrum generators.

Finite-field Hall spectrum invariance: Hall wall crossing preserves the completed spectrum generators.

Gaiotto, Moore, and Neitzke, Introduction and Section 2.2.

Denef, Section 5.3.

Seiberg and Witten, Sections 4 and 6.

The equality above is a model-conditional deduction from a separate finite-field Hall theorem and the explicit per-phase bridge. The physical Witten heat-supertrace item defines a different index interface; no identification of it with the second-helicity trace is used. The Hall proposition's verification record and the current owner proof inspection are kept distinct from the additional physical bridge premise.