Spectral-network, framed BPS, and defect data in a specified class-S model

Assume the Axiom of Countable Choice, ${\mathrm{A}\mathrm{C}}_{\omega }$: for every sequence $(X_n{)}_{n\in \mathbb{N}}$ of nonempty sets, there is a function $f$ with domain $\mathbb{N}$ such that $f(n)\in X_n$ for every $n$. The imported trace-prerequisite chain uses this mathematical premise in several places. The published thm-spectral-theorem-for-compact-self-adjoint-operators applies to compact self-adjoint operators on real or complex Hilbert spaces and uses ${\mathrm{A}\mathrm{C}}_{\omega }$ through thm-countable-union-of-countable to count its nonzero eigenvalues and to select an orthonormal basis of each finite-dimensional eigenspace. The published thm-singular-value-decomposition-for-compact-operators applies to compact operators between real or complex Hilbert spaces and uses ${\mathrm{A}\mathrm{C}}_{\omega }$ to select bases of the positive eigenspaces of $\mathrm{\mid }T\mathrm{\mid }$. The trace-class chain also uses ${\mathrm{A}\mathrm{C}}_{\omega }$ in its compactness argument: the published positive-square-root lemma uses thm-norm-limit-of-compact-operators-is-compact, which states that a norm limit of compact operators into a Banach space is compact under ${\mathrm{A}\mathrm{C}}_{\omega }$; that theorem in turn invokes thm-complete-and-totally-bounded-implies-compact, which states that a complete totally bounded metric space is compact under ${\mathrm{A}\mathrm{C}}_{\omega }$. The latter proof's step 3.1 selects, for each $n$, a finite $1\mathrm{/}(n+1)$-net together with a listing. These are examples of transitive mathematical prerequisites of the imported trace definitions and theorem, not physical premises. No full Axiom of Choice is assumed here.

Fix an integer $K\ge 2$. A Riemann surface here is a connected complex one-manifold; assume $\bar{C}$ compact. A selected class-S input of type $A_{K-1}$ consists of $\bar{C}$, a finite marked set $P\subset \bar{C}$, a collection $D$ of specified puncture defects at $P$, a Coulomb-branch datum $u=({\varphi }_2,\dots ,{\varphi }_K)$, a selected global form with Lie algebra $\mathfrak{s}\mathfrak{u}(K)$, and a four-dimensional rigid $\mathcal{N}=2$ model $\mathcal{T}$ on $M={\mathbb{R}}^4$ with Minkowski metric $\eta =\mathrm{diag}(-1,+1,+1,+1)$, a chosen orientation, and a chosen time orientation. The model is adopted data: its six-dimensional $\mathcal{N}=(2,0)$ construction compactifies on $\bar{C}$ with a specified twist, meaning a chosen background coupling of the six-dimensional internal R-symmetry bundle to the tangent-bundle connection on $\bar{C}$ for which the compactification is taken to preserve four-dimensional $\mathcal{N}=2$ supersymmetry. This definition does not construct a continuum quantum field theory. The selected model also supplies its field bundles, admissible configuration space and field regularity, boundary conditions, observables, normalized correlation functional, and any regulator or measure used; none is deduced here. Put $C=\bar{C}\setminus P$. The holomorphic cotangent line bundle of $\bar{C}$ is $K_{\bar{C}}$. The sections ${\varphi }_r$ of $K_{\bar{C}}^{\otimes r}$, for $2\le r\le K$, are meromorphic, have poles only in $P$, and have pole orders and residues fixed by $D$. The selected global form and allowed line and surface operators are part of the model data. Thus the coefficients have the units of the corresponding powers of mass when the Seiberg–Witten one-form is assigned mass dimension one. Use $c=\mathrm{\hslash }=1$ and retain the model's trace, charge, and central-charge normalizations. A four-dimensional Bogomolny–Prasad–Sommerfield (BPS) state is a state in a model-supplied one-particle sector annihilated by a nonzero subset of the model's supercharges; when the selected supersymmetry algebra supplies a massive rest-fiber mass operator $M$ and central charge $Z$, its BPS-saturation condition is $M=\mathrm{\mid }Z\mathrm{\mid }$. No such state or algebraic identification is implied by the geometric data below.

Let $p:T^{\ast }\bar{C}\to \bar{C}$ be the cotangent-bundle projection. Its tautological section $\xi$ of $p^{\ast }{K}_{\bar{C}}$ is defined by ${\xi }_{(x,\eta )}=\eta \in T_x^{\ast }\bar{C}$. Its Liouville one-form $\lambda$ is instead the one-form on the total space defined by ${\lambda }_{(x,\eta )}(V)=\eta (d{p}_{(x,\eta )}V)$. Thus $\lambda$ is obtained from $\xi$ by the differential of the projection; in a coordinate $z$ with $\eta =tdz$, one has $\xi =t{p}^{\ast }(dz)$ and $\lambda =td(z\circ p)$. The spectral curve ${\mathrm{\Sigma }}_u$ is the compactification of the degree-$K$ curve in $T^{\ast }C$ defined by the section equation in $p^{\ast }{K}_{\bar{C}}^{\otimes K}$

${\xi }^{\otimes K}+{\sum }_{r=2}^K{p}^{\ast }{\varphi }_r\otimes {\xi }^{\otimes (K-r)}=0.$

Write $\pi :{\mathrm{\Sigma }}_u\to \bar{C}$ for the extension of the restricted projection $p$, and also use $\lambda$ for the meromorphic extension to ${\mathrm{\Sigma }}_u$ of the restricted Liouville one-form. The curve is required to be reduced and to define a finite degree-$K$ branched cover. A branch point is a point of $\bar{C}$ over which at least one point of ${\mathrm{\Sigma }}_u$ has local degree greater than one; the branch locus is the set of all such values. Away from the branch locus and $P$, pulling $\xi$ back along the local sheet sections gives one-forms ${\lambda }^{(1)},\dots ,{\lambda }^{(K)}$. Remove the poles of $\lambda$ and write $\tilde{\mathrm{\Sigma }}={\mathrm{\Sigma }}_u\setminus \mathrm{Pol}(\lambda )$. The geometric charge group is the first singular homology ${\mathrm{\Gamma }}_{\mathrm{g}\mathrm{e}\mathrm{o}\mathrm{m}}=H_1(\tilde{\mathrm{\Sigma }};\mathbb{Z})$, namely closed integral one-cycles modulo boundaries. For a marked subset $A\subseteq \tilde{\mathrm{\Sigma }}$, relative one-cycles are chains in $\tilde{\mathrm{\Sigma }}$ whose boundary lies in $A$, modulo boundaries of two-chains in $\tilde{\mathrm{\Sigma }}$ and one-chains contained in $A$; their classes form $H_1(\tilde{\mathrm{\Sigma }},A;\mathbb{Z})$.

For distinct sheets $i,j$, a regular $ij$-trajectory of Wentzel–Kramers–Brillouin (WKB) phase ${\vartheta }_{\mathrm{W}}\in \mathbb{R}\mathrm{/}2\pi \mathbb{Z}$ is a $C^1$ immersed path $z:I\to C\setminus \mathrm{Branch}(\pi )$, for an open interval $I\subseteq \mathbb{R}$, with $\dot{z}\mathrm{\ne }0$ and

$\mathrm{Im}\text{\hspace{0.17em}⁣}(e^{-i{\vartheta }_{\mathrm{W}}}({\lambda }^{(i)}-{\lambda }^{(j)})(\dot{z}))=0.$

Orient it so the real part inside this expression is positive. Reversing the orientation exchanges $ij$ with $ji$. The following geometric seed convention applies at every ramification order. An outgoing seed at a branch value $b\in \bar{C}$, including $b\in P$, is a germ of such an oriented regular $ij$-trajectory $z:(0,ϵ)\to C\setminus \mathrm{Branch}(\pi )$ with $z(s)\to b$ as $s\downarrow 0$, for which both sheet lifts extend continuously to the same ramification point over $b$. Two germs agree when their oriented sheet-labeled images agree sufficiently near that endpoint, up to increasing reparametrization. Take all such germs and extend each maximally through the regular locus. No number of seeds or finite network is assumed for a general cover. For a simple branch point, exactly two sheets ramify and their difference has local form $a(z)z^{1\mathrm{/}2}dz$ with $a(0)\mathrm{\ne }0$; its integral has leading order $z^{3\mathrm{/}2}$, giving the familiar three outgoing directions with the exchanged-sheet labels included. This three-direction statement is a special case, not a restriction to simple ramification.

Starting with these maximal seed trajectories, form successive generations by adjoining, at each regular intersection of an $ij$-trajectory and a $jk$-trajectory with distinct $i,j,k$ and ${\lambda }^{(i)}-{\lambda }^{(k)}\mathrm{\ne }0$, the maximal outgoing $ik$-trajectory through that point satisfying the same phase and positive-real orientation. The geometric WKB network is the union of all generations. This specifies a geometric closure convention at higher ramification as well; it does not assert that the general cover determines physical wall or soliton data.

For the cubic-differential case, use the following explicit conical specialization. Let $\phi$ be a meromorphic cubic differential with simple zeros and poles of order at most two, and canonical cover ${\lambda }^3={\pi }^{\ast }\phi$. A zero or pole of order $m\in \{1,-1,-2\}$ is a conical singularity, with flat cone angle $2\pi (m+3)\mathrm{/}3$ in the metric $\mathrm{\mid }{\phi }^{1\mathrm{/}3}\mathrm{\mid }$. In a local flat coordinate $u=\int {\phi }^{1\mathrm{/}3}$, the KT convention at phase ${\vartheta }_{\mathrm{K}\mathrm{T}}$ seeds every outward critical ray of slope ${\vartheta }_{\mathrm{K}\mathrm{T}}+n\pi \mathrm{/}3$, $n\in \mathbb{Z}$, from every conical singularity, even a marked pole. Even $n$ gives positive rays and odd $n$ negative rays. Extend them maximally; at each regular intersection of two rays of the same sign, adjoin the outgoing ray of the opposite sign, and iterate. This is the conical cubic convention of Kidwai and Tahar, not a simple-branch approximation. For sheets ${\lambda }^{(j)}=\omega {\lambda }^{(i)}$, $\omega =e^{2\pi i\mathrm{/}3}$, their difference is $(1-\omega ){\lambda }^{(i)}$ with $\arg (1-\omega )=-\pi \mathrm{/}6$. Thus the unlabelled seed supports agree with the sheet-pair convention when ${\vartheta }_{\mathrm{W}}={\vartheta }_{\mathrm{K}\mathrm{T}}-\pi \mathrm{/}6$ modulo $\pi \mathrm{/}3$; the ordered pair and orientation fix the exact phase. For T3, the cover is fully ramified over its three double poles and $\lambda$ is nonzero at the corresponding cover points. In its flat cover coordinate there are six outgoing rays with angles ${\vartheta }_{\mathrm{K}\mathrm{T}}+n\pi \mathrm{/}3$ at each such point, projected to the base. No soliton index is assigned by this conical rule.

A physical spectral network also includes the model-supplied soliton data on its walls; in the usual class-S construction an $ij$ wall carries a soliton index $\mu$. A geometric trajectory with no supplied nonzero soliton data is not thereby a physical BPS state. The WKB phase ${\vartheta }_{\mathrm{W}}$ is independent data from the phase ${\vartheta }_{\mathrm{f}\mathrm{r}}$ of a supersymmetric interface unless the selected model supplies a relation.

A finite web is a finite connected metric ribbon graph: each edge has positive finite length and a cyclic order at each vertex; each internal vertex is trivalent and maps to a network joint, and each external univalent vertex maps to a branch point. The metric is part of the declared convention. In the Gaiotto–Moore–Neitzke (GMN) string convention, an $ij$ edge has length $L_{ij}={\int }_a^b\mathrm{\mid }({\lambda }^{(i)}-{\lambda }^{(j)})(\dot{z}(t))\mathrm{\mid }dt$ along a regular parameterization $z:[a,b]\to C$, and its physical string mass is $L_{ij}\mathrm{/}\pi$. In the geometric cubic-differential convention, a lifted T3 edge has flat length $L_{\mathrm{K}\mathrm{T}}=\int \mathrm{\mid }\lambda (\dot{\tilde{z}}(t))\mathrm{\mid }dt$ on the canonical cover. For a local $K=3$ sheet pair with ${\lambda }^{(i)}-{\lambda }^{(j)}=(1-\omega )\lambda$, the GMN edge length is $\sqrt{3}{L}_{\mathrm{K}\mathrm{T}}$; these conventions are not silently identified. The graph's map is isometric in the explicitly selected edge metric, with its sheet-pair labels, to compatible network trajectories. Closed edges are permitted. An open web ending at a surface-defect parameter point is instead a relative web. For a closed web, the chosen sheet-lift rule joins its oriented sheetwise edge lifts into a cycle ${\gamma }_{\mathrm{g}\mathrm{e}\mathrm{o}\mathrm{m}}\in {\mathrm{\Gamma }}_{\mathrm{g}\mathrm{e}\mathrm{o}\mathrm{m}}$. For an open web on a surface defect ${\mathbb{S}}_z$, put $A_z={\pi }^{-1}(z)\subset \tilde{\mathrm{\Sigma }}$; its lifted chain is a class in $H_1(\tilde{\mathrm{\Sigma }},A_z;\mathbb{Z})$ with boundary $z^{(j)}-z^{(i)}$ for its outgoing $ij$ edge, where $z^{(i)}$ and $z^{(j)}$ are the corresponding preimages of $z$. For an interface path $\mathrm{\wp }:z_1\to z_2$, put $A_{12}={\pi }^{-1}(\{z_1,z_2\})\subset \tilde{\mathrm{\Sigma }}$; its framed charge $\bar{a}$ is a class in $H_1(\tilde{\mathrm{\Sigma }},A_{12};\mathbb{Z})$ with boundary $z_2^{(j)}-z_1^{(i)}$. These lifts depend on the stated local branch and junction rules. The physical charge group ${\mathrm{\Gamma }}_{\mathcal{T}}$ is a model-specified quotient of a subgroup ${\mathrm{\Gamma }}_0\le {\mathrm{\Gamma }}_{\mathrm{g}\mathrm{e}\mathrm{o}\mathrm{m}}$, supplied with a surjective quotient map $\rho :{\mathrm{\Gamma }}_0\to {\mathrm{\Gamma }}_{\mathcal{T}}$; it is not identified with all of $H_1(\tilde{\mathrm{\Sigma }};\mathbb{Z})$ by notation alone.

For geometric periods use $Z_{\mathrm{g}\mathrm{e}\mathrm{o}\mathrm{m}}(\gamma )={\int }_{\gamma }\lambda$. The physical central charge is a model-supplied group homomorphism $Z_{\mathcal{T}}:{\mathrm{\Gamma }}_{\mathcal{T}}\to (\mathbb{C},+)$ of mass dimension one; in particular, $Z_{\mathcal{T}}(0)=0$. The selected GMN normalization $Z_{\mathcal{T}}(\rho (\gamma ))={\pi }^{-1}{\int }_{\gamma }\lambda$ for $\gamma \in {\mathrm{\Gamma }}_0$ is available only when ${\int }_{\kappa }\lambda =0$ for every $\kappa \in \ker \rho$, so the period formula is independent of the representative of $\rho (\gamma )$. If this descent condition is not supplied, the physical central charge remains a separate model datum and no equality with geometric periods is asserted. A cubic-differential T3 geometry may use $Z_{\mathrm{g}\mathrm{e}\mathrm{o}\mathrm{m}}$; its geometric normalization is not a physical mass normalization.

For regular points $z,z^{\mathrm{\prime }}\in C$, including $z=z^{\mathrm{\prime }}$, put $A_{z{z}^{\mathrm{\prime }}}={\pi }^{-1}(\{z,z^{\mathrm{\prime }}\})$. For each ordered pair of sheets define the geometric endpoint sector ${\mathrm{\Gamma }}_{ij}^{\mathrm{g}\mathrm{e}\mathrm{o}\mathrm{m}}(z,z^{\mathrm{\prime }})$ to be the fiber of the relative-homology boundary map consisting of classes $a\in H_1(\tilde{\mathrm{\Sigma }},A_{z{z}^{\mathrm{\prime }}};\mathbb{Z})$ with $\mathrm{\partial }a=z^{\mathrm{\prime }(j)}-z^{(i)}$. Empty fibers are permitted. Addition of an absolute cycle acts on each nonempty fiber; the difference of two classes in that fiber is an absolute homology class, so this action of ${\mathrm{\Gamma }}_{\mathrm{g}\mathrm{e}\mathrm{o}\mathrm{m}}$ is free and transitive. For each admitted physical endpoint sector, the model supplies a nonempty ${\mathrm{\Gamma }}_0$-orbit $O_{ij}(z,z^{\mathrm{\prime }})$ in this fiber. Its physical relative-charge sector is ${\mathrm{\Gamma }}_{ij}^{\mathcal{T}}(z,z^{\mathrm{\prime }})=O_{ij}(z,z^{\mathrm{\prime }})\mathrm{/}\ker \rho$, with quotient map $q_{ij}^{z,z^{\mathrm{\prime }}}$. Thus it is a ${\mathrm{\Gamma }}_{\mathcal{T}}$-torsor, meaning a set with a free transitive ${\mathrm{\Gamma }}_{\mathcal{T}}$-action: $q(a)+\rho (\gamma )=q(a+\gamma )$ for $\gamma \in {\mathrm{\Gamma }}_0$. For $z=z^{\mathrm{\prime }}$ and $i=j$, take $O_{ii}(z,z)={\mathrm{\Gamma }}_0$ under the canonical identification of the zero-boundary fiber with absolute homology; then ${\mathrm{\Gamma }}_{ii}^{\mathcal{T}}(z,z)={\mathrm{\Gamma }}_{\mathcal{T}}$ and $q_{ii}^{z,z}=\rho$. The model specifies all admitted sectors; an unadmitted sector supplies no physical charge label. Write ${\mathrm{\Gamma }}^{\mathcal{T}}(z,z^{\mathrm{\prime }})={⨆}_{i,j}{\mathrm{\Gamma }}_{ij}^{\mathcal{T}}(z,z^{\mathrm{\prime }})$, retaining the endpoint labels in this disjoint union. Earlier relative lifts are geometric representatives; the soliton and framed indices below use their physical images $\bar{a}=q_{ij}^{z,z^{\mathrm{\prime }}}(a)$, when admitted, and any further sign-refining lifts specified below.

On each admitted sector the model supplies a relative central-charge map $Z_{ij}^{z,z^{\mathrm{\prime }}}:{\mathrm{\Gamma }}_{ij}^{\mathcal{T}}(z,z^{\mathrm{\prime }})\to \mathbb{C}$, of mass dimension one, satisfying $Z_{ij}^{z,z^{\mathrm{\prime }}}(\bar{a}+\gamma )=Z_{ij}^{z,z^{\mathrm{\prime }}}(\bar{a})+Z_{\mathcal{T}}(\gamma )$ for every $\gamma \in {\mathrm{\Gamma }}_{\mathcal{T}}$. On the diagonal sector this map is $Z_{\mathcal{T}}$ itself. When the model admits composition of endpoint sectors, it supplies compatible geometric orbits under chain addition and requires $Z_{ik}^{z,z^{\mathrm{\prime }\mathrm{\prime }}}(\bar{a}+\bar{b})=Z_{ij}^{z,z^{\mathrm{\prime }}}(\bar{a})+Z_{jk}^{z^{\mathrm{\prime }},z^{\mathrm{\prime }\mathrm{\prime }}}(\bar{b})$ for composable admitted charges. These are supplied compatibility conditions, not existence claims. Under the descended GMN period convention one additionally takes $Z_{ij}^{z,z^{\mathrm{\prime }}}(q(a))={\pi }^{-1}{\int }_a\lambda$, using piecewise smooth representatives. For a fixed relative class the integral is unchanged by boundaries, since $\lambda$ is closed away from its excluded poles and restricts to zero on the finite endpoint set. Passing from $a$ to $a+\kappa$ in the quotient changes the integral by ${\int }_{\kappa }\lambda$, so the same vanishing condition on $\ker \rho$ is required. Affine compatibility then follows from the bulk normalization and additivity of integration. Without this period convention, the affine maps and their endpoint offsets remain independent model data; the bulk map alone does not determine them.

For a generic point $z\in C$ outside the branch and puncture loci, the selected model may supply a canonical half-BPS surface defect ${\mathbb{S}}_z$ in the fundamental representation of the type-$A_{K-1}$ Lie algebra, whose $K$ isolated gapped vacua are labeled by ${\pi }^{-1}(z)$. The model's four-dimensional $\mathcal{N}=2$ Poincare algebra has eight real supercharges; here a half-BPS insertion preserves a specified four-dimensional real subspace. As a defect insertion, it is model-specified data $(j_z,{\lambda }_z)$ in the earlier general defect-insertion sense: $j_z:W_z\hookrightarrow M^{\circ }$ is a smooth embedding of a two-dimensional manifold $W_z$ without boundary with closed image in $M^{\circ }=M\setminus \mathrm{\partial }M$, and ${\lambda }_z$ is the complete field, operator, and boundary-condition data supported there, including any orientations, framings, and label-transport data required by the model. The model declares this insertion admissible; it need not be topological, so no correlator invariance under deformation is assumed. In Cartesian coordinates $x^0,x^1,x^2,x^3$, the GMN interface arrangement has closed support $K_{\mathrm{d}\mathrm{e}\mathrm{f}}=\{x^1=x^2=0\}$. Its two two-dimensional strata are $\{x^1=x^2=0,x^3>0\}$ and $\{x^1=x^2=0,x^3<0\}$, labeled respectively by $z_1$ and $z_2$, and its one-dimensional interface stratum is $\{x^1=x^2=x^3=0\}$. These strata form one model-supplied defect network: the selected model supplies the complete interface or junction operator, label transport, and admissibility data on the shared stratum, as required for a complete insertion input. The two open half-planes are not treated as separate closed defect insertions. A generic interface datum consists of distinct regular points $z_1,z_2$, a path $\mathrm{\wp }:z_1\to z_2$, and a phase ${\vartheta }_{\mathrm{f}\mathrm{r}}$, giving a model-supplied half-BPS interface $L_{\mathrm{\wp },{\vartheta }_{\mathrm{f}\mathrm{r}}}$ between the two surface defects. In the GMN convention used here, ${\vartheta }_{\mathrm{f}\mathrm{r}}$ selects the two supercharges of the interface from the four preserved by the surface defects. Precisely, ${\vartheta }_{\mathrm{f}\mathrm{r}}$ is bulk-generic when $e^{-i{\vartheta }_{\mathrm{f}\mathrm{r}}}{Z}_{\mathcal{T}}(\gamma )\mathrm{\notin }{\mathbb{R}}_{<0}$ for every $\gamma \in {\mathrm{\Gamma }}_{\mathcal{T}}$. The pair $(\mathrm{\wp },{\vartheta }_{\mathrm{f}\mathrm{r}})$ is generic when it is bulk-generic and, for each $\mathrm{\ell }\in \{1,2\}$, every admitted endpoint sector ${\mathrm{\Gamma }}_{ij}^{\mathcal{T}}(z_{\mathrm{\ell }},z_{\mathrm{\ell }})$ and every $\bar{a}$ in that sector satisfy $e^{-i{\vartheta }_{\mathrm{f}\mathrm{r}}}{Z}_{ij}^{z_{\mathrm{\ell }},z_{\mathrm{\ell }}}(\bar{a})\mathrm{\notin }{\mathbb{R}}_{<0}$. Zero values are allowed because the forbidden ray is strictly negative. This tests the two endpoint soliton sectors, rather than the framed sectors from $z_1$ to $z_2$.

For a compact operator $A$ on a real or complex Hilbert space, trace class means that its singular values $s_n(A)$, the positive eigenvalues of $(A^{\ast }A{)}^{1\mathrm{/}2}$ counted with multiplicity and padded with zeros, satisfy ${\sum }_{n\ge 1}{s}_n(A)<+\mathrm{\infty }$. Under the stated ${\mathrm{A}\mathrm{C}}_{\omega }$ premise, the imported trace theorem gives a basis-independent trace $\mathrm{Tr}(A)$; for any supplied complete orthonormal system $E$, it equals the absolutely convergent sum ${\sum }_{e\in E}⟨Ae,e⟩$. The following one-particle BPS sectors, when they exist, are model-supplied complex Hilbert spaces in the stated bulk-charge, defect-soliton, and framed-interface sectors. The selected model must specify the domains of the displayed composite operators; each must be defined on a dense domain and extend to a bounded trace-class operator on its indicated sector. Here and below, $\mathrm{Tr}$ denotes the basis-independent trace of that extension.

The bulk second-helicity index is

${\mathrm{\Omega }}_{\mathcal{T}}(\gamma )=-\frac{1}{2}{\mathrm{Tr}}_{{\mathcal{H}}_{\gamma }^{1,\mathrm{B}\mathrm{P}\mathrm{S}}}\text{\hspace{0.17em}⁣}((-1{)}^{2J_3}(2J_3{)}^2),$

where $J_3$ is the self-adjoint generator of a massive four-dimensional spatial rotation in the rest frame. The unrefined surface-defect soliton index is $\mu (\bar{a})={\mathrm{Tr}}_{{\mathcal{H}}_{{\mathbb{S}}_z,\bar{a}}^{1,\mathrm{B}\mathrm{P}\mathrm{S}}}F{e}^{i\pi F}$, with $F$ the specified fermion-number operator. The framed index for $L_{\mathrm{\wp },{\vartheta }_{\mathrm{f}\mathrm{r}}}$ is

$\bar{\mathrm{\Omega }}(L_{\mathrm{\wp },{\vartheta }_{\mathrm{f}\mathrm{r}}},\bar{a};y)={\mathrm{Tr}}_{{\mathcal{H}}_{L_{\mathrm{\wp },{\vartheta }_{\mathrm{f}\mathrm{r}}},\bar{a}}^{1,\mathrm{B}\mathrm{P}\mathrm{S}}}{e}^{i\pi F}(-y{)}^{\mathcal{J}},\mathcal{J}=2J_{12}+2I_{12},$

where $J_{12}$ and $I_{12}$ are the model-supplied self-adjoint generators of spatial rotations in the selected $1$–$2$ plane and the selected $SU(2{)}_R$, respectively. The $1$–$2$ rotations fix the $x^3$ axis and hence preserve each stratum of $K_{\mathrm{d}\mathrm{e}\mathrm{f}}$; the model must also supply their action on the complete defect labels. On each indicated BPS charge sector, the required symmetry actions are strongly continuous one-parameter unitary groups $U_J(t)=e^{it{J}_{12}}$ and $U_I(t)=e^{it{I}_{12}}$ preserving that sector, with $t\in \mathbb{R}$ and dimensionless self-adjoint generators. The model supplies the self-adjoint realization of $\mathcal{J}$ agreeing with $2J_{12}+2I_{12}$ on its specified common dense domain; support invariance alone does not construct these operators. The model specifies an admissible set $Y_{\mathcal{T}}\subseteq {\mathbb{C}}^{\times }$ of dimensionless $y$-values and a functional-calculus convention defining $(-y{)}^{\mathcal{J}}$ for every $y\in Y_{\mathcal{T}}$; the trace-class extension above is required for each such value. The unrefined specialization uses $y=1\in Y_{\mathcal{T}}$ and is the unrefined framed index in this GMN convention. The model also fixes the additive normalization of $F$ and $\mathcal{J}$ and any sign refinement. Without these choices, the soliton and unrefined framed indices do not have a canonical sign; for general $y$, changing the additive normalization of $\mathcal{J}$ can also change the refined trace by its corresponding $y$-dependent factor. When the chosen sign refinement requires an extended lift of $\bar{a}$, the index is indexed by that lift and it must not be silently identified with a function of $\bar{a}$ alone. The cited arXiv-v3 chapter writes its unrefined framed trace as $\mathrm{Tr}{e}^{i\pi F}$; these are recorded as separate conventions, with no equality asserted without a supplied operator relation. The geometric network does not determine any of these traces without the specified physical Hilbert sectors.

Topological defect insertion in a specified model

The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)

Trace class operator

Trace of a trace class operator

Trace is absolutely convergent and basis independent

Gaiotto, Moore, and Neitzke, Spectral Networks, §§2–5 and §9

Kineider, Kydonakis, Rogozinnikov, Tatitscheff, and Thomas, Spectral Networks: Bridging Higher-Rank Teichmüller Theory and BPS States, arXiv v3 (7 May 2026), Chapter 22

Kidwai and Tahar, Spectral Networks, Finite Webs, and the T3 Cubic Differential, §§2–3

Gaiotto, Moore, and Neitzke state that the general finite-web zero-mode quantization is not complete in their treatment and describe the geometric wall-crossing explanation as “if not quite a proof.” Kidwai and Tahar's 2026 Spectral Networks, Finite Webs, and the T3 Cubic Differential, §§2–3, establishes geometric finite-web properties, not a numerical physical BPS index. These source qualifications do not update any historical open question to a current open status; any such classification needs a separate current source review.