Pure SU(2) physical BPS charge and index data

Use natural units $c=\mathrm{\hslash }=1$ and the selected pure four-dimensional rigid $\mathcal{N}=2$ Yang–Mills model with global gauge group $SU(2)$ and no microscopic matter hypermultiplets. Let $U$ be a simply connected regular patch of its source-proposed Seiberg–Witten Coulomb branch on which the local periods $a,a_D:U\to \mathbb{C}$ are holomorphic and $\mathrm{Im}(\mathrm{\partial }{a}_D\mathrm{/}\mathrm{\partial }a)>0$. The earlier Coulomb-branch data define the formal charge-label lattice ${\mathrm{\Gamma }}_{\mathrm{S}\mathrm{W}}={\mathbb{Z}}^2$, with labels $\gamma =(n_e,n_m)$ and period pairing ${\mathcal{Z}}_{\mathrm{S}\mathrm{W},u}(\gamma )=n_{e}a(u)+n_m{a}_D(u)$. The source-proposed model convention fixes the scalar central charge to be $z_u(\gamma )=\sqrt{2}{\mathcal{Z}}_{\mathrm{S}\mathrm{W},u}(\gamma )$; $a,a_D,{\mathcal{Z}}_{\mathrm{S}\mathrm{W},u},z_u$ have mass dimension one, $u$ has mass dimension two, and charge labels and the DSZ pairing are dimensionless.

For the model-specific physical charge data used here, equip ${\mathrm{\Gamma }}_{\mathrm{S}\mathrm{W}}$ with the integral antisymmetric Dirac pairing

$⟨(n_e,n_m),(n_e^{\mathrm{\prime }},n_m^{\mathrm{\prime }}){⟩}_{\mathrm{D}\mathrm{S}\mathrm{Z}}=2(n_m{n}_e^{\mathrm{\prime }}-n_e{n}_m^{\mathrm{\prime }})\mathrm{.}$

In particular the source-designated monopole label ${\gamma }_m=(0,1)$ and dyon label ${\gamma }_d=(1,-1)$ satisfy $⟨{\gamma }_m,{\gamma }_d{⟩}_{\mathrm{D}\mathrm{S}\mathrm{Z}}=2$, the convention matched by a two-arrow Kronecker quiver with both arrows directed from the dyon vertex to the monopole vertex. The lattice records formal charge labels only: it does not assert that every label is an allowed line operator or a realized particle.

For the local Hall convention, order the quiver vertices as $(m,d)$ and let $x=(x_m,x_d),y=(y_m,y_d)\in {\mathbb{Z}}^2$. With two arrows $d\to m$, define the Euler form and its antisymmetrization by

${\chi }_Q(x,y)=x_m{y}_m+x_d{y}_d-2x_d{y}_m,⟨x,y{⟩}_Q={\chi }_Q(x,y)-{\chi }_Q(y,x)=2(x_m{y}_d-x_d{y}_m)\mathrm{.}$

Thus ${\chi }_Q(e_m,e_d)=0$, ${\chi }_Q(e_d,e_m)=-2$, and $⟨e_m,e_d{⟩}_Q=+2$, where $e_m=(1,0)$ and $e_d=(0,1)$. The map $\rho (x)=x_m{\gamma }_m+x_d{\gamma }_d$ therefore satisfies $⟨\rho (x),\rho (y){⟩}_{\mathrm{D}\mathrm{S}\mathrm{Z}}=⟨x,y{⟩}_Q$. Denef's directed-intersection convention assigns the number of arrows from the first vertex to the second minus the reverse number; on this same directed quiver its bilinear extension is $I_D(x,y)=2(x_d{y}_m-x_m{y}_d)=-⟨x,y{⟩}_Q$. Denef is cited here for the two-arrow magnitude and pure-$SU(2)$ model discussion. The signed Hall identification used here is the explicitly defined $\mathrm{D}\mathrm{S}\mathrm{Z}={\chi }_Q-{\chi }_Q^{\mathsf{T}}$ convention.

For each $u\in U$, the selected model supplies a stable support ${\mathrm{\Gamma }}_u^{\mathrm{s}\mathrm{t}}\subseteq {\mathrm{\Gamma }}_{\mathrm{S}\mathrm{W}}\setminus \{0\}$. For each $\gamma \in {\mathrm{\Gamma }}_u^{\mathrm{s}\mathrm{t}}$, supply a selected unitary four-dimensional super-Poincaré representation in the precise earlier sense and its representation-theoretic one-particle massive rest-fiber Hilbert sector ${\mathcal{H}}_{u,\gamma }^1$ with positive inner product linear in its first variable, a bounded self-adjoint grading involution, a positive rest mass $M_{u,\gamma }$, and the conserved label $\gamma$. On a supplied common dense graded invariant domain ${\mathcal{D}}_{u,\gamma }$, supply the $\mathcal{N}=2$ supertranslation representation of the earlier BPS-state definition: $P^0=M_{u,\gamma }I$, $P^j=0$, $Z^{12}=z_u(\gamma )I$, $Z^{21}=-z_u(\gamma )I$, and closed odd supercharges $Q_{\alpha }^I$ with their actual Hilbert adjoints, with every product in the bracket table defined there. Use metric $\eta =\mathrm{diag}(-1,1,1,1)$, so $P_0=-M_{u,\gamma }I$. The rest-fiber identification with that selected representation is supplied model data; it does not mean a normalizable fixed-momentum subspace of the full representation.

The spin-cover Lorentz action on vectors is characterized by $\mathrm{\Phi }({\mathrm{\Lambda }}_{g}x)=g\mathrm{\Phi }(x)g^{†}$ for $g\in \mathrm{S}\mathrm{L}(2,\mathbb{C})$ and $\mathrm{\Phi }(x)=-{\sigma }^{\mu }{x}_{\mu }$, as in the unitary representation definition. For the rest vector $P_{u,\gamma }=(M_{u,\gamma },0,0,0)$ define its stabilizer $G_{P_{u,\gamma }}:=\{g\in \mathrm{S}\mathrm{L}(2,\mathbb{C}):{\mathrm{\Lambda }}_g{P}_{u,\gamma }=P_{u,\gamma }\}=\mathrm{S}\mathrm{U}(2)\mathrm{.}$ Indeed $\mathrm{\Phi }(P_{u,\gamma })=M_{u,\gamma }{I}_2$, so the stabilizer condition is $g{g}^{†}=I_2$. As part of the rest-fiber identification, supply the selected full representation's little-group action as a grading-preserving unitary representation ${\pi }_{u,\gamma }:\mathrm{S}\mathrm{U}(2)\to \mathcal{U}({\mathcal{H}}_{u,\gamma }^1)$, meaning $\pi (gh)=\pi (g)\pi (h)$, $\pi (I)=I$, and $\pi (g{)}^{\ast }\pi (g)=I=\pi (g)\pi (g{)}^{\ast }$. Require strong continuity: $g\mapsto \pi (g)\psi$ is norm-continuous for every $\psi$. Require $\pi (g){\mathcal{D}}_{u,\gamma }={\mathcal{D}}_{u,\gamma }$ and, on that domain, $\pi (g)Q_{\alpha }^I\pi (g{)}^{-1}={\sum }_{\beta }{g}^{\beta }{}_{\alpha }{Q}_{\beta }^I,\pi (g)(Q_{\alpha }^I{)}^{\ast }\pi (g{)}^{-1}={\sum }_{\beta }\overline{g^{\beta }{}_{\alpha }}(Q_{\beta }^I{)}^{\ast },$ where $g^{\beta }{}_{\alpha }$ is row $\beta$, column $\alpha$; scalar momenta and central charges are unchanged. Require this $\pi$ to be the little-group action induced by the supplied representation-theoretic rest-fiber identification, with the displayed supercharge covariance. It is not an independently chosen unrelated action or a restriction to a normalizable sharp-momentum subspace of the full-space operator $U(g,0)$. Choose the spin axis $3$ and the one-parameter rotation subgroup $r_3(\theta )=\left(\begin{array}{cc}{e}^{-i\theta \mathrm{/}2}& 0\\ 0& e^{i\theta \mathrm{/}2}\end{array}\right),\theta \in \mathbb{R}\mathrm{/}(4\pi \mathbb{Z})\mathrm{.}$ Supply a nonzero closed Hilbert subspace ${\mathcal{H}}_{u,\gamma }^{1,\mathrm{B}\mathrm{P}\mathrm{S}}\subseteq {\mathcal{H}}_{u,\gamma }^1$ invariant under this supplied $\mathrm{S}\mathrm{U}(2)$ action and identified as the selected BPS sector. Its vectors lie in ${\mathcal{D}}_{u,\gamma }$. Require the grading involution to preserve this subspace and, for every $I,\alpha$, require $Q_{\alpha }^I{\mathcal{H}}_{u,\gamma }^{1,\mathrm{B}\mathrm{P}\mathrm{S}}\subseteq {\mathcal{H}}_{u,\gamma }^{1,\mathrm{B}\mathrm{P}\mathrm{S}}$ and $(Q_{\alpha }^I{)}^{\ast }{\mathcal{H}}_{u,\gamma }^{1,\mathrm{B}\mathrm{P}\mathrm{S}}\subseteq {\mathcal{H}}_{u,\gamma }^{1,\mathrm{B}\mathrm{P}\mathrm{S}}$; thus the selected sector includes every supercharge partner of each of its states. Every nonzero vector is BPS-saturated in the exact sense of the earlier BPS-state definition: $M_{u,\gamma }=\mathrm{\mid }{z}_u(\gamma )\mathrm{\mid }$, and the two $\mathcal{N}=2$ annihilating supercharge combinations in that definition vanish on them. To make stability an exact condition, the model also supplies a same-charge rest-energy Hilbert space ${\mathcal{K}}_{u,\gamma }$ and a nonnegative self-adjoint Hamiltonian $H_{u,\gamma }$. Supply an isometric embedding $\iota :{\mathcal{H}}_{u,\gamma }^{1,\mathrm{B}\mathrm{P}\mathrm{S}}\to {\mathcal{K}}_{u,\gamma }$ whose closed image reduces $H_{u,\gamma }$ and on which $H_{u,\gamma }\iota \psi =M_{u,\gamma }\iota \psi$. Supply an orthogonal reducing subspace ${\mathcal{K}}_{u,\gamma }^{\mathrm{m}\mathrm{p}}$ representing the multiparticle channels at this same total conserved charge and zero total spatial momentum, with self-adjoint restriction $H_{u,\gamma }^{\mathrm{m}\mathrm{p}}$. For a self-adjoint operator $T$, write $\sigma (T)$ for its real spectrum, namely the real numbers $e$ for which $T-eI$ has no bounded everywhere-defined inverse. Define the multiparticle threshold $E_{u,\gamma }^{\mathrm{m}\mathrm{p}}=\inf \sigma (H_{u,\gamma }^{\mathrm{m}\mathrm{p}})$, with $\inf \mathrm{\varnothing }=+\mathrm{\infty }$ if the supplied channel space is zero. In a continuum model these supplied channels include its multiparticle continuous spectrum; in a discrete regulator they are the model's designated multiparticle channel subspace. Require $M_{u,\gamma }$ to be isolated in $\sigma (H_{u,\gamma })$: for some $\epsilon >0$, $\sigma (H_{u,\gamma })\cap (M_{u,\gamma }-\epsilon ,M_{u,\gamma }+\epsilon )=\{M_{u,\gamma }\}$. Require also a supplied real $g_{u,\gamma }>0$ with $M_{u,\gamma }+g_{u,\gamma }\le E_{u,\gamma }^{\mathrm{m}\mathrm{p}}$. These isolation and threshold conditions define stability here; they make precise the positive gap below the multiparticle continuum without asserting finite degeneracy. The stable support consists of the charges selected by the model with these data; no completeness of that selection or existence of these data is proved. On ${\mathcal{H}}_{u,\gamma }^{1,\mathrm{B}\mathrm{P}\mathrm{S}}$, supply a self-adjoint operator $J_3$ with $2J_3$-integer spectrum such that the restricted action of $r_3(\theta )$ is $e^{-i\theta J_3}$ for every $\theta \in \mathbb{R}$. Require its spectral calculus to define a bounded trace-class operator $B_{u,\gamma }=(-1{)}^{2J_3}(2J_3{)}^2$, and supply a Hilbert basis $E_{u,\gamma }$ of ${\mathcal{H}}_{u,\gamma }^{1,\mathrm{B}\mathrm{P}\mathrm{S}}$.

Define the four-dimensional $\mathcal{N}=2$ second-helicity supertrace, also called the $B_2$ index here, by

${\mathrm{\Omega }}_u(\gamma )=-\frac{1}{2}{\mathrm{tr}}_{E_{u,\gamma }}(B_{u,\gamma }),$

Here ${\mathrm{tr}}_{E_{u,\gamma }}$ is the absolutely convergent supplied-basis trace of a trace-class operator; basis independence is not needed here. Set ${\mathrm{\Omega }}_u(\gamma )=0$ for $\gamma \mathrm{\notin }{\mathrm{\Gamma }}_u^{\mathrm{s}\mathrm{t}}$. The trace convention uses only ${\mathrm{A}\mathrm{C}}_{\omega }$ through the stated mathematical suppliers, not full Choice. No existence or local constancy of this support is asserted.

Let $\mathcal{C}\subseteq {\mathrm{\Gamma }}_{\mathrm{S}\mathrm{W}}\setminus \{0\}$ be any chosen set of charges, and fix $\alpha \in \mathbb{R}$. At each $u\in U$ for which $e^{-i\alpha }{z}_u(\gamma )\in \mathbb{H}:=\{z\in \mathbb{C}:\mathrm{Im}z>0\}$ for every $\gamma \in \mathcal{C}$, define ${\varphi }_u(\gamma )=\arg (e^{-i\alpha }{z}_u(\gamma ))\mathrm{/}\pi \in (0,1)$ for $\gamma \in \mathcal{C}$. For two charges $\gamma ,\delta$ with nonzero central charges, define their marginal-stability wall by

$W_{\gamma ,\delta }:=\{u\in U:z_u(\gamma )\mathrm{/}{z}_u(\delta )\in {\mathbb{R}}_{>0}\}\mathrm{.}$

Given a specified family $\mathcal{W}$ of such wall subsets of $U$, a chamber relative to $\mathcal{W}$ is a connected component of $U\setminus {\bigcup }_{W\in \mathcal{W}}W$. A formal positive cone ${\mathrm{\Gamma }}_{+}\subseteq {\mathrm{\Gamma }}_{\mathrm{S}\mathrm{W}}$ used for ordered products contains $0$, is closed under addition, and is pointed (${\mathrm{\Gamma }}_{+}\cap (-{\mathrm{\Gamma }}_{+})=\{0\}$). It is supplied with an additive degree $\mathrm{\ell }:{\mathrm{\Gamma }}_{+}\to \mathbb{N}$ that is positive on nonzero charges and has finite sublevel sets, meaning $\{\gamma \in {\mathrm{\Gamma }}_{+}:\mathrm{\ell }(\gamma )\le N\}$ is finite for every $N\in \mathbb{N}$. The definition is conditional on the existence of the Hilbert sectors, trace data, cone, and degree; it constructs no continuum theory and makes no Hall-index identification.

Pure $N=2$ SU(2) Coulomb-branch and period data: Pure $N=2$ SU(2) Coulomb-branch and period data.

Source-proposed pure SU(2) Seiberg–Witten effective data and scalar central-charge normalization: Source-proposed pure SU(2) $N=2$ Seiberg–Witten low-energy data.

Massive-fiber BPS saturation: BPS state and central-charge saturation.

Spin-cover Lorentz action and its $\mathrm{S}\mathrm{L}(2,\mathbb{C})$ representation: Positive unitary super-Poincare representation with spectral-domain covariance.

Trace-class operators: Trace class operator.

The supplied-basis trace: Trace of a trace class operator.

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

Seiberg and Witten, Sections 1, 4, and 6.

Tong, Section 2.4.2.

Gaiotto, Moore, and Neitzke, Sections 1, 2.1, and 2.2.

Denef, printed pages 20 and 29 for the directed-intersection convention, and Section 5.3 for the pure-SU(2) two-arrow magnitude and model.