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Monopole–dyon two-arrow Hall sector

Physical scope

A conditional pure-SU(2) monopole/dyon charge example whose local proof counts finite-field representations and their Hall groupoid weight; the finite-field coefficient is not identified with a physical second-helicity index, and no BPS state or continuum QFT is asserted to exist.

Example

Use the pure-SU(2) physical charge datum with monopole label $γ_{m} = (0, 1)$ and dyon label $γ_{d} = (1, - 1)$ . Their source-normalized central charges are $z_{m} (u) = \sqrt{2} a_{D} (u)$ and $z_{d} (u) = \sqrt{2} (a (u) - a_{D} (u))$ , and their chosen Dirac pairing is $⟨ γ_{m}, γ_{d} ⟩_{D S Z} = 2$ . For a prime power $q > 1$ , take the finite acyclic quiver with vertices $m, d$ and two arrows from $d$ to $m$ :

The charge map sends the two vertex basis vectors to $γ_{m}, γ_{d}$ ; it matches the quiver skew Euler form because both pairings give $2$ on the ordered pair $(m, d)$ . Suppose regular points $u_{-}, u_{+} \in U$ and a real phase rotation $α$ are supplied such that both rotated central charges lie in $H = {z \in C : Im z > 0}$ at both points, with $ϕ_{m} (u_{-}) < ϕ_{d} (u_{-})$ and $ϕ_{d} (u_{+}) < ϕ_{m} (u_{+})$ , where $ϕ_{i} (u) = \arg (e^{- i α} z_{i} (u)) / π$ . Among representations of dimension $(1, 1)$ , the nonzero arrow pairs have $q + 1$ stable isomorphism classes, each with automorphism group of order $q - 1$ , while the zero pair is unstable; the skew-integrated stable-factor coefficient is therefore $(q + 1) / (q - 1)$ because the Euler self-pairing of $(1, 1)$ is zero. At $u_{+}$ , there is no semistable representation of that dimension. The complete finite-field Hall generators nevertheless agree for the two charge assignments. This rational groupoid weight is not a physical second-helicity supertrace. If all hypotheses of the A-page theorem “Conditional physical BPS wall crossing from an exact Hall bridge” hold for these points—including cone-supported integer-valued nonzero index support with phases defined at both points, completed charge-series topologies, and the exact continuous per-phase bridge—then the finite-field Hall equality transfers to the physical KS automorphisms; this example does not assert those additional inputs.

Example references

Physical charge labels, central charge, and DSZ pairing: Pure SU(2) physical BPS charge and index data.

Conditional Hall-to-physical wall-crossing theorem: Conditional physical BPS wall crossing from an exact Hall bridge.

Finite-quiver stability data: Finite-field stability data for an acyclic quiver.

Skew integration and completed finite-quiver Hall generators: Finite-quiver Hall spectrum generators.

Completed finite-quiver Hall spectrum generator: Finite-quiver Hall spectrum generators.

The phase of a sum of upper-half-plane charges: The phase of a sum lies between upper-half-plane phases.

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

Denef, Section 5.3.

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

Facts & Assumptions

Given: ${A C}_{ω}$ , a prime power $q > 1$ , the two-arrow quiver above, a charge map on the two vertex basis vectors, and regular points $u_{-}, u_{+}$ with the phase orders stated in the Example.

[A1]

Assume Countable Choice, ${A C}_{ω}$ , as inherited through the physical charge datum and conditional transport theorem. The finite-field representation enumeration below uses no choice. The Axiom of Countable Choice ( ${A C}_{ω}$ )

[F1]

Finite-field stability data for an acyclic quiver: a finite-quiver representation assigns a finite-dimensional $F_{q}$ vector space to each vertex and a linear map to each arrow; stability means every proper nonzero subrepresentation has strictly smaller phase than the whole representation.

[F2]

Finite-quiver Hall spectrum generators: the skew integration sends each Hall basis class to $J (δ_{M}) = t^{χ_{Q} (d (M), d (M))} ∣ Aut (M) ∣^{- 1} x^{d (M)}$ .

[F3]

Pure SU(2) physical BPS charge and index data: $γ_{m} = (0, 1)$ , $γ_{d} = (1, - 1)$ , the scalar central charges are $z_{m} = \sqrt{2} a_{D}$ and $z_{d} = \sqrt{2} (a - a_{D})$ , and $⟨ γ_{m}, γ_{d} ⟩_{D S Z} = 2$ .

[F4]

The phase of a sum lies between upper-half-plane phases: for two nonzero upper-half-plane charges with strictly ordered phases, the phase of their sum lies strictly between them.

[F5]

Hall wall crossing preserves the completed spectrum generators: for a finite acyclic quiver over $F_{q}$ , the completed skew generators for any two upper-half-plane charge assignments are equal; this is a formal finite-field identity only.

[F6]

Conditional physical BPS wall crossing from an exact Hall bridge: under its cone-support, integer-index, phase-domain, and degree-completion hypotheses, an exact continuous per-phase Hall-to-Hilbert bridge preserving charge, pairing, coefficients, and phase order transfers the finite-field equality to the physical KS automorphisms. This example does not assert that those additional hypotheses or the bridge are supplied.

Verification

Given: The items and hypotheses in the Statement and [A1], [F1]–[F6].

1.1F1constructalgebra

In dimension $(1, 1)$ each vertex space is $F_{q}$ . The two arrow maps are a pair $(r, s) \in F_{q}^{2}$ , and a change of bases $(g_{m}, g_{d}) \in (F_{q}^{\times})^{2}$ sends $(r, s)$ to $(g_{m} r g_{d}^{- 1}, g_{m} s g_{d}^{- 1})$ . Hence the zero pair is one orbit and the nonzero pairs are classified by their one-dimensional spans in $F_{q}^{2}$ , giving $∣ P^{1} (F_{q}) ∣ = (q^{2} - 1) / (q - 1) = q + 1$ nonzero isomorphism classes.

2.1F1step 1.1algebra

For a nonzero pair, an automorphism must satisfy $g_{m} r = r g_{d}$ and $g_{m} s = s g_{d}$ ; at least one arrow scalar is nonzero, so $g_{m} = g_{d}$ and the automorphism group has order $q - 1$ . For the zero pair every pair of base changes is allowed, so its automorphism group has order $(q - 1)^{2}$ .

2.2F1F4step 1.1

The subrepresentation supported only at $m$ exists for every representation because both arrows have source $d$ . The subrepresentation supported only at $d$ exists exactly for the zero pair. At $u_{-}$ , where $ϕ_{m} < ϕ_{d}$ , [F4] places the total phase between the two vertex phases; thus every nonzero pair has only the lower-phase proper subrepresentation at $m$ and is stable, whereas the zero pair has the higher-phase subrepresentation at $d$ and is unstable. At $u_{+}$ , where $ϕ_{d} < ϕ_{m}$ , the subrepresentation at $m$ has higher phase than the total for every pair, so none is semistable.

3.1F2step 1.1step 2.1step 2.2algebra

At $u_{-}$ , the stable groupoid cardinality is the sum of $q + 1$ weights $1 / (q - 1)$ , namely $(q + 1) / (q - 1)$ . For $d = (1, 1)$ and two arrows $d \to m$ , $χ_{Q} (d, d) = 1 + 1 - 2 = 0$ , so the skew integration contributes the factor $t^{0} = 1$ and leaves this exact coefficient.

4.1A1F3F5F6step 3.1algebra∎

The charge map preserves the skew form on the basis pair: the two-arrow orientation gives $⟨ e_{m}, e_{d} ⟩_{Q} = 2$ , equal to $⟨ γ_{m}, γ_{d} ⟩_{D S Z}$ by [F3]. Applying [F5] to the two phase assignments proves equality of the full completed Hall generators. If the bridge in [F6] is supplied with these exact phases and pairings, the physical KS products agree by the conditional theorem. The coefficient $(q + 1) / (q - 1)$ remains the finite-field groupoid weight computed in step 3.1; no step identifies it with the physical index $Ω_{u} (γ_{m} + γ_{d})$ .

Monopole–dyon two-arrow Hall sector

Example

Example references

Facts & Assumptions

Verification

Verification sources

Depends on

Dependency tree · 2 levels

Sources