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$N = 2$ SU(2) central-charge mass bound from Seiberg–Witten periods

Physical scope

Conditional representation-theoretic mass bound for an existing massive state in a selected pure-SU(2) N=2 Seiberg–Witten period patch; it asserts neither state existence nor stability, and does not construct the underlying continuum QFT.

Statement

Assume the Axiom of Countable Choice. Use the four-dimensional rigid $N = 2$ super-Poincare convention $η = diag (- 1, 1, 1, 1)$ , $ϵ_{12} = 1$ , and $c = ℏ = 1$ . Let $U$ be a regular patch of the selected pure-SU(2) Seiberg--Witten period data. Fix $u \in U$ and a formal charge label $(n_{e}, n_{m}) \in Z^{2}$ . Suppose the selected physical model has a nonzero unitary rest-fiber representation on a Hilbert space $K$ , with translation generators $P^{0} = M I_{K}$ and $P^{j} = 0$ for $j = 1, 2, 3$ , where $M > 0$ . Suppose its scalar central-charge matrix is $Z^{12} = z I_{K}$ , $Z^{21} = - z I_{K}$ , and its supercharges and actual Hilbert adjoints have a common dense invariant product domain $D$ on which every product in the $N = 2$ charge anticommutators is defined. Require its charge normalization to be

$z = \sqrt{2} Z_{S W} (u; n_{e}, n_{m}) = \sqrt{2} (n_{e} a (u) + n_{m} a_{D} (u)) .$

Then

$M \geq \sqrt{2} ∣ n_{e} a (u) + n_{m} a_{D} (u) ∣ .$

If equality holds, then $∣ z ∣ = M > 0$ ; set $ω = z / ∣ z ∣$ , so $∣ ω ∣ = 1$ and $\overline{ω} z = ∣ z ∣$ , and define $A = Q_{1}^{1} - ω (Q_{2}^{2})^{*}$ and $A^{♯} = (Q_{1}^{1})^{*} - \overline{ω} Q_{2}^{2}$ on $D$ . Both $A ψ$ and $A^{♯} ψ$ vanish for every $ψ \in D$ . Equality defines algebraic BPS saturation only; the statement asserts no state exists for any chosen charge label and no BPS state is stable.

Statement references

Seiberg and Witten, §1, equations (1.1)–(1.2), and §4, equations (4.5)–(4.7).

Bilal, §4.3, equation (4.14).

Facts & Assumptions

Given: The Axiom of Countable Choice, the period data on $U$ , and the massive rest-fiber representation and common domain in the Statement.

[A1]

Assume ${A C}_{ω}$ : every sequence of nonempty sets has a choice function. The Axiom of Countable Choice ( ${A C}_{ω}$ )

[F1]

Positive central-charge bounds on a massive rest representation fiber: under ${A C}_{ω}$ , in a massive rest-fiber representation with $M > 0$ and common invariant product domain, $M \geq ∥ Z ∥_{o p}$ ; for $N = 2$ , $Z^{12} = z I$ gives $∥ Z ∥_{o p} = ∣ z ∣$ , and equality implies the phase-selected null charge pair annihilates every domain vector.

[F2]

Positive unitary super-Poincare representation with spectral-domain covariance: the inner product is positive, each $\overset{ˉ}{Q}$ is the actual Hilbert adjoint with its actual domain, and all displayed generator products are defined on a common dense invariant domain.

[F3]

Pure $N = 2$ SU(2) Coulomb-branch and period data: on a local regular patch, $a$ and $a_{D}$ are the electric and dual period coordinates, and the source-normalized charge-label pairing is $Z_{S W} = n_{e} a + n_{m} a_{D}$ .

[A2]

Source-proposed pure SU(2) $N = 2$ Seiberg–Witten low-energy data: the adopted model convention identifies the scalar algebraic central charge with $z = \sqrt{2} Z_{S W}$ ; the singularity and low-energy solution data are source-proposed model inputs.

Proof

Given: [F1] applies to the stated nonzero massive $N = 2$ rest fiber with its common domain; [F3] and [A2] fix the period pairing and its central-charge normalization.

1.1F3A2givenalgebra

By [F3], the formal period pairing for the chosen label is $Z_{S W} = n_{e} a (u) + n_{m} a_{D} (u)$ . By [A2], the scalar central-charge entry is therefore $z = \sqrt{2} (n_{e} a (u) + n_{m} a_{D} (u))$ .

2.1A1F1step 1.1algebra

For the antisymmetric $N = 2$ matrix $Z = (\begin{array}{cc} 0 & z \\ - z & 0 \end{array})$ , direct multiplication gives $Z^{*} Z = ∣ z ∣^{2} I_{2}$ , hence $∥ Z ∥_{o p} = ∣ z ∣$ . Apply [F1] to obtain $M \geq ∥ Z ∥_{o p} = ∣ z ∣ = \sqrt{2} ∣ n_{e} a (u) + n_{m} a_{D} (u) ∣$ .

3.1F1F2step 2.1∎

If equality holds, then $M = ∣ z ∣$ . The $N = 2$ saturation conclusion in [F1], with $ω = z / ∣ z ∣$ and $\overline{ω} z = ∣ z ∣$ , gives $A ψ = A^{♯} ψ = 0$ for every $ψ \in D$ . The conclusion remains confined to this represented rest fiber: neither [F1] nor the period pairing asserts that the selected charge label occurs as a stable state.

Remarks

The source papers use the period mass normalization for BPS states; this item derives the inequality from the local mathematical positive-representation theorem and records the equality condition without taking BPS saturation as a premise. The formal lattice may contain labels absent from the stable spectrum.

$N = 2$ SU(2) central-charge mass bound from Seiberg–Witten periods

Statement

Statement references

Facts & Assumptions

Proof

Remarks

Depends on

Dependency tree · 2 levels

Sources

N=2N=2 SU(2) central-charge mass bound from Seiberg–Witten periods

Statement

Statement references

Facts & Assumptions

Proof

Remarks

Depends on

Dependency tree · 2 levels

Sources

$N = 2$ SU(2) central-charge mass bound from Seiberg–Witten periods