Physics theoremStatement: Literature-sourcedProof: AI-generatedPipeline-generatedprecheck passverified 2026-11-05 (gpt-6.1-sol)

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Iib Pq String Bps Tension Covariance

Physical scope

Conditional algebraic covariance of the source-reported classical type-IIB BPS (p,q)-string tension at a constant asymptotic axiodilaton, in the explicit Schwarz charge and Einstein-frame convention. It is not a proof of full quantum S-duality, a quantum Hilbert-space equivalence, or a claim about varying asymptotic moduli.

Statement

Let $τ_{\infty} = C_{0, \infty} + i e^{- Φ_{\infty}} \in H$ be the constant asymptotic type-IIB axiodilaton, and let $Q = (p, q)^{T} \in Z^{2} ∖ {0}$ be a charge label in the ordered NS–NS/RR source basis. Assume the source-reported classical BPS family and its Einstein-frame tension formula

$T_{p, q}^{(E)} = T_{F} \sqrt{e^{Φ_{\infty}} (p - q C_{0, \infty})^{2} + e^{- Φ_{\infty}} q^{2}} = T_{F} \frac{∣ p - q τ_{\infty} ∣}{\sqrt{Im τ_{\infty}}} .$

For every $Λ = (\begin{array}{cc} a & b \\ c & d \end{array}) \in S L (2, Z)$ , transform the asymptotic modulus and charge together by $τ^{'} = (a τ_{\infty} + b) / (c τ_{\infty} + d)$ and $(p^{'}, q^{'})^{T} = Λ (p, q)^{T}$ . Then the normalized central-charge factor transforms by the unit phase $Z_{p^{'}, q^{'}} (τ^{'}) = \frac{∣ c τ_{\infty} + d ∣}{c τ_{\infty} + d} Z_{p, q} (τ_{\infty})$ , and the Einstein-frame tension is invariant. This is a conditional algebraic calculation for the classical protected family, not an assertion that $S L (2, Z)$ is proved to act as an exact equivalence on the complete type-IIB quantum theory.

Statement references

Iib Axiodilaton And Pq String Charge

Schwarz, §2, equations (16)–(21)

Uranga, §14.3.1

Facts & Assumptions

Given: A constant asymptotic $τ_{\infty} \in H$ , an integer source charge $Q = (p, q)^{T}$ , $T_{F} > 0$ , a matrix $Λ \in S L (2, Z)$ , the simultaneous transformations in the Statement, and the source-reported BPS tension formula there.

[F1]

Iib Axiodilaton And Pq String Charge: $τ_{\infty} = C_{0, \infty} + i e^{- Φ_{\infty}}$ , $Im τ_{\infty} > 0$ , $Q = (p, q)^{T}$ in the ordered NS–NS/RR basis, $Q^{'} = Λ Q$ , and $Z_{p, q} = (p - q τ_{\infty}) / \sqrt{Im τ_{\infty}}$ .

[A1]

Adopt Schwarz's source-reported classical source-coupled BPS family: for integer charges $(p, q)$ and constant asymptotic $τ_{\infty}$ , the Einstein-frame tension is $T_{p, q}^{(E)} = T_{F} \sqrt{e^{Φ_{\infty}} (p - q C_{0, \infty})^{2} + e^{- Φ_{\infty}} q^{2}}$ . Primitive charges label the elementary string solutions; a nonprimitive vector denotes the source's neutrally stable multiple-string threshold with multiplicity $\gcd (∣ p ∣, ∣ q ∣)$ . This is a physical premise, not a deduction that the full quantum theory has the proposed duality. Schwarz, §2, equations (16)–(21).

Proof

1.1givenF1A1algebra

Set $τ^{'} = (a τ_{\infty} + b) / (c τ_{\infty} + d)$ and $(p^{'}, q^{'}) = (a p + b q, c p + d q)$ . The lower row $(c, d)$ is nonzero because $a d - b c = 1$ . If $c τ_{\infty} + d = 0$ , then $c = 0$ would force $d = 0$ , while $c \neq 0$ would force $τ_{\infty} = - d / c \in R$ ; both contradict $Im τ_{\infty} > 0$ or the determinant condition. Expanding and using $a d - b c = 1$ gives $(c τ_{\infty} + d) (p^{'} - q^{'} τ^{'}) = (c τ_{\infty} + d) (a p + b q) - (a τ_{\infty} + b) (c p + d q) = p - q τ_{\infty}$ . Hence $p^{'} - q^{'} τ^{'} = (p - q τ_{\infty}) / (c τ_{\infty} + d)$ . Since $Λ$ and $Λ^{- 1}$ have integer entries, a common divisor of $p^{'}, q^{'}$ divides $p, q$ and conversely, so primitivity and the multiple-string gcd are preserved.

2.1givenF1step 1.1algebra

Multiplying numerator and denominator of $τ^{'}$ by $c \overline{τ_{\infty}} + d$ gives $Im τ^{'} = Im ((a τ_{\infty} + b) (c \overline{τ_{\infty}} + d)) / ∣ c τ_{\infty} + d ∣^{2} = (a d - b c) Im τ_{\infty} / ∣ c τ_{\infty} + d ∣^{2} = Im τ_{\infty} / ∣ c τ_{\infty} + d ∣^{2} > 0$ . Thus the transformed modulus remains in $H$ and its positive square-root normalization is defined.

3.1givenF1A1step 1.1step 2.1algebra∎

Combining steps 1.1 and 2.1 yields $Z_{p^{'}, q^{'}} (τ^{'}) = \frac{(p - q τ_{\infty}) / (c τ_{\infty} + d)}{\sqrt{Im τ_{\infty}} / ∣ c τ_{\infty} + d ∣} = \frac{∣ c τ_{\infty} + d ∣}{c τ_{\infty} + d} Z_{p, q} (τ_{\infty})$ . The factor has modulus one, so $∣ Z_{p^{'}, q^{'}} (τ^{'}) ∣ = ∣ Z_{p, q} (τ_{\infty}) ∣$ and [A1] gives $T_{p^{'}, q^{'}}^{(E)} = T_{p, q}^{(E)}$ . Finally, $∣ p - q τ_{\infty} ∣^{2} / Im τ_{\infty} = ((p - q C_{0, \infty})^{2} + q^{2} e^{- 2 Φ_{\infty}}) / e^{- Φ_{\infty}} = e^{Φ_{\infty}} (p - q C_{0, \infty})^{2} + e^{- Φ_{\infty}} q^{2}$ , which verifies the two displayed forms of the tension.

Depends on

Iib Axiodilaton And Pq String Charge