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Iib Self Dual Point S Exchanges F1 And D1

Physical scope

A single exact algebraic check of the source-reported classical IIB (p,q)-string tension at the self-dual asymptotic modulus tau=i. It illustrates the F1/D1 charge exchange and unit phase only; it establishes no complete quantum S-duality.

Example

In the ordered NS–NS/RR source basis, take the primitive F-string charge $Q = (1, 0)^{T}$ and the S generator $S = (\begin{array}{cc} 0 & - 1 \\ 1 & 0 \end{array}) \in S L (2, Z)$ . At the self-dual asymptotic axiodilaton $τ_{\infty} = i$ (equivalently $C_{0, \infty} = 0$ and $Φ_{\infty} = 0$ ), the transformed charge is the D-string label $(0, 1)^{T}$ and the normalized central charge changes from $1$ to $- i$ . Their classical Einstein-frame tensions both equal the positive reference tension $T_{F}$ .

Example references

Iib Pq String Bps Tension Covariance

Schwarz, §2, equations (8)–(21)

Uranga, §14.3.1

Facts & Assumptions

Given: $τ_{\infty} = i$ , $Q = (1, 0)^{T}$ , $S = (\begin{array}{cc} 0 & - 1 \\ 1 & 0 \end{array})$ , and the classical source-family scope of the preceding theorem.

[F1]

Iib Pq String Bps Tension Covariance: for the stated classical source family, $Z_{p, q} = (p - q τ_{\infty}) / \sqrt{Im τ_{\infty}}$ , $Z_{p^{'}, q^{'}} (τ^{'}) = \frac{∣ c τ_{\infty} + d ∣}{c τ_{\infty} + d} Z_{p, q} (τ_{\infty})$ , and the Einstein-frame tension is $T_{F} ∣ Z_{p, q} ∣$ .

[A1]

The selected inputs are the primitive charge $(p, q) = (1, 0)$ , self-dual modulus $τ_{\infty} = i$ , and S matrix $(\begin{array}{cc} 0 & - 1 \\ 1 & 0 \end{array})$ ; this is one classical point in the source's protected family.

Verification

1.1givenA1algebra

The determinant of $S$ is $1$ , $S (1, 0)^{T} = (0, 1)^{T}$ , and $- 1 / i = i$ , so the transformed charge is primitive and $τ^{'} = i$ .

2.1givenF1A1step 1.1algebra∎

Since $Im i = 1$ , $Z_{1, 0} (i) = 1$ and $Z_{0, 1} (i) = - i$ . The phase factor is $∣ i ∣ / i = - i$ , whose modulus is one; therefore [F1] gives $T_{1, 0}^{(E)} = T_{F} = T_{0, 1}^{(E)}$ . This exact one-point calculation does not prove full quantum equivalence.

Depends on

Iib Pq String Bps Tension Covariance