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IIA D0 and First Kaluza-Klein Momentum Mass Match

Physical scope

Conditional equality of the leading single-D0 rest energy and the first Kaluza-Klein momentum magnitude on a flat asymptotic IIA/M-theory circle in hbar=c=1 units; assumes the stated D0/KK sector identification and says nothing about completeness or exact duality.

Statement

Assume $ℏ = c = 1$ , positive string length $ℓ_{s} = \sqrt{α^{'}}$ , positive dimensionless coupling $g_{s}$ , and a flat asymptotic circle of radius $R_{11} = g_{s} ℓ_{s}$ with angle $θ \sim θ + 2 π$ . Put $y = R_{11} θ$ and define the $n$ th Kaluza-Klein mode by its single-valued Fourier wave $e^{i n θ}$ with momentum operator $p_{11} = - i \partial_{y}$ . For a free massless eleven-dimensional mode, adopt the flat-space mass shell $E^{2} = ∣ p ∣^{2} + p_{11}^{2}$ , where $p \in R^{9}$ is the momentum in the noncompact spatial directions. Define its ten-dimensional rest mass to be the nonnegative rest energy $m_{K K, n} : = E ∣_{p = 0} \geq 0$ , so the mass shell gives $m_{K K, n} = ∣ p_{11}^{(n)} ∣$ . Adopt the leading Abelian D-brane tension formula $T_{p} = μ_{p} / g_{s}$ with $μ_{p} = ((2 π)^{p} α^{' (p + 1) / 2})^{- 1}$ , and identify the single unit D0 sector with the first Kaluza-Klein mode. Then

T_{0} = m_{K K, 1} = ∣ p_{11}^{(1)} ∣ = \frac{1}{g_{s} ℓ_{s}} = \frac{1}{R_{11}} .

This conditional mass match proves neither a complete Kaluza–Klein tower, full M/IIA equivalence, nor a nonperturbative M-theory construction.

Statement references

Witten, String Theory Dynamics in Various Dimensions, §§1–2.2

Banks, Fischler, Shenker, and Susskind, M Theory As A Matrix Model: A Conjecture, §§1, 3

Polchinski, Chaudhuri, and Johnson, Notes on D-Branes, Lecture III §3.3

Townsend, Four Lectures on M-theory, §§1–3

Facts & Assumptions

Given: The positive circle radius, positive string length and coupling, periodic angle, leading D-brane normalization, free massless eleven-dimensional mass shell, and adopted unit D0/first-KK sector identification in the statement.

[F1]

The adopted M/IIA circle dictionary states $R_{11} = g_{s} ℓ_{s}$ with $g_{s}, ℓ_{s} > 0$ (M Theory Type IIA Circle Limit Dictionary).

[F2]

The leading Abelian D $p$ -brane action uses $α^{'} > 0$ , $μ_{p} = ((2 π)^{p} α^{' (p + 1) / 2})^{- 1}$ and $T_{p} = μ_{p} / g_{s}$ (Type II Dp Brane DBI WZ Action).

[A1]

The asymptotic eleven-dimensional mode is free and massless, so its flat-space energy satisfies $E^{2} = ∣ p ∣^{2} + p_{11}^{2}$ ; in the ten-dimensional rest frame $p = 0$ , its rest mass is $∣ p_{11} ∣$ . The single D0/first-mode sector identification is adopted, not derived.

Proof

technique · direct arithmetic

1.1givenA1F1algebra

Periodicity makes $e^{i n θ}$ single-valued exactly for $n \in Z$ . With $y = R_{11} θ$ , applying $p_{11} = - i \partial_{y}$ gives $p_{11} e^{i n θ} = (n / R_{11}) e^{i n θ}$ . Thus the unit mode has $∣ p_{11}^{(1)} ∣ = 1 / R_{11} = 1 / (g_{s} ℓ_{s})$ by [F1], with inverse-length units.

1.2givenF2algebra

Set $p = 0$ in [F2]. Then $(2 π)^{0} = 1$ and $α^{' (0 + 1) / 2} = \sqrt{α^{'}} = ℓ_{s}$ , so $μ_{0} = 1 / ℓ_{s}$ and $T_{0} = μ_{0} / g_{s} = 1 / (g_{s} ℓ_{s})$ .

2.1givenA1step 1.1algebra

In the ten-dimensional rest frame $p = 0$ , [A1] reduces the eleven-dimensional mass shell to $E^{2} = p_{11}^{2}$ . The positive rest energy is therefore $m_{K K, 1} = ∣ p_{11}^{(1)} ∣ = 1 / R_{11}$ by step 1.1.

3.1givenA1step 2.1step 1.2algebra∎

Steps 2.1 and 1.2 give $T_{0} = m_{K K, 1} = ∣ p_{11}^{(1)} ∣ = 1 / (g_{s} ℓ_{s}) = 1 / R_{11}$ . This equality is conditional on the sector identification in [A1]; it neither proves that identification nor any tower-completeness or duality claim.

IIA D0 and First Kaluza-Klein Momentum Mass Match

Statement

Statement references

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources