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Wrapped M2 and M5 tensions match the IIA string and D4 tensions

Physical scope

Conditional leading-tension identities for one M2 or M5 wrapped once on the same flat circle, with negligible backreaction; no full M/IIA equivalence or compactification solution is proved.

Statement

Assume the low-energy circle dictionary $R_{11} = g_{s} ℓ_{s}, ℓ_{p}^{3} = g_{s} ℓ_{s}^{3}, ℓ_{s}^{2} = α^{'},$ the adopted M2/M5 tensions, and the leading constant-dilaton tensions $T_{F} = \frac{1}{2 π α^{'}}, T_{D 4} = \frac{1}{(2 π)^{4} g_{s} ℓ_{s}^{5}} .$ For a single M2 or M5 wrapped once on the same flat circle of radius $R_{11} > 0$ , the wrapped tensions obey $2 π R_{11} T_{M 2} = T_{F}, 2 π R_{11} T_{M 5} = T_{D 4} .$ These are conditional parameter identities for the leading tensions. They correspond to double dimensional reduction but do not prove full M/IIA equivalence.

Statement References

Witten, §§1–2.2

Pasti–Sorokin–Tonin, §4

Facts & Assumptions

Given: $g_{s} > 0$ , $ℓ_{s} > 0$ , $α^{'} = ℓ_{s}^{2}$ , one flat circle of circumference $2 π R_{11}$ , and the exact planned/current supplier interfaces below.

[F1]

The current circle-dictionary carrier adopts $R_{11} = g_{s} ℓ_{s}$ , $ℓ_{p}^{3} = g_{s} ℓ_{s}^{3}$ , and $ℓ_{s} = \sqrt{α^{'}}$ , with a flat-circle sector identification and an explicit non-equivalence caveat (M Theory Type IIA Circle Limit Dictionary).

[F2]

The adopted normalization postulate fixes $T_{M 2} = (2 π)^{- 2} ℓ_{p}^{- 3}$ and $T_{M 5} = (2 π)^{- 5} ℓ_{p}^{- 6}$ (M2 M5 tension normalizations).

[F3]

The current Dp action states $μ_{p} = ((2 π)^{p} α^{' (p + 1) / 2})^{- 1}$ and $T_{p} = μ_{p} / g_{s}$ at constant dilaton; this proof uses only its DBI tension clause, not its Wess–Zumino coupling or its explicit local RR-potential crosswalk (Type II Dp Brane DBI WZ Action).

[F4]

The string-tension definition gives $ℓ_{s}^{2} = α^{'}$ and $T_{F} = 1 / (2 π α^{'})$ (String Tension Alpha Prime).

[A1]

Both wrapped objects use the leading tensions, wrap the same flat circle exactly once, and have negligible backreaction.

Proof

Given: [F1]–[F4], $g_{s} > 0$ , $ℓ_{s} > 0$ , and [A1].

1.1givenF1F2F4algebra

Using $R_{11} = g_{s} ℓ_{s}$ , $ℓ_{p}^{3} = g_{s} ℓ_{s}^{3}$ , and [F2], $2 π R_{11} T_{M 2} = (2 π g_{s} ℓ_{s}) (2 π)^{- 2} (g_{s} ℓ_{s}^{3})^{- 1} = (2 π)^{- 1} ℓ_{s}^{- 2} = \frac{1}{2 π α^{'}} = T_{F} .$ The positive $g_{s}$ and $ℓ_{s}$ hypotheses justify the cancellations and inverses.

1.2givenF1F2F3algebra

Since $ℓ_{p}^{6} = (ℓ_{p}^{3})^{2} = g_{s}^{2} ℓ_{s}^{6}$ , the M5 product is $2 π R_{11} T_{M 5} = (2 π g_{s} ℓ_{s}) (2 π)^{- 5} (g_{s}^{2} ℓ_{s}^{6})^{- 1} = (2 π)^{- 4} g_{s}^{- 1} ℓ_{s}^{- 5} .$ By [F3] with $p = 4$ and $α^{'} = ℓ_{s}^{2}$ , $T_{D 4} = (2 π)^{- 4} g_{s}^{- 1} ℓ_{s}^{- 5}$ . Hence the second equality holds.

2.1A1F3step 1.1step 1.2∎

The two equalities are therefore established under the stated source-reported dictionary and leading-tension assumptions; no RR Wess–Zumino term or full compactification equivalence was used.

Remarks

The Dp item has a current escalation for its RR convention, but its own statement separately authorizes the DBI tension clause used here; this proof does not use its RR Wess–Zumino conversion. The result remains a conditional leading-tension identity, not a proof of full M/IIA equivalence.

Wrapped M2 and M5 tensions match the IIA string and D4 tensions

Statement

Statement References

Facts & Assumptions

Proof

Remarks

Depends on

Dependency tree · 2 levels

Sources