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Heterotic And Type I Tree Weight Ratios

Physical scope

A symbolic evaluation of the adopted heterotic and Type-I tree-level dilaton multipliers. It is not a solution of either theory, a comparison of fully normalized action coefficients, or evidence for S-duality.

Example

For the adopted branch weights, set the heterotic dilaton value to $Φ_{H} = 0$ and the Type-I dilaton value to $Φ_{I} = \log 2$ . The heterotic closed gravity/dilaton and gauge multipliers are both $1$ . The Type-I closed gravity/dilaton multiplier is $1 / 4$ and its open-string gauge multiplier is $1 / 2$ , so the Type-I open-string gauge multiplier divided by the Type-I closed gravity/dilaton multiplier is $2$ . The two field values are evaluated independently; this does not identify heterotic and Type-I dilatons or compare their fully normalized actions.

Statement references

Washington Taylor, §§2.2 and 2.4

Mohammad R. Garousi, §1

Facts & Assumptions

Given: Evaluate only the positive dimensionless multiplier functions at $Φ_{H} = 0$ and $Φ_{I} = \log 2$ .

[F1]

Heterotic Type I Ten Dimensional N1 Supergravity Action: the heterotic closed and gauge multipliers are $e^{- 2 Φ_{H}}$ ; the Type-I closed multiplier is $e^{- 2 Φ_{I}}$ and the Type-I open gauge multiplier is $e^{- Φ_{I}}$ . The postulate treats the branches separately and does not fix common action coefficients.

Verification

1.1givenF1algebra

Substituting $Φ_{H} = 0$ into both heterotic multiplier functions gives $e^{- 2 \cdot 0} = 1$ , so their ratio is $1 / 1 = 1$ .

1.2givenF1algebra

Substituting $Φ_{I} = \log 2$ gives the Type-I closed multiplier $e^{- 2 \log 2} = 2^{- 2} = 1 / 4$ and open gauge multiplier $e^{- \log 2} = 2^{- 1} = 1 / 2$ .

2.1step 1.2algebra∎

Since $1 / 4 > 0$ , the Type-I open-string gauge multiplier divided by the Type-I closed gravity/dilaton multiplier is $(1 / 2) / (1 / 4) = 2$ . These are only dilaton multipliers; the calculation makes no statement about unlisted coefficients or a duality map.

Depends on

Heterotic Type I Ten Dimensional N1 Supergravity Action