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String Einstein Frame NS Action Calculation

Physical scope

Pointwise conformal rewrite of the adopted source-free ten-dimensional tree-level two-derivative NS–NS action. The full divergence term is retained; removing its integrated value requires an actual zero-flux boundary condition. The result does not add RR fields, sources, loops, or higher-derivative corrections.

Statement

Let $G_{s}$ and $G_{E}$ be smooth Lorentzian metrics on the same oriented ten-manifold, let $Φ$ be a smooth real scalar, and let $B_{2}$ be a smooth local two-form potential with $H_{3} = d B_{2}$ . Use the supergravity conventions fixed by the earlier definition def-supergravity-conventions-and-normalizations, set $G_{s} = e^{Φ / 2} G_{E}, X_{E} : = G_{E}^{M N} \partial_{M} Φ \partial_{N} Φ, H_{E}^{2} : = H_{M N P} H_{Q R S} G_{E}^{M Q} G_{E}^{N R} G_{E}^{P S},$ and $□_{E} Φ : = G_{E}^{M N} \nabla_{M}^{(E)} \nabla_{N}^{(E)} Φ$ . Use the signed form norm $∣ F_{p} ∣_{G}^{2} = (1 / p!) F_{M_{1} \dots M_{p}} F^{M_{1} \dots M_{p}}$ and the curvature convention with Ricci contraction $R_{Q N} = R^{P}_{Q P N}$ from the earlier supergravity conventions definition. In length-valued coordinates, the adopted action normalization makes $B_{M N}$ dimensionless and $H_{M N P}$ of dimension length $^{- 1}$ . Then $\sqrt{- G_{s}} = e^{5 Φ / 2} \sqrt{- G_{E}}, R_{s} = e^{- Φ / 2} (R_{E} - \frac{9}{2} □_{E} Φ - \frac{9}{2} X_{E}), ∣ d Φ ∣_{G_{s}}^{2} = e^{- Φ / 2} X_{E}, H_{M N P} H_{s}^{M N P} = e^{- 3 Φ / 2} H_{E}^{2},$ so that $\sqrt{- G_{s}} e^{- 2 Φ} = e^{Φ / 2} \sqrt{- G_{E}}$ . Substitution into the adopted action post-ten-dimensional-ns-ns-supergravity-action gives the full local Einstein-frame density $S_{N S} = \frac{1}{2 κ_{0}^{2}} \int d^{10} x \sqrt{- G_{E}} (R_{E} - \frac{1}{2} X_{E} - \frac{1}{12} e^{- Φ} H_{E}^{2} - \frac{9}{2} □_{E} Φ),$ on every region where the action integrals are defined. The last term is retained in this identity. In local coordinates, $\sqrt{- G_{E}} □_{E} Φ = \partial_{M} ⁣ (\sqrt{- G_{E}} G_{E}^{M N} \partial_{N} Φ) .$ It is a local total divergence; its integrated value may be removed only when its flux integral vanishes. Its first variation vanishes for compactly supported field variations, so it does not change the bulk Euler–Lagrange equations.

Statement references

Supergravity Conventions And Normalizations

Ten Dimensional NS NS Supergravity Action

Facts & Assumptions

Given: The smooth ten-dimensional fields, conformal frame relation, action and norm conventions, and the curvature convention stated above.

[F1]

In ten dimensions the convention is $G_{E} = e^{- Φ / 2} G_{s}$ , $H = d B$ , $X_{E} = ∣ d Φ ∣_{G_{E}}^{2}$ , and $∣ H ∣_{G}^{2} = H_{M N P} H^{M N P} / 3!$ (Supergravity Conventions And Normalizations).

[A1]

The adopted string-frame action has density $\sqrt{- G_{s}} e^{- 2 Φ} [R_{s} + 4 ∣ d Φ ∣_{G_{s}}^{2} - \frac{1}{12} H_{M N P} H_{s}^{M N P}]$ with coefficient $(2 κ_{0}^{2})^{- 1}$ (Ten Dimensional NS NS Supergravity Action).

[F2]

For $\tilde{g} = e^{2 σ} g$ in dimension $n$ , the scalar curvature satisfies $R_{\tilde{g}} = e^{- 2 σ} (R_{g} - 2 (n - 1) □_{g} σ - (n - 1) (n - 2) ∣ d σ ∣_{g}^{2})$ (Scalar Curvature Under Conformal Rescaling).

[F3]

The chart integral of a compactly supported smooth coordinate divergence is zero (Compactly Supported Metric Variation of a Weighted Scalar-Curvature Action).

Proof

technique · direct; scale the determinant and inverse metrics, apply the general Weyl-curvature identity, substitute each transformed term, and keep track of the boundary divergence

1.1givenF1algebra

From $G_{s} = e^{Φ / 2} G_{E} = e^{2 σ} G_{E}$ with $σ = Φ / 4$ , each inverse metric contributes $e^{- Φ / 2}$ and the square-root determinant contributes $e^{10 Φ / 4} = e^{5 Φ / 2}$ . Therefore $\sqrt{- G_{s}} = e^{5 Φ / 2} \sqrt{- G_{E}}$ , $∣ d Φ ∣_{G_{s}}^{2} = e^{- Φ / 2} X_{E}$ , and the three raised indices in $H_{M N P} H_{s}^{M N P}$ give $e^{- 3 Φ / 2} H_{E}^{2}$ . Multiplication by $e^{- 2 Φ}$ gives the measure factor $e^{Φ / 2} \sqrt{- G_{E}}$ .

2.1F1F2step 1.1algebra

Apply [F2] with $n = 10$ , $g = G_{E}$ , and $σ = Φ / 4$ . Then $e^{- 2 σ} = e^{- Φ / 2}$ , $2 (n - 1) □_{E} σ = 18 (□_{E} Φ) / 4 = (9 / 2) □_{E} Φ$ , and $(n - 1) (n - 2) ∣ d σ ∣_{E}^{2} = 72 X_{E} / 16 = (9 / 2) X_{E}$ . This gives the stated scalar-curvature transformation with its signs fixed by [F1].

3.1A1F1step 1.1step 2.1algebra

Substitute steps 1.1 and 2.1 into [A1]. The curvature term acquires unit net exponential factor, the dilaton kinetic term contributes $+ 4 X_{E}$ , and the three-form term acquires $e^{- Φ}$ . Combining the curvature gradient contribution $- \frac{9}{2} X_{E}$ with $+ 4 X_{E}$ gives $- \frac{1}{2} X_{E}$ ; the curvature divergence remains $- \frac{9}{2} □_{E} Φ$ . Thus the complete transformed density is exactly the one in the Statement, including the coefficient $- 1 / 12$ of $e^{- Φ} H_{E}^{2}$ .

4.1givenF1F3step 3.1algebra∎

The Levi-Civita density identity is $\sqrt{- G_{E}} □_{E} Φ = \partial_{M} (\sqrt{- G_{E}} G_{E}^{M N} \partial_{N} Φ)$ , obtained by expanding $□_{E} Φ = G_{E}^{M N} (\partial_{M} \partial_{N} Φ - Γ^{P}_{M N} \partial_{P} Φ)$ and using $Γ^{M}_{M P} = \partial_{P} \log \sqrt{- G_{E}}$ . Hence the final term is pointwise a coordinate divergence. Under a smooth compactly supported variation of $G_{E}, Φ$ , the variation of its vector-density components $\sqrt{- G_{E}} G_{E}^{M N} \partial_{N} Φ$ is compactly supported; its divergence integrates to zero by [F3]. This proves the asserted bulk-variation statement. The value of the unvaried divergence integral is removable only when its flux integral is actually zero; no assumption of boundarylessness alone is used for a noncompact region.

String Einstein Frame NS Action Calculation

Statement

Statement references

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources