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Strong Constraint DFT NS NS Action And Section Reduction

Physical scope

Smooth ten-dimensional massless NS–NS fields in a local doubled chart, subject to the pairwise strong constraint and a fixed supergravity section; no RR fields, localized sources, loops, higher derivatives, strong-constraint-violating twists, global T-fold existence, or full string-CFT equivalence.

Statement

Let $D = 10$ and use doubled coordinates $X^{M} = (x^{i}, {\tilde{x}}_{i})$ , $i = 0, \dots, 9$ , all with length dimension, with dimensionless $η_{M N} = (\begin{array}{cc} 0 & I_{10} \\ I_{10} & 0 \end{array})$ and inverse-length derivatives. Let $H_{M N}$ be a symmetric generalized metric satisfying $H_{M P} η^{P Q} H_{Q N} = η_{M N}$ , and let $d$ be the dimensionless generalized dilaton. Require every field, gauge parameter, and product of such objects $A, B$ to obey the strong constraint $η^{M N} \partial_{M} \partial_{N} A = 0$ and $η^{M N} (\partial_{M} A) (\partial_{N} B) = 0$ . Whenever a trivial parameter $ξ^{M} = \partial^{M} χ$ is used in the gauge-algebra statement, also require its scalar potential $χ$ to belong to this strongly constrained collection. With compactly supported gauge parameters or vanishing boundary flux, the two-derivative NS–NS DFT action

S_{D F T} = \frac{1}{2 κ_{0}^{2}} \int d^{20} X e^{- 2 d} [\frac{1}{8} H^{M N} \partial_{M} H^{K L} \partial_{N} H_{K L} - \frac{1}{2} H^{M N} \partial_{N} H^{K L} \partial_{L} H_{M K} - 2 \partial_{M} d \partial_{N} H^{M N} + 4 H^{M N} \partial_{M} d \partial_{N} d]

is invariant under generalized diffeomorphisms, and their gauge algebra closes modulo trivial parameters. Its integrand has dimension length $^{- 2}$ , so the full doubled integral has dimension length $^{10}$ after the factor $κ_{0}^{- 2}$ ; the section statement below divides by the dual-coordinate volume of dimension length $^{10}$ . On a fixed local supergravity section ${\tilde{\partial}}^{i} = 0$ , set $G : = G^{(s)}$ in the string-frame convention, parameterize $H$ by $G$ and $B_{2}$ , and set $e^{- 2 d} = \sqrt{- G} e^{- 2 Φ}$ . The action per unit dual-coordinate volume then reduces, up to a total derivative, to the dimensionless source-free leading tree-level two-derivative string-frame NS–NS action. This is a local massless-field result; it does not assert global T-fold existence, independent dependence on all doubled coordinates, RR/source terms, higher-derivative corrections, string loops, or an uplift for every gauged supergravity.

Statement references

Hohm, Hull, and Zwiebach, §§3–4.3

Aldazabal, Marqués, and Núñez, §§3.3–3.7

Facts & Assumptions

Given: Smooth ten-dimensional Lorentzian $G^{(s)}$ , smooth $B_{2}$ and $Φ$ , the DFT field and transformation conventions, the pairwise strong constraint (including any scalar potential $χ$ used for a trivial parameter), a fixed local physical section, and, for the integrated gauge-invariance claim, a compactly supported gauge parameter or vanishing boundary flux for the gauge-variation divergence. The section reduction is stated modulo the total derivatives specified in [F4].

[F1]

Strong Constraint DFT Generalized Metric Data: $X^{M} = (x^{i}, {\tilde{x}}_{i})$ , $η$ is off-diagonal, $H$ is the block generalized metric built from a local metric $g$ and $B_{2}$ , and $e^{- 2 d} = \sqrt{- g} e^{- 2 Φ}$ on the physical section; the stated strong constraint applies to all fields, gauge parameters, and products.

[F2]

Strong Constraint DFT Generalized Gauge Algebra Closure: under [F1]'s strong constraint, generalized diffeomorphisms close with the defined C-bracket modulo parameters $ξ^{M} = \partial^{M} χ$ whose potentials are themselves in the strongly constrained scalar collection and which act trivially.

[F3]

Strong Constraint DFT Scalar and Action Gauge Invariance: the generalized-metric action is $\int e^{- 2 d} R$ up to an explicit total divergence, and is invariant up to boundary terms because $δ_{ξ} R = ξ^{M} \partial_{M} R$ .

[F4]

Strong Constraint DFT Supergravity Section Reduction: on a product coordinate domain $U_{x} \times \tilde{U}$ with $0 < V_{\tilde{U}} : = \int_{\tilde{U}} d^{10} \tilde{x} < \infty$ , the normalized action $S_{D F T} = (2 κ_{0}^{2})^{- 1} \int L_{D F T}$ divided by $V_{\tilde{U}}$ reduces on ${\tilde{\partial}}^{i} = 0$ to the string-frame NS–NS action, modulo the supplier's explicit total derivatives and wherever the integrals exist. The NS–NS action is adopted in [A2].

[A1]

Supergravity Conventions And Normalizations: in ten dimensions $G^{(s)}$ is the string-frame metric, the signature is mostly plus, $ℏ = c = 1$ , and $2 κ_{0}^{2} = (2 π)^{7} α^{' 4}$ .

[A2]

Ten Dimensional NS NS Supergravity Action: adopt the source-free leading tree-level two-derivative NS–NS action in terms of $G^{(s)}$ , $Φ$ , and $H_{3} = d B_{2}$ , with no RR fields, localized sources, loops, or higher powers of $α^{'}$ .

Proof

1.1givenF1F2F3algebra

By [F2], the generalized diffeomorphism commutator closes modulo trivial parameters on the DFT data in [F1]. By [F3], the action density changes only by a total divergence under the same strong constraint. The compact-support or boundary condition in the Given clause removes the integrated divergence, so the DFT action is invariant.

2.1givenA1A2F4step 1.1algebra∎

On the section ${\tilde{\partial}}^{i} = 0$ , evaluate the local action per unit dual-coordinate volume on a product $U_{x} \times \tilde{U}$ with $0 < V_{\tilde{U}} < \infty$ , wherever the integrals exist. The section density is independent of $\tilde{x}$ , so $S_{D F T} / V_{\tilde{U}} = (2 κ_{0}^{2})^{- 1} \int_{U_{x}} L_{D F T} ∣_{\tilde{\partial} = 0}$ . This expression is independent of the chosen finite positive dual volume; it is the meaning of the local per-unit-volume action, without an infinite-volume quotient. Apply [F4]'s section calculation to this expression with [A1]–[A2]. It gives the string-frame action density and normalization in the Statement, with the stated total-derivative qualification. Combining this result with step 1.1 establishes the DFT claim under the adopted NS–NS action premise; this proof does not derive that physical premise or claim independent proof approval for its supplier.

Strong Constraint DFT NS NS Action And Section Reduction

Statement

Statement references

Facts & Assumptions

Proof

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