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Strong Constraint DFT Supergravity Section Reduction

Physical scope

Local two-derivative NS–NS DFT action evaluated on the strong-constraint supergravity section in the stated string-frame and curvature conventions; no global compactification, sources, RR fields, higher derivatives, loops, or doubled-coordinate dependence is asserted.

Statement

Let $D = 10$ and use the generalized-metric DFT data of the preceding definition, with ${\tilde{\partial}}^{i} = 0$ . On a product coordinate domain $U_{x} \times \tilde{U}$ , with $0 < V_{\tilde{U}} : = \int_{\tilde{U}} d^{10} \tilde{x} < \infty$ , fix the action normalization $S_{D F T} : = (2 κ_{0}^{2})^{- 1} \int_{U_{x} \times \tilde{U}} d^{20} X L_{D F T}$ , where $L_{D F T}$ is the preceding density and $2 κ_{0}^{2} = (2 π)^{7} α^{' 4}$ . All integrals are asserted where defined. Set $G : = G^{(s)}$ , parameterize $H$ by $G, B_{2}$ , and set $e^{- 2 d} = \sqrt{- G} e^{- 2 Φ}$ . Under the adopted source-free leading tree-level two-derivative NS–NS action, the action $S_{D F T} / V_{\tilde{U}}$ reduces, modulo the integrated total derivatives below, to $(2 κ_{0}^{2})^{- 1} \int_{U_{x}} d^{10} x$ times the density $\sqrt{- G^{(s)}} e^{- 2 Φ} [R_{s} + 4 ∣ d Φ ∣_{G^{(s)}}^{2} - (1 / 12) H_{i j k} H_{G^{(s)}}^{i j k}]$ , where $H_{3} = d B_{2}$ . Here $R_{s}$ uses $Γ^{P}_{M N} = \frac{1}{2} G^{P Q} (\partial_{M} G_{N Q} + \partial_{N} G_{M Q} - \partial_{Q} G_{M N})$ , $R^{P}_{Q M N} = \partial_{M} Γ^{P}_{N Q} - \partial_{N} Γ^{P}_{M Q} + Γ^{P}_{M R} Γ^{R}_{N Q} - Γ^{P}_{N R} Γ^{R}_{M Q}$ , $R_{Q N} = R^{P}_{Q P N}$ , and $R_{s} = G^{(s) Q N} R_{Q N}$ . This is a local section calculation; the action remains an adopted physical premise.

Statement references

Hohm, Hull, and Zwiebach, §§4.1–4.2

Aldazabal, Marqués, and Núñez, §§3.3 and 3.6

Facts & Assumptions

Given: Smooth fields on the local section, the DFT generalized metric and density definitions, the normalized action and finite dual-coordinate volume specified in the Statement, ${\tilde{\partial}}^{i} = 0$ , and, when asserting equality of integrated actions, vanishing boundary flux for the DFT divergence in [F2] and the dilaton divergence below. Compact support of variations is used only for variational statements and does not by itself remove these field-dependent boundary fluxes.

[F1]

Strong Constraint DFT Generalized Metric Data: $H_{M N}$ is the block generalized metric built from a local metric $g$ , $B_{2}$ , and $e^{- 2 d} = \sqrt{- g} e^{- 2 Φ}$ on the physical section; this item specializes $g$ to $G^{(s)}$ using [A1].

[F2]

Strong Constraint DFT Scalar and Action Gauge Invariance: the density identity is $L_{D F T} = e^{- 2 d} R + \partial_{M} ⁣ [e^{- 2 d} (\partial_{N} H^{M N} - 4 H^{M N} \partial_{N} d)]$ ; its integrated use requires that flux to vanish at the boundary.

[A1]

Supergravity Conventions And Normalizations: in ten dimensions $G^{(s)}$ is the mostly-plus string-frame metric, $ℏ = c = 1$ , $2 κ_{0}^{2} = (2 π)^{7} α^{' 4}$ , $Γ^{P}_{M N} = \frac{1}{2} G^{P Q} (\partial_{M} G_{N Q} + \partial_{N} G_{M Q} - \partial_{Q} G_{M N})$ , and $R_{s}$ is formed from $R^{P}_{Q M N} = \partial_{M} Γ^{P}_{N Q} - \partial_{N} Γ^{P}_{M Q} + Γ^{P}_{M R} Γ^{R}_{N Q} - Γ^{P}_{N R} Γ^{R}_{M Q}$ by $R_{Q N} = R^{P}_{Q P N}$ and $R_{s} = G^{(s) Q N} R_{Q N}$ .

[A2]

Ten Dimensional NS NS Supergravity Action: adopt the source-free leading tree-level two-derivative 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

1.1givenF1algebra

On the section ${\tilde{\partial}}^{i} = 0$ , every doubled field depends only on $x^{i}$ , so $η^{M N} \partial_{M} \partial_{N} A = 0$ and $η^{M N} \partial_{M} A \partial_{N} B = 0$ for all fields and products. Thus this section satisfies both clauses of the strong constraint in [F1].

1.2givenF1A1algebra

Put $G : = G^{(s)}$ by [A1] and define $H_{i j k} = 3 \partial_{[i} B_{j k]}$ . Taking logarithms of [F1]'s density relation gives $d = Φ - \frac{1}{4} \log ∣ \det G ∣$ . Raising both generalized-metric indices with $η$ gives $H^{i j} = G^{i j}$ and the full block matrix $H^{M N} = (\begin{array}{cc} G^{- 1} & - G^{- 1} B_{2} \\ B_{2} G^{- 1} & G - B_{2} G^{- 1} B_{2} \end{array})$ . On the section $\partial_{M} = (\partial_{i}, 0)$ , so the first four terms of $R$ are exactly $- \partial_{i} \partial_{j} G^{i j} + 4 (\partial_{i} G^{i j}) \partial_{j} d + 4 G^{i j} \partial_{i} \partial_{j} d - 4 G^{i j} \partial_{i} d \partial_{j} d$ . Direct multiplication of the blocks in the last two terms gives $\frac{1}{8} H^{i j} \partial_{i} H^{K L} \partial_{j} H_{K L} - \frac{1}{2} H^{i j} \partial_{i} H^{K L} \partial_{K} H_{j L} = \frac{1}{4} G^{i j} \partial_{i} G^{k ℓ} \partial_{j} G_{k ℓ} - \frac{1}{2} G^{i j} \partial_{j} G^{k ℓ} \partial_{ℓ} G_{i k} - \frac{1}{12} G^{i a} G^{j b} G^{k c} H_{i j k} H_{a b c} .$ Thus the block substitution yields the complete intermediate scalar $R = \frac{1}{4} G^{i j} \partial_{i} G^{k ℓ} \partial_{j} G_{k ℓ} - \frac{1}{2} G^{i j} \partial_{j} G^{k ℓ} \partial_{ℓ} G_{i k} - \partial_{i} \partial_{j} G^{i j} + 4 (\partial_{i} G^{i j}) \partial_{j} d + 4 G^{i j} \partial_{i} \partial_{j} d - 4 G^{i j} \partial_{i} d \partial_{j} d - \frac{1}{12} G^{i a} G^{j b} G^{k c} H_{i j k} H_{a b c} .$

2.1givenF1A1step 1.2algebra

Put $q_{i} : = \partial_{i} \log \sqrt{- \det G} = \frac{1}{2} G^{a b} \partial_{i} G_{a b}$ , so $d = Φ - \frac{1}{2} \log \sqrt{- \det G}$ , $\partial_{i} d = \partial_{i} Φ - \frac{1}{2} q_{i}$ , and $\partial_{i} \partial_{j} d = \partial_{i} \partial_{j} Φ - \frac{1}{2} \partial_{i} q_{j}$ . Differentiating $G^{i a} G_{a j} = δ_{j}^{i}$ gives $\partial_{k} G^{i j} = - G^{i a} G^{j b} \partial_{k} G_{a b}$ ; metric compatibility and $q_{i} = Γ_{k i}^{k}$ give $\partial_{i} G^{i j} + G^{i j} q_{i} + G^{a b} Γ_{a b}^{j} = 0$ . Expanding the Ricci scalar formula in [A1] with the Christoffel symbols $Γ_{i j}^{k} = \frac{1}{2} G^{k ℓ} (\partial_{i} G_{j ℓ} + \partial_{j} G_{i ℓ} - \partial_{ℓ} G_{i j})$ yields the exact metric identity $R_{s} = \frac{1}{4} G^{i j} \partial_{i} G^{k ℓ} \partial_{j} G_{k ℓ} - \frac{1}{2} G^{i j} \partial_{j} G^{k ℓ} \partial_{ℓ} G_{i k} - \partial_{i} \partial_{j} G^{i j} - 2 (\partial_{i} G^{i j}) q_{j} - 2 G^{i j} \partial_{i} q_{j} - G^{i j} q_{i} q_{j} .$ Substitution of $\partial_{i} d$ and $\partial_{i} \partial_{j} d$ into the metric/dilaton terms of step 1.2, followed by this identity, groups the terms independent of $Φ$ into $R_{s}$ . The remaining terms are $4 \partial_{i} G^{i j} \partial_{j} Φ + 4 G^{i j} \partial_{i} \partial_{j} Φ + 4 G^{i j} q_{i} \partial_{j} Φ - 4 G^{i j} \partial_{i} Φ \partial_{j} Φ .$ The metric-compatibility identity above turns the first and third terms into $- 4 G^{a b} Γ_{a b}^{j} \partial_{j} Φ$ . Thus, defining $□_{s} f : = G^{i j} (\partial_{i} \partial_{j} f - Γ_{i j}^{k} \partial_{k} f)$ , the metric/dilaton part is $R_{s} + 4 □_{s} Φ - 4 ∣ d Φ ∣_{G^{(s)}}^{2}$ . The last term of step 1.2 is $- \frac{1}{12} H_{i j k} H_{G^{(s)}}^{i j k}$ . Therefore $R = R_{s} + 4 □_{s} Φ - 4 ∣ d Φ ∣_{G^{(s)}}^{2} - \frac{1}{12} H_{i j k} H_{G^{(s)}}^{i j k}$ .

3.1givenF2A1A2step 2.1algebra∎

Using the action normalization in the Statement and independence of the dual coordinates gives $S_{D F T} / V_{\tilde{U}} = (2 κ_{0}^{2})^{- 1} \int_{U_{x}} d^{10} x L_{D F T}$ . By [F2], this equals $(2 κ_{0}^{2})^{- 1} \int_{U_{x}} d^{10} x e^{- 2 d} R$ modulo the same normalized integral of its divergence. Multiplying step 2.1 by $e^{- 2 d} = \sqrt{- G^{(s)}} e^{- 2 Φ}$ gives the physical density, where $\sqrt{- G} : = \sqrt{- \det (G_{i j})}$ in this mostly-plus Lorentzian chart. The Christoffel formula gives $\sqrt{- G} □_{s} Φ = \partial_{i} (\sqrt{- G} G^{i j} \partial_{j} Φ)$ . For a coordinate domain $U_{x}$ , define the oriented boundary element $d Σ_{i}$ by $\int_{\partial U_{x}} V^{i} d Σ_{i} : = \int_{\partial U_{x}} ι_{V} (d x^{0} \land \dots \land d x^{9})$ for every smooth vector field $V$ ; integration by parts then has the boundary term $\int_{U_{x}} \sqrt{- G} e^{- 2 Φ} □_{s} Φ = \int_{\partial U_{x}} d Σ_{i} \sqrt{- G} e^{- 2 Φ} G^{i j} \partial_{j} Φ + 2 \int_{U_{x}} \sqrt{- G} e^{- 2 Φ} ∣ d Φ ∣_{G^{(s)}}^{2} .$ Under the boundary-flux condition in the Given clause, the integrated dilaton contribution $4 □_{s} Φ - 4 ∣ d Φ ∣_{G^{(s)}}^{2}$ therefore becomes $+ 4 ∣ d Φ ∣_{G^{(s)}}^{2}$ . Without that condition, the same conclusion is an equality of densities modulo the displayed divergence. The prefactor $1 / (2 κ_{0}^{2})$ is unchanged throughout; by [A1]–[A2], the resulting normalized physical-coordinate integral is exactly the adopted NS–NS action. This proves the local section reduction under the adopted physical action, without deriving that action itself.

Strong Constraint DFT Supergravity Section Reduction

Statement

Statement references

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources