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Kaluza–Klein product and warped reduction data

Physical scope

Classical two-derivative metric reduction on smooth compact internal spaces, with separate unwarped fixed-background and warped Einstein-frame ansätze. Source, flux, dilaton, higher-derivative, loop, and retained-modulus effects are included only when explicitly declared.

Definition

Let $d, n \geq 1$ , $D = d + n$ , let $M_{d}$ be a smooth Lorentzian $d$ -manifold of signature $(-, +, \dots, +)$ , and let $(Y_{n}, {\hat{g}}_{Y})$ be a smooth compact boundaryless Riemannian $n$ -manifold. Write ${Vol}_{\hat{g}} (Y) = \int_{Y} \sqrt{\det {\hat{g}}_{Y}} d^{n} y$ . In coordinate expressions use $h_{Y} : = {\hat{g}}_{Y}$ and $h_{m n} (y) : = {\hat{g}}_{m n} (y)$ for this same internal metric. The fixed unwarped product metric ansatz is $G_{D} = g_{d} \oplus h_{Y}, d s_{D}^{2} = g_{μ ν} (x) d x^{μ} d x^{ν} + h_{m n} (y) d y^{m} d y^{n} .$ For an internal-only warped ansatz, let $A, B \in C^{\infty} (Y, R)$ and set $G_{D}^{(E)} = e^{2 A (y)} g_{d} + e^{2 B (y)} {\hat{g}}_{Y}, d s_{D, E}^{2} = e^{2 A (y)} g_{μ ν} (x) d x^{μ} d x^{ν} + e^{2 B (y)} {\hat{g}}_{m n} (y) d y^{m} d y^{n} .$ The metrics are required to be nondegenerate; their signatures are respectively Lorentzian on $M_{d}$ and Riemannian on $Y$ . The symbols $h_{Y}$ and $h_{m n}$ are only coordinate aliases of the already specified ${\hat{g}}_{Y}$ and ${\hat{g}}_{m n}$ ; no second internal metric is introduced.

For the coefficient calculations below, additionally take the $D$ -dimensional action to be $S_{D} [G] = \frac{1}{2 κ_{D}^{2}} \int_{M_{d} \times Y_{n}} d^{D} X \sqrt{- \det G_{D}} R [G_{D}],$ with $κ_{D} > 0$ and, in $ℏ = c = 1$ units, $[κ_{D}^{2}] = (length)^{D - 2}$ . The coefficient calculation excludes other action terms. A variable-volume product ansatz, when used, is the block-diagonal metric $G_{D} (x, y) = g_{J} (x) \oplus γ_{Y} (x, y),$ where $γ_{Y} (x, \cdot)$ is a smooth Riemannian metric on $Y$ for each $x$ and there are no mixed $d x^{μ} d y^{m}$ components. Its Jordan-frame internal volume is $V_{J} (x) : = \int_{Y} d^{n} y \sqrt{\det γ_{Y} (x, y)} .$ When $d > 2$ and $V_{J} > 0$ , the corresponding lower-dimensional Einstein metric is defined by $g_{E} = (V_{J} / V_{*})^{2 / (d - 2)} g_{J}$ , for a fixed reference volume $V_{*} > 0$ .

For a specified smooth finite-rank real or complex vector bundle $F \to Y$ , write $Γ (F) : = C^{\infty} (Y, F)$ for its space of smooth sections. For a specified linear internal kinetic operator $L_{Y} : Γ (F) \to Γ (F)$ , an internal mode is a smooth section $u$ satisfying $L_{Y} u = λ u$ for a declared eigenvalue $λ$ . A finite mode ansatz is a field expansion $Ψ (x, y) = \sum_{j \in J} q^{j} (x) u_{j} (y)$ for a declared finite index set $J$ and declared sections $u_{j}$ . Here a Laplace-type operator is a second-order differential operator with smooth coefficients and principal symbol $σ_{2} (L_{Y}) (y, ξ) = ∣ ξ ∣_{{\hat{g}}_{Y}}^{2} {id}_{F_{y}}$ , using the symbol convention $\partial_{y^{a}} \mapsto i ξ_{a}$ . Equivalently, in a local frame its second-order part is $- {\hat{g}}_{Y}^{a b} \partial_{y^{a}} \partial_{y^{b}} id$ ; the remaining terms have order at most one. This fixes the positive-principal-symbol sign; the scalar model is $- div grad$ . Formal self-adjointness or nonnegativity is additional data when required, not part of the principal-symbol condition alone. For this operator a zero or harmonic mode means a section in $\ker L_{Y}$ . This definition makes no claim that such a finite family is complete or that a finite mode ansatz is a consistent truncation.

These ansätze are declared data, not consequences of compactness, harmonicity, or the Einstein equations. Every reduction must also specify any coordinate periods, field normalizations, frame and dilaton convention, retained-modulus dependence, source localization or smearing, boundary conditions, and retained orders in $α^{'}$ , the string coupling, derivatives, and field modes. In ten dimensions use the declared convention $G_{E} = e^{- Φ / 2} G_{s}$ , with $g_{s t r} = e^{Φ_{\infty}}$ and $2 κ_{0}^{2} = (2 π)^{7} α^{' 4}$ ; when the constant asymptotic dilaton is absorbed into the physical coupling, $2 κ_{10, p h y s}^{2} = g_{s t r}^{2} 2 κ_{0}^{2}$ . These are the conventions of the supplied supergravity-definition item.

Definition references

Supergravity Conventions And Normalizations

Facts & Assumptions

Given: The manifold, metric, action, boundary conditions, frame, dilaton shift, and field dependence specified in the statement.

[F1]

Supergravity Conventions And Normalizations: in natural units the item declares mostly-plus signature, $2 κ_{0}^{2} = (2 π)^{7} α^{' 4}$ , $2 κ_{10, p h y s}^{2} = g_{s t r}^{2} 2 κ_{0}^{2}$ when the constant asymptotic dilaton is absorbed, $G^{(E)} = e^{- Φ / 2} G^{(s)}$ , and $M_{P}^{- 2} = κ_{4}^{2} = 8 π G_{N}$ . Here $G^{(s)}$ is a metric and $g_{s t r} = e^{Φ_{\infty}}$ is a dimensionless scalar; they are distinct objects.

[A1]

$D = d + n$ , $Y$ has no boundary, the metrics are smooth and nondegenerate, and the warp factors depend only on internal coordinates for the fixed-background warped calculation. For the ten-dimensional constant-dilaton comparison set $Φ_{0} = Φ_{\infty}$ , so $g_{s} = g_{s t r} = e^{Φ_{0}}$ in the convention of [F1].

[A2]

For the moduli Weyl calculation, $d > 2$ , the variable-volume metric is the declared $γ_{Y} (x, y)$ , $V_{J} (x) > 0$ is smooth, and external boundary terms vanish or are discarded under the declared boundary conditions.

[A3]

For the ten-dimensional frame conversion in step 1.3, $D = 10$ , $d = 4$ , $n = 6$ , $Φ = Φ_{0} = Φ_{\infty}$ is constant, the metric is an unwarped product, and $κ_{10, p h y s}^{2} = e^{2 Φ_{0}} κ_{0}^{2}$ .

Proof

1.1givenA1algebra

In the fixed product case the metric is block diagonal, so its determinant is $\det G_{D} = (\det g_{d}) (\det {\hat{g}}_{Y})$ ; the Lorentzian square-root convention gives $\sqrt{- \det G_{D}} = \sqrt{- \det g_{d}} \sqrt{\det {\hat{g}}_{Y}}$ . Thus internal and external volume factors separate in the stated Einstein–Hilbert action.

1.2givenA1algebra

In the warped case the external block is multiplied by $e^{2 A}$ in each of $d$ directions and the internal block by $e^{2 B}$ in each of $n$ directions. Therefore $\sqrt{- \det G_{D}} = e^{d A + n B} \sqrt{- \det g_{d}} \sqrt{\det {\hat{g}}_{Y}}$ . The part of the scalar curvature multiplying the external scalar curvature is $e^{- 2 A} R [g_{d}]$ ; terms containing derivatives of $A$ or $B$ contain no factor $R [g_{d}]$ . Hence the coefficient of $\int_{M_{d}} \sqrt{- \det g_{d}} R [g_{d}]$ is $\int_{Y} d^{n} y \sqrt{\det {\hat{g}}_{Y}} e^{(d - 2) A + n B} / (2 κ_{D}^{2})$ .

1.3givenF1A3algebra

Under [A3], $G_{E} = e^{- Φ_{0} / 2} G_{s}$ implies $G_{s} = e^{Φ_{0} / 2} G_{E}$ . In four external dimensions a constant Weyl rescaling gives $\sqrt{- g_{s}} R [g_{s}] = e^{Φ_{0} / 2} \sqrt{- g_{E}} R [g_{E}]$ . On six internal dimensions $V_{s} = e^{3 Φ_{0} / 2} V_{E}$ . Since $κ_{10, p h y s}^{2} = e^{2 Φ_{0}} κ_{0}^{2}$ , the string-frame coefficient $V_{s} e^{Φ_{0} / 2} / κ_{10, p h y s}^{2}$ equals $V_{E} / κ_{0}^{2}$ . This is only a constant-dilaton frame conversion; it makes no statement about terms involving derivatives or dynamics of $Φ$ .

1.4givenA2algebra

For the variable-volume ansatz in the Definition, the coefficient of $R [g_{J}]$ is $V_{J} (x) / (2 κ_{D}^{2})$ . Under $g_{E} = Ω^{2} g_{J}$ with $Ω = (V_{J} / V_{*})^{1 / (d - 2)}$ , the curvature term transforms with coefficient $V_{J} Ω^{2 - d} / (2 κ_{D}^{2}) = V_{*} / (2 κ_{D}^{2})$ . The conformal transformation also produces derivatives of $Ω$ ; they are modulus-derivative terms and are not removed by making the Einstein coefficient constant.

2.1givenA1F1algebra∎

The identity $\int_{Y} \sqrt{\hat{h}} e^{(d - 2) A + n B}$ is a coefficient calculation for the displayed ansatz and two-derivative Einstein term only. A varying warp factor, localized sources, flux constraints, dilaton dependence, higher- $α^{'}$ corrections, loops, or nonlinear mode couplings can change the equations and source omitted modes. Therefore harmonicity alone does not establish that the ansatz uplifts every reduced solution.

Depends on

Supergravity Conventions And Normalizations