Solution-preserving consistent truncation

Let $M_D$ and $M_d$ be specified spacetime manifolds, and let ${\mathcal{C}}_D$ and ${\mathcal{C}}_d$ be specified $C^1$ Banach manifolds of admissible $C^k$ full and retained field configurations on them, with $k$ large enough for the chosen action and equations. Fields are sections of declared bundles. Boundary data and either a gauge fixing or a smooth gauge quotient are part of each configuration space. Let $S_D:{\mathcal{C}}_D\to \mathbb{R}$ be the specified $C^1$ full action and $I:{\mathcal{C}}_d\to {\mathcal{C}}_D$ the specified $C^1$ uplift ansatz. The reduced action is the restricted variational principle $S_d:=S_D\circ I$, with internal integration already performed when appropriate; it is not an independently chosen action. Fix the action normalization and boundary terms before this equality. Removing a field-independent nonzero overall factor or a term with identically vanishing admissible first variation gives the same stationary configurations, but that change must be declared. Define the variational equation maps by ${\mathcal{E}}_D(\mathrm{\Psi }):=D{S}_D{\mathrm{\mid }}_{\mathrm{\Psi }}\in T_{\mathrm{\Psi }}^{\ast }{\mathcal{C}}_D,{\mathcal{E}}_d(\varphi ):=D{S}_d{\mathrm{\mid }}_{\varphi }\in T_{\varphi }^{\ast }{\mathcal{C}}_d\mathrm{.}$ Their variational solution sets are ${\mathrm{S}\mathrm{o}\mathrm{l}}_D:=\{\mathrm{\Psi }\in {\mathcal{C}}_D:{\mathcal{E}}_D(\mathrm{\Psi })=0\}$ and ${\mathrm{S}\mathrm{o}\mathrm{l}}_d:=\{\varphi \in {\mathcal{C}}_d:{\mathcal{E}}_d(\varphi )=0\}$. For a local action with vanishing boundary variation and a nondegenerate test-field pairing, these conditions are equivalently the corresponding Euler–Lagrange equations in the weak sense; no particular differential expression is assumed here.

The uplift ansatz $I$ is a consistent truncation for the specified actions and boundary/gauge data precisely when $I({\mathrm{S}\mathrm{o}\mathrm{l}}_d)\subseteq {\mathrm{S}\mathrm{o}\mathrm{l}}_D\mathrm{.}$ Thus every retained solution must satisfy every full variational equation, including equations in field directions omitted by the ansatz. The map $I$ need not be linear and may encode omitted fields as functions of retained fields. If gauge-equivalence classes are used, $I$ must be well-defined on those classes and the equation maps must descend to the chosen quotient.

For tensor fields on an internal manifold $Y$, a symmetry-restricted or first-type reduction means choosing a smooth group action on $Y$, with pointwise independent fundamental vector fields $K_a$ satisfying the Killing equations ${\mathcal{L}}_{K_a}{\widehat{g}}_Y=0$, and restricting tensor fields to the invariant sector ${\mathcal{L}}_{K_a}\mathrm{\Psi }=0$. If the orbit space is used as a smooth quotient, the action is also required to be free and proper. The precise bosonic first-type result cited from Pons–Talavera §§2.1–2.4 uses local product coordinates $(x^{\mu },y^{\alpha })$, linearly independent internal Killing fields $K_a=K_a^{\alpha }(y){\mathrm{\partial }}_{\alpha }$ with constant Lie brackets, and an $x$-independent invariant frame $Y_a$ dual to one-forms ${\omega }^a$ with $d{\omega }^a=\frac{1}{2}{C}^a{}_{bc}{\omega }^b\wedge {\omega }^c$. In that frame every retained bosonic tensor component $\mathrm{\Phi }$ depends only on $x$, all such components are retained, and the local density has the form $\mathcal{L}=\mathrm{\mid }\omega \mathrm{\mid }\tilde{\mathcal{L}}$, where $\mathrm{\mid }\omega \mathrm{\mid }=\mathrm{\mid }\det ({\omega }^a{}_{\alpha })\mathrm{\mid }$ is nonzero and fixed under field variations. The density $\tilde{\mathcal{L}}$ depends on the components and their external and frame derivatives, with no explicit internal-coordinate dependence after imposing $Y_a\mathrm{\Phi }=0$; the displayed source derivation treats derivatives through order two. Define ${\mathcal{L}}_R:=\tilde{\mathcal{L}}{\mathrm{\mid }}_{Y_a\mathrm{\Phi }=Y_a{Y}_b\mathrm{\Phi }=0}$, rather than choosing a new reduced density. Boundary conditions remove all integration-by-parts terms. If the internal integral is used for an action, its finite nonzero field-independent factor $\int \mathrm{\mid }\omega \mathrm{\mid }{d}^{n}y$ is retained in $S_d$ (or explicitly absorbed into the normalization). Under these hypotheses and unimodularity $C^c{}_{ac}=0$ for every $a$, the source's equation (2.24) identifies the restricted full Euler–Lagrange derivatives with $\mathrm{\mid }\omega \mathrm{\mid }$ times those of ${\mathcal{L}}_R$, and §2.4 gives solution uplift. This is the cited local first-type case, not a claim that arbitrary symmetry restrictions or independent Killing fields alone guarantee consistency; fermionic reductions require their separately specified invariant vielbein and spinor conditions. A constraint-driven or second-type restriction is instead specified by a constraint map $C:{\mathcal{C}}_D\to Z$ into a declared space $Z$ and the constrained configurations $\{\mathrm{\Psi }:C(\mathrm{\Psi })=0\}$, without adding Lagrange-multiplier terms to the original action. Such a restriction is consistent only if it satisfies the same solution-inclusion criterion above.

At a background $\bar{\mathrm{\Psi }}$ where $S_D$ is $C^2$, a quadratic-mode comparison is the decomposition of the Hessian $D^2{S}_D{\mathrm{\mid }}_{\bar{\mathrm{\Psi }}}$, or its associated linearized operator, into declared retained and omitted directions. It tests the linearized spectrum only. Solving the omitted equations for a branch and substituting it into the action is a separate operation; a generic harmonic zero-mode ansatz or quadratic diagonalization alone does not establish consistent truncation. This item asserts no exact supergravity reduction.

Kaluza–Klein product and warped reduction data

Restricted variation and the full field equations