Consistent Truncations

A truncation is consistent when the retained lower-dimensional equations form a closed sector of the higher-dimensional theory. Every lower-dimensional solution must uplift through the declared ansatz to a solution of all higher-dimensional equations, including those for discarded fields. A consistent truncation is not the same as a low-energy approximation: consistency concerns sourcing, whereas effective-field-theory control concerns the size of omitted effects.

Required background. Gauge Theory and String-Theory Preparation supplies compactification and mode expansions; Gravitational Actions and Boundary Terms supplies the action, variational principle, and boundary data whose order of reduction and variation is tested.

Helpful background. Quantum Gravity frames effective descriptions; Quantum Information supports later holographic uses; and Gravity Beyond General Relativity develops the resulting scalar, vector, and higher-derivative sectors.

The nonlinear uplift criterion

Proposition. Let the higher-dimensional fields be parametrized by an ansatz

$$\widehat\Phi^A(x,y)=\mathcal U^A[\phi^i(x);y].$$

Assume the ansatz specifies all retained and discarded components, respects the declared gauge symmetries and constraints, and has compatible boundary conditions and regularity. If a closed lower-dimensional system $E_i[\phi]=0$ implies every higher-dimensional equation after substitution of the ansatz, then every solution $\phi^i$ uplifts to a solution $\widehat\Phi^A$ of the complete higher-dimensional system. Discarded equations may vanish identically or follow from the retained equations; they must impose no additional condition on a lower-dimensional solution.

The conclusion is exact within the ansatz. It does not state that every higher-dimensional solution lies in the truncation, that omitted modes are heavy, or that quantum corrections preserve the sector.

The commuting test

Denote the full equations by $\widehat E_A[\widehat\Phi]=0$. Consistency requires

$$\widehat E_A[\mathcal U(\phi)] =M_A{}^i[\phi,y]\,E_i[\phi]$$

for every component $A$, with no independent residual. Thus solving $E_i=0$ before uplifting or substituting the ansatz into the full equations before solving gives the same result. This is the commuting diagram behind the definition.

At action level there is an additional danger. “Substitute and then vary” can lose equations if the ansatz fixes fields whose variations enforce constraints. One must either prove that the reduced action yields all restricted full equations or work directly with the higher-dimensional equations. Boundary and Chern–Simons terms must follow the same variational problem. Precise criteria and counterexamples are developed in Pons and Talavera 2004, §§1–2.4 and §§4–5.1, pp. 428–440 and 445–449.

Why circle zero modes close

For fields on a circle, Fourier momentum is additive. Products of zero modes carry zero momentum, so a translation-invariant ansatz cannot source a nonzero Fourier mode. In pure gravity, the metric ansatz retaining $g_{\mu\nu}$, the Kaluza–Klein vector, and the radius scalar is closed under the circle isometry. Substituting it into every Einstein component yields the lower-dimensional Einstein, Maxwell, and scalar equations.

This argument uses the full zero-mode multiplet. Keeping $g_{\mu\nu}$ while freezing a radius scalar that is sourced by $F_{\mu\nu}F^{\mu\nu}$ is generally inconsistent. Symmetry helps identify a closed sector only when every retained configuration is invariant and all singlet discarded equations vanish.

Sphere reductions are subtler because products of retained harmonics can contain discarded harmonics. Special nonlinear ansätze, gauge symmetry, and flux identities can nevertheless produce consistent reductions, as in the type-IIB $S^5$ reduction relevant to AdS$_5$ black holes. The historical status and black-hole use of such reductions are discussed in Aharony et al. 2000, §2.2.5, pp. 52–54; modern uses must rely on the completed nonlinear uplift formula appropriate to the chosen sector.

A decisive inconsistent example

Consider

$$S=\int d^dx\sqrt{-g} \left[-\frac12(\partial\phi)^2-\frac12(\partial\chi)^2 -\frac12m_\chi^2\chi^2-g\chi\phi^2\right].$$

Setting $\chi=0$ in the action appears to leave a free theory for $\phi$. But the discarded equation is

$$(\Box-m_\chi^2)\chi=g\phi^2.$$

At $\chi=0$, it demands $g\phi^2=0$. Generic solutions of the reduced free equation therefore do not uplift. The ansatz is inconsistent even if $m_\chi$ is very large. At energies well below $m_\chi$, one may instead solve $\chi\simeq-g\phi^2/m_\chi^2$ and obtain an approximate effective action. That controlled integration-out is a low-energy procedure, not a consistent truncation.

Conversely, a consistent truncation need not be low energy: it may retain a massive symmetry singlet while discarding lighter nonsinglets. Exact closure and spectral hierarchy answer different questions.

Proof architecture and limits

The proof follows three steps.

1. Substitute the full nonlinear ansatz into every Bianchi identity, constraint, gauge equation, and dynamical field equation.
2. Show that discarded equations vanish once the retained equations hold, and that the latter reduce to one closed $y$-independent set with matching boundary data.
3. Verify that any lower-dimensional solution satisfying those equations makes every full residual zero and that the uplift is regular in its declared domain.

A finite collection of successful solutions is not a proof; consistency quantifies over all solutions in the reduced theory. Dropping the boundary hypothesis can also fail: an uplift may solve local equations yet violate periodicity, flux quantization, or asymptotic conditions.

Exercises

For the two-scalar model above, integrate out $\chi$ at leading order in $m_\chi^{-2}$ and compare the resulting low-energy theory with the naive truncation $\chi=0$.

Solution

Neglecting derivatives relative to $m_\chi$, the $\chi$ equation gives $\chi\simeq-g\phi^2/m_\chi^2$. Substituting this solution into the action produces an effective interaction proportional to $+g^2\phi^4/(2m_\chi^2)$, together with derivative corrections at higher order. The naive action obtained by setting $\chi=0$ misses this term and, more seriously, its solutions violate the exact discarded equation whenever $\phi\ne0$. Therefore the naive sector is not consistent. The integrated-out theory is instead an approximation whose residual is suppressed by powers of energy over $m_\chi$; its validity follows from a hierarchy, not an exact uplift statement.

References

• Aharony, Ofer, Steven S. Gubser, Juan Maldacena, Hirosi Ooguri, and Yaron Oz. “Large N Field Theories, String Theory and Gravity.” Physics Reports 323 (2000): 183–386. Open manuscript.
• Pons, Josep M., and Pere Talavera. “Consistent and Inconsistent Truncations: Some Results and the Issue of the Correct Uplifting of Solutions.” Nuclear Physics B 678 (2004): 427–454. Open manuscript.