Conformal Galileons in Field Theory
- Conformal Galileons are scalar field theories featuring derivative self-interactions that remain invariant under non-linearly realized conformal transformations and yield second-order equations of motion.
- They are constructed via the coset method from SO(4,2) to ISO(3,1), producing key invariant Lagrangians including a unique Wess–Zumino term that captures anomaly-related effects.
- Their structure connects ghost-free higher-curvature terms and dual realizations—Weyl (dilaton) and AdS/DBI—offering robust frameworks for cosmological models and effective field theory extensions.
Conformal Galileons are a class of scalar field theories characterized by derivative self-interactions that are invariant under non-linearly realized conformal transformations and yield equations of motion of at most second order. Arising from both algebraic coset methods and higher-dimensional brane constructions, these theories are central in cosmology, effective field theory, and high-energy physics due to their ability to stably violate the null energy condition and their deep geometrical and symmetry properties. Their structure tightly interlaces with general relativity analogues (notably Lovelock invariants), and they admit extensions to supersymmetric and gravitationally-dressed variants as well as dual realizations connected via explicit equivalence maps.
1. Coset Construction and the Conformal Galileon Lagrangians
The conformal Galileons are constructed via the coset method for non-linearly realized symmetries, specifically the breaking of the conformal group to the Poincaré subgroup . The relevant coset is parameterized as
where is the dilaton (conformal mode) and are Goldstones for the special conformal transformations . The Maurer–Cartan one-form yields a set of basic invariant differential forms (vielbein , dilatation , and "special conformal" connection ) from which invariant Lagrangians are built. Imposing the inverse Higgs constraint (IHC),
expresses the Goldstone fields in terms of derivatives of .
Five algebraically independent invariant "coset terms" can be constructed in four dimensions, built as wedge products of and , and integrating the resulting 4-form over spacetime: plus the independent Wess–Zumino (WZ) term
which can be shown, via a limiting procedure, to be a genuine WZ term (Goon et al., 2012). Upon implementing the IHC, these invariants reduce to a minimal basis for the conformal Galileon Lagrangians in terms of and its derivatives, e.g. , , and so on.
2. Wess–Zumino Terms, Trace Anomaly, and Dimensional Structure
The crucial feature distinguishing conformal Galileons from their DBI counterparts is the presence of a Wess–Zumino term in even dimensions (the "middle" Galileon in ). While most coset-invariant terms can be constructed from algebraic combinations of the Maurer–Cartan one-forms, the WZ term arises as the potential for a closed but non-exact -form in Lie algebra cohomology:
selected by a nontrivial element in (Goon et al., 2012). Gravitational dressing of this WZ term produces the effective action for the trace (conformal) anomaly under Weyl transformations,
where is the Euler (Gauss–Bonnet) density (Gabadadze et al., 2020). In odd dimensions, this WZ structure is absent, all Galileons can be constructed from symmetric polynomials of the coset matrix, and no conformal anomaly arises (Hinterbichler et al., 30 Jul 2025).
3. Equivalence Transformations and Dual Realizations: Weyl vs. AdS/DBI Representations
Conformal Galileons admit two distinct, but dynamically equivalent, realizations: the Weyl dilaton (metric) realization and the AdS brane (DBI) realization. In the former, the dilaton appears via the conformally flat metric , whereas in the latter a brane embedded in is described by a scalar field parameterizing the brane bending (Creminelli et al., 2013).
The AdS/CFT equivalence transformation is a nontrivial field redefinition mapping the building blocks of the two realizations: and similarly in the reverse direction (Hinterbichler et al., 30 Jul 2025). Critically, this transformation preserves the set of Galileon interactions (including Wess–Zumino terms) and the second-order property of the equations of motion in any dimension. Under this correspondence, the DBI galileons constructed as induced curvature invariants on the brane are mapped to conformal Galileons constructed with the Weyl dilaton, and vice versa. The transformation properly accounts for the presence or absence of Wess–Zumino terms, depending on the spacetime dimension.
4. Gravitational Dressing, Lovelock Invariants, and Higher-Dimensional Origins
Five of the six invariant conformal Galileon Lagrangians in four dimensions can be shown to descend from higher-dimensional Lovelock invariants via a "higher-dimensional reduction" (HDR) algorithm. This process systematically projects the Lovelock invariants—unique higher-curvature invariants with second-order equations of motion—onto four-dimensional terms that are invariant under nonlinearly realized conformal symmetry (Gabadadze et al., 2020). For instance, after HDR and proper regularization, the quartic Lovelock term in higher dimensions reduces to the quartic conformal Galileon term as a function of the conformal mode. The Wess–Zumino term, by contrast, is not obtainable from higher Lovelock terms—reflecting its anomalous (cohomological) origin.
The full cosmological action is therefore a sum of: (i) the coset/Galileon terms (cosmological, Einstein–Hilbert, GB, and higher invariants), (ii) the WZ/anomaly term, and (iii) operators involving at least one Weyl tensor, which do not contribute to the dynamics of the conformal mode.
5. Cosmological Dynamics and EFT Embedding
When the scalar sector of gravity is isolated (e.g., in FLRW cosmology), the conformal Galileon action reduces effectively to a minisuperspace model for the scale factor. The modified Friedmann equation takes the schematic form
with , determined by higher-curvature terms. This allows for bouncing and non-standard cosmological solutions—for example, the solution
represents a non-singular bounce. However, a plausible implication is that such solutions typically require curvatures near the UV cutoff (i.e., Planck scale), mandating fine-tuning and constraining their regime of validity within the EFT (Gabadadze et al., 2020). The higher-derivative structure, ensured by Lovelock descent, guarantees that the scalar sector remains ghost-free below the cutoff.
6. Dimensional Generalization and Structure of Invariant Basis
In arbitrary dimension , the number of conformal Galileon terms (including Wess-Zumino when present) is , constructed systematically via symmetric polynomials of a key coset-derived matrix. For even , the "middle" Galileon is Wess–Zumino, requiring limiting procedures to define it; for odd , all terms admit manifest invariant definitions and the Wess–Zumino structure is absent (Hinterbichler et al., 30 Jul 2025). The AdS/CFT equivalence transformation preserves this organization, mapping the symmetric polynomial basis in the Weyl representation to the corresponding basis built from extrinsic curvature polynomials in the AdS brane realization.
Dimension | Number of Terms | Wess–Zumino Term? |
---|---|---|
Even | Yes (middle term, e.g. cubic in 4d) | |
Odd | No |
This structure matches the counting from Lie algebra cohomology and underpins the unique status of conformal Galileons among scalar EFTs with nonlinearly realized conformal symmetry.
7. Summary and Implications
Conformal Galileons serve as a nexus for the intersection of symmetry, geometry, and dynamics in scalar effective field theory. Their construction via the coset method, their relation to ghost-free higher-curvature terms (Lovelock invariants), their dual realizations (Weyl and AdS/DBI), and the resolution of puzzle cases (like the Wess–Zumino terms in odd dimensions) provide a unified framework for understanding nonlinearly realized conformal symmetry and its dynamical consequences. Their mathematical structure is tightly linked to anomaly matching in quantum field theory, to cosmological model-building (especially NEC-violating scenarios), and to possible UV completions—particularly via brane constructions in string/M-theory. Their continued generalization and the explicit mapping between representations stand as key tools for developing robust cosmological EFTs and for exploring new interactions compatible with both symmetry and stability (Goon et al., 2012, Gabadadze et al., 2020, Hinterbichler et al., 30 Jul 2025).