Higher-dimensional foliated Euler descendants with zero local bulk modes

Establish whether analogous foliated scalar-tensor constructions exist in odd spacetime dimensions D=2m+1, with 2m-dimensional scalar leaves carrying the Euler density, as descendants or boundary sectors of the (2m+2)-dimensional Euler invariant, and determine whether they possess disformal closure, leaf-shape gauge symmetry, and nonlinear constraints that eliminate all local bulk modes.

Background

The paper develops a three-dimensional theory whose two-dimensional scalar leaves carry an intrinsic Euler density and relates this structure, in restricted settings, to a four-dimensional Gauss–Bonnet invariant. This suggests a dimensional pattern in which an even-dimensional Euler invariant produces a foliated theory in one lower odd dimension, whose scalar leaves carry the corresponding lower-dimensional Euler density.

For higher odd dimensions, however, the intrinsic leaf Euler functional alone does not determine the dynamics of the associated extrinsic-curvature terms. The authors therefore leave unresolved whether the proposed higher-dimensional descendants retain the special disformal closure, leaf-shape gauge symmetry, and complete elimination of local bulk modes found in the three-dimensional case.