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Register a new userUnfolding Conformal Geometry
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Conformal geometry is studied using the unfolded formulation `a la Vasiliev. Analyzing the first-order consistency of the unfolded equations, we identify the content of zero-forms as the spin-two off-shell Fradkin-Tseytlin module of $mathfrak{so}(2,d)$. We sketch the nonlinear structure of the equations and explain how Weyl invariant densities, which Type-B Weyl anomaly consist of, could be systematically computed within the unfolded formulation. The unfolded equation for conformal geometry is also shown to be reduced to various on-shell gravitational systems by requiring additional algebraic constraints.
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Systematic understanding for classes of inflationary models is investigated from the viewpoint of the local conformal symmetry and the slightly broken global symmetry in the framework of the metric-affine geometry. In the metric-affine geometry, which is a generalisation of the Riemannian one adopted in the ordinary General Relativity, the affine connection is an independent variable of the metric rather than given e.g. by the Levi-Civita connection as its function. Thanks to this independency, the metric-affine geometry can preserve the local conformal symmetry in each term of the Lagrangian contrary to the Riemannian geometry, and then the local conformal invariance can be compatible with much more kinds of global symmetries. As simple examples, we consider the two-scalar models with the broken $mathrm{SO}(1,1)$ or $mathrm{O}(2)$, leading to the well-known $alpha$-attractor or natural inflation, respectively. The inflaton can be understood as their pseudo Nambu-Goldstone boson.
A first-order differential equation is provided for a one-form, spin-s connection valued in the two-row, width-(s-1) Young tableau of GL(5). The connection is glued to a zero-form identified with the spin-s Cotton tensor. The usual zero-Cotton equation for a symmetric, conformal spin-s tensor gauge field in 3D is the flatness condition for the sum of the GL(5) spin-s and background connections. This presentation of the equations allows to reformulate in a compact way the cohomological problem studied in 1511.07389, featuring the spin-s Schouten tensor. We provide full computational details for spin 3 and 4 and present the general spin-s case in a compact way.
Weyl conformal geometry may play a role in early cosmology where effective theory at short distances becomes conformal. Weyl conformal geometry also has a built-in geometric Stueckelberg mechanism: it is broken spontaneously to Riemannian geometry after a Weyl gauge transformation (of gauge fixing) while Stueckelberg mechanism re-arranges the degrees of freedom, conserving their number ($n_{df}$). The Weyl gauge field ($omega_mu$) of local scale transformations acquires a mass after absorbing a compensator (dilaton), decouples, and Weyl connection becomes Riemannian. Mass generation has thus a dynamic origin, as a transition from Weyl to Riemannian geometry. We show that a gauge fixing symmetry transformation of the original Weyl quadratic gravity action in its Weyl geometry formulation immediately gives the Einstein-Proca action for the Weyl gauge field and a positive cosmological constant, plus matter action (if present). As a result, the Planck scale is an {it emergent} scale, where Weyl gauge symmetry is spontaneously broken and Einstein action is the broken phase of Weyl action. This is in contrast to local scale invariant models (no gauging) where a negative kinetic term (ghost dilaton) remains present and $n_{df}$ is not conserved when this symmetry is broken. The mass of $omega_mu$, setting the non-metricity scale, can be much smaller than $M_text{Planck}$, for ultraweak values of the coupling ($q$). If matter is present, a positive contribution to the Planck scale from a scalar field ($phi_1$) vev induces a negative (mass)$^2$ term for $phi_1$ and spontaneous breaking of the symmetry under which it is charged. These results are immediate when using a Weyl geometry formulation of an action instead of its Riemannian picture. Briefly, Weyl gauge symmetry is physically relevant and its role in high scale physics should be reconsidered.
We motivate a minimal realization of slow-roll $k$-inflation by incorporating the local conformal symmetry and the broken global $mathrm{SO}(1,1)$ symmetry in the metric-affine geometry. With use of the metric-affine geometry where both the metric and the affine connection are treated as independent variables, the local conformal symmetry can be preserved in each term of the Lagrangian and thus higher derivatives of scalar fields can be easily added in a conformally invariant way. Predictions of this minimal slow-roll $k$-inflation, $n_mathrm{s}sim 0.96$, $rsim 0.005$, and $c_mathrm{s}sim 0.03$, are not only consistent with current observational data but also have a prospect to be tested by forthcoming observations.
We study the Standard Model (SM) in Weyl conformal geometry. This embedding is truly minimal, {it with no new fields} beyond the SM spectrum and Weyl geometry. The action inherits a gauged scale symmetry $D(1)$ (known as Weyl gauge symmetry) from the underlying geometry. The associated Weyl quadratic gravity undergoes spontaneous breaking of $D(1)$ by a geometric Stueckelberg mechanism in which the Weyl gauge field ($omega_mu$) acquires mass by absorbing the spin-zero mode of the $tilde R^2$ term in the action. This mode also generates the Planck scale. The Einstein-Hilbert action emerges in the broken phase. In the presence of the SM, this mechanism receives corrections (from the Higgs) and it can induce electroweak (EW) symmetry breaking. The Higgs field has direct couplings to the Weyl gauge field while the SM fermions only acquire such couplings following the kinetic mixing of the gauge fields of $D(1)times U(1)_Y$. One consequence is that part of the mass of $Z$ boson is not due to the usual Higgs mechanism, but to its mixing with massive $omega_mu$. Precision measurements of $Z$ mass set lower bounds on the mass of $omega_mu$ which can be light (few TeV), depending on the mixing angle and Weyl gauge coupling. The Higgs mass and the EW scale are proportional to the vev of the Stueckelberg field. In the early Universe the Higgs field can have a geometric origin, by Weyl vector fusion, and the Higgs potential can drive inflation. The dependence of the tensor-to-scalar ratio $r$ on the spectral index $n_s$ is similar to that in Starobinsky inflation but mildly shifted to lower $r$ by the Higgs non-minimal coupling to Weyl geometry.
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