--- title: Weyl-Invariant Einstein–Cartan Gravity url: https://www.emergentmind.com/topics/weyl-invariant-einstein-cartan-gravity type: topic --- # Weyl-Invariant Einstein–Cartan Gravity Weyl-invariant Einstein–Cartan gravity is a class of torsionful gravitational theories in which the Einstein–Cartan variables—the tetrad \(e^a{}_\mu\) and an independent Lorentz connection \(\omega^{ab}{}_\mu\)—are supplemented by a local scale symmetry. In this setting, the torsion trace can play the role of a Weyl gauge field, nonminimal scalar couplings can replace an explicit Planck mass, and quadratic curvature invariants can generate effective scalar or pseudoscalar degrees of freedom after auxiliary-field rewriting, gauge fixing, and algebraic elimination of torsion. Across the literature, the subject includes Higgs–dilaton Einstein–Cartan models, one-field locally Weyl-invariant theories in which the dilaton is gauge, and purely gravitational quadratic constructions whose Einstein-frame description is standard general relativity plus a single scalar or axion-like pseudoscalar [2108.05897]. ## 1. Geometric structure and Weyl realization In the Einstein–Cartan formulation, the fundamental gravitational variables are the tetrad \(e^a{}_\mu(x)\) and an independent Lorentz connection \(\omega^{ab}{}_\mu(x)=-\omega^{ba}{}_\mu(x)\). From them one constructs the spacetime metric, \[ g_{\mu\nu}=\eta_{ab}\,e^a{}_\mu e^b{}_\nu\,,\qquad \eta_{ab}={\rm diag}(-,+,+,+)\,, \] the torsion two-form, \[ T^a{}_{\mu\nu}=\partial_\mu e^a{}_\nu-\partial_\nu e^a{}_\nu+\omega^a{}_{b\,\mu}\,e^b{}_\nu-\omega^a{}_{b\,\nu}\,e^b{}_\mu\,, \] and the curvature two-form, \[ R^{ab}{}_{\mu\nu}=\partial_\mu\omega^{ab}{}_\nu-\partial_\nu\omega^{ab}{}_\mu+\omega^a{}_{c\,\mu}\,\omega^{cb}{}_\nu-\omega^a{}_{c\,\nu}\,\omega^{cb}{}_\mu\,. \] The torsion tensor is commonly decomposed into irreducible \(SO(1,3)\) pieces, \[ v_\mu\equiv T^\nu{}_{\nu\mu},\qquad a_\mu\equiv\epsilon_{\mu\nu\rho\sigma}\,T^{\nu\rho\sigma},\qquad \tau_{\mu\nu\rho}\,, \] with \(v_\mu\) the trace vector, \(a_\mu\) the axial vector, and \(\tau_{\mu\nu\rho}\) totally traceless. Equivalent formulations use the parity-even Einstein–Cartan scalar \(F\) and the parity-odd Holst term \(\tilde F\), or, in affine notation, the torsionful scalars \(R(\Gamma)\) and \(\tilde R(\Gamma)\) [2108.05897]. The defining observation for local scale symmetry is that, under a local Weyl rescaling, the torsion trace shifts by a gradient. In one commonly used convention, \[ e^a{}_\mu\to q^{-1}(x)e^a{}_\mu,\qquad h\to q(x)h,\qquad \omega^{ab}{}_\mu\to\omega^{ab}{}_\mu, \] and \[ v_\mu\to v_\mu+3\,\partial_\mu\ln q\,. \] This permits the identification \[ W_\mu\equiv \frac13\,v_\mu \] and the Weyl-covariant derivative \[ D^W_\mu h=\partial_\mu h-W_\mu h\,. \] Related papers write the rescaling with opposite weight conventions, for example \(g_{\mu\nu}\to e^{2\alpha(x)}g_{\mu\nu}\) or \(g_{\mu\nu}\to \Omega^{-2}(x)g_{\mu\nu}\), while keeping the underlying local scale principle intact. This suggests that part of the variation across the literature is conventional rather than substantive [2506.11847]. ## 2. Action principles and model classes A first class of constructions starts from Einstein–Cartan gravity with nonpropagating torsion and nonminimally coupled scalars. For rigid scale invariance, the Planck mass is replaced by \(\sqrt{\xi_\chi\chi^2+\xi_h h^2}\), yielding a two-scalar Higgs–dilaton action, \[ \begin{aligned} S_{\rm SI}&=\int d^4x\sqrt{g}\,\Bigg\{\frac{\xi_\chi\,\chi^2+\xi_h\,h^2}{2}\,\mathring R -\frac12(\partial\chi)^2-\frac12(\partial h)^2-U(\chi,h) \\ &\qquad +\frac{\chi^2}{2}\Big[ G_{vv}\,v_\mu v^\mu+2\,G_{va}\,v_\mu a^\mu +G_{aa}\,a_\mu a^\mu +G_{\tau\tau}\,\tau_\mu\tau^\mu +\tilde G_{\tau\tau}\,\epsilon^{\mu\nu\rho\sigma}\,\tau_{\mu\nu\lambda}\tau_\rho{}^\lambda{}_\sigma \Big] \\ &\qquad +\chi^2\Big(J^v(\chi,h)\,\nabla_\mu v^\mu+J^a(\chi,h)\,\nabla_\mu a^\mu\Big)\Bigg\}, \end{aligned} \] with \[ U(\chi,h)=\frac\lambda4\bigl(h^2-\alpha\,\chi^2\bigr)^2+\beta^4\,. \] For local Weyl symmetry and no new propagating fields, the theory reduces to a one-field Weyl-invariant Einstein–Cartan action in which the allowed torsion couplings are fixed functions of \(h/\chi\) [2108.05897]. A second class uses purely gravitational, quadratic, Weyl-invariant actions. In one formulation the most general purely gravitational Weyl-invariant quadratic action is written in terms of the Cartan curvature scalars \(\mathcal R\) and \(\widetilde{\mathcal R}\), \[ S =\int d^4x\,\sqrt{-g}\,\Bigl\{ \tfrac{\gamma}{4}\,\Phi^{-4}\,\mathcal R^2 +\tfrac{\delta}{4}\,\Phi^{-4}\,\widetilde{\mathcal R}^2 +\tfrac{\epsilon}{2}\,\Phi^{-4}\,\mathcal R\,\widetilde{\mathcal R} \Bigr\} +S_{\rm Weyl}+S_{\rm torsion}\,. \] In a closely related Einstein–Cartan form, the pure-gravity sector is built from \(F^2\), \(\tilde F^2\), and \(F\tilde F\), or rewritten with auxiliary fields of Weyl weight \(1\), \[ S_{\rm EC,4} =\tfrac12\!\int\!d^4x\det e\, \Bigl[F\,\chi^2+\tilde F\,a^2-\alpha\,\chi^4-\beta\,a^4+2\gamma\,\chi^2a^2\Bigr]\,. \] The 2024 minimal construction further emphasizes that, if one demands exact local Weyl invariance with at most two derivatives, the pure-gravity sector contains quadratic Lorentz-curvature invariants and, after gauge fixing, produces one extra scalar degree of freedom with axion-like properties [2506.11847]. | Realization | Basic ingredients | Effective content after torsion elimination / gauge fixing | |---|---|---| | Global scale Einstein–Cartan | \(h,\chi,\mathring R\), algebraic torsion, \(U(\chi,h)\) | Two-field metric theory; genuine dilaton remains | | Local Weyl Einstein–Cartan | \(W_\mu=\tfrac13 v_\mu\), \(D_\mu^W\), fixed Weyl couplings | One-field theory; dilaton can be removed | | Quadratic geometric Weyl-invariant EC | \(R^2,\tilde R^2,R\tilde R\) or \(F^2,\tilde F^2,F\tilde F\) | GR plus one scalar or pseudoscalar | ## 3. Torsion elimination, gauge fixing, and metric equivalence A central structural property is that torsion is frequently nonpropagating. In the Einstein–Cartan models with Higgs and dilaton, one varies the action with respect to the torsion components \((v_\mu,a_\mu,\tau_{\mu\nu\rho})\), solves the resulting algebraic equations of motion, and substitutes back. In the Einstein frame, with \[ \Omega^2=\frac{\xi_\chi\,\chi^2+\xi_h\,h^2}{M_P^2}\,, \] the two-field globally scale-invariant theory becomes \[ S_{\rm SI}=\int d^4x\sqrt{g}\,\Bigl[ \tfrac{M_P^2}{2}\,\mathring R -\tfrac12\,\mathcal G_{ab}(Z)\,\partial_\mu\varphi^a\,\partial^\mu\varphi^b -\widetilde U(Z)\Bigr], \] where \(\varphi^a=(\ln(\chi/M_P),\,Z=h^2/\chi^2)\). After diagonalizing the field-space metric by redefinitions, \[ \Phi\equiv\sqrt{\chi^2+\xi_h\,h^2},\qquad Z\equiv\frac{h^2}{\chi^2}, \] one obtains a sigma-model form with one free shift direction and one potential direction [2108.05897]. In the one-field Weyl case, gauge fixing \[ \chi(x)=M_P \] removes the dilaton entirely. After eliminating torsion, the equivalent metric theory is \[ S_{\rm WI} =\int d^4x\sqrt{g}\, \Bigl[\tfrac{M_P^2}{2}\,\mathring R -\tfrac12\,K(h)\,(\partial h)^2 -\frac{U(h)}{\Omega^4(h)}\Bigr], \qquad \Omega^2(h)=1+\xi_h\,h^2/M_P^2\,. \] This is the standard pattern in locally Weyl-invariant Einstein–Cartan constructions: the local symmetry fixes one scalar redundancy, and algebraic torsion modifies the kinetic sector rather than introducing additional propagating tensorial modes [2108.05897]. The quadratic purely gravitational theories admit a parallel reduction. In one affine formulation, auxiliary fields \(\chi\) and \(\phi\) linearize the \(R^2\) and \(\tilde R^2\) terms; after gauge fixing \(\chi=M_P/\sqrt2\), decomposing the connection, and solving the torsion equations, one finds \[ v_\mu = (q+\phi)\,a_\mu,\qquad a_\mu = -24\,\partial_\mu\phi /[1+16(q+\phi)^2],\qquad \tau^\rho{}_{\mu\nu}=0\,, \] which leads to a canonical Einstein-frame scalar \(\Phi\) with a nontrivial potential. In another formulation, algebraic elimination of torsion and fixing \(\Phi(x)=M_P\) yield \[ S_E =\!\!\int d^4x\,\sqrt{-g}\,\biggl[ \frac{M_P^2}{2}\,R(g) -\tfrac12\,(\partial\phi)^2 -V(\phi)\biggr] \] with a single pseudoscalar \(\phi\) [2501.16416]. ## 4. Dilaton, Higgs, and axion-like sectors The distinction between global scale invariance and local Weyl invariance is sharp. In the global Higgs–dilaton Einstein–Cartan theory, both \(h\) and \(\chi\) are genuine fields, and spontaneous breaking of dilatations yields a Nambu–Goldstone dilaton which couples only derivatively. In the local Weyl case, gauge fixing \(\chi=M_P\) removes the dilaton from the spectrum, leaving only one scalar \(h\). The statement that a dilaton is always physical is therefore incorrect for locally Weyl-invariant Einstein–Cartan models [2108.05897]. A different sector arises in the minimal quadratic Weyl-invariant Einstein–Cartan framework coupled to the Standard Model. There, the theory contains just one extra scalar degree of freedom beyond the graviton and the Standard Model fields, and that scalar has the properties of an axion-like particle. In the effective action obtained after gauge fixing and torsion elimination, \[ S_{\rm eff} =\int d^4x\sqrt{-g}\Big\{ \tfrac{M_P^2}{2}R -\tfrac12(\nabla a)^2 -\tfrac12\,m_{a,{\rm grav}}^2\,a^2 -\tfrac1{f_a}\,\partial_\mu a\,J_A^\mu +\cdots\Big\}, \] with \[ m_{a,{\rm grav}}=\frac{\tilde f\,M_P}{4\sqrt3}\,. \] The same pseudoscalar couples through the QCD anomaly, \[ S_{a\,GG} =-\int d^4x\sqrt{-g}\;\frac{g_s^2}{32\pi^2\,f_a}\;a\;\epsilon^{\mu\nu\rho\sigma}G^a_{\mu\nu}G^a_{\rho\sigma}\,, \] so that, provided the gravitationally induced mass is much smaller than the QCD-induced mass, the usual Peccei–Quinn relaxation mechanism follows without postulating a new PQ symmetry [2406.11956]. The scalar content depends on the branch of model building. In one 2025 analysis the gravitational pseudoscalar is heavy, \(m_\phi^2=(\tilde f^2/8)\,M_P^2\), so that \(\phi=0\) during inflation and the Higgs drives the dynamics. In another 2025 treatment the same axion-like field is approximately massless in the early Universe and acts as a spectator while successful Higgs inflation requires a loop-induced nonminimal coupling. A plausible implication is that Weyl-invariant Einstein–Cartan gravity does not define a unique low-energy scalar sector; rather, the spectrum depends on the quadratic invariants retained and on the hierarchy of Lorentz-gauge couplings [2507.15927]. ## 5. Inflationary realizations Inflationary model building is one of the main applications of Weyl-invariant Einstein–Cartan gravity. In the 2021 Higgs-sector construction, the one-field Weyl-invariant theory yields an Einstein-frame action with noncanonical kinetic function \(K(h)\) and flattened potential \(U(h)/\Omega^4(h)\). For large field values, \[ h\gg \frac{M_P}{\sqrt{\xi_h}}\,, \] the kinetic function develops a pole, flattening the potential, and the slow-roll predictions become \[ n_s\simeq1-\frac2N,\qquad r\simeq\frac{12}{N^2}\,, \] with the scalar amplitude requiring \(\xi_h\sim10^4\text{--}10^5\) [2108.05897]. Purely geometric quadratic models realize inflation without introducing a fundamental inflaton by hand. One affine construction leads, after auxiliary-field rewriting and canonical normalization, \[ V(\Phi)=V_0\,[4q-\sinh(\arsinh(4q)-\sqrt{2/3}\,\Phi/M_P)]^2\,, \] and in the large-\(q\) plateau limit reproduces \[ n_s\simeq1-\frac2N,\qquad r\simeq\frac{12}{N^2}\,. \] For \(q=10^3\) and \(N=60\), the paper reports \(n_s\simeq0.9673\) and \(r\simeq0.003\), while for \(q\gtrsim30\) it finds \(0.96071/3\) increases \(N_k\) and raises \(n_s\), while \(w_{\rm reh}<1/3\) lowers \(N_k\) and lowers \(n_s\). In the Starobinsky limit \((\theta\gg1)\), current data favor stiff reheating \((w_{\rm reh}\simeq1)\) and relatively low \(T_{\rm reh}\), whereas for smaller \(\theta\) such as \(\theta=15\), softer reheating with \(w_{\rm reh}<1/3\) and \(T_{\rm reh}\lesssim10^{10\!-\!13}\,\rm GeV\) is preferred [2602.00317]. Outside inflation, the framework has been used for particle-physics model building. In the 2021 Higgs–dilaton Einstein–Cartan analysis, torsion-induced four-fermion operators scale as \(g_{ij}/M_P^2\), so order-one coefficients remain well below current collider bounds. In the minimal Weyl-invariant Standard Model plus Einstein–Cartan theory, tiny Lorentz-gauge couplings \(f,\tilde f\ll1\) simultaneously suppress the cosmological constant, the tree-level Higgs mass, and the gravitational axion mass. The same framework has been proposed as a setting in which nonperturbative gravitational effects generate the electroweak scale and Majorana masses for right-handed neutrinos [2108.05897]. The literature also contains broader and partly distinct generalizations. Gauging the Maxwell–Weyl algebra leads to an Einstein–Cartan–Weyl theory with a compensating scalar \(\phi\), a Weyl gauge field \(\chi\), an additional antisymmetric gauge field \(B^{ab}\), and a shifted curvature \(\mathcal J^{ab}=R^{ab}+2\gamma\phi^2F^{ab}\). In four dimensions this yields an Einstein equation with a dynamical cosmological term \(\Lambda_{\rm eff}=3\gamma\phi^2\). A later broken-phase construction uses the coset formalism for gauged Poincaré \(\times\) dilatations, treats the dilaton as a Stueckelberg field, and shows that after unitary gauge the Weyl field acquires a Proca mass and can be exchanged for the propagating torsion trace \(T_\mu=3A_\mu\). This is not the same regime as the nonpropagating-torsion models emphasized above. A common misconception is therefore that Weyl-invariant Einstein–Cartan gravity always contains a propagating Weyl vector; in many of the inflationary and Higgs constructions the torsion sector is algebraic, whereas propagating vector torsion emerges only in distinct broken-Weyl realizations [1404.3969]. The field remains technically heterogeneous. Some formulations identify the Weyl gauge field directly with the torsion trace, some introduce an explicit compensator and separate Weyl connection, some make the Higgs the compensator, and some derive the effective scalar from \(R^2\), \(\tilde R^2\), or \(R\tilde R\) terms. What unifies them is the use of local scale symmetry in a torsionful first-order gravitational framework and the recurrent result that, after gauge fixing and torsion elimination, the theory reduces to an Einstein-frame model with a sharply constrained scalar or pseudoscalar sector. Source: https://www.emergentmind.com/topics/weyl-invariant-einstein-cartan-gravity