Spinfoams, $\gamma$-duality and parity violation in primordial gravitational waves

The Barbero-Immirzi parameter $\gamma$ plays a central role in Loop Quantum Gravity (LQG) [1, 2, 3]. The parameter sets the typical scale of the spectrum of geometric observables such as the volume of a quantum of space, the area of the interface between two quanta and the length of a curvature defect line [4, 5, 6, 7, 8]. The elementary area operator has eigenvalues

$$A_{j}=8\pi G\hbar\,\gamma\,\sqrt{j(j+1)}\,,\qquad j\in\left\{0,\,\tfrac{1}{2},\,1,\,\tfrac{3}{2},\ldots\right\}\,.$$

(1)

In particular, the theory predicts an area gap $a_{*}=4\sqrt{3}\,\pi\,\gamma\,\ell_{P}^{2}\,$, where $\ell_{P}=\sqrt{G\hbar/c^{3}}\simeq 1.6\times 10^{-35}\,\mathrm{m}$ is the Planck length and the Barbero-Immirzi parameter $\gamma>0$ is a dimensionless coupling constant that, in principle, is to be determined experimentally. At the classical level [9, 10, 11, 12], the parameter $\gamma$ can be understood as a coupling constant in the Einstein-Cartan-Holst action for gravity. This action is invariant under orientation-preserving diffeomorphisms but not under orientation-reversing ones, and therefore is parity violating:

$$S=\frac{1}{16\pi G}\!\int\frac{1}{2}\epsilon_{IJKL}\,e^{I}\wedge e^{J}\wedge F^{KL}-\frac{1}{\gamma}\,e_{I}\wedge e_{J}\wedge F^{IJ}\,,$$

(2)

where the first term in the integral is the parity-odd Einstein-Cartan density and the second is the parity-even Holst density. At the quantum level, spinfoams provide a non-perturbative definition of the covariant dynamics of LQG [13]. The spinfoam dynamics is not defined in terms of an action for fields on a manifold, but instead by a spinfoam vertex amplitude. The Engle-Pereira-Rovelli-Livine (EPRL) [14] vertex amplitude $W_{\gamma}$ depends on the Barbero-Immirzi parameter $\gamma$ and belongs to a one-parameter family of spinfoam models related by a duality rotation with an angle $\theta$. In the limit $\theta\to 0$ at fixed eigenvalues of the area, the EPRL model reduces to the Barrett-Crane model (BC) [15], which is parity invariant. On the other hand, the EPRL model with finite value of $\gamma$ is parity-violating [16] and defined by

$$\tan\theta=\frac{1}{\gamma}\,.$$

(3)

At the semiclassical level, this one-parameter family of theories can be understood in terms of a duality rotation of the curvature

	$\displaystyle F^{IJ}\quad$	$\displaystyle\overset{\theta}{\longrightarrow}\quad+\cos\theta\;\,F{{}^{IJ}}\;+\;\sin\theta\;\;{}^{*}\!F^{IJ}\,,$		(4)
	$\displaystyle{}^{*}\!F^{IJ}\quad$	$\displaystyle\overset{\theta}{\longrightarrow}\quad-\sin\theta\;\>F^{IJ}\;+\;\cos\theta\;\;{}^{*}\!F^{IJ}\,,$

where ${}^{*}\!F_{IJ}=\frac{1}{2}\epsilon_{IJKL}\,F^{KL}$ is the Hodge dual in the internal Lorentz indices. Starting from an effective action for gravity that is parity non-violating, we can produce a parity-violating effective action via a duality rotation of the curvature. For instance, the action (2) can be obtained by a duality rotation of the Einstein-Cartan density with angle $\theta$ satisfying the condition (3). We note that the duality rotation is not a symmetry of the action, or of its solutions [17, 18, 19], but a relation between theories with different values of their coupling constants.

When the effective action is organized as a derivative expansion [20], each term in the expansion is a Lagrangian density that comes with a coupling constant which, in principle, needs to be fixed by observations. Remarkably, the requirement that the effective action is obtained by a duality rotation of a parity non-violating effective action introduces a non-trivial relation between these coupling constants — a single parameter $\gamma$ sets the scale of all parity violating terms in the action, modeling at the semiclassical level an exact property of the spinfoam dynamics $W_{\gamma}$. An effective field theory that satisfies this requirement is described by an action $S_{\gamma}$ that we call $\gamma$-dual.

While a top-down derivation of the effective action directly from the non-perturbative spinfoam dynamics $W_{\gamma}$ is still missing, the condition of $\gamma$-duality allows us to relate coupling constants of parity-even and parity-odd terms in the effective action $S_{\gamma}$. Therefore the observation of gravitational parity violation would provide an indirect measurement of the Barbero-Immirzi parameter $\gamma$, and therefore of the LQG area gap. In this paper we investigate a $\gamma$-dual model of cosmological inflation and show how the spectrum of primordial gravitational waves depends on the coupling constant $\gamma$.

The effect of gravitational parity violation in primordial inflation has been studied for the first time in [21] where a Chern-Simons modification of General Relativity was considered and a non-vanishing polarization $\Pi\equiv(A_{T+}-A_{T-})/(A_{T+}+A_{T-})$ is shown to arise for the amplitude $A_{T\pm}$ of primordial tensor modes with circular polarization $(\pm)$. We refer to [22] for a review and to [23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38] for older and more recent developments. Our analysis is motivated by a relation between parity violation in primordial gravitational waves and the Barbero-Immirzi parameter in LQG originally proposed in [26]. We develop this proposal by introducing a new crucial ingredient — spinfoam $\gamma$-duality. In general, at the quadratic order in curvature, the Lagrangian for gravity and an inflaton scalar field $\phi$ can include both a parity-even term $L_{GB}$ proportional to the Gauss-Bonnet density and a parity-odd term $L_{CS}$ proportional to the Pontryagin density (also know as Chern-Simons term) [27, 28], with

	$\displaystyle L_{GB}=$	$\displaystyle\,f_{GB}(\phi)\,(R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma}\!-\!4R_{\mu\nu}R^{\mu\nu}\!+\!R^{2})\,,$		(5)
	$\displaystyle L_{CS}=$	$\displaystyle\,f_{CS}(\phi)\;\big{(}\tfrac{-1}{\sqrt{-g}}\epsilon^{\mu\nu\rho\sigma}\,R^{\alpha}{}_{\beta\mu\nu}\,R^{\beta}{}_{\alpha\rho\sigma}\big{)}\,.$		(6)

The two functions $f_{GB}(\phi)$ and $f_{CS}(\phi)$ are in principle independent. The requirement of $\gamma$-duality of the action imposes a relation between the two functions: $f_{GB}(\phi)/f_{CS}(\phi)=\gamma-\frac{1}{\gamma}$. In this paper we show that, in a $\gamma$-dual model of slow-roll inflation, the asymmetry $\Pi$ of the amplitude of circular polarizations, the tensor-to-scalar ratio $r\equiv(A_{T+}+A_{T-})/A_{S}$ and the tensor tilt $n_{T}$ are related to the parameter $\gamma$ by the equation

$$\frac{1}{\gamma}-\gamma=\frac{\pi}{8}\,\frac{r+8\,n_{T}}{\Pi}.$$

(7)

Therefore the observation of a primordial parity violation $\Pi\neq 0$ and a violation of the $(r,n_{T})$ ‘consistency relations’, $r\neq-8\,n_{T}$, would provide a measurement of the Barbero-Immirzi parameter $\gamma$, and thus fix the scale $a_{*}$ of the area gap.

The analysis presented here is complementary to the framework of loop quantum cosmology [39, 40, 41, 42, 43] where Planck scale effects on the background dynamics and on metric perturbations during the pre-inflationary era are found to leave phenomenological imprints in cosmic microwave background observables. The analysis is also complementary to the approach of spinfoam cosmology [44, 45, 46] where one investigates the cosmological regime of the full spinfoam dynamics. The phenomenological effects considered in this paper have their origin in the inflationary era, where the curvature is already far from the Planck scale and an effective field theory motivated by spinfoams can be used. In this sense, the analysis follows the same logic used in the study of parity violation in the coupling of fermions to gravity and torsion in an effective field theory [47, 48], with the additional new ingredient of $\gamma$-duality introduced here.

Specifically, we adopt the point of view that $\gamma$ is a coupling constant that needs to be determined experimentally, as any other coupling constant. This viewpoint is adopted also in recent analysis in loop quantum cosmology [49, 50], where observational bounds on the area gap $a_{*}$ — and therefore on $\gamma$ — are investigated. This point of view differs from the one (adopted for instance in [42]) where $\gamma$ is set to a fixed exact value $\gamma_{0}=\log(2)/\pi\sqrt{3}$ [51] or $\gamma_{M}$ [52], which is obtained by imposing that the microscopic entropy of the horizon-area ensemble matches the thermodynamics entropy of a black hole of the same horizon area. Other analysis of the derivation of the Bekenstein-Hawking entropy formula show that the coefficient of the area law is determined by semiclassical infrared phenomena and is independent of the value of the Barbero-Immirzi parameter [53, 54, 55, 56], and therefore needs to be determined experimentally.

The paper is organized as follows. In section II we describe $\gamma$-duality in spinfoams, both at the level of the non-perturbative definition of the vertex amplitude and at the semiclassical level. In section III we discuss the link between parity, orientation-reversing diffeomorphisms and internal Lorentz reversals, and describe how different gravitational terms in the effective action transform. In section IV we use duality rotations to construct a spinfoam-motivated effective action for inflation and discuss the assumptions of the model. In section V we compute the primordial power spectrum of primordial gravitational waves, and in section VI we discuss prospects for observations.

We briefly discuss the EPRL spinfoam model [1, 13], illustrating its parity-violating nature [16, 57, 58, 59] and the role of $\gamma$-duality both in its non-perturbative definition and in the semiclassical limit.

The Barbero-Immirzi parameter appears as a dimensionless coupling constant in the Einstein-Cartan-Holst action $S_{ECH}$, (2). This action can be obtained starting from the Einstein-Cartan action $S_{EC}$,

	$\displaystyle S_{EC}=\frac{1}{2\kappa}\,\int\frac{1}{2}\epsilon_{IJKL}\,e^{I}\wedge e^{J}\wedge F^{KL}$		(8)
	$\displaystyle\overset{\theta}{\longrightarrow}\quad S_{ECH}=\frac{\cos\theta}{2\kappa}\,\int\frac{1}{2}\epsilon_{IJKL}\,e^{I}\wedge e^{J}\wedge F^{KL}$
	$\displaystyle\hskip 102.00012pt-\tan\theta\;e_{I}\wedge e_{J}\wedge F^{IJ}\,,$		(9)

via the duality rotation (4) of the curvature. Comparing the expressions (2) and (9), we observe that the Barbero-Immirzi parameter is related to the duality rotation angle $\theta$ by the relation (3) and $\kappa$ is related to the observed value of Newton’s constant $G$ by $8\pi G=\kappa/\cos\theta$.

In spinfoam quantum gravity one finds a similar relation at the non-perturbative level. The relation can be expressed in terms of the labels $(p,j)$ of the unitary irreducible representations of the principal series of the Lorentz group $SL(2,\mathbb{C})$,

$$g\in SL(2,\mathbb{C})\qquad\longrightarrow\quad D^{(p,j)}(g)\,.$$

(10)

The Engle-Pereira-Rovelli-Livine spinfoam model (EPRL) with Barbero-Immirzi parameter $\gamma$ [14] is defined by the representation $D^{(\gamma j,j)}(g)$, while the Barrett-Crane model (BC) [15] by the representation $D^{(p_{0},0)}(g)$. The two models are related by the duality rotation

$$p\;=\;\cos\theta\;p_{0}\;,\qquad j\;=\;\sin\theta\;p_{0}\;,$$

(11)

where the angle $\theta$ satisfies (3), i.e.,

$$p\,=\,\gamma\,j\,.$$

(12)

The BC model is obtained from the EPRL model as the limit $\theta\to 0$ at fixed $p_{0}$. Equivalently, this duality rotation defines a one-parameter family of parity-violating EPRL models starting from the parity-non-violating BC model as a seed. Specifically, the spinfoam wedge amplitude $\mathcal{A}_{w}$ for a coherent boundary state labeled by the spinors $\zeta$ and $\zeta^{\prime}$ takes the form [60]

$$\mathcal{A}_{w}(g,\zeta^{\prime},\zeta^{\prime\prime})=[\,j_{w},\zeta^{\prime}|D^{(p_{w},j_{w})}(g)|j_{w},\zeta^{\prime\prime}\rangle=\!\!\int_{\mathbb{C}P^{1}}\!\!\!d\mu\;\;\mathrm{e}^{\,\mathrm{i}S_{w}},$$

(13)

where the wedge action $S_{w}$ consists of the sum of two terms proportional to the quantum numbers $p_{w}$ and $j_{w}$ that label the representation, i.e., $S_{w}=p_{w}\,\Theta_{w}\;+\;j_{w}\,\chi_{w}$. In the semiclassical limit for a Lorentzian $4$-simplex, a saddle-point analysis shows that, for $p_{0}\to\infty$, the integral is dominated by the exponential of the sum of the wedge actions evaluated at each saddle point,

$$S_{v}=\sum_{w}\bar{S}_{w}\;=\;\sum_{w}p_{w}\,\bar{\Theta}_{w}\;+\;\sum_{w}j_{w}\,\bar{\chi}_{w}\,.$$

(14)

The two terms appearing in the semiclassical action for a spinfoam vertex can be understood as a discretization on a Lorentzian $4$-simplex of the two terms in the Einstein-Cartan-Holst action (9). The first of the two terms reproduces the Regge action for a Lorentzian $4$-simplex [61],

$$\frac{1}{8\pi G\hbar}S_{\mathrm{Regge}}\;=\;\sum_{w}\frac{A_{w}}{8\pi G\hbar}\bar{\Theta}_{w}\,,$$

(15)

where $A_{w}$ is the area of the triangle where the wedge $w$ hinges, and $\bar{\Theta}_{w}$ is the boost angle between the two space-like tetrahedra that define the wedge. This relation is consistent with the area spectrum in loop quantum gravity in the large spin limit,

$$A_{w}=8\pi G\hbar\;p_{w}\;=\;8\pi G\hbar\,\gamma\,j_{w}\,.$$

(16)

On the other hand, the second term in (14) includes a twist angle $\bar{\chi}_{w}$ that depends on the phase of the coherent boundary state spinors and does not affect the dynamics of the model. This term is analogous to the Holst term in (9) and has the same transformation properties under parity¹¹1Their parity can be read for instance in the asymptotics of the vertex amplitude for Lorentzian boundary data, where a sum of saddle points results in the two terms of different parity $A_{v}\sim\mathrm{e}^{+\mathrm{i}\sum_{w}p_{w}\,\bar{\Theta}_{w}\;+\;\mathrm{i}\sum_{w}j_{w}\,\bar{\chi}_{w}}+\mathrm{e}^{-\mathrm{i}\sum_{w}p_{w}\,\bar{\Theta}_{w}\;+\;\mathrm{i}\sum_{w}j_{w}\,\bar{\chi}_{w}}$.. Note also that the Barrett-Crane model is obtained as the limit $\theta\to 0$ where this second term is not present. When changing the angle $\theta$, the geometric relation between the area $A_{w}$ and the representation $p_{w}$ can be preserved by rescaling $\kappa$ by $\cos\theta$ as discussed after (9) at the level of the classical action.

The argument described above shows that the duality rotation (4) relates the semiclassical limit of spinfoam models with different values of the Barbero-Immirzi parameter. Remarkably, this is an exact relation that holds beyond the semiclassical limit $p_{0}\gg 1$. The defining equations of the EPRL spinfoam model with Barbero-Immirzi parameter $\gamma$ is the linear simplicity constraint [1]

$$\vec{K}=\gamma\vec{L}$$

(17)

where $\vec{K}$ and $\vec{L}$ are the generators of boosts and rotations for the Lorentz group.²²2Specifically, let us denote $J^{IJ}$ the generators of the Lorentz group $SL(2,\mathbb{C})$ and introducing an orthonormal frame $e^{I}_{\mu}=(u^{I},\xi_{i}^{I})$ with $\eta_{IJ}u^{I}u^{J}=-1$. We can then define the generators of the $SU(2)$ rotations group $L_{i}=\frac{1}{2}\epsilon_{IJKL}J^{IJ}u^{K}\xi^{L}_{i}$ that preserve the timelike direction $u^{I}$, and the generators of boosts $K_{i}=J_{IJ}\xi^{I}_{i}u^{J}$. In terms of the self-dual and the anti-self-dual generators,

$$J_{i}^{(\pm)}=\frac{L_{i}\pm\mathrm{i}\,K_{i}}{2}\,,$$

(18)

the linear simplicity constraint takes the form

$$J_{i}^{(\pm)}=\frac{1\pm\mathrm{i}\,\gamma}{\sqrt{1+\gamma^{2}}}\,J^{(0)}_{i}\,$$

(19)

where $J^{(0)}_{i}$ is defined by the parity-even limit $\gamma\to\infty$. We can now introduce a duality rotation [62]

$$J_{i}^{(\pm)}\;\overset{\theta}{\longrightarrow}\;\;\mathrm{e}^{\pm\mathrm{i}\theta}\,J_{i}^{(\pm)}\,,$$

(20)

and notice again that the EPRL and the BC models are consistently related by a duality rotation with angle $\theta$ given by (3). In a (formal) connection representation $\Psi[\omega_{i}^{(+)},\omega_{i}^{(-)}]$ where the self-dual/anti-selfdual generators act as a derivative, $J_{i}^{(\pm)}\Psi[\omega_{i}^{(+)},\omega_{i}^{(-)}]=-\mathrm{i}\frac{\delta}{\delta\omega_{i}^{\pm}}\Psi[\omega_{i}^{(+)},\omega_{i}^{(-)}]$, the duality transformation relates the amplitude of states for theories with different value of the Barbero-Immirzi parameter $\gamma$, with the requirement that, in the limit $\gamma\to\infty$ at fixed elementary areas, the theory is parity invariant. Any spinfoam vertex that satisfies this condition, and therefore implements the relation (9) at the non-perturbative level, is called $\gamma$-dual. The EPRL model is a specific and concrete proposal in this class. In the rest of this paper we take this property, $\gamma$-duality, as fundamental and we explore its consequences in a semiclassical limit described by an effective action.

In this section we discuss the notion of parity in a general relativistic theory where geometry is dynamical. In particular we show how orientation-reversing diffeomorphisms generalize the notion of parity defined in a special relativistic theory, and clarify the link between orientation-reversing diffeomorphisms and orientation reversals in the internal Lorentz group in the Einstein-Cartan formalism. We then describe how different gravitational terms in the effective action transform.

In 3-dimensional Euclidean space, a parity transformation $P\,:\,\{x,y,z\}\mapsto\{-x,-y,-z\}$ is defined as the simultaneous reversal of the three spatial Cartesian axes, that is the product of the three spatial-inversion operators with, e.g., $P_{x}:\,x\mapsto-x$. Similarly, a time reversal $T:\,t\mapsto-t$ is a flip of the temporal axis. In a special relativistic theory, these specific transformations clearly depend on a choice of an inertial rest frame in Minkowski space. It is then useful to think of them in terms of Lorentz transformations that are not connected to the identity. The Lorentz group $O(1,3)$ is the group of transformations $\Lambda^{I}{}_{J}$ that preserve the Minkowski metric $\eta_{IJ}$. The parameter space of Lorentz transformations splits into four connected components $O(1,3)\simeq L_{+}^{\uparrow}\cup L_{+}^{\downarrow}\cup L_{-}^{\uparrow}\cup L_{-}^{\downarrow}$, with the subgroup of proper orthochronous Lorentz transformations $O_{0}(1,3)=SO^{\uparrow}(1,3)\simeq L_{+}^{\uparrow}$ being the components connected to the identity [63, 64]. We can then consider Lorentz transformations modulo transformations connected to the identity,

$$O(1,3)/O_{0}(1,3)\;=\;\{\mathds{1},\,P,\,T,\,PT\}\,,$$

(21)

with the quotient space generated by parity $P$ and time reversal $T$. Orientation reversals of Minkowski space are Lorentz transformations $\Lambda$ with parameters in $L_{+}^{\downarrow}\cup L_{-}^{\uparrow}\,=\,TL_{+}^{\uparrow}\cup PL_{+}^{\uparrow}$ and are defined by $\det(\Lambda)=-1$.

The notions of parity and time reversal described above require a fixed background metric such as the Minkowski metric. Here we are interested in gravitational parity violation, where the metric and its causal structure are not fixed as a background structure but are dynamical instead. Before defining orientation reversals for a spacetime manifold, let us start from a vector space, a linear space of dimension $d$ that is not equipped with any additional structure. In this case, given an ordered basis of $d$ linearly independent vectors $E_{I}=\{E_{1},\ldots,E_{d}\}$, we can consider a change of basis given by a matrix $M$. The requirement that $\tilde{E}_{I}\,=\,M_{I}{}^{J}\,E_{J}$ is still a linear basis imposes that $\det M\neq 0$. Therefore we have two distinct equivalence classes of bases: If $\det(M)>0$ then the two bases belong to the same class; if $\det(M)<0$, they belong to different classes. Each one of such equivalence classes is an orientation $\Xi$ for the vector space.

In the case of a differentiable $d$-dimensional manifold $\mathcal{M}$, an orientation $\Xi$ can be equivalently defined as either [65, 66, 67]: (i) A continuous point-wise orientation on the manifold; (ii) An equivalence class of nowhere-vanishing top-forms (or $d$-forms in a $d$-dimensional manifold); or, (iii) An equivalence class of oriented atlases. A manifold is called orientable if it is possible to make a choice of orientation. If $\mathcal{M}$ is connected and orientable, there are again only two possible orientations $\Xi=\pm 1$.

Definition (i) is the generalization of the concept of orientation of a vector space introduced above, where one considers a frame field $E_{I}$ (i.e., a set of bases $E_{I}(p)$, one at each point $p\in\mathcal{M}$), with the additional condition that the frame must have a point-wise continuous class: at every point $p$ there is a neighborhood with the same choice of orientation $\Xi_{p}$ of its tangent space.³³3We note that this definition of orientation of a manifold does not require the existence of a global continuous basis. The frame can be discontinuous, as long as the point-wise orientation is continuous.

Definition (ii) is the one that is of the most relevance here. Let us discuss briefly its equivalence to (i). Considering a nowhere-vanishing top-form $\Omega$, and an orientation $\Xi=[E_{I}]$ for which $E_{I}$ is a representing frame, we can check if the inner product $\langle\Omega,E_{1}\cdots E_{d}\rangle$ is either positive or negative as, by definition, it cannot be zero. This defines an equivalence relation for nowhere-vanishing top-forms, splitting them into two disjoint sets. The orientation $\Xi$ of the manifold $\mathcal{M}$ is given by an equivalence class of smooth nowhere-vanishing top-forms $\Omega$ and we have $\Xi=[E_{I}]=[\Omega]$ with $\langle\Omega,E_{1}\cdots E_{d}\rangle>0$.

Definition (iii) in terms of an oriented atlas is specifically useful once we introduce coordinates $x^{\mu}$ and dynamical fields over the manifold, as done in General Relativity. We set $d=4$ and assume that the $4$-dimensional manifold $\mathcal{M}$ representing spacetime is orientable. The orientation $\Xi=[\partial_{\mu}]$ in each chart of the atlas is given by the coordinate frame $\partial_{\mu}=\partial/\partial x^{\mu}$. A spacetime diffeomorphism $\varphi\in\mathrm{Diff}(\mathcal{M})$ is then a smooth function $y^{\mu}=\varphi^{\mu}(x)$ with smooth inverse. Under a spacetime diffeomorphism, the nowhere vanishing $4$-form $dx^{0}\wedge\cdots\wedge dx^{3}$ transforms as

$$dy^{0}\wedge dy^{1}\wedge dy^{2}\wedge dy^{3}\,=\,\det(J^{\mu}{}_{\nu})\,dx^{0}\wedge dx^{1}\wedge dx^{2}\wedge dx^{3}\,,$$

(22)

where $J^{\mu}{}_{\nu}=\partial y^{\mu}/\partial x^{\nu}$ is the Jacobian of the transformation. An orientation-preserving diffeomorphism is characterized by $\det(J)>0$, an orientation-reversing diffeomorphism by $\det(J)<0$. Diffeomorphisms that are connected to the identity, denoted by $\mathrm{Diff}_{0}(\mathcal{M})$, form a subgroup and the action of infinitesimal diffeomorphisms on fields is represented by the Lie derivative $\mathcal{L}_{\xi}$ with respect to the vector field $\xi^{\mu}$. If we assume that the spacetime manifold $\mathcal{M}$ is orientable and has a topology that is trivial [68], we have then that the mapping class group of spacetime diffeomorphisms modulo transformations connected to the identity,

$$\mathrm{Diff}(\mathcal{M})/\mathrm{Diff}_{0}(\mathcal{M})\;=\;\{\mathds{1},\mathcal{R}\}\,,$$

(23)

is generated by the manifold orientation reversal $\mathcal{R}:\Xi\mapsto-\Xi$. As we have introduced no Lorentzian metric up to this point, there is no distinction between spatial parity and time reversal, only a notion of orientation reversal $\mathcal{R}$.

In the Einstein-Cartan formulation of General Relativity, the fundamental dynamical variables are a Lorentz connection $\omega_{\mu}^{IJ}(x)dx^{\mu}$ and a coframe field $e^{I}_{\mu}(x)dx^{\mu}$. The Lorentz connection allows us to parallel transport Lorentz fields. The coframe field allows us to define a Lorentzian metric as a derived quantity. The ordered basis $E_{I}=E_{I}^{\mu}(x)\partial_{\mu}$ discussed earlier, the frame field, is also a derived quantity defined by $\langle e^{I},E_{J}\rangle=\delta^{I}{}_{J}$, i.e., the change of basis $E_{I}^{\mu}$ from the coordinate frame $\partial_{\mu}$ to the non-coordinate frame $E_{I}$ is given by the inverse of the coframe field, $E_{I}^{\mu}=(e_{\mu}^{I})^{-1}$. The requirement that the two basis represent the same orientation implies that the determinant of the coframe field is positive,

$$\Xi=[\partial_{\mu}]=[E_{I}]\quad\Longrightarrow\quad\det(e_{\mu}^{I})>0\,.$$

(24)

Equivalently, we can consider the volume $4$-form

$$\Omega_{\mathrm{v}}\,=\,\frac{1}{4!}\,\epsilon_{IJKL}\,e^{I}\wedge e^{J}\wedge e^{K}\wedge e^{L}\,,$$

(25)

together with the coordinate $4$-form

$$\differential^{4}x\,=\,\frac{1}{4!}\,\epsilon_{\mu\nu\rho\sigma}\,dx^{\mu}\wedge dx^{\nu}\wedge dx^{\rho}\wedge dx^{\sigma}\,,$$

(26)

where $\epsilon_{IJKL}$ and $\epsilon_{\mu\nu\rho\sigma}$ are alternating symbols (with $\epsilon_{0123}=+1$). As they are both nowhere-vanishing top forms and $\Omega_{\mathrm{v}}=\det(e_{\mu}^{I})\,\differential^{4}x$, we have again

$$\Xi=[\differential^{4}x]=[\Omega_{\mathrm{v}}]\quad\Longrightarrow\quad\det(e_{\mu}^{I})>0\,.$$

(27)

We turn now to the discussion of diffeomorphisms, and in particular of orientation-reversing diffeomorphisms, in the Einstein-Cartan formulation. Let us consider a Lorentz vector-valued one-form $\alpha^{I}=\alpha^{I}_{\mu}(x)dx^{\mu}$. Given a spacetime diffeomorphism $\varphi\in\mathrm{Diff}(\mathcal{M})$, we need to define a Kosmann lift to the principle Lorentz bundle [69, 70, 71, 72],

$$\hat{\varphi}\;\in\;O(1,3)_{\mathcal{M}}\rtimes\mathrm{Diff}(\mathcal{M})\,,$$

(28)

that allows us to compare Lorentz vectors at different spacetime points, i.e.,

$$\alpha^{I}_{\mu}(x)\quad\longrightarrow\quad(\Lambda_{\varphi}(x))^{I}{}_{K}\;\alpha^{K}_{\nu}(\varphi(x))\;(J_{\varphi}(x))^{\nu}{}_{\mu}\,.$$

(29)

At the infinitesimal level, the lift of a spacetime diffeomorphism to Lorentz vectors is given by the Lorentz-covariant Lie derivative $\hat{\mathcal{L}}_{\xi}$. This notion requires the use of the Lorentz connection $\omega_{\mu}^{IJ}$, with

$$\hat{\mathcal{L}}_{\xi}\alpha^{I}_{\mu}\;=\;\xi^{\nu}\partial^{\vphantom{I}}_{\nu}\alpha^{I}_{\mu}+\alpha^{I}_{\nu}\partial^{\vphantom{I}}_{\mu}\xi^{\nu}\;+\;\xi^{\nu}\omega^{\,I}_{\nu\,J}\,\alpha^{J}_{\mu}\,.$$

(30)

For large diffeomorphims that are not connected to the identity, the natural lift $\hat{\varphi}$ extends consistently to reversals $\mathcal{R}$ of the orientation defined in a coordinate basis $\Xi=[\partial_{\mu}]=[\differential^{4}x]$ or equivalently in a non-coordinate basis $\Xi=[E_{I}]=[\Omega_{\mathrm{v}}]$, i.e.,

	$\displaystyle\hat{\mathcal{R}}:\;\;$	$\displaystyle[\differential^{4}x]\,\mapsto-[\differential^{4}x]\,,$		(31)
		$\displaystyle[\Omega_{\mathrm{v}}]\;\>\mapsto-[\Omega_{\mathrm{v}}]\,.$		(32)

This natural lift of the spacetime orientation reversal requires that the local Lorentz transformation $\Lambda_{\varphi}$ has the same determinant sign as the Jacobian $(J_{\varphi})^{\mu}{}_{\nu}=\partial\varphi^{\mu}/\partial x^{\nu}$,

$$\det(\Lambda_{\varphi})\;=\;\operatorname{sgn}\det(J_{\varphi})\,.$$

(33)

Specifically, an orientation reversing diffeomorphism $\det(J_{\varphi})<0$ always comes together with an internal space Lorentz reversal, $\det(\Lambda_{\varphi})=-1$. In particular, we have that under a spacetime orientation-reversing diffeomorphism $\hat{\varphi}$, the coframe field transforms as $e_{\mu}^{I}\to(\Lambda_{\varphi})^{I}{}_{K}\;e^{K}_{\nu}\;(J_{\varphi})^{\nu}{}_{\mu}$ and, as a result, $\det(e)>0$ is sent to $\det(\Lambda_{\varphi}\,e\,J_{\varphi})>0$ . With these preliminaries, we are ready to discuss how this lift provides a consistent definition of spacetime orientation reversals both in the first-order and in the second-order formulations of General Relativity.

Einstein’s general covariance is realized in General Relativity as the invariance of the action $S=\int\Omega$ under spacetime diffeomorphisms. While diffeomorphisms connected to the identity are assumed to be an exact symmetry also at the quantum level, transformation in the mapping class group (23) are expected to be only accidental approximate symmetries in the semiclassical regime. This is what happens for instance in $2+1$ quantum gravity [73] and in LQG in four dimensions [16]. As we assume here that the effective action is invariant only under $\mathrm{Diff}_{0}(\mathcal{M})$, it is useful to list the transformation properties of various $4$-forms $\Omega$ that can appear in the action for gravity. We start from the coframe field $e^{I}$, the coframe torsion $\tau^{I}$, and the Lorentz curvature $F^{IJ}$,

	$\displaystyle\tau^{I}=$	$\displaystyle\;\,\nabla e^{I}\,=\,de^{I}+\omega^{I}{}_{J}\,e^{J}\,,$		(34)
	$\displaystyle F^{I}{}_{J}=$	$\displaystyle\;\,d\omega^{I}{}_{J}+\omega^{I}{}_{K}\wedge\omega^{K}{}_{J}\,,$		(35)

and we raise and lower Lorentz indices $I,J,\ldots$ with the internal Minkowski metric $\eta_{IJ}$. The only $4$-form that is Lorentz invariant, polynomial in $e^{I}$ and $\omega^{IJ}$, and contains no exterior derivatives is the volume $4$-form $\Omega_{\mathrm{v}}$ (25). With one single exterior derivative we have the Einstein-Cartan and the Holst $4$-forms

	$\displaystyle\Omega_{EC}\;=\;$	$\displaystyle\;\frac{1}{2}\epsilon_{IJKL}\,e^{I}\wedge e^{J}\wedge F^{KL}\,,$		(36)
	$\displaystyle\Omega_{H}\;\;=\;$	$\displaystyle\;e_{I}\wedge e_{J}\wedge F^{IJ}\,,$		(37)

With two exterior derivatives, we have the torsion-squared $4$-form $\Omega_{TT}$, the Gauss-Bonnet (also known as Euler) $4$-form $\Omega_{GB}$ and the Chern-Simons (also known as Pontryagin) $4$-form $\Omega_{CS}$ [74, 75, 76, 77, 78]:

	$\displaystyle\Omega_{TT}\;=\;$	$\displaystyle\;\nabla e_{I}\wedge\nabla e^{I}\,,$		(38)
	$\displaystyle\Omega_{GB}\;=\;$	$\displaystyle\;\frac{1}{2}\epsilon_{IJKL}\,F^{IJ}\wedge F^{KL}\,,$		(39)
	$\displaystyle\Omega_{CS}\;=\;$	$\displaystyle\;F_{IJ}\wedge F^{IJ}\,.$		(40)

Under orientation-reversing diffeomorphisms $\mathcal{R}$ we have the following transformation properties:

	$\displaystyle\Omega_{\mathrm{v}}\;\;\;\quad$	$\displaystyle\overset{\mathcal{R}}{\longrightarrow}\quad-\Omega_{\mathrm{v}}\,,$		(41)
	$\displaystyle\Omega_{EC}\quad$	$\displaystyle\overset{\mathcal{R}}{\longrightarrow}\quad-\Omega_{EC}\,,$		(42)
	$\displaystyle\Omega_{H}\;\;\quad$	$\displaystyle\overset{\mathcal{R}}{\longrightarrow}\quad+\Omega_{H}\,,$		(43)
	$\displaystyle\Omega_{TT}\quad$	$\displaystyle\overset{\mathcal{R}}{\longrightarrow}\quad+\Omega_{TT}\,,$		(44)
	$\displaystyle\Omega_{GB}\quad$	$\displaystyle\overset{\mathcal{R}}{\longrightarrow}\quad-\Omega_{GB}\,,$		(45)
	$\displaystyle\Omega_{CS}\quad$	$\displaystyle\overset{\mathcal{R}}{\longrightarrow}\quad+\Omega_{CS}\,.$		(46)

Therefore, the Einstein-Cartan action with a cosmological constant, $S_{EC\Lambda}=\frac{1}{2\kappa}\int(\Omega_{EC}-2\Lambda\,\Omega_{\mathrm{v}})$ is odd under orientation reversals, $S_{EC\Lambda}\to-S_{EC\Lambda}$. On the other hand, including also a Holst term or a Chern-Simons term results in an action that is neither odd nor even under orientation reversals. We will call an action that is neither odd nor even under orientation-reversing diffeomorphisms a parity-violating action for short.

The conditions for a parity-violating action described above apply also to the metric formulation of General Relativity. Let us assume that the spacetime torsion vanishes, $\tau^{I}=\nabla e^{I}=0$, and that there is no coupling to spinor fields that can source torsion. We can then formulate the theory purely in terms the spacetime metric $g_{\mu\nu}(x)$ taken as a fundamental field, instead of considering it a derived quantity

$$g_{\mu\nu}(x)\,=\,\eta_{IJ}^{\vphantom{I}}\,e^{I}_{\mu}(x)e^{J}_{\nu}(x)\,,$$

(47)

as done in the Einstein-Cartan formulation. Moreover the condition of zero torsion $\tau^{I}=0$ allows us to solve for the metric-compatible Lorentz connection that defines the Christoffel connection $\Gamma^{\rho}_{\mu\nu}=\Gamma^{\rho}_{\mu\nu}(g)$ and the Riemann tensor $R^{\mu}{}_{\nu\rho\sigma}(g)$. In the metric formulation, the volume $4$-form (25) can be expressed as

$$\Omega_{\mathrm{v}}=\sqrt{-g}\,\differential^{4}x$$

(48)

in terms of the coordinate $4$-form $\differential^{4}x=dx^{0}\wedge dx^{1}\wedge dx^{2}\wedge dx^{3}$ and the volume density $\sqrt{-g}=\sqrt{-\det g_{\mu\nu}}$. Note that we have already assumed that the coordinate basis and the coframe field define the same orientation, and therefore $\sqrt{-g}=\det(e_{\mu}^{I})>0$. We can now write the $4$-forms described earlier in terms of the metric, under the assumption of vanishing torsion. In particular we have that the Einstein-Cartan $4$-form reduces to the Ricci scalar times the volume $4$-form (also known as the Einstein-Hilbert density), while the Holst $4$-form vanishes:

	$\displaystyle\Omega_{EC}^{(\tau=0)}\;=\;$	$\displaystyle\;R\,\sqrt{-g}\,\differential^{4}x\,,$		(49)
	$\displaystyle\Omega_{H}^{(\tau=0)}\;=\;$	$\displaystyle\;0\,.$		(50)

The second equation follows from the identity $\Omega_{H}=\Omega_{TT}-\Omega_{NY}$ where $\Omega_{NY}=d(e_{I}\wedge\tau^{I})$ is the Nieh-Yan $4$-form. Similarly, the Gauss-Bonnet and the Chern-Simons $4$-forms in the absence of torsion reduce to

	$\displaystyle\Omega_{GB}^{(\tau=0)}=$	$\displaystyle\;\tfrac{1}{2}\big{(}R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma}-4R_{\mu\nu}R^{\mu\nu}+R^{2}\big{)}\,\sqrt{-g}\,\differential^{4}x\,,$
	$\displaystyle\Omega_{CS}^{(\tau=0)}=$	$\displaystyle\;\tfrac{1}{4}\,\epsilon^{\mu\nu\rho\sigma}\,R^{\alpha}{}_{\beta\mu\nu}\,R^{\beta}{}_{\alpha\rho\sigma}\;\differential^{4}x\,.$		(51)

In the metric formulation, the Lagrangian $L$ associated to a $4$-form $\Omega$ is defined as

$$\Omega\,=\,L\,\sqrt{-g}\,\differential^{4}x\,.$$

(52)

Orientation-reversing diffeomorphisms in the metric formulation do not require a lift from spacetime to the Lorentz bundle (28). The transformation properties of the metric $4$-forms (49-51) coincide with (41-46). Note that, as $\differential^{4}x\to-\differential^{4}x$ under orientation reversals, we have that the action $S=\int\Omega$ and the associated Lagrangian $L$ have opposite transformation properties

$$S\,\overset{\mathcal{R}}{\longrightarrow}\,\mp S\quad\Longrightarrow\quad L\,\overset{\mathcal{R}}{\longrightarrow}\,\pm L\,,$$

(53)

i.e., the Lagrangian $L$ transforms as a scalar or a pseudoscalar. In particular, the Einstein-Hilbert action $S_{EH}=\frac{1}{2\kappa}\int R\,\sqrt{-g}\,\differential^{4}x$ is odd under orientation reversals, while the Einstein-Hilbert Lagrangian $L_{EH}=\frac{1}{2\kappa}R$ is even and therefore a scalar.

In this analysis, we have focused on orientation-reversing spacetime diffeomorphisms and their lift to the principle Lorentz bundle. We have not discussed purely internal Lorentz transformations that are Lorentz reversals, such as the internal parity $P$ and internal time reversal $T$ considered in [16]. These transformations change the determinant of the frame field $e_{\mu}^{I}$ without changing the orientation of the spacetime manifold, and therefore they do not have a corresponding transformation in the metric formulation.

In formal manipulations in metric variables, it is useful to adopt the metric-dependent antisymmetric tensor [79],

$$\varepsilon_{(g)}^{\mu\nu\rho\sigma}=\frac{-1}{\sqrt{-g}}\,\epsilon^{\mu\nu\rho\sigma}\,,$$

(54)

which transforms as a tensor and is defined in terms of the Levi-Civita alternating symbol $\epsilon^{\mu\nu\rho\sigma}$ (with $\epsilon^{0123}=+1$) appearing for instance in (51).

We include here a remark about the Gauss-Bonnet and Chern-Simons densities that will be relevant in the next section. The integral of $\Omega_{GB}$ and of $\Omega_{CS}$ defines topological invariants of a manifold as they are given by the exterior derivatives of $3$-forms,

	$\displaystyle\Omega_{GB}=$	$\displaystyle\;d\big{(}\tfrac{1}{2}\epsilon_{IJKL}\,\omega^{IJ}\wedge\big{(}F^{KL}-\tfrac{1}{3}\omega^{K}{}_{M}\wedge\omega^{ML}\big{)}\big{)},$		(55)
	$\displaystyle\Omega_{CS}=$	$\displaystyle\;d\big{(}\omega_{IJ}\wedge\big{(}F^{IJ}-\tfrac{1}{3}\omega^{I}{}_{M}\wedge\omega^{MJ}\big{)}\big{)}\,.$		(56)

In metric variables

	$\displaystyle L_{GB}=\nabla_{\mu}K_{GB}^{\mu}$		(57)
	$\displaystyle L_{CS}=\nabla_{\mu}K_{CS}^{\mu}$		(58)

where $K_{GB}^{\mu}$ is given by [80]

$$K_{GB}^{\mu}=\varepsilon_{(g)}^{\mu\nu\rho\sigma}\varepsilon_{(g)}^{\alpha\beta}{}_{\gamma\delta}\,\Gamma^{\gamma}_{\alpha\nu}\big{(}\tfrac{1}{2}R^{\delta}{}_{\beta\rho\sigma}+\tfrac{1}{3}\Gamma^{\delta}_{\lambda\rho}\Gamma^{\lambda}_{\beta\sigma}\big{)}\,,$$

(59)

and $K_{CS}^{\mu}$ is the Chern-Simons current, which in the absence of torsion can be written as [22, 23]

$$K_{CS}^{\mu}=\varepsilon_{(g)}^{\mu\nu\rho\sigma}\,\Gamma^{\alpha}_{\nu\beta}\big{(}\partial_{\rho}\Gamma^{\beta}_{\sigma\alpha}+\tfrac{2}{3}\Gamma^{\beta}_{\rho\lambda}\Gamma^{\lambda}_{\sigma\alpha}\big{)},$$

(60)

Even though by themselves they do not contribute to the equations of motion as they are total derivatives, a non-minimal coupling of the form (5), (6) makes them no longer trivial.

In this section we describe the construction of an effective action for inflation motivated by the $\gamma$-duality of the spinfoam dynamics. The $\gamma$-dual action $S_{\gamma}$ is obtained via a duality rotation of a seed action $S_{\mathrm{odd}}$ that is parity-non-violating (odd under orientation-reversing diffeomorphisms $\mathcal{R}$). While, the action $S_{\mathrm{odd}}$ we start with here is chosen with the requirement that it provides a minimal model of inflation compatible with current observations [81], the construction of a $\gamma$-dual action via a duality rotation is more general and applies also to other possible seed actions.

We consider an action for gravity and a single scalar field with a potential that drives inflation, written in first order formalism:

$$S_{\mathrm{odd}}^{(1)}=\int\tfrac{1}{2\kappa}\Omega_{EC}\,+\,L_{m}\,\Omega_{\mathrm{v}}\,+\,\Omega_{S}\,.$$

(61)

The action depends on $e^{I}$ and $\omega^{IJ}$, the Lorentz coframe and the Lorentz connection; and on $\phi$, and $\pi^{I}$, the scalar field and its covariant momentum. It is given by the integral of the Einstein-Cartan $4$-form $\Omega_{EC}$ (36), the volume $4$-form $\Omega_{\mathrm{v}}$ (25), and the $4$-form

$$\Omega_{S}=\frac{1}{3!}\epsilon_{IJKL}\,\pi^{I}\,d\phi\wedge e^{J}\wedge e^{K}\wedge e^{L}\,,$$

(62)

that depends both on the scalar field $\phi$ and its covariant momentum, the vector valued scalar $\pi^{I}$. The dynamics of the scalar field is encoded in the Lagrangian $L_{m}$,

$$L_{m}=+\tfrac{1}{2}\pi^{I}\pi_{I}-V(\phi)\,,$$

(63)

with a potential $V(\phi)$ with a plateau that can drive slow-roll inflation [81], such as the Starobinsky potential [82]. This first-order action is equivalent to the one generally considered in single-field slow-roll inflation in the metric formalism: the stationarity of the action under variations of the Lorentz connection $\omega^{IJ}$ and of the scalar momentum $\pi^{I}$ imposes that the torsion vanishes $\tau^{I}=\nabla e^{I}=0$ and the momentum is given by $\pi_{I}=e_{I}^{\mu}\partial_{\mu}\phi$. Working in the Einstein-Cartan formalism, though, allows us to highlight three properties of this action that are relevant for our construction:

(i) The action $S_{\mathrm{odd}}^{(1)}$ is polynomial in the one-forms $e^{I}$ and $\omega^{IJ}$. In particular, it does not require that the coframe field $e^{I}_{\mu}(x)dx^{\mu}$ has an inverse $e^{\mu}_{I}=(e^{I}_{\mu})^{-1}$. As such, it allows spinfoam-like $2$d configurations of the geometry.

(ii) The action defines a theory that is invariant under spacetime diffeomorphisms $\mathrm{Diff}(\mathcal{M})$, including orientation-reversing diffeomorphisms $\mathcal{R}$ as the action is parity-odd, $S_{\mathrm{odd}}^{(1)}\overset{\mathcal{R}}{\longrightarrow}-S_{\mathrm{odd}}^{(1)}$.

(iii) In an expansion in the number of exterior derivatives $d$ of fields, the term $L_{m}\,\Omega_{\mathrm{v}}$ is of zero-th order, while the two terms $\Omega_{EC}$ and $\Omega_{S}$ are of first order as they depend on $d\omega^{IJ}$ and $d\phi$.

Following Weinberg’s construction of an effective field theory of inflation [20], we consider now an extension $S_{\mathrm{odd}}^{(2)}$ of the action $S_{\mathrm{odd}}^{(1)}$ that includes terms of second order in exterior derivatives, with the additional conditions that $S_{\mathrm{odd}}^{(2)}$ is (i) polynomial in the one-forms $e^{I}$ and $\omega^{IJ}$, and (ii) it is parity odd under orientation-reversing diffeomorphisms $\mathcal{R}$. These conditions exclude $\Omega_{TT}$ and $\Omega_{CS}$, the torsion-squared and the Chern-Simons $4$-forms which are parity-even (44), (46). We are left with the Gauss-Bonnet $4$-form and $S_{\mathrm{odd}}^{(2)}=\int f(\phi)\,\Omega_{GB}$, where $f(\phi)$ is a function of the scalar field $\phi$. Note that we do not consider possible couplings to the covariant momentum $\pi^{I}$ which effectively correspond to a modified kinetic term for the scalar field as in models of $K$-inflation [83].

The parity non-violating action $S_{\mathrm{odd}}=S_{\mathrm{odd}}^{(1)}+S_{\mathrm{odd}}^{(2)}$,

$$S_{\mathrm{odd}}=\int\tfrac{1}{2\kappa}\Omega_{EC}\,+\,L_{m}\,\Omega_{\mathrm{v}}\,+\,\Omega_{S}\,+\,f(\phi)\,\Omega_{GB}\,,$$

(64)

is taken here as the starting point, or seed, for the construction of a $\gamma$-dual action for inflation. Under a duality rotation (4) of the Lorentz curvature $F^{IJ}$, the $4$-forms appearing in $S_{\mathrm{odd}}$ transform as⁴⁴4These transformation properties can equivalently be derived using the self-dual and antiself-dual variables $F^{(\pm)}_{IJ}=\frac{1}{2}(F_{IJ}\mp\mathrm{i}\,{}^{*}\!F_{IJ})$ which transform as $F^{(\pm)}_{IJ}\overset{\theta}{\longrightarrow}\mathrm{e}^{\pm\mathrm{i}\theta}F^{(\pm)}_{IJ}$.

	$\displaystyle\Omega_{\mathrm{v}}\;\;\,\quad$	$\displaystyle\overset{\theta}{\longrightarrow}\quad\Omega_{\mathrm{v}}\,,$		(65)
	$\displaystyle\Omega_{S}\;\;\quad$	$\displaystyle\overset{\theta}{\longrightarrow}\quad\Omega_{S}\,,$		(66)
	$\displaystyle\Omega_{EC}\quad$	$\displaystyle\overset{\theta}{\longrightarrow}\quad\cos\theta\;\;\,\Omega_{EC}\;+\;\sin\theta\;\;\,\Omega_{H}\,,$		(67)
	$\displaystyle\Omega_{GB}\quad$	$\displaystyle\overset{\theta}{\longrightarrow}\quad\cos 2\theta\;\Omega_{GB}\;+\;\sin 2\theta\;\Omega_{CS}\,,$		(68)

and define a one-parameter family of duality-rotated actions $S_{\mathrm{odd}}\overset{\theta}{\longrightarrow}S_{\theta}$.

We note that duality rotations of the curvature tensor have been considered multiple times in the literature with the different goal of relating solutions of Einstein equations [84, 85].⁵⁵5Note that here we consider only duality rotations in the Lorentz internal indices of $F^{IJ}$. Alternatively, in the metric formalism, one can also consider a duality rotation of the Riemann tensor on spacetime manifold indices (via a tangent-space Hodge dual operator). These two definitions are not trivially equivalent; for a discussion on this comparison, see [86]. Recently, the duality rotation (67) of the Einstein-Cartan action has been considered in [17, 18] as an exact symmetry of vacuum Einstein gravity, but shown in [19] to be only a symmetry of a duality-preserving sector of vacuum solutions. The transformation considered here is not a symmetry of the action $S_{\mathrm{odd}}$, but the construction of a one-parameter family of actions $S_{\theta}$ that reproduces the relation between the EPRL spinfoam model and the parity-non-violating limit $\theta\to 0$ given by the BC model discussed in Sec. II. By setting the parameter $\theta$ to the value $\tan\theta=1/\gamma$ that relates it to the Barbero-Immirzi parameter $\gamma$ (3) and normalizing the constant $\kappa/\cos\theta=8\pi G$ to the observed value of Newton’s constant $G$ as discussed in Sec. II, we obtain the spinfoam-motivated $\gamma$-dual action,

$$S_{\gamma}=S_{\gamma}^{(1)}+S_{\gamma}^{(2)}$$

(69)

with

	$\displaystyle S_{\gamma}^{(1)}[e^{I},\omega^{IJ},\phi\,]$	$\displaystyle=\frac{1}{16\pi G}\!\int\left(\frac{1}{2}\epsilon_{IJKL}\,e^{I}\wedge e^{J}\wedge F^{KL}-\frac{1}{\gamma}\,e_{I}\wedge e_{J}\wedge F^{IJ}\right)$
		$\displaystyle\quad+\int\left(-\frac{1}{2}\,\eta^{IJ}e^{\mu}_{I}\partial_{\mu}\phi\;e^{\nu}_{J}\partial_{\nu}\phi-V(\phi)\right)\frac{1}{4!}\,\epsilon_{IJKL}\,e^{I}\wedge e^{J}\wedge e^{K}\wedge e^{L}\,$		(70)
	$\displaystyle S_{\gamma}^{(2)}[e^{I},\omega^{IJ},\phi\,]$	$\displaystyle=\int f(\phi)\left(\frac{\gamma^{2}-1}{\gamma^{2}+1}\;\frac{1}{2}\epsilon_{IJKL}\,F^{IJ}\wedge F^{KL}\,+\frac{2\gamma}{\gamma^{2}+1}F_{IJ}\wedge F^{IJ}\right)\,,$		(71)

where we have already solved for the covariant momentum $\pi^{I}$. Furthermore, imposing the zero-torsion condition $\tau^{I}=0$ as a constraint on the Lorentz connection,⁶⁶6Alternatively, one can implement a reduction of order procedure where one substitutes the solutions of the equations of motion of the first-order action $S_{\gamma}^{(1)}$ into the second order action $S_{\gamma}^{(2)}$ as discussed in [20]. we can write the $\gamma$-dual action in the metric formalism

	$\displaystyle S_{\gamma}[g_{\mu\nu},\phi]=$	$\displaystyle\;\;\frac{1}{16\pi G}\!\int R\,\sqrt{-g}\,\differential^{4}x\;\;+\int\Big{(}-\frac{1}{2}\,g^{\mu\nu}\partial_{\mu}\phi\,\partial_{\nu}\phi-V(\phi)\Big{)}\sqrt{-g}\,\differential^{4}x$		(72)
		$\displaystyle+\int f(\phi)\left(\frac{\gamma^{2}-1}{2(\gamma^{2}+1)}\left(R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma}-4R_{\mu\nu}R^{\mu\nu}+R^{2}\right)\,+\frac{\gamma}{2(\gamma^{2}+1)}\varepsilon_{(g)}^{\alpha\beta\rho\sigma}R_{\mu\nu\alpha\beta}R^{\mu\nu}{}_{\rho\sigma}\right)\sqrt{-g}\,\differential^{4}x\,.$

In the remaining part of the paper we study primordial cosmological perturbations described by this parity-violating $\gamma$-dual action and the possibility of determining the Barbero-Immirzi parameter from observations.