# The Full Quadratic Metric-Affine Gravity (Including Parity Odd Terms): Exact
solutions for the Affine-Connection

Damianos Iosifidis  Institute of Theoretical Physics, Department of Physics
Aristotle University of Thessaloniki, 54124 Thessaloniki, Greece
<EMAIL_ADDRESS>
###### Abstract

We consider the most general Quadratic Metric-Affine Gravity setup in the
presence of generic matter sources with non-vanishing hypermomentum. The
gravitational action consists of all $17$ quadratic invariants (both parity
even and odd) in torsion and non-metricity as well as their mixings, along
with the terms that are linear in the curvature namely the Ricci scalar and
the totally antisymmetric Riemann piece. Adding also a matter sector to the
latter we first obtain the field equations for the generalized quadratic
Theory. Then, using a recent Theorem, we successfully find the exact form of
the affine connection under some quite general non-degeneracy conditions.
Having obtained the exact and unique solution of the affine connection we
subsequently derive the closed forms of spacetime torsion and non-metricity
and also recast the metric field equations into a GR form with modified source
terms that are quadratic in the hypermomentum and linear in its derivatives.
We also study the vacuum quadratic Theory and prove that in this instance, or
more generally for vanishing hypermomentum, the connection becomes the Levi-
Civita one. Therefore, we also find exactly to what does the quadratic vacuum
Theory correspond to. Finally, we generalize our result even further and also
discuss the physical consequences and applications of our study.

###### Contents

  1. I Introduction
  2. II The Setup
    1. II.1 Canonical and Metrical Energy Momentum and Hypermomentum Tensors
  3. III Full Quadratic Theory
  4. IV The full Quadratic MAG Theory
  5. V Solving for the Affine Connection
    1. V.1 Expressions for Torsion and Non-metricity
  6. VI Post Riemannian Expansion of the Metric Field Equations
  7. VII Conclusions
  8. VIII Acknowledgments
  9. A Variations
    1. A.1 g-Variations
    2. A.2 $\Gamma$-Variations
    3. A.3 More Variations
  10. B Expansions
    1. B.1 Distortion vectors in terms of torsion and non-metricity vectors
    2. B.2 Post-Riemannian expansions
  11. C The $\gamma_{ij}$’s and $\alpha_{ij}$’s
    1. C.1 The $\gamma_{ij}$’s
    2. C.2 The $\alpha_{ij}$’s

##  I Introduction

General Relativity (GR) is one of the pillars of modern Theoretical Physics.
Apart from its rigorous mathematical formulation and the solid predictions it
provides, the geometrization of gravity that is associated to it makes the
latter particularly appealing. However, GR has been challenged by many
observational data and the very fact that it cannot be quantized makes it
clear that the latter is not the final answer to the puzzle of gravitation.
Driven mostly by Cosmological data many alternative Theories of gravity have
been proposed over the years clifton2012modified ; CANTATA:2021ktz . From the
great plethora of alternatives to GR, of particular interest are those
extensions/modifications that are in this spirit of geometrization of
gravitational phenomena that GR preaches. In this direction, one can extend
the geometric arena upon which GR is based by enriching it with non-Riemannian
structures as well. Namely, one can alleviate the Riemannian constraints of
vanishing torsion and non-metricity and be in the realm of non-Riemannian
geometry eisenhart2012non . In this extended geometry, the linear connection
is, in general, neither symmetric (i.e. it processes torsion) nor is it
compatible with the metric (non-metricity). The gravity Theory that is
formulated on such a non-Riemannian manifold is widely known as Metric-Affine
Gravity (MAG) hehl1995metric ; iosifidis2019metric .

In MAG one starts off with a general affine connection that is independent of
the metric and it possesses both torsion and non-metricity. The field
equations are then obtained by independent variations with respect to the
metric and the affine connection. There are many reasons that make MAG an
appealing gravity Theory111See hehl1995metric  for an extended list of motives
for studying MAG. but probably the most astonishing one is the link it
provides in relation to the microstructure of matter hehl1991spacetime ;
puetzfeld2008probing . In recent years, there is an ever-increasing interest
in the MAG formulation iosifidis2018raychaudhuri ; iosifidis2019exactly ;
iosifidis2019scale ; vitagliano2011dynamics ; sotiriou2007metric ;
capozziello2010metric ; percacci2020new ; Jimenez:2020dpn ;
BeltranJimenez:2019acz ; delhom2018observable ; aoki2019scalar ;
Cabral:2020fax ; Ariwahjoedi:2020wmo ; Ariwahjoedi:2021yth ; Yang:2021fjy ;
hohmann2020metric ; bahamonde2020new  with a special emphasis given in
Cosmological applications Iosifidis:2021fnq ; iosifidis2020cosmic ;
Iosifidis:2020gth ; Iosifidis:2021iuw ; jimenez2016spacetimes ;
beltran2017modified ; kranas2019friedmann ; barragan2009bouncing ;
shimada2019metric ; Yang:2021fjy ; kubota2021cosmological ; Mikura:2020qhc ;
Mikura:2021ldx ; Babourova:2002fn ; Puetzfeld:2001hk ; Minkevich:1998cv 222For
the importance of non-Riemannian effects in the early Universe see
Puetzfeld:2004yg  and references therein. . Let us also mention that MAG has
some promising features in regards to the Quantization of Gravity (see ref
Percacci:2020bzf ). From the above discussion it becomes clear that MAG is
worth studying and has a lot to offer in relation to our understanding of
Gravity and the effects of microstructure of matter. Our concern here will be
to study the most general MAG Theory consisting of all possible (both parity
even and parity odd) quadratic invariants333Here we wish to generalize the
findings of Iosifidis:2021tvx  where only the parity even quadratic terms were
included. in torsion and non-metricity along with terms that are linear in the
curvature. That is our gravitational action will be comprised of the usual
Einstein-Hilbert term (i.e Ricci scalar) and the $17$ quadratic invariants of
torsion and non-metricity. To this action we shall also add matter which will,
in general, depend also on the connection, that is it will possess a non-
vanishing hypermomentum. Our task will then be to find the general form of the
connection for the quadratic Theory in the presence of matter but for the
vacuum case as well.

More precisely, the paper is organized as follows. Firstly, we set up
conventions and the definitions we are going to use throughout and briefly
discuss some geometrical as well as physical aspects of MAG. We then consider
the general $17$ parameter quadratic MAG Theory in the presence of matter.
After varying independently with respect to the metric and the affine
connection we derive the field equations of the quadratic Theory. Then,
focusing on the connection field equations, by performing a simple trick and
using a recent result (see Iosifidis:2021tvx ) on how to solve complicated
linear tensor equations, we obtain the exact a unique solution of the affine
connection. In our step by step proof, we show that the latter is expressed in
terms of the usual Levi-Civita part plus modifications coming from various
combinations/dualizations and contractions of the hypermomentum tensor.
Subsequently we compute the associated forms of torsion and non-metricity as
they emerge from the microstructure of matter (i.e. hypermomentum). With these
results we are then able to perform a post-Riemannian expansion on the metric
field equations and recast them as an effective GR with modified source terms
emerging from the intrinsic characteristics of matter. Finally, by switching-
off the sources we show that the vacuum version of the quadratic Theory,
despite its complexity, is always equivalent to GR given that there are no
degeneracies. The same result continuous to hold true also for matter that
does not couple to the connection (i.e vanishing hypermomentum). We then
generalize our result even further and finally conclude and discuss future
applications.

##  II The Setup

We start off by considering a $4-dim$ differentiable manifold endowed with a
symmetric metric and an independent affine connection ($\mathcal{M}$, g,
$\nabla$). We will use the definitions and notations of Iosifidis:2020gth  and
therefore will go through them rather briefly here. On this non-Riemannian
manifold endowed with the metric $g_{\mu\nu}$ and an independent affine
connection with components $\Gamma^{\lambda}_{\;\;\;\mu\nu}$, we define the
curvature, torsion and non-metricity tensors according to

$R^{\mu}_{\;\;\;\nu\alpha\beta}:=2\partial_{[\alpha}\Gamma^{\mu}_{\;\;\;|\nu|\beta]}+2\Gamma^{\mu}_{\;\;\;\rho[\alpha}\Gamma^{\rho}_{\;\;\;|\nu|\beta]}$
(1) $S_{\mu\nu}^{\;\;\;\lambda}:=\Gamma^{\lambda}_{\;\;\;[\mu\nu]}$ (2)
$Q_{\alpha\mu\nu}:=-\nabla_{\alpha}g_{\mu\nu}$ (3)

Of great importance is the the so-called distortion tensor which measures how
much the general affine connection $\Gamma^{\lambda}_{\;\;\;\mu\nu}$ deviates
away from the Levi-Civita one schouten1954ricci , namely

$\displaystyle
N^{\lambda}_{\;\;\;\;\mu\nu}:=\Gamma^{\lambda}_{\;\;\;\mu\nu}-\widetilde{\Gamma}^{\lambda}_{\;\;\;\mu\nu}=\frac{1}{2}g^{\alpha\lambda}(Q_{\mu\nu\alpha}+Q_{\nu\alpha\mu}-Q_{\alpha\mu\nu})-g^{\alpha\lambda}(S_{\alpha\mu\nu}+S_{\alpha\nu\mu}-S_{\mu\nu\alpha})$
(4)

where $\widetilde{\Gamma}^{\lambda}_{\;\;\;\mu\nu}$ is the usual Levi-Civita
connection calculated only by the metric and its first derivatives. Once the
distortion is given, torsion and non-metricity are readily computed through
(see for instance iosifidis2019metric )

$S_{\mu\nu\alpha}=N_{\alpha[\mu\nu]}\;\;,\;\;\;Q_{\nu\alpha\mu}=2N_{(\alpha\mu)\nu}$
(5)

Out of torsion we can construct a vector as well as a pseudo-vector. Our
definitions for the torsion vector and pseudo-vector are

$S_{\mu}:=S_{\mu\lambda}^{\;\;\;\;\lambda}\;\;,\;\;\;t_{\mu}:=\epsilon_{\mu\alpha\beta\gamma}S^{\alpha\beta\gamma}$
(6)

respectively. Note that the former is defined for any dimension while the
latter only for the $n=4$ case we are considering here. Continuing with non-
metricity, we define the Weyl and the second non-metricity vector according to

$Q_{\alpha}:=Q_{\alpha\mu\nu}g^{\mu\nu}\;,\;\;q_{\nu}=Q_{\alpha\mu\nu}g^{\alpha\mu}$
(7)

Finally, from the Riemann tensor we can construct the three contractions

$R_{\nu\beta}:=R^{\mu}_{\;\;\nu\mu\beta}$ (8)
$\hat{R}_{\alpha\beta}:=R^{\mu}_{\;\;\mu\alpha\beta}$ (9)
$\breve{R}^{\mu}_{\;\;\beta}:=R^{\mu}_{\;\;\nu\alpha\beta}g^{\nu\alpha}$ (10)

As usual, the first one is the Ricci tensor (which is not symmetric in
general), the second one is the homothetic curvature and the last one is the
the co-Ricci tensor. Note that the first two aforementioned tensors can be
formed without the use of any metric while for the latter a metric is
required. As for the generalized Ricci scalar, the latter is still uniquely
defined since

$R:=g^{\mu\nu}R_{\mu\nu}=-g^{\mu\nu}\breve{R}_{\mu\nu}\;\;,\;\;\;g^{\mu\nu}\hat{R}_{\mu\nu}=0$
(11)

Of prominent importance and of great use in our proceeding discussion is the
decomposition (4). Indeed, by virtue of (4) each quantity can be split into
its Riemannian part (i.e. computed with respect to the Levi-Civita connection)
plus non-Riemannian contributions. For instance inserting the connection
decomposition (4) into the definition (1) we obtain for the Riemann tensor
444Quantities with $\widetilde{}$ will always denote Riemannian parts unless
otherwise stated.

${R^{\mu}}_{\nu\alpha\beta}=\widetilde{R}^{\mu}_{\phantom{\mu}\nu\alpha\beta}+2\widetilde{\nabla}_{[\alpha}{N^{\mu}}_{|\nu|\beta]}+2{N^{\mu}}_{\lambda|\alpha}{N^{\lambda}}_{|\nu|\beta]}\,,$
(12)

The last decomposition is very useful and we are going to be using it later on
in order to express the metric field equations of our Theory in Einstein-like
form plus modified matter sources. For instance, with the use of the above the
post-Riemannian expansion of the Ricci scalar reads

$\displaystyle
R=\tilde{R}+\frac{1}{4}Q_{\alpha\mu\nu}Q^{\alpha\mu\nu}-\frac{1}{2}Q_{\alpha\mu\nu}Q^{\mu\nu\alpha}-\frac{1}{4}Q_{\mu}Q^{\mu}+\frac{1}{2}Q_{\mu}q^{\mu}+S_{\mu\nu\alpha}S^{\mu\nu\alpha}-2S_{\mu\nu\alpha}S^{\alpha\mu\nu}-4S_{\mu}S^{\mu}$
$\displaystyle+2Q_{\alpha\mu\nu}S^{\alpha\mu\nu}+2S_{\mu}(q^{\mu}-Q^{\mu})+\tilde{\nabla}_{\mu}(q^{\mu}-Q^{\mu}-4S^{\mu})$
(13)

Before closing this section let us introduce some useful definitions that we
are going to use throughout.

###### Definition 1

Consider the components $N_{\alpha\mu\nu}$ of a rank-3 tensor field.555Of
course these definition hold true for arbitrary rank $3$ tensors but in our
discussion here this tensor will be the distortion. We define the $1^{st}$,
$2^{nd}$ and $3^{rd}$ contractions of $N_{\alpha\mu\nu}$ according to

$N^{(1)}_{\mu}:=N_{\alpha\beta\mu}g^{\alpha\beta}\;\;,\;\;N^{(2)}_{\mu}:=N_{\alpha\mu\beta}g^{\alpha\beta}\;\;,\;\;N^{(3)}_{\mu}:=N_{\mu\alpha\beta}g^{\alpha\beta}$
(14)

respectively.

###### Definition 2

Contracting $N_{\alpha\mu\nu}$ with the Levi-Civita pseudotensor we form the
$3$ parity odd combinations as follows

$M^{(1)}_{\lambda\alpha\beta}:=N_{\mu\nu\lambda}\varepsilon^{\mu\nu}_{\;\;\;\;\alpha\beta}\;\;,\;\;M^{(2)}_{\nu\alpha\beta}:=N_{\mu\nu\lambda}\varepsilon^{\mu\lambda}_{\;\;\;\;\alpha\beta}\;\;,\;\;M^{(3)}_{\mu\alpha\beta}:=N_{\mu\nu\lambda}\varepsilon^{\nu\lambda}_{\;\;\;\;\alpha\beta}$
(15)

Note that by construction each of the $M^{(i)}$’s is antisymmetric in its last
pair of indices. In addition, we can construct a $4^{th}$ pseudo-trace as

$M^{\alpha}=N^{(4)\alpha}:=\varepsilon^{\alpha\mu\nu\lambda}N_{\mu\nu\lambda}$
(16)

Note also that further contractions of the $M^{(i)}$’s do not give any new
traces since
$g^{\mu\nu}M^{(1)}_{\mu\nu\alpha}=-g^{\mu\nu}M^{(2)}_{\mu\nu\alpha}=g^{\mu\nu}M^{(3)}_{\mu\nu\alpha}=-M_{\alpha}$
and the rest are either identically vanishing or proportional to the latter.

Let us now touch upon some of the physical aspects of MAG.

###  II.1 Canonical and Metrical Energy Momentum and Hypermomentum Tensors

In the holonomic666In an anholonomic description we have three independent
fields $g_{ab},\vartheta^{b}$ and $\Gamma^{a}_{\;\;b}$, that is the tangent
space metric, the co-frame and the spin connection. However, the metric field
equations are redundant in this case and therefore we have only two
independent fields in this case as well (see hehl1995metric ).  formulation of
MAG we are considering here, one regards the affine connection
$\Gamma^{\lambda}_{\;\;\;\mu\nu}$ and the metric $g_{\mu\nu}$ as independent
fundamental fields. Then, the field equations of the Theory are obtained by
varying the total action independently with respect the aforementioned fields.
Consequently, the variations of the matter part of the action would be the
sources of Gravity. Let $\mathcal{L}_{m}$ be the matter Lagrangian of the
Theory. As usual we define the energy-momentum tensor of matter by the metric
variation of the matter sector, namely

$T^{\alpha\beta}:=+\frac{2}{\sqrt{-g}}\frac{\delta(\sqrt{-g}\mathcal{L}_{M})}{\delta
g_{\alpha\beta}}$ (17)

Additionally, since we now have an independent affine connection, the
variation of the matter part of the action with respect to the latter defines
a new tensor called hypermomentum hehl1976hypermomentum

$\Delta_{\lambda}^{\;\;\;\mu\nu}:=-\frac{2}{\sqrt{-g}}\frac{\delta(\sqrt{-g}\mathcal{L}_{M})}{\delta\Gamma^{\lambda}_{\;\;\;\mu\nu}}$
(18)

Note that the hypermomentum can be split into the three irreducible pieces of
spin, dilation and shear as follows

$\Delta_{\mu\nu\alpha}=\tau_{\mu\nu\alpha}+\frac{1}{n}\Delta_{\alpha}g_{\mu\nu}+\hat{\Delta}_{\mu\nu\alpha}$
(19)

where $\tau_{\mu\nu\alpha}:=\Delta_{[\mu\nu]\alpha}$ is the spin part,
$\Delta_{\alpha}:=\Delta_{\mu\nu\alpha}g^{\mu\nu}$ the dilation (trace) and
$\hat{\Delta}_{\mu\nu\alpha}=\Delta_{(\mu\nu)\alpha}-\frac{1}{n}\Delta_{\alpha}g_{\mu\nu}$
the shear (symmetric traceless part). The above sources are not totally
independent but rather they are subject to the conservation law

$\sqrt{-g}(2\tilde{\nabla}_{\mu}T^{\mu}_{\;\;\alpha}-\Delta^{\lambda\mu\nu}R_{\lambda\mu\nu\alpha})+\hat{\nabla}_{\mu}\hat{\nabla}_{\nu}(\sqrt{-g}\Delta_{\alpha}^{\;\;\mu\nu})+2S_{\mu\alpha}^{\;\;\;\;\lambda}\hat{\nabla}_{\nu}(\sqrt{-g}\Delta_{\lambda}^{\;\;\;\mu\nu})=0\;\;,\;\;\hat{\nabla}_{\mu}:=2S_{\mu}-\nabla_{\mu}$
(20)

which serves as the generalization of the conservation law of energy-momentum
for matter with microstructure and comes about from the diffeomorphism
invariance of the matter part of the action Iosifidis:2020gth .

##  III Full Quadratic Theory

As we already mentioned the mere inclusion of the Ricci scalar as the
gravitational Lagrangian for MAG not only is it problematic due to its
projective invariance777In the sense that it imposes a vanishing dilation
current. However in certain instances (see BeltranJimenez:2019acz ;
aoki2019scalar ) projective invariance is a key requirement in establishing
healthy Theories. , but it also is not the only invariant with dimensions
$L^{-2}$. Indeed, since now we also have torsion and non-metricity, any
quadratic combination of them would be of the same dimension as $R$ and since
there is no fundamental principle that excludes them, their presence on the
gravitational action is indispensable. It turns out (see for instance
pagani2015quantum ; iosifidis2019scale ) that in any dimension there are $5$
pure on-metricity, $3$ pure torsion, and $3$ mixed invariants which are
quadratic in torsion and non-metricity and parity even. So, in total (and for
any dimension) we have the following $11$ parity even quadratic invariants:

Pure Non-Metricity

$\displaystyle A_{1}=Q_{\alpha\mu\nu}Q^{\alpha\mu\nu}$ (21) $\displaystyle
A_{2}=Q_{\alpha\mu\nu}Q^{\mu\nu\alpha}$ (22) $\displaystyle
A_{3}=Q_{\mu}Q^{\mu}$ (23) $\displaystyle A_{4}=q_{\mu}q^{\mu}$ (24)
$\displaystyle A_{5}=Q_{\mu}q^{\mu}$ (25)

Pure Torsion

$\displaystyle B_{1}=S_{\alpha\mu\nu}S^{\alpha\mu\nu}$ (26) $\displaystyle
B_{2}=S_{\alpha\mu\nu}S^{\mu\nu\alpha}$ (27) $\displaystyle
B_{3}=S_{\mu}S^{\mu}$ (28)

Mixed

$\displaystyle C_{1}=Q_{\alpha\mu\nu}S^{\alpha\mu\nu}$ (29) $\displaystyle
C_{2}=Q_{\mu}S^{\mu}$ (30) $\displaystyle C_{3}=q_{\mu}S^{\mu}$ (31)

The above invariants will then supplement the Einstein-Hilbert term (i.e. $R$)
into the gravitational action to built the parity even quadratic MAG
hehl1995metric ; pagani2015quantum ; iosifidis2019scale . The aforementioned
inclusion is fairly general and holds true for any dimension. Now,
interestingly enough, in our physical $n=4$ spacetime and with the use of the
Levi-Civita pseudo-tensor $\varepsilon^{\mu\nu\alpha\beta}$ additional
quadratic (pseudo) scalars can be built. Indeed, in our four-dimensional
spacetime we find the $6$ independent pseudoscalars (see pagani2015quantum ;
iosifidis2019scale )

Parity Odd Invariants

$\displaystyle
A_{6}=\epsilon^{\alpha\beta\gamma\delta}Q_{\alpha\beta\mu}Q_{\gamma\delta}^{\;\;\;\;\mu}$
(32) $\displaystyle B_{5}=S_{\mu}t^{\mu}$ (33) $\displaystyle
B_{6}=\epsilon^{\alpha\beta\gamma\delta}S_{\alpha\beta\mu}S_{\gamma\delta}^{\;\;\;\;\mu}$
(34) $\displaystyle C_{4}=Q_{\mu}t^{\mu}$ (35) $\displaystyle
C_{5}=q^{\mu}t_{\mu}$ (36) $\displaystyle
C_{6}=\epsilon^{\alpha\beta\gamma\delta}Q_{\alpha\beta\mu}S_{\gamma\delta}^{\;\;\;\;\mu}$
(37)

Note that other quadratic combinations could be considered, such as
$t_{\mu}t^{\mu}$ for instance. However, these can be shown to be related to
the ones written above and so there are exactly $6$ parity odd quadratic
invariants in torsion and non-metricity.

Now, is this all there is to it? Well, our initial intention was to built a
gravitational action consisting of scalars that are linear in the Riemann
tensor and quadratic in torsion and non-metricity since all of these will be
of the same dimension. The above exposition has exhausted the possibilities
for the quadratic terms so the question is how many scalars can we built out
of the Riemann tensor? Of course, the scalar that is constructed by (and is
linear in) the Riemann tensor is the Ricci scalar $R$. For a symmetric
connection this is the one and only possibility. Interestingly, for a general
affine connection, as the one we are considering here, in $n=4$ and given the
existence of the Levi-Civita pseudotensor one can include the totally
antisymmetric part of the Riemann tensor, which is given by the
pseudoscalar888Here $\varepsilon^{\mu\nu\alpha\beta}$ is the Levi-Civita
pesudo-tensor which is related to the Levi-Civita symbol
$\epsilon^{\mu\nu\rho\sigma}$ through
$\varepsilon^{\mu\nu\rho\sigma}:=\frac{\epsilon^{\mu\nu\rho\sigma}}{\sqrt{-g}}$.

$\varepsilon^{\mu\nu\alpha\beta}R_{\mu\nu\alpha\beta}$ (38)

and which is of purely non-Riemannian nature999More precisely, the emergence
of this term has to do with torsion a is clearly demonstrated in an identity
we give below. In other words, for vanishing torsion
$\varepsilon^{\mu\nu\alpha\beta}R_{\mu\nu\alpha\beta}$ vanishes as well.. The
above totally antisymmetric part is sometimes referred to as as the Hojman, or
most commonly called the Holst term101010This term was first studied by Hojman
et al in PhysRevD.22.1915  and then later by Holst holst1996barbero . Its
inclusion in Gravity Theories has regained a lot of attention recently and has
been studied in various contexts (see Iosifidis:2020dck ; Bombacigno:2021bpk ;
Boudet:2020eyr ; Obukhov:2020zal ; Bombacigno:2019nua ; laangvik2021higgs )..
Therefore, the most general gravitational MAG action (in $n=4$) that is linear
in curvature and quadratic in torsion and non-metricity is given by

$\displaystyle S=\frac{1}{2\kappa}\int
d^{4}x\sqrt{-g}\Big{[}R+\lambda\varepsilon^{\alpha\beta\gamma\delta}R_{\alpha\beta\gamma\delta}+b_{1}S_{\alpha\mu\nu}S^{\alpha\mu\nu}+b_{2}S_{\alpha\mu\nu}S^{\mu\nu\alpha}+b_{3}S_{\mu}S^{\mu}$
$\displaystyle
a_{1}Q_{\alpha\mu\nu}Q^{\alpha\mu\nu}+a_{2}Q_{\alpha\mu\nu}Q^{\mu\nu\alpha}+a_{3}Q_{\mu}Q^{\mu}+a_{4}q_{\mu}q^{\mu}+a_{5}Q_{\mu}q^{\mu}$
$\displaystyle+c_{1}Q_{\alpha\mu\nu}S^{\alpha\mu\nu}+c_{2}Q_{\mu}S^{\mu}+c_{3}q_{\mu}S^{\mu}$
$\displaystyle+a_{6}\varepsilon^{\alpha\beta\gamma\delta}Q_{\alpha\beta\mu}Q_{\gamma\delta\;\;\;\;}^{\mu}+b_{5}S_{\mu}t^{\mu}+b_{6}\varepsilon^{\alpha\beta\gamma\delta}S_{\alpha\beta\mu}S_{\gamma\delta}^{\;\;\;\;\mu}$
$\displaystyle
c_{4}Q_{\mu}t^{\mu}+c_{5}q^{\mu}t_{\mu}+c_{6}\varepsilon^{\alpha\beta\gamma\delta}Q_{\alpha\beta\mu}S_{\gamma\delta}^{\;\;\;\;\mu}\Big{]}$
(39)

However, it is important to stress out that there exists the identity (for the
proof see for instance iosifidis2019metric )

$\varepsilon^{\mu\nu\alpha\beta}R_{\mu\nu\alpha\beta}=2\tilde{\nabla}_{\alpha}\tilde{S}^{\alpha}+2\varepsilon^{\mu\nu\alpha\beta}S_{\alpha\beta}^{\;\;\;\;\lambda}(Q_{\mu\nu\lambda}+S_{\mu\nu\lambda})\,,$
(40)

where we have used the decomposition (4) for the Riemann tensor, the fact that
$\varepsilon^{\mu\nu\alpha\beta}\tilde{R}_{\mu\nu\alpha\beta}=0$. From the
above identity we see that the first term will result in a total derivative
term which will not contribute to the field equations. Furthermore, we see
that the second and third terms are already included in the parity odd term
classification we outlined earlier. More precisely, these terms are
respectively equal (up to a factor of $2$) to the invariants $C_{6}$ and
$B_{6}$ we presented above. Therefore, the inclusion of (38) into the
Gravitational action would simply amount to a shift on the coefficients
$c_{6}$ and $b_{6}$ of the terms
$\varepsilon^{\alpha\beta\gamma\delta}Q_{\alpha\beta\mu}S_{\gamma\delta}^{\;\;\;\;\mu}$
and
$\varepsilon^{\alpha\beta\gamma\delta}S_{\alpha\beta\mu}S_{\gamma\delta}^{\;\;\;\;\mu}$
respectively. Therefore the inclusion of $(\ref{eR})$ is redundant and we can
always safely set $\lambda=0$ in (39) without altering the physics of the
Theory at classical level.

##  IV The full Quadratic MAG Theory

As we already mentioned the main intention of this work is to analyze the full
$17$-parameter quadratic MAG Theory including all possible, parity even and
parity odd, quadratic terms in torsion and non-metricity in addition to the
Einstein-Hilbert term. Even though of great importance is to answer what
exactly the vacuum Theory corresponds to, even more interesting and
challenging is to add matter and study the full Theory in the presence of
sources. Of course, having done so we can always turn off the sources
afterwards and focus on the vacuum case. As we shall show, regardless of the
complexity of the $17$ parameter full quadratic Theory, we will be able to
answer to both of the aforementioned questions. Our $4$ dimensional $17$
parameter full quadratic Theory reads

$\displaystyle S[g,\Gamma,\Phi]=\frac{1}{2\kappa}\int
d^{4}x\sqrt{-g}\Big{[}R+b_{1}S_{\alpha\mu\nu}S^{\alpha\mu\nu}+b_{2}S_{\alpha\mu\nu}S^{\mu\nu\alpha}+b_{3}S_{\mu}S^{\mu}$
$\displaystyle
a_{1}Q_{\alpha\mu\nu}Q^{\alpha\mu\nu}+a_{2}Q_{\alpha\mu\nu}Q^{\mu\nu\alpha}+a_{3}Q_{\mu}Q^{\mu}+a_{4}q_{\mu}q^{\mu}+a_{5}Q_{\mu}q^{\mu}$
$\displaystyle+c_{1}Q_{\alpha\mu\nu}S^{\alpha\mu\nu}+c_{2}Q_{\mu}S^{\mu}+c_{3}q_{\mu}S^{\mu}$
$\displaystyle+a_{6}\varepsilon^{\alpha\beta\gamma\delta}Q_{\alpha\beta\mu}Q_{\gamma\delta}^{\;\;\;\;\mu}+b_{5}S_{\mu}t^{\mu}+b_{6}\varepsilon^{\alpha\beta\gamma\delta}S_{\alpha\beta\mu}S_{\gamma\delta}^{\;\;\;\;\mu}$
$\displaystyle
c_{4}Q_{\mu}t^{\mu}+c_{5}q^{\mu}t_{\mu}+c_{6}\varepsilon^{\alpha\beta\gamma\delta}Q_{\alpha\beta\mu}S_{\gamma\delta}^{\;\;\;\;\mu}\Big{]}+S_{M}[g,\Gamma,\Phi]$
(41)

where we have set $\lambda=0$ since, as demonstrated previously, this term is
already included in the quadratic terms. In the above, the first, second and
third lines contain (apart from the usual Einstein-Hilbert term) the pure
torsion, pure non-metricity and mixed quadratic parity even parts while the
last two lines contain all the parity odd terms. Let us highlight that the
matter action we are considering here can (and will) in general depend on the
connection which will result in a non-vanishing hypermomentum. Varying the
above action independently with respect to the metric and the affine
connection, after some long calculations we obtain the field equations

$\displaystyle
R_{(\mu\nu)}-\frac{R}{2}g_{\mu\nu}-\frac{\mathcal{L}^{(2)}_{even}}{2}g_{\mu\nu}-\frac{1}{\sqrt{-g}}(\nabla_{\alpha}-2S_{\alpha})\Big{[}\sqrt{-g}\Big{(}c_{1}S^{\alpha}_{\;\;(\mu\nu)}+c_{2}g_{\mu\nu}S^{\alpha}+c_{3}\delta^{\alpha}_{(\mu}S_{\nu)}\Big{)}\Big{]}$
$\displaystyle-\frac{1}{\sqrt{-g}}(\nabla_{\alpha}-2S_{\alpha})\Big{[}\sqrt{-g}\Big{(}2a_{1}Q^{\alpha}_{\;\;\mu\nu}+2a_{2}Q_{(\mu\nu)}^{\;\;\;\;\alpha}+(2a_{3}Q^{\alpha}+a_{5}q^{\alpha})g_{\mu\nu}+(2a_{4}q_{(\mu}+a_{5}Q_{(\mu})\delta^{\alpha}_{\nu)}\Big{)}\Big{]}$
$\displaystyle+a_{1}(Q_{\mu\alpha\beta}Q_{\nu}^{\;\;\alpha\beta}-2Q_{\alpha\beta\mu}Q^{\alpha\beta}_{\;\;\;\;\nu})-a_{2}Q_{\alpha\beta(\mu}Q^{\beta\alpha}_{\;\;\;\;\nu)}+a_{3}(Q_{\mu}Q_{\nu}-2Q^{\alpha}Q_{\alpha\mu\nu})-a_{4}q_{\mu}q_{\nu}-a_{5}q^{\alpha}Q_{\alpha\mu\nu}$
$\displaystyle+b_{1}(2S_{\nu\alpha\beta}S_{\mu}^{\;\;\;\alpha\beta}-S_{\alpha\beta\mu}S^{\alpha\beta}_{\;\;\;\;\nu})-b_{2}S_{(\mu}^{\;\;\;\beta\alpha}S_{\nu)\alpha\beta}+b_{3}S_{\mu}S_{\nu}$
$\displaystyle+c_{1}(Q_{(\mu}^{\;\;\;\alpha\beta}S_{\nu)\alpha\beta}-S_{\alpha\beta(\mu}Q^{\alpha\beta}_{\;\;\;\;\nu)})+c_{2}(S_{(\mu}Q_{\nu)}-S^{\alpha}Q_{\alpha\mu\nu})$
$\displaystyle-
b_{6}\varepsilon^{\alpha\beta\gamma\delta}S_{\alpha\beta\nu}S_{\gamma\delta\mu}+a_{6}\varepsilon^{\alpha\beta\gamma\delta}Q_{\alpha\beta\mu}Q_{\gamma\delta\nu}$
$\displaystyle-
g_{\beta(\mu}g_{\nu)\lambda}\frac{1}{\sqrt{-g}}(\nabla_{\alpha}-2S_{\alpha})\Big{(}\sqrt{-g}\varepsilon^{\alpha\beta\gamma\delta}(2a_{6}Q_{\gamma\delta}^{\;\;\;\;\lambda}+c_{6}S_{\gamma\delta}^{\;\;\;\;\lambda})\Big{)}$
$\displaystyle-
S_{\alpha\beta(\mu}\varepsilon^{\lambda\alpha\beta}_{\;\;\;\;\;\;\nu)}(b_{5}S_{\lambda}+c_{4}Q_{\lambda}+c_{5}q_{\lambda})-c_{4}g_{\mu\nu}\Big{(}\partial_{\lambda}(\sqrt{-g}t^{\lambda})\Big{)}$
$\displaystyle-
c_{5}g_{\alpha(\mu}g_{\nu)\beta}\frac{1}{\sqrt{-g}}\nabla_{\lambda}(\sqrt{-g}t^{\alpha}g^{\lambda\beta})+c_{5}2S_{(\mu}t_{\nu)}+c_{5}t^{\alpha}Q_{(\mu\nu)\alpha}=\kappa
T_{\mu\nu}$ (42)

$\Gamma$-Variation

$\displaystyle\left(\frac{Q_{\lambda}}{2}+2S_{\lambda}\right)g^{\mu\nu}-Q_{\lambda}^{\;\;\mu\nu}-2S_{\lambda}^{\;\;\mu\nu}+\left(q^{\mu}-\frac{Q^{\mu}}{2}-2S^{\mu}\right)\delta_{\lambda}^{\nu}+4a_{1}Q^{\nu\mu}_{\;\;\;\;\lambda}+2a_{2}(Q^{\mu\nu}_{\;\;\;\;\lambda}+Q_{\lambda}^{\;\;\;\mu\nu})+2b_{1}S^{\mu\nu}_{\;\;\;\;\lambda}$
$\displaystyle+2b_{2}S_{\lambda}^{\;\;\;[\mu\nu]}+c_{1}\Big{(}S^{\nu\mu}_{\;\;\;\;\lambda}-S_{\lambda}^{\;\;\;\nu\mu}+Q^{[\mu\nu]}_{\;\;\;\;\;\lambda}\Big{)}+\delta_{\lambda}^{\mu}\Big{(}4a_{3}Q^{\nu}+2a_{5}q^{\nu}+2c_{2}S^{\nu}\Big{)}+\delta_{\lambda}^{\nu}\Big{(}a_{5}Q^{\mu}+2a_{4}q^{\mu}+c_{3}S^{\mu}\Big{)}$
$\displaystyle+g^{\mu\nu}\Big{(}a_{5}Q_{\lambda}+2a_{4}q_{\lambda}+c_{3}S_{\lambda}\Big{)}+\Big{(}c_{2}Q^{[\mu}+c_{3}q^{[\mu}+2b_{3}S^{[\mu}\Big{)}\delta^{\nu]}_{\lambda}$
$\displaystyle+(-2a_{6}+c_{6})\varepsilon^{\mu\nu\alpha\beta}Q_{\alpha\beta\lambda}+(2b_{6}-c_{6})\varepsilon^{\mu\nu\alpha\beta}S_{\alpha\beta\lambda}-2a_{6}\varepsilon_{\lambda}^{\;\;\nu\alpha\beta}Q_{\alpha\beta}^{\;\;\;\;\mu}-c_{6}\varepsilon_{\lambda}^{\;\;\nu\alpha\beta}S_{\alpha\beta}^{\;\;\;\;\mu}$
$\displaystyle+\varepsilon^{\alpha\mu\nu}_{\;\;\;\;\;\;\lambda}(b_{5}S_{\alpha}+c_{4}Q_{\alpha}+c_{5}q_{\alpha})+\Big{(}\frac{b_{5}}{2}+c_{5}\Big{)}t^{\mu}\delta_{\lambda}^{\nu}+\Big{(}-\frac{b_{5}}{2}+2c_{4}\Big{)}t^{\nu}\delta_{\lambda}^{\mu}+c_{5}g^{\mu\nu}t_{\lambda}=\kappa\Delta_{\lambda}^{\;\;\;\mu\nu}$
(43)

We see that both of the above sets of field equations look somewhat
complicated. However, as we shall show, using a recent result (see
iosifidis2021solving ), quite remarkably, we will be able to solve completely
(and exactly) for the affine connection and subsequently recast the metric
field equations as an effective Einstein’s GR with modified/extended source
terms that are associated to the microstructure of matter. We do so in what
follows.

##  V Solving for the Affine Connection

We shall now undertake the challenge of solving for the affine connection.
Even though the situation seems a little messy, using a certain Theorem, we
will provide the unique and exact solution of (43) given some general non-
degeneracy conditions. Of paramount importance in arriving at such a solution
is the following result which we give here without proof. The detailed proof
can be found in iosifidis2021solving .

###### Theorem 1

In a $4$-dimensional space of signature $s$, consider the $30$ parameter
tensor equation

$\displaystyle
a_{1}N_{\alpha\mu\nu}+a_{2}N_{\nu\alpha\mu}+a_{3}N_{\mu\nu\alpha}+a_{4}N_{\alpha\nu\mu}+a_{5}N_{\nu\mu\alpha}+a_{6}N_{\mu\alpha\nu}+\sum_{i=1}^{3}\Big{(}a_{7i}N^{(i)}_{\mu}g_{\alpha\nu}+a_{8i}N^{(i)}_{\nu}g_{\alpha\mu}+a_{9i}N^{(i)}_{\alpha}g_{\mu\nu}\Big{)}$
$\displaystyle+b_{11}M^{(1)}_{\alpha\mu\nu}+b_{12}M^{(1)}_{\nu\alpha\mu}+b_{13}M^{(1)}_{\mu\nu\alpha}+b_{21}M^{(2)}_{\alpha\mu\nu}+b_{22}M^{(2)}_{\nu\alpha\mu}+b_{23}M^{(2)}_{\mu\nu\alpha}+b_{31}M^{(3)}_{\alpha\mu\nu}+b_{32}M^{(3)}_{\nu\alpha\mu}+b_{33}M^{(3)}_{\mu\nu\alpha}$
$\displaystyle+\varepsilon_{\rho\alpha\mu\nu}\Big{(}b_{1}N^{(1)\rho}+b_{2}N^{(2)\rho}+b_{3}N^{(3)\rho}\Big{)}+c_{1}M_{\mu}g_{\alpha\nu}+c_{2}M_{\nu}g_{\alpha\mu}+c_{3}M_{\alpha}g_{\mu\nu}=B_{\alpha\mu\nu}$
(44)

where $a_{i}$, $a_{ji}$ $i=1,2,...,6$, $j=7,8,9$ are scalars, $b_{kl}$,
$c_{m}$ are pseudo-scalars, $B_{\alpha\mu\nu}$ is a given (known) tensor and
$N_{\alpha\mu\nu}$ are the components of the unknown tensor111111Of course the
result holds true even when $N_{\alpha\mu\nu}$ are the components of a tensor
density instead or even of a connection given that $B_{\alpha\mu\nu}$ are also
of the same kind.  $N$. Define the matrices

$A:=\setcounter{MaxMatrixCols}{15}\begin{pmatrix}\alpha_{11}&\alpha_{12}&\alpha_{13}&\alpha_{14}&\alpha_{15}&\alpha_{16}&\alpha_{17}&\alpha_{18}&\alpha_{19}&\alpha_{1,10}&\alpha_{1,11}&\alpha_{1,12}&\alpha_{1,13}&\alpha_{1,14}&\alpha_{1,15}\\\
\alpha_{21}&\alpha_{22}&\alpha_{23}&\alpha_{24}&\alpha_{25}&\alpha_{26}&\alpha_{27}&\alpha_{28}&\alpha_{29}&\alpha_{2,10}&\alpha_{2,11}&\alpha_{1,12}&\alpha_{2,13}&\alpha_{2,14}&\alpha_{2,15}\\\
\alpha_{31}&\alpha_{32}&\alpha_{33}&\alpha_{34}&\alpha_{35}&\alpha_{36}&\alpha_{37}&\alpha_{38}&\alpha_{39}&\alpha_{3,10}&\alpha_{3,11}&\alpha_{1,12}&\alpha_{1,13}&\alpha_{1,14}&\alpha_{1,15}\\\
\alpha_{41}&\alpha_{42}&\alpha_{43}&\alpha_{44}&\alpha_{45}&\alpha_{46}&\alpha_{47}&\alpha_{48}&\alpha_{49}&\alpha_{4,10}&\alpha_{4,11}&\alpha_{4,12}&\alpha_{4,13}&\alpha_{4,14}&\alpha_{4,15}\\\
\alpha_{51}&\alpha_{52}&\alpha_{53}&\alpha_{54}&\alpha_{55}&\alpha_{56}&\alpha_{57}&\alpha_{58}&\alpha_{59}&\alpha_{5,10}&\alpha_{5,11}&\alpha_{5,12}&\alpha_{5,13}&\alpha_{5,14}&\alpha_{5,15}\\\
\alpha_{61}&\alpha_{62}&\alpha_{63}&\alpha_{64}&\alpha_{65}&\alpha_{66}&\alpha_{67}&\alpha_{68}&\alpha_{69}&\alpha_{6,10}&\alpha_{6,11}&\alpha_{6,12}&\alpha_{6,13}&\alpha_{6,14}&\alpha_{6,15}\\\
\alpha_{71}&\alpha_{72}&\alpha_{73}&\alpha_{74}&\alpha_{75}&\alpha_{76}&\alpha_{77}&\alpha_{78}&\alpha_{79}&\alpha_{7,10}&\alpha_{7,11}&\alpha_{7,12}&\alpha_{7,13}&\alpha_{7,14}&\alpha_{7,15}\\\
\alpha_{81}&\alpha_{82}&\alpha_{83}&\alpha_{84}&\alpha_{85}&\alpha_{86}&\alpha_{87}&\alpha_{88}&\alpha_{89}&\alpha_{8,10}&\alpha_{8,11}&\alpha_{8,12}&\alpha_{8,13}&\alpha_{8,14}&\alpha_{8,15}\\\
\alpha_{91}&\alpha_{92}&\alpha_{93}&\alpha_{94}&\alpha_{95}&\alpha_{96}&\alpha_{97}&\alpha_{98}&\alpha_{99}&\alpha_{9,10}&\alpha_{9,11}&\alpha_{9,12}&\alpha_{9,13}&\alpha_{9,14}&\alpha_{9,15}\\\
\alpha_{10,1}&\alpha_{10,2}&\alpha_{10,3}&\alpha_{10,4}&\alpha_{10,5}&\alpha_{10,6}&\alpha_{10,7}&\alpha_{10,8}&\alpha_{10,9}&\alpha_{10,10}&\alpha_{10,11}&\alpha_{10,12}&\alpha_{10,13}&\alpha_{10,14}&\alpha_{10,15}\\\
\alpha_{11,1}&\alpha_{11,2}&\alpha_{11,3}&\alpha_{11,4}&\alpha_{11,5}&\alpha_{11,6}&\alpha_{11,7}&\alpha_{11,8}&\alpha_{11,9}&\alpha_{11,10}&\alpha_{11,11}&\alpha_{11,12}&\alpha_{11,13}&\alpha_{11,14}&\alpha_{11,15}\\\
\alpha_{12,1}&\alpha_{12,2}&\alpha_{12,3}&\alpha_{12,4}&\alpha_{12,5}&\alpha_{12,6}&\alpha_{12,7}&\alpha_{12,8}&\alpha_{12,9}&\alpha_{12,10}&\alpha_{12,11}&\alpha_{12,12}&\alpha_{12,13}&\alpha_{12,14}&\alpha_{12,15}\\\
\alpha_{13,1}&\alpha_{13,2}&\alpha_{13,3}&\alpha_{14,4}&\alpha_{14,5}&\alpha_{14,6}&\alpha_{14,7}&\alpha_{14,8}&\alpha_{14,9}&\alpha_{1,10}&\alpha_{1,11}&\alpha_{1,12}&\alpha_{1,13}&\alpha_{1,14}&\alpha_{1,15}\\\
\alpha_{11}&\alpha_{12}&\alpha_{13}&\alpha_{14}&\alpha_{15}&\alpha_{16}&\alpha_{17}&\alpha_{18}&\alpha_{19}&\alpha_{1,10}&\alpha_{14,11}&\alpha_{14,12}&\alpha_{14,13}&\alpha_{14,14}&\alpha_{14,15}\\\
\alpha_{15,1}&\alpha_{15,2}&\alpha_{15,3}&\alpha_{15,4}&\alpha_{15,5}&\alpha_{15,6}&\alpha_{15,7}&\alpha_{15,8}&\alpha_{15,9}&\alpha_{15,10}&\alpha_{15,11}&\alpha_{15,12}&\alpha_{15,13}&\alpha_{15,14}&\alpha_{15,15}\\\
\end{pmatrix}$ (45)

and

$\Gamma:=\begin{pmatrix}\gamma_{11}&\gamma_{12}&\gamma_{13}&\gamma_{14}\\\
\gamma_{21}&\gamma_{22}&\gamma_{23}&\gamma_{24}\\\
\gamma_{31}&\gamma_{32}&\gamma_{33}&\gamma_{34}\\\
\gamma_{41}&\gamma_{42}&\gamma_{43}&\gamma_{44}\\\ \end{pmatrix}$ (46)

where the $\alpha_{ij}$ and $\gamma_{ij}$’s are some linear combinations of
the original $a_{i}$, $a_{ji}$, $b_{j}$ and $c_{k}$’s (see appendix). Then
given that both of these matrices are non-singular, that is if

$det(A)\neq 0\;\;and\;\;det(\Gamma)\neq 0$ (47)

holds true, then the general and unique solution of (44) reads

$\displaystyle
N_{\alpha\mu\nu}=\tilde{\alpha}_{11}\hat{B}_{\alpha\mu\nu}+\tilde{\alpha}_{12}\hat{B}_{\nu\alpha\mu}+\tilde{\alpha}_{13}\hat{B}_{\mu\nu\alpha}+\tilde{\alpha}_{14}\hat{B}_{\alpha\nu\mu}+\tilde{\alpha}_{15}\hat{B}_{\nu\mu\alpha}+\tilde{\alpha}_{16}\hat{B}_{\mu\alpha\nu}+\tilde{\alpha}_{17}\breve{B}_{\alpha\mu\nu}+\tilde{\alpha}_{18}\breve{B}_{\mu\nu\alpha}+\tilde{\alpha}_{19}\breve{B}_{\nu\alpha\mu}$
$\displaystyle+\tilde{\alpha}_{1,10}\bar{B}_{\alpha\mu\nu}+\tilde{\alpha}_{1,11}\bar{B}_{\mu\nu\alpha}+\tilde{\alpha}_{1,12}\bar{B}_{\nu\alpha\mu}+\tilde{\alpha}_{1,13}\mathring{B}_{\alpha\mu\nu}+\tilde{\alpha}_{1,14}\mathring{B}_{\mu\nu\alpha}+\tilde{\alpha}_{1,15}\mathring{B}_{\nu\alpha\mu}$
(48)

where the $\tilde{\alpha}_{1i}$ ${}^{\prime}s$ are the first row elements of
the inverse matrix $A^{-1}$ and

$\hat{B}_{\alpha\mu\nu}=B_{\alpha\mu\nu}-\sum_{i=1}^{4}\sum_{j=1}^{4}\Big{(}a_{7i}\tilde{\gamma}_{ij}B^{(j)}_{\mu}g_{\alpha\nu}+a_{8i}\tilde{\gamma}_{ij}B^{(j)}_{\nu}g_{\alpha\mu}+a_{9i}\tilde{\gamma}_{ij}B^{(j)}_{\alpha}g_{\mu\nu}\Big{)}-\varepsilon^{\rho}_{\;\;\alpha\mu\nu}\sum_{i=1}^{3}\sum_{j=1}^{4}b_{i}\tilde{\gamma}_{ij}B^{(j)}_{\rho}$
(49)
$\breve{B}_{\alpha\mu\nu}=\varepsilon^{\beta\gamma}_{\;\;\;\;\alpha\mu}\hat{B}_{\beta\gamma\nu}-2(-1)^{s}\sum_{j=1}^{4}\Big{[}(b_{21}+b_{23}+b_{31}+b_{33})\tilde{\gamma}_{1j}+(b_{11}+b_{13}-b_{31}-b_{33})\tilde{\gamma}_{2j}-(b_{11}+b_{13}+b_{21}+b_{23})\tilde{\gamma}_{3j}\Big{]}B^{(j)}_{[\alpha}g_{\mu]\nu}$
(50)
$\bar{B}_{\alpha\mu\nu}:=\varepsilon^{\beta\gamma}_{\;\;\;\;\alpha\nu}\hat{B}_{\beta\mu\gamma}-2(-1)^{s}\sum_{j=1}^{4}\Big{[}(b_{21}+b_{22}+b_{31}+b_{32})\tilde{\gamma}_{1j}+(b_{11}+b_{12}-b_{31}-b_{32})\tilde{\gamma}_{2j}-(b_{11}+b_{12}+b_{21}+b_{22})\tilde{\gamma}_{3j}\Big{]}B^{(j)}_{[\alpha}g_{\mu]\nu}$
(51)
$\mathring{B}_{\alpha\mu\nu}:=\varepsilon^{\beta\gamma}_{\;\;\;\;\mu\nu}\mathring{B}_{\alpha\beta\gamma}-2(-1)^{s+1}\sum_{j=1}^{4}\Big{[}(b_{22}+b_{23}+b_{32}+b_{33})\tilde{\gamma}_{1j}+(b_{12}+b_{13}-b_{32}-b_{33})\tilde{\gamma}_{2j}-(b_{22}+b_{23}+b_{12}+b_{13})\tilde{\gamma}_{3j}\Big{]}B^{(j)}_{[\mu}g_{\nu]\alpha}$
(52)

$\tilde{\gamma}_{ij}^{\prime}s$ being the elements of the inverse matrix
$\Gamma^{-1}$.

From the above, it also immediately follows that when $B_{\alpha}=0$ and
$det(A)\neq 0$ along with $det(\Gamma)\neq 0$ we have that the unique solution
of $(\ref{eq1})$ is always $N_{\alpha\mu\nu}=0$. More precisely, we have the
following iosifidis2021solving

###### Corollary 1

If $B_{\alpha\mu\nu}=0$ and both matrices $A$ and $\Gamma$ are non-singular,
then the unique solution of $(\ref{eq1})$ is $N_{\alpha\mu\nu}=0$.

With the above result, despite the seemingly complicated form of the
connection field equations (43), as we shall show below the exact form of the
affine connection in therms of the hypermomentum can be readily obtained.

The crucial point now is to use equations (5) and also their contractions (see
appendix) in order eliminate torsion and non-metricity from (43) and express
everything in terms of the distortion tensor and its associated vectors. After
some lengthy but rather straightforward algebra we finally arrive at

$\displaystyle(4a_{1}+b_{1}-c_{1})N_{\alpha\mu\nu}+\Big{(}-1+2a_{2}+\frac{c_{1}+b_{2}}{2}\Big{)}N_{\nu\alpha\mu}+\Big{(}-1+2a_{2}+\frac{c_{1}+b_{2}}{2}\Big{)}N_{\mu\nu\alpha}$
$\displaystyle+(2a_{2}-b_{1}+c_{1})N_{\alpha\nu\mu}+\Big{(}2a_{2}-\frac{b_{2}}{2}\Big{)}N_{\nu\mu\alpha}+\Big{(}4a_{1}-\frac{b_{2}}{2}-c_{1}\Big{)}N_{\mu\alpha\nu}$
$\displaystyle+\left(2a_{5}+c_{2}-\frac{b_{3}+c_{3}}{2}\right)g_{\nu\alpha}N^{(1)}_{\mu}+\left(2a_{4}+\frac{b_{3}}{2}+c_{3}\right)g_{\nu\alpha}N^{(2)}_{\mu}+\left(1+2a_{4}+\frac{c_{3}}{2}\right)g_{\nu\alpha}N^{(3)}_{\mu}$
$\displaystyle+\left(8a_{3}-2c_{2}+\frac{b_{3}}{2}\right)g_{\mu\alpha}N^{(1)}_{\nu}+\left(2a_{5}+c_{2}-\frac{c_{3}+b_{3}}{2}\right)g_{\mu\alpha}N^{(2)}_{\mu}+\left(2a_{5}-\frac{c_{3}}{2}\right)g_{\mu\alpha}N^{(3)}_{\nu}$
$\displaystyle+\left(2a_{5}-\frac{c_{3}}{2}\right)g_{\mu\nu}N^{(1)}_{\alpha}+\left(2a_{4}+\frac{c_{3}}{2}+1\right)g_{\mu\nu}N^{(2)}_{\alpha}+2a_{4}g_{\mu\nu}N^{(3)}_{\alpha}$
$\displaystyle-(-2a_{6}+c_{6})\varepsilon^{\mu\nu\alpha\beta}N_{\alpha\lambda\beta}-(-2a_{6}+c_{6})\varepsilon^{\mu\nu\alpha\beta}N_{\lambda\alpha\beta}+(2b_{6}-c_{6})\varepsilon^{\mu\nu\alpha\beta}N_{\alpha\beta\lambda}$
$\displaystyle+2a_{6}\varepsilon_{\lambda}^{\;\;\nu\alpha\beta}\Big{(}N_{\alpha\;\;\;\;\beta}^{\;\;\mu}+N^{\mu}_{\;\;\alpha\beta}\Big{)}-c_{6}\varepsilon_{\lambda}^{\;\;\nu\alpha\beta}N^{\mu}_{\;\;\alpha\beta}+\varepsilon^{\alpha\mu\nu}_{\;\;\;\;\;\;\lambda}\Big{(}2c_{4}-\frac{b_{5}}{2}\Big{)}N^{(1)}_{\alpha}$
$\displaystyle+\Big{(}c_{5}+\frac{b_{5}}{2}\Big{)}\varepsilon^{\alpha\mu\nu}_{\;\;\;\;\;\;\lambda}N^{(2)}_{\alpha}+c_{5}\varepsilon^{\alpha\mu\nu}_{\;\;\;\;\;\;\lambda}N^{(3)}_{\alpha}+\Big{(}\frac{b_{5}}{2}+c_{5}\Big{)}M^{\mu}\delta_{\lambda}^{\nu}+\Big{(}-\frac{b_{5}}{2}+2c_{4}\Big{)}M^{\nu}\delta_{\lambda}^{\mu}+c_{5}g^{\mu\nu}M_{\lambda}=\kappa\Delta_{\alpha\mu\nu}$
(53)

Now we see that the latter equation falls in the case of $Theorem-1$ with the
obvious identifications among the parameters $\alpha_{i}$ and
$a_{i},b_{j},c_{k}$ etc. Therefore, in applying $Theorem-1$ to the latter and
setting $B_{\alpha\mu\nu}=\kappa\Delta_{\alpha\mu\nu}$ we solve for the
distortion

$\displaystyle
N_{\alpha\mu\nu}=\kappa\tilde{\alpha}_{11}\hat{\Delta}_{\alpha\mu\nu}+\kappa\tilde{\alpha}_{12}\hat{\Delta}_{\nu\alpha\mu}+\kappa\tilde{\alpha}_{13}\hat{\Delta}_{\mu\nu\alpha}+\kappa\tilde{\alpha}_{14}\hat{\Delta}_{\alpha\nu\mu}+\kappa\tilde{\alpha}_{15}\hat{\Delta}_{\nu\mu\alpha}+\kappa\tilde{\alpha}_{16}\hat{\Delta}_{\mu\alpha\nu}+\kappa\tilde{\alpha}_{17}\breve{\Delta}_{\alpha\mu\nu}+\kappa\tilde{\alpha}_{18}\breve{\Delta}_{\mu\nu\alpha}$
$\displaystyle+\kappa\tilde{\alpha}_{19}\breve{\Delta}_{\nu\alpha\mu}+\kappa\tilde{\alpha}_{1,10}\bar{\Delta}_{\alpha\mu\nu}+\kappa\tilde{\alpha}_{1,11}\bar{\Delta}_{\mu\nu\alpha}+\kappa\tilde{\alpha}_{1,12}\bar{\Delta}_{\nu\alpha\mu}+\kappa\tilde{\alpha}_{1,13}\mathring{\Delta}_{\alpha\mu\nu}+\kappa\tilde{\alpha}_{1,14}\mathring{\Delta}_{\mu\nu\alpha}+\kappa\tilde{\alpha}_{1,15}\mathring{\Delta}_{\nu\alpha\mu}$
(54)

The latter is the exact form of the distortion tensor for the full quadratic
Theory. Substituting this back at the decomposition (4) we obtain the exact
and unique solution for the affine connection. We may then readily compute the
associated forms of spacetime torsion and non-metricity which we give below.

###  V.1 Expressions for Torsion and Non-metricity

Using the solution (54) and the relations (5) we easily derive the forms of
torsion and non-metricity which read121212Here vertical bars $||$ around an
index indicate that the latter has been left out of the (anti-)symmetrization
process.

$\displaystyle
S_{\mu\nu\alpha}=\kappa\Big{[}(\tilde{\alpha}_{11}-\tilde{\alpha}_{14})\hat{\Delta}_{\alpha[\mu\nu]}+\tilde{\alpha}_{17}\breve{\Delta}_{\alpha[\mu\nu]}+\tilde{\alpha}_{1,10}\bar{\Delta}_{\alpha[\mu\nu]}\Big{]}$
$\displaystyle+\kappa\Big{[}(\tilde{\alpha}_{16}-\tilde{\alpha}_{12})\hat{\Delta}_{[\mu|\alpha|\nu]}-\tilde{\alpha}_{19}\breve{\Delta}_{[\mu|\alpha|\nu]}-\tilde{\alpha}_{1,15}\mathring{\Delta}_{[\mu|\alpha|\nu]}\Big{]}$
$\displaystyle+\kappa\Big{[}(\tilde{\alpha}_{13}-\tilde{\alpha}_{15})\hat{\Delta}_{[\mu\nu]\alpha}+\tilde{\alpha}_{1,11}\bar{\Delta}_{[\mu\nu]\alpha}+\tilde{\alpha}_{1,14}\mathring{\Delta}_{[\mu\nu]\alpha}\Big{]}$
$\displaystyle+\kappa\Big{[}\tilde{\alpha}_{18}\breve{\Delta}_{\mu\nu\alpha}+\tilde{\alpha}_{1,12}\bar{\Delta}_{\nu\alpha\mu}+\tilde{\alpha}_{1,13}\mathring{\Delta}_{\alpha\mu\nu}\Big{]}$
(55)

and

$\displaystyle
Q_{\nu\alpha\mu}=2\kappa\Big{[}(\tilde{\alpha}_{11}+\tilde{\alpha}_{16})\hat{\Delta}_{(\alpha\mu)\nu}+\tilde{\alpha}_{1,10}\bar{\Delta}_{(\alpha\mu)\nu}+\tilde{\alpha}_{1,13}\mathring{\Delta}_{(\alpha\mu)\nu}\Big{]}$
$\displaystyle+2\kappa\Big{[}(\tilde{\alpha}_{12}+\tilde{\alpha}_{15})\hat{\Delta}_{\nu(\alpha\mu)}+\tilde{\alpha}_{19}\breve{\Delta}_{\nu(\alpha\mu)}+\tilde{\alpha}_{1,12}\bar{\Delta}_{\nu(\alpha\mu)}\Big{]}$
$\displaystyle+2\kappa\Big{[}(\tilde{\alpha}_{13}+\tilde{\alpha}_{14})\hat{\Delta}_{(\mu|\nu|\alpha)}+\tilde{\alpha}_{18}\breve{\Delta}_{(\mu|\nu|\alpha)}+\tilde{\alpha}_{1,14}\mathring{\Delta}_{(\mu|\nu|\alpha)}\Big{]}$
(56)

respectively. In the above we have also exploited the anti-symmetry properties
of $\breve{\Delta}_{\mu\nu\alpha}$, $\bar{\Delta}_{\mu\nu\alpha}$ and
$\mathring{\Delta}_{\mu\nu\alpha}$ which follow immediately by their very
definitions. Let us note that in the equation for torsion all elements
$\tilde{\alpha}_{ij}$ of the inverse matrix appear, while in the expression
for non-metricity the elements $\tilde{\alpha}_{17}$, $\tilde{\alpha}_{1,11}$
and $\tilde{\alpha}_{1,15}$ are absent. Again, the above expressions are exact
and hold true for any kind of hypermomentum. Of course, much more intuition
will be gained once we specify the material content giving us the explicit
form of the hypermomentum tensor.

##  VI Post Riemannian Expansion of the Metric Field Equations

Let us now elaborate on the metric field equations. Firstly, we take the trace
of (42) to arrive at

$R+\mathcal{L}^{(2)}_{even}+\mathcal{L}^{(2)}_{odd}+\frac{1}{\sqrt{-g}}\partial_{\lambda}\Big{(}\sqrt{-g}\xi^{\lambda}\Big{)}=-\kappa
T$ (57)

where we have set

$\xi^{\alpha}:=(c_{1}+nc_{2}+c_{3})S^{\alpha}+(2a_{1}+2na_{3}+a_{5})Q^{\alpha}+(2a_{2}+2a_{4}+na_{5})q^{\alpha}+(c_{6}+4c_{4}+c_{5})t^{\alpha}$
(58)

In addition, using the post-Riemannian expansion of the Ricci scalar, the
above trace equation becomes

$\displaystyle\tilde{R}+\Big{(}a_{1}+\frac{1}{4}\Big{)}Q_{\alpha\mu\nu}Q^{\alpha\mu\nu}+\Big{(}a_{2}-\frac{1}{2}\Big{)}Q_{\alpha\mu\nu}Q^{\mu\nu\alpha}+\Big{(}a_{3}-\frac{1}{4}\Big{)}Q_{\mu}Q^{\mu}+a_{4}q_{\mu}q^{\mu}+\Big{(}a_{5}+\frac{1}{2}\Big{)}Q_{\mu}q^{\mu}$
$\displaystyle+(b_{1}+1)S_{\alpha\mu\nu}S^{\alpha\mu\nu}+(b_{2}-2)S_{\alpha\mu\nu}S^{\mu\nu\alpha}+(b_{3}-4)S_{\mu}S^{\mu}+(c_{1}+2)Q_{\alpha\mu\nu}S^{\alpha\mu\nu}+(c_{2}-2)Q_{\mu}S^{\mu}+(c_{3}+2)q_{\mu}S^{\mu}$
$\displaystyle+a_{6}\varepsilon^{\alpha\beta\gamma\delta}Q_{\alpha\beta\mu}Q_{\gamma\delta}^{\;\;\;\;\mu}+b_{6}\varepsilon^{\alpha\beta\gamma\delta}S_{\alpha\beta\mu}S_{\gamma\delta}^{\;\;\;\;\mu}+c_{6}\varepsilon^{\alpha\beta\gamma\delta}Q_{\alpha\beta\mu}S_{\gamma\delta}^{\;\;\;\;\mu}+b_{5}S_{\mu}t^{\mu}+c_{4}Q_{\mu}t^{\mu}+c_{5}q^{\mu}t_{\mu}$
$\displaystyle+\frac{1}{\sqrt{-g}}\partial_{\mu}\Big{[}\sqrt{-g}\Big{(}(2a_{2}+2a_{4}+na_{5}+1)q^{\mu}+(2a_{1}+2na_{3}+a_{5}-1)Q^{\mu}$
$\displaystyle+(c_{1}+nc_{2}+c_{3}-4)S^{\mu}+(c_{6}+4c_{4}+c_{5})t^{\mu}\Big{)}\Big{]}=-\kappa
T$ (59)

Notice that there exists non-trivial parameter space for which all the parity
even quadratic terms disappear, this is of course the case of the teleparallel
equivalents of GR jimenez2019general . On the other hand, there is no
parameter choice (except the trivial of vanishing coefficients) for which the
parity odd terms cancel. Now, using a post-Riemannian expansion for every term
in (42), after some lengthy calculations (see appendix for details) we finally
arrive at

$\displaystyle\Big{(}\tilde{R}_{\mu\nu}-\frac{\tilde{R}}{2}g_{\mu\nu}\Big{)}=\kappa
T_{\mu\nu}-\tilde{\nabla}_{\alpha}N^{\alpha}_{\;\;\;(\mu\nu)}+\tilde{\nabla}_{(\nu}N^{(2)}_{\mu)}-N^{(2)}_{\lambda}N^{\lambda}_{\;\;\;(\mu\nu)}+N^{\alpha}_{\;\;\;\lambda(\nu}N^{\lambda}_{\;\;\;\mu)\alpha}$
$\displaystyle+\frac{1}{2}g_{\mu\nu}\Big{[}\tilde{\nabla}_{\alpha}(N^{(3)\alpha}-N^{(2)\alpha})+N^{(3)}_{\alpha}N^{(2)\alpha}-N_{\alpha\beta\gamma}N^{\beta\gamma\alpha}\Big{]}$
$\displaystyle+\frac{g_{\mu\nu}}{2}\Big{[}\Big{(}\frac{b_{1}}{2}+2a_{1}-c_{1}\Big{)}N_{\alpha\beta\gamma}N^{\alpha\beta\gamma}+\Big{(}-\frac{b_{1}}{2}+a_{2}+c_{1}\Big{)}N_{\alpha\beta\gamma}N^{\alpha\gamma\beta}$
$\displaystyle+\Big{(}\frac{b_{2}}{2}+2a_{2}+c_{1}\Big{)}N_{\alpha\beta\gamma}N^{\beta\gamma\alpha}+\Big{(}-\frac{b_{2}}{2}+2a_{1}+a_{2}-c_{1}\Big{)}N_{\kappa\lambda\alpha}N^{\lambda\kappa\alpha}$
$\displaystyle+\Big{(}\frac{b_{3}}{4}+4a_{3}-c_{2}\Big{)}N^{(1)}_{\alpha}N^{(1)}_{\beta}g^{\alpha\beta}+\Big{(}\frac{b_{3}}{4}+a_{4}+\frac{c_{3}}{4}\Big{)}N^{(2)}_{\alpha}N^{(2)}_{\beta}g^{\alpha\beta}+a_{4}N^{(3)}_{\alpha}N^{(3)}_{\beta}g^{\alpha\beta}$
$\displaystyle+\Big{(}2a_{5}-\frac{c_{3}}{2}+c_{2}-\frac{b_{3}}{2}\Big{)}N^{(1)}_{\alpha}N^{(2)}_{\beta}g^{\alpha\beta}+\Big{(}2a_{4}+\frac{c_{3}}{2}\Big{)}N^{(2)}_{\alpha}N^{(3)}_{\beta}g^{\alpha\beta}+\Big{(}2a_{5}-\frac{c_{3}}{2}\Big{)}N^{(1)}_{\alpha}N^{(3)}_{\beta}g^{\alpha\beta}\Big{]}$
$\displaystyle-\Big{(}4a_{1}-\frac{c_{1}}{2}\Big{)}\frac{1}{\sqrt{-g}}\hat{\nabla}_{\beta}\Big{(}\sqrt{-g}g^{\alpha\beta}N_{(\mu\nu)\alpha}\Big{)}-\Big{(}2a_{2}+\frac{c_{1}}{2}\Big{)}\frac{1}{\sqrt{-g}}\hat{\nabla}_{\beta}\Big{(}\sqrt{-g}g^{\alpha\beta}N_{(\mu|\alpha|\nu)}\Big{)}-2a_{2}\frac{1}{\sqrt{-g}}\hat{\nabla}_{\beta}\Big{(}\sqrt{-g}g^{\alpha\beta}N_{\alpha(\mu\nu)}\Big{)}$
$\displaystyle-\frac{1}{\sqrt{-g}}\hat{\nabla}_{\alpha}\left[\Big{(}4a_{3}-\frac{c_{2}}{2}\Big{)}N^{\alpha(1)}g_{\mu\nu}+\Big{(}a_{5}+\frac{c_{2}}{2}\Big{)}N^{\alpha(2)}g_{\mu\nu}+a_{5}N^{\alpha(3)}g_{\mu\nu}\right]$
$\displaystyle-\frac{1}{\sqrt{-g}}\hat{\nabla}_{(\mu}\left[\Big{(}2a_{5}-\frac{c_{3}}{2}\Big{)}N_{\nu)}^{(1)}+\Big{(}2a_{4}+\frac{c_{3}}{2}\Big{)}N_{\nu)}^{(2)}+2a_{4}N_{\nu)}^{(3)}\right]$
$\displaystyle-\Big{(}2a_{1}-\frac{c_{1}}{2}\Big{)}N^{\alpha\beta}_{\;\;\;\;\mu}N_{\alpha\beta\nu}-\Big{(}2a_{1}-\frac{c_{1}}{2}\Big{)}N^{\alpha\beta}_{\;\;\;\;\mu}N_{\beta\alpha\nu}-\frac{c_{1}}{2}N^{\alpha\beta}_{\;\;\;\;(\mu}(N_{\beta|\nu)\alpha}+N_{\alpha|\nu)\beta})$
$\displaystyle+(N^{\alpha\;\;\beta}_{\;\;(\nu}+N_{(\nu}^{\;\;\alpha\beta})\left[2a_{1}N_{\alpha|\mu)\beta}+a_{2}N_{\beta|\mu)\alpha}+\Big{(}2a_{1}-\frac{c_{1}}{2}\Big{)}N_{\mu)\alpha\beta}+\Big{(}a_{2}+\frac{c_{1}}{2}\Big{)}N_{\mu)\beta\alpha}\right]$
$\displaystyle-\frac{1}{4}(N_{\beta(\nu|\alpha}-N_{\beta\alpha(\nu})\Big{(}2b_{1}(N^{\beta\;\;\alpha}_{\;\;\mu)}-b_{2}N^{\alpha\;\;\beta}_{\;\;\mu)}-2b_{1}N^{\beta\alpha}_{\;\;\;\mu)}+b_{2}N^{\alpha\beta}_{\;\;\;\mu)}\Big{)}+b_{1}N_{\mu\alpha\beta}N_{\nu}^{\;\;[\alpha\beta]}$
$\displaystyle+(N_{\mu\nu\alpha}+N_{\nu\mu\alpha})\left[\Big{(}4a_{3}-\frac{c_{2}}{2}\Big{)}N^{(1)\alpha}+\Big{(}a_{5}+\frac{c_{2}}{2}\Big{)}N^{(2)\alpha}+a_{5}N^{(3)\alpha}\right]$
$\displaystyle-\Big{(}4a_{3}-\frac{c_{3}}{2}\Big{)}N_{\mu}^{(1)}N_{\nu}^{(1)}-\Big{(}\frac{b_{3}}{4}-a_{4}\Big{)}N_{\mu}^{(2)}N_{\nu}^{(2)}+a_{4}N_{\mu}^{(3)}N_{\nu}^{(3)}-\Big{(}c_{3}-\frac{b_{3}}{2}\Big{)}N_{(\mu}^{(1)}N_{\nu)}^{(2)}+2a_{4}N_{(\mu}^{(2)}N_{\nu)}^{(3)}$
$\displaystyle-
b_{6}\varepsilon^{\alpha\beta\gamma\delta}N_{\alpha\beta\nu}N_{\gamma\delta\mu}+2a_{6}\varepsilon^{\alpha\beta\gamma\delta}N_{(\alpha\mu)\beta}(N_{\gamma\nu\delta}+N_{\nu\gamma\delta})$
$\displaystyle-
g_{\beta(\mu}g_{\nu)\lambda}\frac{1}{\sqrt{-g}}(\nabla_{\alpha}-2S_{\alpha})\Big{(}\sqrt{-g}(c_{6}-2a_{6})\varepsilon^{\alpha\beta\gamma\delta}N^{\lambda}_{\;\;\gamma\delta}-2a_{6}\sqrt{-g}\varepsilon^{\alpha\beta\gamma\delta}N_{\gamma\;\;\;\;\delta}^{\;\;\lambda}\Big{)}$
$\displaystyle-\varepsilon^{\lambda\alpha\beta}_{\;\;\;\;\;\;(\nu}N_{\mu)\alpha\beta}\left[\Big{(}2c_{4}-\frac{b_{5}}{2}\Big{)}N^{(1)}_{\lambda}+\Big{(}c_{5}+\frac{b_{5}}{2}\Big{)}N^{(2)}_{\lambda}+c_{5}N^{(3)}_{\lambda}\right]-c_{4}g_{\mu\nu}\Big{(}\partial_{\lambda}(\sqrt{-g}M^{\lambda})\Big{)}$
$\displaystyle-
c_{5}g_{\alpha(\mu}g_{\nu)\beta}\frac{1}{\sqrt{-g}}\nabla_{\lambda}(\sqrt{-g}M^{\alpha}g^{\lambda\beta})+c_{5}\Big{(}N^{(2)}_{(\mu}-N^{(1)}_{(\mu}\Big{)}M_{\nu)}+\frac{c_{5}}{2}\Big{(}N_{\nu\alpha\mu}+N_{\alpha\nu\mu}+N_{\mu\alpha\nu}+N_{\alpha\mu\nu}\Big{)}M^{\alpha}$
(60)

and recall that

$\displaystyle
N_{\alpha\mu\nu}=\kappa\tilde{\alpha}_{11}\hat{\Delta}_{\alpha\mu\nu}+\kappa\tilde{\alpha}_{12}\hat{\Delta}_{\nu\alpha\mu}+\kappa\tilde{\alpha}_{13}\hat{\Delta}_{\mu\nu\alpha}+\kappa\tilde{\alpha}_{14}\hat{\Delta}_{\alpha\nu\mu}+\kappa\tilde{\alpha}_{15}\hat{\Delta}_{\nu\mu\alpha}+\kappa\tilde{\alpha}_{16}\hat{\Delta}_{\mu\alpha\nu}+\kappa\tilde{\alpha}_{17}\breve{\Delta}_{\alpha\mu\nu}+\kappa\tilde{\alpha}_{18}\breve{\Delta}_{\mu\nu\alpha}$
$\displaystyle+\kappa\tilde{\alpha}_{19}\breve{\Delta}_{\nu\alpha\mu}+\kappa\tilde{\alpha}_{1,10}\bar{\Delta}_{\alpha\mu\nu}+\kappa\tilde{\alpha}_{1,11}\bar{\Delta}_{\mu\nu\alpha}+\kappa\tilde{\alpha}_{1,12}\bar{\Delta}_{\nu\alpha\mu}+\kappa\tilde{\alpha}_{1,13}\mathring{\Delta}_{\alpha\mu\nu}+\kappa\tilde{\alpha}_{1,14}\mathring{\Delta}_{\mu\nu\alpha}+\kappa\tilde{\alpha}_{1,15}\mathring{\Delta}_{\nu\alpha\mu}$
(61)

where the various $\Delta_{\alpha\mu\nu}^{\prime}s$ are given by the
expressions $(\ref{B1})$-$(\ref{B4})$ by setting
$B_{\alpha\mu\nu}=\kappa\Delta_{\alpha\mu\nu}$. The field equations
$(\ref{GReff})$ represent a modified version of GR with additional matter
couplings that are associated to the microscopic characteristics of matter. It
is obvious then, by direct substitution of the latter form of the distortion
into $(\ref{GReff})$ that apart from the usual energy-momentum tensor
quadratic and derivative terms of the hypermomentum serve also as sources
producing gravitational fields. Our result above serves as a generalization of
the earlier findings of Iosifidis:2021tvx  where now the additional $6$ parity
odd quadratic scalars have been added into the action along with the $11$
parity even ones. Our study now covers the whole spectrum of quadratic Metric-
Affine Gravity Theories, that is we have found the exact form of the
connection and consequently that of torsion and non-metricity for the most
general $17$ parameter quadratic MAG action.

Of course, given the complexity of the right-hand side of the above field
equations, no concrete results can be inferred. More information can be
extracted once a specific matter configuration is adopted giving rise to a
certain form of the hypermomentum tensor. However, one thing one can be sure
about is that the effective energy-momentum tensor as given by the right-hand
side of (60), that is, the sum of usual energy-momentum tensor $+$
hypermomentum contributions is covariantly conserved with respect to the Levi-
Civita connection. Indeed, since the Riemannian contracted Bianchi identities
ensure that the left hand side of the above field equations is covariantly
conserved with respect to the Levi-Civita connection so must be for the right-
hand side as well.

Now, it is interesting to study what happens when the matter fields do not
couple to the connection, the so-called Palatini case. In this instance
$\Delta_{\alpha\mu\nu}\equiv 0$ and from corollary $1$ we also conclude the
following

###### Corollary 2

In the Palatini version of (41), that is when the matter sector of the action
is independent of the connection (i.e. $\Delta_{\alpha\mu\nu}\equiv 0$) and
given that both $det(A)\neq 0$ and $det(\Gamma)\neq 0$ the connection solution
is always the Levi-Civita. In other words, torsion and non-metricity vanish
identically and the $17$ parameter quadratic Theory is equivalent to GR.

A special case of the latter is of course the vacuum Theory and therefore we
have also the following result holding true.

###### Corollary 3

In vacuum and given that both matrices $A$ and $\Gamma$ are non-singular, the
$17$ parameter quadratic Theory always reduces to GR.

It is rather remarkable that despite the complexity of the full quadratic
Theory, in vacuum the latter is equivalent to GR. Important deviations occur
only when matter with intrinsic structure is present, that is matter with non-
vanishing hypermomentum.

Now, let us stress out that in applying the results of $Theorem-1$ in order to
solve for the distortion, nowhere did we assume that the coefficients $a_{i}$,
$b_{j}$ and $c_{k}$ are constants. Consequently,, the main result for the
affine connection will continue to hold true even if we promote the
aforementioned constants to be spacetime functions. In particular, the
following generalized Theorem holds true.

###### Theorem 2

Let $\Phi$ be a collection of matter fields (collectively denoted) and $d\Phi$
their exterior derivatives. Furthermore, let
$a_{i}(\Phi,d\Phi)=a_{i}(\Phi,\partial\Phi)$, $b_{i}(\Phi,\partial\Phi)$ and
$c_{i}(\Phi,\partial\Phi)$ $i=1,2,...,6$ be arbitrary functions continuous and
differentiable in their arguments131313These functions can just as well depend
also on the metric tensor and its derivatives, but not on the connection..
Then, for the general class of Theories given by the action

$\displaystyle S[g,\Gamma,\Phi]=\frac{1}{2\kappa}\int
d^{n}x\sqrt{-g}\Big{[}R+b_{1}(\Phi,\partial\Phi)S_{\alpha\mu\nu}S^{\alpha\mu\nu}+b_{2}(\Phi,\partial\Phi)S_{\alpha\mu\nu}S^{\mu\nu\alpha}+b_{3}(\Phi,\partial\Phi)S_{\mu}S^{\mu}$
$\displaystyle
a_{1}(\Phi,\partial\Phi)Q_{\alpha\mu\nu}Q^{\alpha\mu\nu}+a_{2}(\Phi,\partial\Phi)Q_{\alpha\mu\nu}Q^{\mu\nu\alpha}+a_{3}(\Phi,\partial\Phi)Q_{\mu}Q^{\mu}+a_{4}(\Phi,\partial\Phi)q_{\mu}q^{\mu}$
$\displaystyle+a_{5}(\Phi,\partial\Phi)Q_{\mu}q^{\mu}+c_{1}(\Phi,\partial\Phi)Q_{\alpha\mu\nu}S^{\alpha\mu\nu}+c_{2}(\Phi,\partial\Phi)Q_{\mu}S^{\mu}+c_{3}(\Phi,\partial\Phi)q_{\mu}S^{\mu}$
$\displaystyle+a_{6}(\Phi,\partial\Phi)\varepsilon^{\alpha\beta\gamma\delta}Q_{\alpha\beta\mu}Q_{\gamma\delta}^{\;\;\;\;\mu}+b_{5}(\Phi,\partial\Phi)S_{\mu}t^{\mu}+b_{6}(\Phi,\partial\Phi)\varepsilon^{\alpha\beta\gamma\delta}S_{\alpha\beta\mu}S_{\gamma\delta}^{\;\;\;\;\mu}$
$\displaystyle
c_{4}(\Phi,\partial\Phi)Q_{\mu}t^{\mu}+c_{5}(\Phi,\partial\Phi)q^{\mu}t_{\mu}+c_{6}\varepsilon^{\alpha\beta\gamma\delta}Q_{\alpha\beta\mu}S_{\gamma\delta}^{\;\;\;\;\mu}\Big{]}+S_{M}[g,\Gamma,\Phi]$
(62)

the exact solution for the distortion tensor reads

$\displaystyle
N_{\alpha\mu\nu}=\tilde{\alpha}_{11}\hat{B}_{\alpha\mu\nu}+\tilde{\alpha}_{12}\hat{B}_{\nu\alpha\mu}+\tilde{\alpha}_{13}\hat{B}_{\mu\nu\alpha}+\tilde{\alpha}_{14}\hat{B}_{\alpha\nu\mu}+\tilde{\alpha}_{15}\hat{B}_{\nu\mu\alpha}+\tilde{\alpha}_{16}\hat{B}_{\mu\alpha\nu}+\tilde{\alpha}_{17}\breve{B}_{\alpha\mu\nu}+\tilde{\alpha}_{18}\breve{B}_{\mu\nu\alpha}+\tilde{\alpha}_{19}\breve{B}_{\nu\alpha\mu}$
$\displaystyle+\tilde{\alpha}_{1,10}\bar{B}_{\alpha\mu\nu}+\tilde{\alpha}_{1,11}\bar{B}_{\mu\nu\alpha}+\tilde{\alpha}_{1,12}\bar{B}_{\nu\alpha\mu}+\tilde{\alpha}_{1,13}\mathring{B}_{\alpha\mu\nu}+\tilde{\alpha}_{1,14}\mathring{B}_{\mu\nu\alpha}+\tilde{\alpha}_{1,15}\mathring{B}_{\nu\alpha\mu}$
(63)

and consequently the affine connection is given by

$\displaystyle\Gamma^{\lambda}_{\;\;\;\mu\nu}=\tilde{\Gamma}^{\lambda}_{\;\;\;\mu\nu}+\tilde{\alpha}_{11}\hat{B}_{\alpha\mu\nu}+\tilde{\alpha}_{12}\hat{B}_{\nu\alpha\mu}+\tilde{\alpha}_{13}\hat{B}_{\mu\nu\alpha}+\tilde{\alpha}_{14}\hat{B}_{\alpha\nu\mu}+\tilde{\alpha}_{15}\hat{B}_{\nu\mu\alpha}+\tilde{\alpha}_{16}\hat{B}_{\mu\alpha\nu}+\tilde{\alpha}_{17}\breve{B}_{\alpha\mu\nu}+\tilde{\alpha}_{18}\breve{B}_{\mu\nu\alpha}+\tilde{\alpha}_{19}\breve{B}_{\nu\alpha\mu}$
$\displaystyle+\tilde{\alpha}_{1,10}\bar{B}_{\alpha\mu\nu}+\tilde{\alpha}_{1,11}\bar{B}_{\mu\nu\alpha}+\tilde{\alpha}_{1,12}\bar{B}_{\nu\alpha\mu}+\tilde{\alpha}_{1,13}\mathring{B}_{\alpha\mu\nu}+\tilde{\alpha}_{1,14}\mathring{B}_{\mu\nu\alpha}+\tilde{\alpha}_{1,15}\mathring{B}_{\nu\alpha\mu}$
(64)

where now $\tilde{\alpha}_{ij}=\tilde{\alpha}_{ij}(\Phi,\partial\Phi)$.

Proof. Following an identical procedure with that of section $V$ and by simply
promoting the $a_{i}^{\prime}s,b_{j}^{\prime}s$ and $c_{j}^{\prime}s$ to be
functions of $\Phi$ and $d\Phi$ we arrive at the stated result. The last
action covers a wide range of Metric-Affine Gravity Theories and includes as
special cases non-minimally coupled scalar-tensor and vector-tensor Theories.

##  VII Conclusions

We have studied the most general MAG Theory that is quadratic in torsion and
non-metricity (and their mixings) and linear in the curvature. Without
restricting ourselves to the parity even case, we included all the parity odd
terms as well. For this $17$ parameter Theory, we first derived the field
equations by varying independently with respect to the metric as well as the
connection. Focusing first on the connection field equations, we performed a
simple trick to recast the set of these equations solely in terms of the
distortion instead of combinations of torsion and non-metricity. Then, using a
recent result on solving linear $30$ parameter tensor equations (see ref.
Iosifidis:2021tvx ) we were able to find the exact and unique form of the
distortion and subsequently of the affine connection in terms of the
hypermomentum and its various contractions. The exact forms of torsion and
non-metricity can then be derived readily. Continuing, we turned our attention
to the metric field equations and by using a post Riemannian expansion for all
relevant pieces we recast the latter into an effective GR with modified source
terms. These extended sources, apart from the usual energy-momentum
contribution, are comprised by quadratic combinations of the hypermomentum
tensor, its dual constructions as well as linear derivatives of both the
former and the latter tensors and their contractions. Their physical role is
tightly connected to the microstructure of matter. The exact effect is given
by equation $(\ref{GReff})$ but given its complexity no concrete conclusions
can be inferred without knowing more about the form of the sources. Equation
(60) , as it stands, provides a generalization of Einstein-Cartan Theory where
on top of spacetime torsion sourced by the spin part of hypermomentum, the
dilation and shear parts of the latter excite also non-metricity producing,
therefore, a fully non-Riemannian arena for Gravity.

Now, even though we are able to obtain the full solution in the presence of
the sources, of equal importance is to know what the vacuum case corresponds
to. In other words, the question: ”What the full quadratic Theory (as given by
action $(\ref{S})$) corresponds to in vacuum? ” was not known till know. In
the current research we were able to answer to this question as well and as it
turns out the vacuum Theory, despite all appearances, is equivalent to GR. As
a result there are no new (and possible harmful) degrees of freedom and the
new interactions are tied to matter. In addition, not only the vacuum Theory
but also the less restricted Palatini case where the matter does not depend on
the connection (i.e. identically vanishing hypermomentum) but does depend on
the metric, yields again GR. It also becomes apparent that in order to fully
appreciate, and be able to deeply study the role of Metric-Affine Theories of
Gravity, the inclusion of hypermomentum is all essential and indispensable. In
other words, the connection-matter couplings are of prominent importance and
have to be taken seriously Hehl:1977gn . After all it is exactly the role of
the latter couplings that distinguish MAG from the rest of modified gravity
Theories, which also provides the astonishing relation between the
microstructure of matter and the excitations of spacetime torsion and non-
metricity. The intrinsic structure of matter is fully described by the
hypermomentum tensor.

In addition, as we explicitly showed, our result for the connection solution
continues to hold true even for more general couplings for the quadratic
invariants that are not necessary constant. In particular, we have also found
the exact connection solution, as given by equation (64), for the fairly wide
class of Theories (62). Subsequently, the next logical step to apply the
results we obtained here would be to consider specific forms for the
hypermomentum and find the exact form of these extra source terms. Then, the
forms of torsion and non-metricity can be obtained and also the new terms on
the right-hand side of $(\ref{GReff})$ could be interpreted. One could then
perhaps consider the manifield construction of Cant:1985yg . Additionally, it
would also be very interesting to study the $17$ parameter quadratic Theory in
the presence of an isotropic hyperfluid iosifidis2021perfect  or even the
homogeneous Cosmology of the quadratic Theory sourced by the Perfect
Cosmological Hyperfluid Iosifidis:2020gth . Some of these aspects are under
investigation now.

##  VIII Acknowledgments

This research is co-financed by Greece and the European Union (European Social
Fund- ESF) through the Operational Programme ’Human Resources Development,
Education and Lifelong Learning’ in the context of the project “Reinforcement
of Postdoctoral Researchers - 2 nd Cycle” (MIS-5033021), implemented by the
State Scholarships Foundation (IKY).

##  Appendix A Variations

Let us gather here the variations of the parity odd terms with respect to the
metric as well as the affine connection. Recall that the parity odd quadratic
Lagrangian reads

$\displaystyle\mathcal{L}_{odd}=a_{6}\varepsilon^{\alpha\beta\gamma\delta}Q_{\alpha\beta\mu}Q_{\gamma\delta}^{\;\;\;\;\mu}+b_{6}\varepsilon^{\alpha\beta\gamma\delta}S_{\alpha\beta\mu}S_{\gamma\delta}^{\;\;\;\;\mu}+c_{6}\varepsilon^{\alpha\beta\gamma\delta}Q_{\alpha\beta\mu}S_{\gamma\delta}^{\;\;\;\;\mu}+b_{5}S_{\mu}t^{\mu}+c_{4}Q_{\mu}t^{\mu}+c_{5}q^{\mu}t_{\mu}$
(65)

We then find the following.

###  A.1 g-Variations

$\delta_{g}\Big{(}\sqrt{-g}\varepsilon^{\alpha\beta\gamma\delta}Q_{\alpha\beta\mu}Q_{\gamma\delta}^{\;\;\;\;\mu}\Big{)}=-2g_{\beta(\mu}g_{\nu)\lambda}(\nabla_{\alpha}-2S_{\alpha})\Big{(}\sqrt{-g}\varepsilon^{\alpha\beta\gamma\delta}Q_{\gamma\delta}^{\;\;\;\;\lambda}\Big{)}(\delta
g^{\mu\nu})+\sqrt{-g}\varepsilon^{\alpha\beta\gamma\delta}Q_{\alpha\beta\mu}Q_{\gamma\delta\nu}(\delta
g^{\mu\nu})$ (66)
$\delta_{g}\Big{(}\sqrt{-g}\varepsilon^{\alpha\beta\gamma\delta}S_{\alpha\beta\mu}S_{\gamma\delta}^{\;\;\;\;\mu}\Big{)}=-\sqrt{-g}\varepsilon^{\alpha\beta\gamma\delta}S_{\alpha\beta\nu}S_{\gamma\delta\mu}(\delta
g^{\mu\nu})$ (67)
$\delta_{g}\Big{(}\sqrt{-g}\varepsilon^{\alpha\beta\gamma\delta}Q_{\alpha\beta\mu}S_{\gamma\delta}^{\;\;\;\;\mu}\Big{)}=-g_{\beta(\mu}g_{\nu)\lambda}(\nabla_{\alpha}-2S_{\alpha})\Big{(}\sqrt{-g}\varepsilon^{\alpha\beta\gamma\delta}S_{\gamma\delta}^{\;\;\;\;\lambda}\Big{)}(\delta
g^{\mu\nu})$ (68)
$\delta_{g}(\sqrt{-g}t^{\mu}S_{\mu})=-\sqrt{-g}S_{\alpha\beta\mu}\varepsilon^{\lambda\alpha\beta}_{\;\;\;\;\;\;\nu}S_{\lambda}(\delta
g^{\mu\nu})$ (69)

$\delta_{g}(\sqrt{-g}t^{\lambda}Q_{\lambda})=-\sqrt{-g}S_{\alpha\beta\mu}\varepsilon^{\lambda\alpha\beta}_{\;\;\;\;\;\;\nu}Q_{\lambda}(\delta
g^{\mu\nu})-g_{\mu\nu}\Big{(}\partial_{\lambda}(\sqrt{-g}t^{\lambda})\Big{)}(\delta
g^{\mu\nu})+t.d.$ (70)
$\displaystyle\delta_{g}(\sqrt{-g}t^{\lambda}q_{\lambda})=\Big{[}-\sqrt{-g}S_{\alpha\beta\mu}\varepsilon^{\lambda\alpha\beta}_{\;\;\;\;\;\;\nu}q_{\lambda}-g_{\mu\alpha}g_{\nu\beta}\nabla_{\lambda}(\sqrt{-g}t^{\alpha}g^{\lambda\beta})+\sqrt{-g}2S_{\mu}t_{\nu}+\sqrt{-g}t^{\alpha}Q_{\mu\nu\alpha}\Big{]}(\delta
g^{\mu\nu})$ (71)

So up to total derivatives we find

$\displaystyle\delta_{g}\Big{(}\sqrt{-g}\mathcal{L}^{(2)}_{odd}\Big{)}=\Big{[}-b_{6}\sqrt{-g}\varepsilon^{\alpha\beta\gamma\delta}S_{\alpha\beta\nu}S_{\gamma\delta\mu}-2g_{\beta(\mu}g_{\nu)\lambda}(\nabla_{\alpha}-2S_{\alpha})\Big{(}\sqrt{-g}a_{6}\varepsilon^{\alpha\beta\gamma\delta}Q_{\gamma\delta}^{\;\;\;\;\lambda}\Big{)}$
$\displaystyle+\sqrt{-g}a_{6}\varepsilon^{\alpha\beta\gamma\delta}Q_{\alpha\beta\mu}Q_{\gamma\delta\nu}-g_{\beta(\mu}g_{\nu)\lambda}(\nabla_{\alpha}-2S_{\alpha})\Big{(}\sqrt{-g}c_{6}\varepsilon^{\alpha\beta\gamma\delta}S_{\gamma\delta}^{\;\;\;\;\lambda}\Big{)}$
$\displaystyle-\sqrt{-g}b_{5}S_{\alpha\beta\mu}\varepsilon^{\lambda\alpha\beta}_{\;\;\;\;\;\;\nu}S_{\lambda}-\sqrt{-g}c_{4}S_{\alpha\beta\mu}\varepsilon^{\lambda\alpha\beta}_{\;\;\;\;\;\;\nu}Q_{\lambda}-c_{4}g_{\mu\nu}\Big{(}\partial_{\lambda}(\sqrt{-g}t^{\lambda})\Big{)}$
$\displaystyle-\sqrt{-g}c_{5}S_{\alpha\beta\mu}\varepsilon^{\lambda\alpha\beta}_{\;\;\;\;\;\;\nu}q_{\lambda}-c_{5}g_{\mu\alpha}g_{\nu\beta}\nabla_{\lambda}(\sqrt{-g}t^{\alpha}g^{\lambda\beta})+\sqrt{-g}c_{5}2S_{\mu}t_{\nu}+\sqrt{-g}c_{5}t^{\alpha}Q_{\mu\nu\alpha}\Big{]}(\delta
g^{\mu\nu})$ (72)

and after some rearranging we finally arrive at

$\displaystyle\frac{1}{\sqrt{-g}}\frac{\Big{(}\sqrt{-g}\mathcal{L}^{(2)}_{odd}\Big{)}}{\delta
g^{\mu\nu}}=-b_{6}\varepsilon^{\alpha\beta\gamma\delta}S_{\alpha\beta\nu}S_{\gamma\delta\mu}+a_{6}\varepsilon^{\alpha\beta\gamma\delta}Q_{\alpha\beta\mu}Q_{\gamma\delta\nu}$
$\displaystyle-
g_{\beta(\mu}g_{\nu)\lambda}\frac{1}{\sqrt{-g}}(\nabla_{\alpha}-2S_{\alpha})\Big{(}\sqrt{-g}\varepsilon^{\alpha\beta\gamma\delta}(2a_{6}Q_{\gamma\delta}^{\;\;\;\;\lambda}+c_{6}S_{\gamma\delta}^{\;\;\;\;\lambda})\Big{)}$
$\displaystyle-
S_{\alpha\beta(\mu}\varepsilon^{\lambda\alpha\beta}_{\;\;\;\;\;\;\nu)}(b_{5}S_{\lambda}+c_{4}Q_{\lambda}+c_{5}q_{\lambda})-c_{4}g_{\mu\nu}\Big{(}\partial_{\lambda}(\sqrt{-g}t^{\lambda})\Big{)}$
$\displaystyle-
c_{5}g_{\alpha(\mu}g_{\nu)\beta}\frac{1}{\sqrt{-g}}\nabla_{\lambda}(\sqrt{-g}t^{\alpha}g^{\lambda\beta})+c_{5}2S_{(\mu}t_{\nu)}+c_{5}t^{\alpha}Q_{(\mu\nu)\alpha}$
(73)

###  A.2 $\Gamma$-Variations

$\delta_{\Gamma}\Big{(}\varepsilon^{\alpha\beta\gamma\delta}Q_{\alpha\beta\mu}Q_{\gamma\delta}^{\;\;\;\;\mu}\Big{)}=-2\Big{(}\varepsilon^{\mu\nu\alpha\beta}Q_{\alpha\beta\lambda}+\varepsilon_{\lambda}^{\;\;\nu\alpha\beta}Q_{\alpha\beta}^{\;\;\;\;\mu}\Big{)}\delta\Gamma^{\lambda}_{\;\;\;\mu\nu}$
(74)
$\delta_{\Gamma}\Big{(}\varepsilon^{\alpha\beta\gamma\delta}S_{\alpha\beta\mu}S_{\gamma\delta}^{\;\;\;\;\mu}\Big{)}=2\varepsilon^{\mu\nu\alpha\beta}S_{\alpha\beta\lambda}\delta\Gamma^{\lambda}_{\;\;\;\mu\nu}$
(75)
$\delta_{\Gamma}\Big{(}\varepsilon^{\alpha\beta\gamma\delta}Q_{\alpha\beta\mu}S_{\gamma\delta}^{\;\;\;\;\mu}\Big{)}=\Big{(}-\varepsilon^{\mu\nu\alpha\beta}S_{\alpha\beta\lambda}-\varepsilon_{\lambda}^{\;\;\nu\alpha\beta}S_{\alpha\beta}^{\;\;\;\;\mu}+\varepsilon^{\mu\nu\alpha\beta}Q_{\alpha\beta\lambda}\Big{)}\delta\Gamma^{\lambda}_{\;\;\;\mu\nu}$
(76)

Let

$B:=(b_{5}S_{\mu}+c_{4}Q_{\mu}+c_{5}q_{\mu})t^{\mu}$ (77)

Then

$\delta_{\Gamma}B=\Big{[}\varepsilon^{\alpha\mu\nu}_{\;\;\;\;\;\;\lambda}(b_{5}S_{\alpha}+c_{4}Q_{\alpha}+c_{5}q_{\alpha})+b_{5}t^{[\mu}\delta^{\nu]}_{\lambda}+2c_{4}t^{\nu}\delta^{\mu}_{\lambda}+c_{5}(g^{\mu\nu}t_{\lambda}+t^{\mu}\delta^{\nu}_{\lambda})\Big{]}\delta\Gamma^{\lambda}_{\;\;\;\mu\nu}$
(78)

and as a result the $\Gamma$ variation of the parity odd Lagrangian reads

$\displaystyle\delta_{\Gamma}\mathcal{L}_{odd}=\delta\Gamma^{\lambda}_{\;\;\;\mu\nu}\Big{[}(-2a_{6}+c_{6})\varepsilon^{\mu\nu\alpha\beta}Q_{\alpha\beta\lambda}+(2b_{6}-c_{6})\varepsilon^{\mu\nu\alpha\beta}S_{\alpha\beta\lambda}-2a_{6}\varepsilon_{\lambda}^{\;\;\nu\alpha\beta}Q_{\alpha\beta}^{\;\;\;\;\mu}-c_{6}\varepsilon_{\lambda}^{\;\;\nu\alpha\beta}S_{\alpha\beta}^{\;\;\;\;\mu}$
$\displaystyle+\varepsilon^{\alpha\mu\nu}_{\;\;\;\;\;\;\lambda}(b_{5}S_{\alpha}+c_{4}Q_{\alpha}+c_{5}q_{\alpha})+\Big{(}\frac{b_{5}}{2}+c_{5}\Big{)}t^{\mu}\delta_{\lambda}^{\nu}+\Big{(}-\frac{b_{5}}{2}+2c_{4}\Big{)}t^{\nu}\delta_{\lambda}^{\mu}+c_{5}g^{\mu\nu}t_{\lambda}\Big{]}$
(79)

###  A.3 More Variations

We compute

$g^{\mu\nu}\frac{\delta\Big{(}\sqrt{-g}\varepsilon^{\alpha\beta\gamma\delta}S_{\alpha\beta\mu}S_{\gamma\delta}^{\;\;\;\;\mu}\Big{)}}{\delta
g^{\mu\nu}}=-\sqrt{-g}\varepsilon^{\alpha\beta\gamma\delta}S_{\alpha\beta\mu}S_{\gamma\delta}^{\;\;\;\;\mu}$
(80)
$g^{\mu\nu}\frac{\delta\Big{(}\sqrt{-g}\varepsilon^{\alpha\beta\gamma\delta}Q_{\alpha\beta\mu}S_{\gamma\delta}^{\;\;\;\;\mu}\Big{)}}{\delta
g^{\mu\nu}}=-\sqrt{-g}\varepsilon^{\alpha\beta\gamma\delta}Q_{\alpha\beta\mu}S_{\gamma\delta}^{\;\;\;\;\mu}-\partial_{\lambda}(\sqrt{-g}t^{\lambda})$
(81)
$g^{\mu\nu}\frac{\delta\Big{(}\sqrt{-g}\varepsilon^{\alpha\beta\gamma\delta}Q_{\alpha\beta\mu}Q_{\gamma\delta}^{\;\;\;\;\mu}\Big{)}}{\delta
g^{\mu\nu}}=-\sqrt{-g}\varepsilon^{\alpha\beta\gamma\delta}Q_{\alpha\beta\mu}Q_{\gamma\delta}^{\;\;\;\;\mu}$
(82)
$g^{\mu\nu}\frac{\delta\Big{(}\sqrt{-g}t^{\alpha}S_{\alpha}\Big{)}}{\delta
g^{\mu\nu}}=-\sqrt{-g}t^{\alpha}S_{\alpha}$ (83)
$g^{\mu\nu}\frac{\delta\Big{(}\sqrt{-g}t^{\alpha}q_{\alpha}\Big{)}}{\delta
g^{\mu\nu}}=-\sqrt{-g}t^{\alpha}q_{\alpha}-\partial_{\lambda}(\sqrt{-g}t^{\lambda})$
(84)
$g^{\mu\nu}\frac{\delta\Big{(}\sqrt{-g}t^{\alpha}Q_{\alpha}\Big{)}}{\delta
g^{\mu\nu}}=-\sqrt{-g}t^{\alpha}Q_{\alpha}-4\partial_{\lambda}(\sqrt{-g}t^{\lambda})$
(85)

With these we easily find that

$g^{\mu\nu}\frac{\delta\Big{(}\sqrt{-g}\mathcal{L}^{(2)}_{odd}\Big{)}}{\delta
g^{\mu\nu}}=-\sqrt{-g}\mathcal{L}^{(2)}_{odd}-(c_{6}+4c_{4}+c_{5})\partial_{\lambda}(\sqrt{-g}t^{\lambda})$
(86)

Note that for the parameter choice $c_{6}+4c_{4}+c_{5}=0$ the derivative term
disappears.

##  Appendix B Expansions

###  B.1 Distortion vectors in terms of torsion and non-metricity vectors

Contracting $N_{\alpha\mu\nu}$ in all possible ways with the metric we find

$N_{(1)}^{\mu}=\frac{1}{2}Q^{\mu}\;\;,\;\;N_{(2)}^{\mu}=\frac{1}{2}Q^{\mu}+2S^{\mu}\;\;,\;\;N_{(3)}^{\mu}=q^{\mu}-\frac{1}{2}Q^{\mu}-2S^{\mu}$
(87)

which relate the distortion vectors in terms of those of torsion and non-
metricity. Inverting them we have

$Q^{\mu}=2N_{(1)}^{\mu}\;\;,\;\;q^{\mu}=N_{(2)}^{\mu}+N_{(3)}^{\mu}\;\;,\;\;S^{\mu}=\frac{1}{2}\Big{(}N_{(2)}^{\mu}-N_{(1)}^{\mu}\Big{)}$
(88)

Then, upon using (5) along with the above relations and the definition of the
Palatini tensor

$P_{\lambda}^{\;\;\;\mu\nu}=-\frac{\nabla_{\lambda}(\sqrt{-g}g^{\mu\nu})}{\sqrt{-g}}+\frac{\nabla_{\sigma}(\sqrt{-g}g^{\mu\sigma})\delta^{\nu}_{\lambda}}{\sqrt{-g}}\\\
+2(S_{\lambda}g^{\mu\nu}-S^{\mu}\delta_{\lambda}^{\nu}+g^{\mu\sigma}S_{\sigma\lambda}^{\;\;\;\;\nu})$
(89)

we can express the latter solely in terms of the distortion as

$P^{\alpha\mu\nu}=g^{\mu\nu}N_{(2)}^{\alpha}+g^{\nu\alpha}N_{(3)}^{\mu}-(N^{\mu\nu\alpha}+N^{\nu\alpha\mu})$
(90)

###  B.2 Post-Riemannian expansions

The Ricci tensor is expanded as

$R_{\nu\beta}=\tilde{R}_{\nu\beta}+\tilde{\nabla}_{\mu}N^{\mu}_{\;\;\;\nu\beta}-\tilde{\nabla}_{\beta}N^{(2)}_{\nu}+N^{(2)}_{\lambda}N^{\lambda}_{\;\;\;\nu\beta}-N^{\mu}_{\;\;\;\rho\beta}N^{\rho}_{\;\;\;\nu\mu}$
(91)

and the Ricci scalar as

$R=\tilde{R}+\tilde{\nabla}_{\mu}(N^{(3)\mu}-N^{(2)\mu})+N^{(3)}_{\mu}N^{(2)\mu}-N_{\alpha\mu\nu}N^{\mu\nu\alpha}$
(92)

As a result for the Einstein-like combination we find

$\displaystyle
R_{(\mu\nu)}-\frac{R}{2}g_{\mu\nu}=\Big{(}\tilde{R}_{(\mu\nu)}-\frac{\tilde{R}}{2}g_{\mu\nu}\Big{)}+\tilde{\nabla}_{\alpha}N^{\alpha}_{\;\;\;(\mu\nu)}-\tilde{\nabla}_{(\nu}N^{(2)}_{\mu)}+N^{(2)}_{\lambda}N^{\lambda}_{\;\;\;(\mu\nu)}-N^{\alpha}_{\;\;\;\lambda(\nu}N^{\lambda}_{\;\;\;\mu)\alpha}$
$\displaystyle-\frac{1}{2}g_{\mu\nu}\Big{[}\tilde{\nabla}_{\alpha}(N^{(3)\alpha}-N^{(2)\alpha})+N^{(3)}_{\alpha}N^{(2)\alpha}-N_{\alpha\beta\gamma}N^{\beta\gamma\alpha}\Big{]}$
(93)

Continuing we calculate

$\displaystyle
b_{1}S_{\alpha\mu\nu}S^{\alpha\mu\nu}+b_{2}S_{\alpha\mu\nu}S^{\mu\nu\alpha}+a_{1}Q_{\alpha\mu\nu}Q^{\alpha\mu\nu}+a_{2}Q_{\alpha\mu\nu}Q^{\mu\nu\alpha}++c_{1}Q_{\alpha\mu\nu}S^{\alpha\mu\nu}=$
$\displaystyle\Big{(}\frac{b_{1}}{2}+2a_{1}-c_{1}\Big{)}N_{\alpha\mu\nu}N^{\alpha\mu\nu}+\Big{(}-\frac{b_{1}}{2}+a_{2}+c_{1}\Big{)}N_{\alpha\mu\nu}N^{\alpha\nu\mu}$
$\displaystyle+\Big{(}\frac{b_{2}}{2}+2a_{2}+c_{1}\Big{)}N_{\alpha\mu\nu}N^{\mu\nu\alpha}+\Big{(}-\frac{b_{2}}{2}+2a_{1}+a_{2}-c_{1}\Big{)}N_{\mu\nu\alpha}N^{\nu\mu\alpha}$
(94)

and also

$\displaystyle
b_{3}S_{\mu}S^{\mu}+a_{3}Q_{\mu}Q^{\mu}+a_{4}q_{\mu}q^{\mu}+a_{5}Q_{\mu}q^{\mu}+c_{2}Q_{\mu}S^{\mu}+c_{3}q_{\mu}S^{\mu}=$
$\displaystyle=\Big{(}\frac{b_{3}}{4}+4a_{3}-c_{2}\Big{)}N^{(1)}_{\mu}N^{(1)}_{\nu}g^{\mu\nu}+\Big{(}\frac{b_{3}}{4}+a_{4}+\frac{c_{3}}{4}\Big{)}N^{(2)}_{\mu}N^{(2)}_{\nu}g^{\mu\nu}+a_{4}N^{(3)}_{\mu}N^{(3)}_{\nu}g^{\mu\nu}$
$\displaystyle+\Big{(}2a_{5}-\frac{c_{3}}{2}+c_{2}-\frac{b_{3}}{2}\Big{)}N^{(1)}_{\mu}N^{(2)}_{\nu}g^{\mu\nu}+\Big{(}2a_{4}+\frac{c_{3}}{2}\Big{)}N^{(2)}_{\mu}N^{(3)}_{\nu}g^{\mu\nu}+\Big{(}2a_{5}-\frac{c_{3}}{2}\Big{)}N^{(1)}_{\mu}N^{(3)}_{\nu}g^{\mu\nu}$
(95)

such that

$\displaystyle\mathcal{L}_{2}=\Big{(}\frac{b_{1}}{2}+2a_{1}-c_{1}\Big{)}N_{\alpha\mu\nu}N^{\alpha\mu\nu}+\Big{(}-\frac{b_{1}}{2}+a_{2}+c_{1}\Big{)}N_{\alpha\mu\nu}N^{\alpha\nu\mu}$
$\displaystyle+\Big{(}\frac{b_{2}}{2}+2a_{2}+c_{1}\Big{)}N_{\alpha\mu\nu}N^{\mu\nu\alpha}+\Big{(}-\frac{b_{2}}{2}+2a_{1}+a_{2}-c_{1}\Big{)}N_{\mu\nu\alpha}N^{\nu\mu\alpha}$
$\displaystyle=\Big{(}\frac{b_{3}}{4}+4a_{3}-c_{2}\Big{)}N^{(1)}_{\mu}N^{(1)}_{\nu}g^{\mu\nu}+\Big{(}\frac{b_{3}}{4}+a_{4}+\frac{c_{3}}{4}\Big{)}N^{(2)}_{\mu}N^{(2)}_{\nu}g^{\mu\nu}+a_{4}N^{(3)}_{\mu}N^{(3)}_{\nu}g^{\mu\nu}$
$\displaystyle+\Big{(}2a_{5}-\frac{c_{3}}{2}+c_{2}-\frac{b_{3}}{2}\Big{)}N^{(1)}_{\mu}N^{(2)}_{\nu}g^{\mu\nu}+\Big{(}2a_{4}+\frac{c_{3}}{2}\Big{)}N^{(2)}_{\mu}N^{(3)}_{\nu}g^{\mu\nu}+\Big{(}2a_{5}-\frac{c_{3}}{2}\Big{)}N^{(1)}_{\mu}N^{(3)}_{\nu}g^{\mu\nu}$
(96)

Defining

$W^{\alpha}_{\;\;(\mu\nu)}=2a_{1}Q^{\alpha}_{\;\;\mu\nu}+2a_{2}Q_{(\mu\nu)}^{\;\;\;\;\alpha}+(2a_{3}Q^{\alpha}+a_{5}q^{\alpha})g_{\mu\nu}+(2a_{4}q_{(\mu}+a_{5}Q_{(\mu})\delta^{\alpha}_{\nu)}$
(97)
$\Pi^{\alpha\mu\nu}=c_{1}S^{\alpha\mu\nu}+c_{2}g^{\mu\nu}S^{\alpha}+c_{3}g^{\alpha\mu}S^{\nu}$
(98)
$A_{\mu\nu}=a_{1}(Q_{\mu\alpha\beta}Q_{\nu}^{\;\;\alpha\beta}-2Q_{\alpha\beta\mu}Q^{\alpha\beta}_{\;\;\;\;\nu})-a_{2}Q_{\alpha\beta(\mu}Q^{\beta\alpha}_{\;\;\;\;\nu)}+a_{3}(Q_{\mu}Q_{\nu}-2Q^{\alpha}Q_{\alpha\mu\nu})-a_{4}q_{\mu}q_{\nu}-a_{5}q^{\alpha}Q_{\alpha\mu\nu}$
(99)
$B_{\mu\nu}=b_{1}(2S_{\nu\alpha\beta}S_{\mu}^{\;\;\;\alpha\beta}-S_{\alpha\beta\mu}S^{\alpha\beta}_{\;\;\;\;\nu})-b_{2}S_{\nu\alpha\beta}S_{\mu}^{\;\;\;\beta\alpha}+b_{3}S_{\mu}S_{\nu}$
(100)
$C_{\mu\nu}=\Pi_{\mu\alpha\beta}Q_{\nu}^{\;\;\;\alpha\beta}-(c_{1}S_{\alpha\beta\nu}Q^{\alpha\beta}_{\;\;\;\;\mu}+c_{2}S^{\alpha}Q_{\alpha\mu\nu}+c_{3}S^{\alpha}Q_{\mu\nu\alpha})=c_{1}(Q_{\mu}^{\;\;\;\alpha\beta}S_{\nu\alpha\beta}-S_{\alpha\beta\mu}Q^{\alpha\beta}_{\;\;\;\;\nu})+c_{2}(S_{\mu}Q_{\nu}-S^{\alpha}Q_{\alpha\mu\nu})$
(101)

we calculate

$\displaystyle
W^{\alpha}_{\;\;\;(\mu\nu)}+\Pi^{\alpha}_{\;\;\;(\mu\nu)}=\Big{(}4a_{1}-\frac{c_{1}}{2}\Big{)}N_{(\mu\nu)\alpha}+\Big{(}2a_{2}+\frac{c_{1}}{2}\Big{)}N_{(\mu|\alpha|\nu)}+2a_{2}N_{\alpha(\mu\nu)}$
$\displaystyle\left[\Big{(}4a_{3}-\frac{c_{2}}{2}\Big{)}N_{\alpha}^{(1)}+\Big{(}a_{5}+\frac{c_{2}}{2}\Big{)}N_{\alpha}^{(2)}+a_{5}N_{\alpha}^{(3)}\right]g_{\mu\nu}$
$\displaystyle+g_{\alpha(\mu}\left[\Big{(}2a_{5}-\frac{c_{3}}{2}\Big{)}N_{\nu)}^{(1)}+\Big{(}2a_{4}+\frac{c_{3}}{2}\Big{)}N_{\nu)}^{(2)}+2a_{4}N_{\nu)}^{(3)}\right]$
(102)

as well as

$\displaystyle
A_{(\mu\nu)}+B_{(\mu\nu)}+C_{(\mu\nu)}=Q_{(\mu}^{\;\;\;\;\alpha\beta}\Big{(}a_{1}Q_{\nu)\alpha\beta}+c_{1}S_{\nu)\alpha\beta}\Big{)}-\Big{(}2a_{1}Q_{\alpha\beta(\mu}+c_{1}S_{\alpha\beta(\mu}+a_{2}Q_{\beta\alpha(\mu}\Big{)}Q^{\alpha\beta}_{\;\;\;\;\nu)}$
$\displaystyle+\Big{(}2b_{1}S_{(\mu}^{\;\;\;\alpha\beta}-b_{2}S_{(\mu}^{\;\;\;\beta\alpha}\Big{)}S_{\nu)\alpha\beta}-b_{1}S_{\alpha\beta\mu}S^{\alpha\beta}_{\;\;\;\nu}-Q_{\alpha\mu\nu}\Big{(}2a_{3}Q^{\alpha}+a_{5}q^{\alpha}+c_{2}S^{\alpha}\Big{)}$
$\displaystyle+a_{3}Q_{\mu}Q_{\nu}-a_{4}q_{\mu}q_{\nu}+b_{3}S_{\mu}S_{\nu}+c_{2}S_{(\mu}Q_{\nu)}=$
$\displaystyle=\Big{(}2a_{1}-\frac{c_{1}}{2}\Big{)}N^{\alpha\beta}_{\;\;\;\;\mu}N_{\alpha\beta\nu}+\Big{(}2a_{1}-\frac{c_{1}}{2}\Big{)}N^{\alpha\beta}_{\;\;\;\;\mu}N_{\beta\alpha\nu}+\frac{c_{1}}{2}N^{\alpha\beta}_{\;\;\;\;(\mu}(N_{\beta|\nu)\alpha}+N_{\alpha|\nu)\beta})$
$\displaystyle-(N^{\alpha\;\;\beta}_{\;\;(\nu}+N_{(\nu}^{\;\;\alpha\beta})\left[2a_{1}N_{\alpha|\mu)\beta}+a_{2}N_{\beta|\mu)\alpha}+\Big{(}2a_{1}-\frac{c_{1}}{2}\Big{)}N_{\mu)\alpha\beta}+\Big{(}a_{2}+\frac{c_{1}}{2}\Big{)}N_{\mu)\beta\alpha}\right]$
$\displaystyle+\frac{1}{4}(N_{\beta(\nu|\alpha}-N_{\beta\alpha(\nu})\Big{(}2b_{1}(N^{\beta\;\;\alpha}_{\;\;\mu)}-b_{2}N^{\alpha\;\;\beta}_{\;\;\mu)}-2b_{1}N^{\beta\alpha}_{\;\;\;\mu)}+b_{2}N^{\alpha\beta}_{\;\;\;\mu)}\Big{)}-b_{1}N_{\mu\alpha\beta}N_{\nu}^{\;\;[\alpha\beta]}$
$\displaystyle-(N_{\mu\nu\alpha}+N_{\nu\mu\alpha})\left[\Big{(}4a_{3}-\frac{c_{2}}{2}\Big{)}N^{(1)\alpha}+\Big{(}a_{5}+\frac{c_{2}}{2}\Big{)}N^{(2)\alpha}+a_{5}N^{(3)\alpha}\right]$
$\displaystyle+\Big{(}4a_{3}-\frac{c_{3}}{2}\Big{)}N_{\mu}^{(1)}N_{\nu}^{(1)}+\Big{(}\frac{b_{3}}{4}-a_{4}\Big{)}N_{\mu}^{(2)}N_{\nu}^{(2)}-a_{4}N_{\mu}^{(3)}N_{\nu}^{(3)}$
$\displaystyle+\Big{(}c_{3}-\frac{b_{3}}{2}\Big{)}N_{(\mu}^{(1)}N_{\nu)}^{(2)}-2a_{4}N_{(\mu}^{(2)}N_{\nu)}^{(3)}$
(103)

We now turn our attention to the parity odd quadratic part. In terms of the
distortion, the latter is expressed as

$\displaystyle\mathcal{L}_{odd}^{(2)}=a_{6}(N_{\alpha\mu\beta}+N_{\mu\alpha\beta})N_{\gamma\;\;\;\delta}^{\;\;\mu}\varepsilon^{\alpha\beta\gamma\delta}+(a_{6}-c_{6})N_{\alpha\mu\beta}N^{\mu}_{\;\;\gamma\delta}\varepsilon^{\alpha\beta\gamma\delta}+(a_{6}+b_{6}-c_{6})N_{\mu\alpha\beta}N^{\mu}_{\;\;\gamma\delta}\varepsilon^{\alpha\beta\gamma\delta}$
$\displaystyle+\left[\Big{(}2c_{4}-\frac{b_{5}}{2}\Big{)}N_{\mu}^{(1)}+\Big{(}\frac{b_{5}}{2}+c_{5}\Big{)}N_{\mu}^{(2)}+c_{5}N_{\mu}^{(3)}\right]M^{\mu}$
(104)

Furthermore, each of its metric-variation terms (as they appear in()) can be
written as

$\varepsilon^{\alpha\beta\gamma\delta}S_{\alpha\beta\nu}S_{\gamma\delta\mu}=\varepsilon^{\alpha\beta\gamma\delta}N_{\alpha\beta\nu}N_{\gamma\delta\mu}$
(105)
$\varepsilon^{\alpha\beta\gamma\delta}Q_{\alpha\beta\mu}Q_{\gamma\delta\nu}=2\varepsilon^{\alpha\beta\gamma\delta}N_{(\alpha\mu)\beta}(N_{\gamma\nu\delta}+N_{\nu\gamma\delta})$
(106)
$\varepsilon^{\alpha\beta\gamma\delta}(2a_{6}Q_{\gamma\delta}^{\;\;\;\;\lambda}+c_{6}S_{\gamma\delta}^{\;\;\;\;\lambda})=(c_{6}-2a_{6})\varepsilon^{\alpha\beta\gamma\delta}N^{\lambda}_{\;\;\gamma\delta}-2a_{6}\varepsilon^{\alpha\beta\gamma\delta}N_{\gamma\;\;\;\;\delta}^{\;\;\lambda}$
(107)
$S_{\alpha\beta(\mu}\varepsilon^{\lambda\alpha\beta}_{\;\;\;\;\;\;\nu)}(b_{5}S_{\lambda}+c_{4}Q_{\lambda}+c_{5}q_{\lambda})=\varepsilon^{\lambda\alpha\beta}_{\;\;\;\;\;\;(\nu}N_{\mu)\alpha\beta}\left[\Big{(}2c_{4}-\frac{b_{5}}{2}\Big{)}N^{(1)}_{\lambda}+\Big{(}c_{5}+\frac{b_{5}}{2}\Big{)}N^{(2)}_{\lambda}+c_{5}N^{(3)}_{\lambda}\right]$
(108)
$S_{(\mu}t_{\nu)}=\frac{1}{2}\Big{(}N^{(2)}_{(\mu}-N^{(1)}_{(\mu}\Big{)}M_{\nu)}$
(109)
$t^{\alpha}Q_{(\mu\nu)\alpha}=\frac{1}{2}\Big{(}N_{\nu\alpha\mu}+N_{\alpha\nu\mu}+N_{\mu\alpha\nu}+N_{\alpha\mu\nu}\Big{)}M^{\alpha}$
(110)

With these we can also re-express (73) solely in terms of distortion
as141414Also recalling that $t^{\alpha}=M^{\alpha}=N_{(4)}^{\alpha}$.

$\displaystyle\frac{1}{\sqrt{-g}}\frac{\Big{(}\sqrt{-g}\mathcal{L}^{(2)}_{odd}\Big{)}}{\delta
g^{\mu\nu}}=-b_{6}\varepsilon^{\alpha\beta\gamma\delta}N_{\alpha\beta\nu}N_{\gamma\delta\mu}+2a_{6}\varepsilon^{\alpha\beta\gamma\delta}N_{(\alpha\mu)\beta}(N_{\gamma\nu\delta}+N_{\nu\gamma\delta})$
$\displaystyle-
g_{\beta(\mu}g_{\nu)\lambda}\frac{1}{\sqrt{-g}}(\nabla_{\alpha}-2S_{\alpha})\Big{(}\sqrt{-g}(c_{6}-2a_{6})\varepsilon^{\alpha\beta\gamma\delta}N^{\lambda}_{\;\;\gamma\delta}-2a_{6}\sqrt{-g}\varepsilon^{\alpha\beta\gamma\delta}N_{\gamma\;\;\;\;\delta}^{\;\;\lambda}\Big{)}$
$\displaystyle-\varepsilon^{\lambda\alpha\beta}_{\;\;\;\;\;\;(\nu}N_{\mu)\alpha\beta}\left[\Big{(}2c_{4}-\frac{b_{5}}{2}\Big{)}N^{(1)}_{\lambda}+\Big{(}c_{5}+\frac{b_{5}}{2}\Big{)}N^{(2)}_{\lambda}+c_{5}N^{(3)}_{\lambda}\right]-c_{4}g_{\mu\nu}\Big{(}\partial_{\lambda}(\sqrt{-g}M^{\lambda})\Big{)}$
$\displaystyle-
c_{5}g_{\alpha(\mu}g_{\nu)\beta}\frac{1}{\sqrt{-g}}\nabla_{\lambda}(\sqrt{-g}M^{\alpha}g^{\lambda\beta})+c_{5}\Big{(}N^{(2)}_{(\mu}-N^{(1)}_{(\mu}\Big{)}M_{\nu)}+\frac{c_{5}}{2}\Big{(}N_{\nu\alpha\mu}+N_{\alpha\nu\mu}+N_{\mu\alpha\nu}+N_{\alpha\mu\nu}\Big{)}M^{\alpha}$
(111)

As for the combinations that appear on the connection field equations we
readily find

$\varepsilon^{\mu\nu\alpha\beta}Q_{\alpha\beta\lambda}=-\varepsilon^{\mu\nu\alpha\beta}N_{\alpha\lambda\beta}-\varepsilon^{\mu\nu\alpha\beta}N_{\lambda\alpha\beta}$
(112)
$\varepsilon^{\mu\nu\alpha\beta}S_{\alpha\beta\lambda}=\varepsilon^{\mu\nu\alpha\beta}N_{\alpha\beta\lambda}$
(113)
$\varepsilon_{\lambda}^{\;\;\nu\alpha\beta}Q_{\alpha\beta}^{\;\;\;\;\mu}=-\varepsilon_{\lambda}^{\;\;\nu\alpha\beta}\Big{(}N_{\alpha\;\;\;\;\beta}^{\;\;\mu}+N^{\mu}_{\;\;\alpha\beta}\Big{)}$
(114)
$\varepsilon_{\lambda}^{\;\;\nu\alpha\beta}S_{\alpha\beta}^{\;\;\;\;\mu}=\varepsilon_{\lambda}^{\;\;\nu\alpha\beta}N^{\mu}_{\;\;\alpha\beta}$
(115)
$(b_{5}S_{\lambda}+c_{4}Q_{\lambda}+c_{5}q_{\lambda})=\left[\Big{(}2c_{4}-\frac{b_{5}}{2}\Big{)}N^{(1)}_{\lambda}+\Big{(}c_{5}+\frac{b_{5}}{2}\Big{)}N^{(2)}_{\lambda}+c_{5}N^{(3)}_{\lambda}\right]$
(116)

and as a result

$\displaystyle\delta_{\Gamma}\mathcal{L}_{odd}=\delta\Gamma^{\lambda}_{\;\;\;\mu\nu}\Big{[}-(-2a_{6}+c_{6})\varepsilon^{\mu\nu\alpha\beta}N_{\alpha\lambda\beta}-(-2a_{6}+c_{6})\varepsilon^{\mu\nu\alpha\beta}N_{\lambda\alpha\beta}+(2b_{6}-c_{6})\varepsilon^{\mu\nu\alpha\beta}N_{\alpha\beta\lambda}$
$\displaystyle+2a_{6}\varepsilon_{\lambda}^{\;\;\nu\alpha\beta}\Big{(}N_{\alpha\;\;\;\;\beta}^{\;\;\mu}+N^{\mu}_{\;\;\alpha\beta}\Big{)}-c_{6}\varepsilon_{\lambda}^{\;\;\nu\alpha\beta}N^{\mu}_{\;\;\alpha\beta}+\varepsilon^{\alpha\mu\nu}_{\;\;\;\;\;\;\lambda}\Big{(}2c_{4}-\frac{b_{5}}{2}\Big{)}N^{(1)}_{\alpha}$
$\displaystyle+\Big{(}c_{5}+\frac{b_{5}}{2}\Big{)}\varepsilon^{\alpha\mu\nu}_{\;\;\;\;\;\;\lambda}N^{(2)}_{\alpha}+c_{5}\varepsilon^{\alpha\mu\nu}_{\;\;\;\;\;\;\lambda}N^{(3)}_{\alpha}+\Big{(}\frac{b_{5}}{2}+c_{5}\Big{)}M^{\mu}\delta_{\lambda}^{\nu}+\Big{(}-\frac{b_{5}}{2}+2c_{4}\Big{)}M^{\nu}\delta_{\lambda}^{\mu}+c_{5}g^{\mu\nu}M_{\lambda}\Big{]}$
(117)

##  Appendix C The $\gamma_{ij}$’s and $\alpha_{ij}$’s

###  C.1 The $\gamma_{ij}$’s

The relations between the elements of $\Gamma$ and the $30$ initial parameters
read

$\displaystyle\gamma_{11}=a_{1}+a_{3}+a_{71}+4a_{81}+a_{91}\;\;,\;\;\gamma_{12}=a_{2}+a_{4}+a_{72}+4a_{82}+a_{92}\;\;,\;\;\gamma_{13}=a_{5}+a_{6}+a_{73}+4a_{83}+a_{93}$
$\displaystyle\gamma_{21}=a_{2}+a_{5}+4a_{71}+a_{81}+a_{91}\;\;,\;\;\gamma_{22}=a_{1}+a_{6}+4a_{72}+a_{82}+a_{92}\;\;,\;\;\gamma_{23}=a_{3}+a_{4}+4a_{73}+a_{83}+a_{93}$
$\displaystyle\gamma_{31}=a_{5}+a_{6}+a_{71}+a_{81}+4a_{91}\;\;,\;\;\gamma_{32}=a_{3}+a_{4}+a_{72}+a_{82}+4a_{92}\;\;,\;\;\gamma_{31}=a_{1}+a_{2}+a_{73}+a_{83}+4a_{93}$
$\displaystyle\gamma_{41}=-2(-1)^{s}\Big{(}b_{21}+b_{22}+b_{23}+b_{31}+b_{32}+b_{33}-3b_{1}\Big{)}\;\;\;,\;\;\;\;\gamma_{42}=-2(-1)^{s}\Big{(}b_{11}+b_{12}+b_{13}-b_{31}-b_{32}-b_{33}-3b_{2}\Big{)}$
$\displaystyle\gamma_{43}=2(-1)^{s}\Big{(}b_{11}+b_{12}+b_{13}+b_{21}+b_{22}+b_{23}+3b_{3}\Big{)}\;\;\;,\;\;\;\;\gamma_{44}=a_{1}+a_{2}+a_{3}-a_{4}-a_{5}-a_{6}$
$\displaystyle\gamma_{14}=c_{1}+4c_{2}+c_{3}-b_{12}-b_{13}-b_{21}-b_{23}-b_{31}-b_{33}\;\;,\;\;\gamma_{24}=4c_{1}+c_{2}+c_{3}+b_{11}+b_{22}-b_{12}-b_{21}-b_{31}-b_{32}$
$\displaystyle\gamma_{34}=c_{1}+c_{2}+4c_{3}+b_{12}-b_{13}-b_{22}+b_{23}+b_{32}-b_{33}$
(118)

These are the elements of the $4\times 4$ matrix $\Gamma$.

###  C.2 The $\alpha_{ij}$’s

The coefficients appearing in $(\ref{MasterN})$ are the first row elements of
$A$, namely

$\displaystyle\alpha_{11}=a_{1}\;\;,\;\;\alpha_{12}=a_{2}\;\;,\;\;\alpha_{13}=a_{3}\;\;,\;\;\alpha_{14}=a_{4}\;\;,\;\;\alpha_{15}=a_{5}\;\;,\;\;\alpha_{16}=a_{6}\;\;,\;\;\alpha_{17}=b_{11}\;\;,\;\;\alpha_{18}=b_{12},\;\;\alpha_{19}=b_{13}$
$\displaystyle\alpha_{1,10}=b_{21}\;\;,\;\;\alpha_{1,11}=b_{22},\;\;\alpha_{1,12}=b_{23}\;\;,\;\;\alpha_{1,13}=b_{31}\;\;,\;\;\alpha_{1,14}=b_{32}\;\;,\;\;\alpha_{1,15}=b_{33}$
(119)

the same goes for every other row with the last one being

$\displaystyle\alpha_{15,1}=(-1)^{s}(2b_{11}-b_{12}-b_{13})\;\;,\;\;\alpha_{15,2}=(-1)^{s}(2b_{31}-b_{32}-b_{33})\;\;,\;\;\alpha_{15,3}=-(-1)^{s}(2b_{21}-b_{22}-b_{23})$
$\displaystyle\alpha_{15,4}=(-1)^{s}(2b_{21}-b_{22}-b_{23})\;\;,\;\;\alpha_{15,5}=-(-1)^{s}(2b_{31}-b_{32}-b_{33})\;\;,\;\;\alpha_{15,6}=-(-1)^{s}(2b_{11}-b_{12}-b_{13})$
$\displaystyle\alpha_{15,7}=0\;\;,\;\;\alpha_{15,8}=a_{3}-a_{5}\;\;,\;\;\alpha_{15,9}=0\;\;,\;\;\alpha_{15,10}=0\;\;,\;\;\alpha_{15,11}=a_{6}-a_{2}\;\;,\;\;\alpha_{15,12}=0$
$\displaystyle\alpha_{15,3}=0\;\;,\;\;\alpha_{15,14}=a_{1}-a_{4}\;\;,\;\;\alpha_{15,15}=0$
(120)

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