monthyeardate[]

# Universal Gibbons-Hawking-York term for
theories with curvature, torsion and non-metricity

Johanna Erdmenger  Institute for Theoretical Physics and Astrophysics, Julius-
Maximilians-Universität Würzburg, Am Hubland, D-97074 Würzburg, Germany
Würzburg-Dresden Cluster of Excellence on Complexity and Topology in Quantum
Matter ct.qmat  Bastian Heß  Institute for Theoretical Physics and
Astrophysics, Julius-Maximilians-Universität Würzburg, Am Hubland, D-97074
Würzburg, Germany  Würzburg-Dresden Cluster of Excellence on Complexity and
Topology in Quantum Matter ct.qmat  Ioannis Matthaiakakis Corresponding
author<EMAIL_ADDRESS>Dipartimento di Fisica, Università
di Genova, via Dodecaneso 33, I-16146, Genova, Italy  I.N.F.N. - Sezione di
Genova, via Dodecaneso 33, I-16146, Genova, Italy  Institute for Theoretical
Physics and Astrophysics, Julius-Maximilians-Universität Würzburg, Am Hubland,
D-97074 Würzburg, Germany  Würzburg-Dresden Cluster of Excellence on
Complexity and Topology in Quantum Matter ct.qmat  René Meyer  Institute for
Theoretical Physics and Astrophysics, Julius-Maximilians-Universität Würzburg,
Am Hubland, D-97074 Würzburg, Germany  Würzburg-Dresden Cluster of Excellence
on Complexity and Topology in Quantum Matter ct.qmat

###### Abstract

Motivated by establishing holographic renormalization for gravitational
theories with non-metricity and torsion, we present a new and efficient
general method for calculating Gibbons-Hawking-York (GHY) terms. Our method
consists of linearizing any nonlinearity in curvature, torsion or non-
metricity by introducing suitable Lagrange multipliers. Moreover, we use a
split formalism for differential forms, writing them in $(n-1)+1$ dimensions.
The boundary terms of the action are manifest in this formalism by means of
Stokes’ theorem, such that the compensating GHY term for the Dirichlet problem
may be read off directly. We observe that only those terms in the Lagrangian
that contain curvature contribute to the GHY term. Terms polynomial solely in
torsion and non-metricity do not require any GHY term compensation for the
variational problem to be well-defined. We test our method by confirming
existing results for Einstein-Hilbert and four-dimensional Chern-Simons
modified gravity. Moreover, we obtain new results for Lovelock-Chern-Simons
and metric-affine gravity. For all four examples, our new method and results
contribute to a new approach towards a systematic hydrodynamic expansion for
spin and hypermomentum currents within AdS/CFT.

toc.sectiontoc.sectionContentsContentstoc.section

###### Contents

  1. 1 Introduction
  2. 2 Geometric setup and summary of the main results
  3. 3 Examples for Gibbons-Hawking-York terms
    1. 3.1 Einstein-Hilbert gravity $\mathcal{L}\propto R$
    2. 3.2 4d Chern-Simons modified gravity $\mathcal{L}\propto\Omega^{2}$
    3. 3.3 Lovelock-Chern-Simons gravity
  4. 4 Derivation of the Gibbons-Hawking-York term
    1. 4.1 Frame decomposition
    2. 4.2 Adapted frame decomposition
    3. 4.3 Decomposition of metric and connection
    4. 4.4 Foundations of the 3+1 formalism in general frames
    5. 4.5 Decomposition of the field strengths
    6. 4.6 Post-Riemannian Gibbons-Hawking-York term
      1. 4.6.1 Gibbons-Hawking-York term for metric compatible theories
  5. 5 Summary and discussion
  6. 6 Acknowledgments
  7. A GHY term for metric-affine gravity (MAG)
    1. A.1 Variation of Hodge stars and interior products
    2. A.2 Variation of the MAG Lagrangian
    3. A.3 GHY term for metric compatible MAG
  8. B GHY term for metric compatible theories

##  1 Introduction

Curved Riemannian and pseudo-Riemannian spacetimes are of immediate relevance
for astrophysics, cosmology and high energy physics. In most cases, the
spacetime metric is the only dynamical field associated to the geometry, while
further geometric structures are argued to be irrelevant by appealing to
current experimental evidence. Recently, however, general spacetimes with non-
vanishing curvature111$\Omega\indices{{}^{\mu}_{\nu}}$ is the curvature two-
form, whose components are the Riemann tensor
$R\indices{{}^{\mu}_{\nu}{}_{\rho}{}_{\sigma}}$.
$\Omega\indices{{}^{\mu}_{\nu}}$ and torsion $T^{\mu}$ found applications in
condensed matter systems—where lattice deformations generate non-trivial
$\Omega\indices{{}^{\mu}_{\nu}}$ and $T^{\mu}$ directly coupled to the
electronic degrees of freedom—as a means to simulate high-energy phenomena in
tabletop experiments [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16].
In particular, torsion has been shown to alter the the hydrodynamic expansion
by introducing terms relevant for both heavy-ion physics and electron flows in
condensed matter [15, 17, 18, 19, 16, 20, 21].

We may, however, go even further beyond spacetimes with curvature and torsion
to the more general metric-affine spacetimes hosting a non-trivial non-
metricity one-form $Q_{\mu\nu}$. In condensed matter systems, this may be
achieved by the introduction of dislocations and cuts in the lattice of a
given material [22]. For our purposes, the importance of considering metric-
affine spacetimes arises from the coupling of torsion and non-metricity to
matter fields. Both torsion and non-metricity may be seen as the sources of
conserved matter currents. In particular, $T^{\mu}$ is the source of the spin
tensor $S_{\mu}^{\leavevmode\nobreak\ \nu\rho}$, while $Q_{\mu\nu}$ is the
source of the hypermomentum tensor $\Delta_{\rho\mu\nu}$ [23]. The spin tensor
is well-known and has been studied in depth: In particular, spin transport was
recently suggested to play an important role in hadron-hadron collisions at
LHC [24, 25, 18], as well as in tabletop experiments on electronic systems
[26, 27]. In contrast, the hypermomentum tensor is not widely studied in the
physics literature. The trace of this tensor,
$J^{D}_{\mu}=\Delta\indices{{}_{\mu}{}_{\nu}^{\nu}}$, however, is referred to
as dilation current and has found applications in the context of trace
anomalies in condensed matter systems [3].

The source interpretation of torsion and non-metricity is the most relevant in
the present context, due to its role in the AdS/CFT correspondence [28, 29,
30]. The AdS/CFT correspondence or holography is a duality between a weakly
curved theory of gravity and a strongly coupled quantum field theory (QFT)
living in one dimension lower, at the boundary of the curved spacetime. The
sources coupled to the QFT determine the boundary conditions of the dynamical
fields of the gravitational theory, while the on-shell gravitational action
acts as the QFT’s generating functional. Holography has widely been used to
shed light on the transport properties of matter, for instance of the quark-
gluon plasma or of graphene [31, 32]. These transport properties, however,
concern only non-trivial energy-momentum and electric charge transfer in the
QFT, i.e. only a dynamical metric and gauge field in the gravitational theory.
It is our goal to extend the correspondence to include spin and hypermomentum
transport in strongly coupled systems. The former arises from the intrinsic
spin of particles, while the latter arises from the modification of the causal
structure of spacetime due to matter [33]. Causality may exclude hypermomentum
transport in relativistic systems of particle and astrophysics, since it may
modify the light-cone structure and turn spacelike separated events into
timelike separated ones under parallel transport. Condensed matter systems are
not bound to such constraints, since their causal structure arises only as an
emergent description of the system’s electronic bands. We therefore expect
hypermomentum to lead to new transport phenomena and perhaps novel phase
transitions in this context.

In order to extend holography to the realm of spin and hypermomentum
transport, the dual gravitational theory must contain dynamical torsion and
non-metricity tensors, i.e. we must consider dynamical theories of metric-
affine gravity (MAG) on spacetimes with boundary. The introduction of
boundaries into the spacetime considered raises issues of fundamental
importance that need to be addressed before applications can be considered. In
particular, we have to ensure that the boundary conditions of the MAG fields
are compatible with a well-defined variational problem. This is achieved in
general relativity by the inclusion of the Gibbons-Hawking-York (GHY) boundary
term.

The importance of the GHY boundary term in theories of gravity cannot be
overstated. In the simplest of applications of gravitational theories, in
terms of a Lagrangian, the GHY term makes the variational formulation well-
posed [34, 35, 36]. In addition, within the Hamiltonian formalism, the GHY
term allows us to correctly define the Hamiltonian as well as the asymptotic
conserved charges of the theory, such as energy [36]. The GHY terms also play
a crucial role within holography: First, the asymptotic charges evaluated at
the boundary are precisely the charges of the dual QFT. Second, the on-shell
gravitational action (and QFT generating functional) suffers from divergences.
The GHY terms act as counterterms and provide (part of) the regularization
necessary to define a finite gravitational action [37]. Thus, in order to use
holography to derive QFT observables and understand spin and hypermomentum
transport, we have to derive the GHY term for every MAG Lagrangian. We carry
out this derivation in the present paper.

The GHY term we find provides the starting point for holographic
renormalization [38] that we plan to address in the future in the present
context. Holographic renormalization in turn will be the starting point for a
fluid-gravity hydrodynamic expansion [39] including torsion and non-metricity.
Also beyond holography, our results are of interest to both the general
relativity and condensed matter communities. The first may use our results as
a stepping stone towards analyzing extensions of general relativity to the MAG
framework and their compatibility with cosmological data. For the second, in
addition to forthcoming results on spin hydrodynamics that we expect to be
obtainable based on the analysis presented here, our terms become important
when considering the dynamics of defects that give rise to torsion and non-
metricity on systems with boundaries. In this context, our GHY terms describe
the dynamics of the defects on the boundary which are consistent with the
dynamics in the bulk. Such terms may also become important when considering
topological field theories on spaces with boundaries, perhaps with quantum
anomalies, since they describe the emergent degrees of freedom at the
boundary.

The main result of this work is a new and efficient method for deriving GHY
terms for actions formulated in the language of differential forms. In
particular, our generalized GHY term applies for any theory which is allowed
to have curved, torsional and non-metric degrees of freedom in arbitrary
polynomial combinations. We achieve this generality by formulating our result
in terms of auxiliary fields which are calculated for generic theories by
taking variations of their Lagrangian $n$-form with respect to suitable
Lagrange multipliers. One of our main findings is that our generalized GHY
term receives contributions from the variation of only those terms in the
Lagrangian that contain curvature. Terms that are solely built from torsion
and non-metricity do not contribute a GHY term, assuming that any derivatives
of torsion and non-metricity have been converted into curvature polynomials by
means of the Bianchi identities. Using our method we confirm the GHY term for
the Einstein-Hilbert action in arbitrary dimensions. We furthermore verify a
result for 4d Chern-Simons modified gravity up to a factor of 2. Moreover, we
present new results for Lovelock-Chern-Simons and metric-affine gravity. These
new results have the form which we expect from comparison of the Lagrangians
with those of Einstein-Hilbert and 4d Chern-Simons modified gravity.

We begin the main part of this paper with section 2, where we briefly set up
the geometric framework used. This allows us to present the main results of
our work in a concise manner in the same section. Subsequently, we apply these
general results to the special cases of Einstein-Hilbert and four-dimensional
Chern-Simons modified gravity in section 3 to find familiar results as a check
of our method. In the same section, we apply our method to the case of
Lovelock-Chern-Simons gravity and derive new results for its GHY term. An
explicit derivation of our results is given in section 4. The more involved
case of MAG is considered in appendix A.

##  2 Geometric setup and summary of the main results

In the current section, we present our main result for the GHY term for any
MAG theory in (12). To fully grasp the meaning of each term in (12), however,
we must first give a lightning review of our geometric setup. For the
interested reader, more details are given in section 4.

As mentioned in the introduction, the geometry that we consider in a dynamical
setting is that of a metric-affine spacetime. This spacetime is an
$n$-dimensional manifold $\mathcal{M}$ equipped with a coframe basis
$\theta^{\mu}$, a metric
$\mathrm{d}s^{2}=g_{\mu\nu}\theta^{\mu}\otimes\theta^{\nu}$ and a connection
one-form $\omega^{\mu}_{\nu}=\Gamma^{\mu}_{\rho\nu}\theta^{\rho}$, where Greek
indices take values in $\left\\{0,\dots,n-1\right\\}$. These fields are
independent of each other and may be thought of as the kinematic variables of
the spacetime in the same sense that a U(1) gauge field provides the kinematic
variables for a U(1) gauge theory. Following this analogy further, we may
define field strengths for $\theta^{\mu}$, $g_{\mu\nu}$ and
$\omega^{\mu}_{\nu}$. To preserve diffeomorphism invariance, we construct
these field strengths in terms of the exterior derivative $\mathrm{d}$ and the
exterior covariant derivative
$D=\mathrm{d}+\omega^{\mu}_{\nu}\rho(L)^{\nu}_{\mu}\wedge$, where
$\rho(L)^{\nu}_{\mu}$ is the appropriate representation of the
GL(n,$\mathbb{R}$) generators $L$. In particular, the field strengths for the
fields $\omega^{\mu}_{\nu}$, $\theta^{\mu}$, $g_{\mu\nu}$ are

$\begin{aligned} &\text{the curvature two-form}\vphantom{\frac{1}{2}}\\\
&\text{the torsion two-form}\vphantom{\frac{1}{2}}\\\ &\text{the non-metricity
one-form}\vphantom{\frac{1}{2}}\end{aligned}\quad\begin{aligned}
&\Omega\indices{{}^{\mu}_{\nu}}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\mathrm{d}\omega^{\mu}_{\nu}+\omega^{\mu}_{\rho}\wedge\omega^{\rho}_{\nu}=\frac{1}{2}R\indices{{}^{\mu}_{\nu}{}_{\rho}{}_{\sigma}}\theta^{\rho}\wedge\theta^{\sigma}\,,\\\
&T^{\mu}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=D\theta^{\mu}=\mathrm{d}\theta^{\mu}+\omega^{\mu}_{\nu}\wedge\theta^{\nu}=\frac{1}{2}T\indices{{}^{\mu}_{\rho}{}_{\sigma}}\theta^{\rho}\wedge\theta^{\sigma}\,,\\\
&Q_{\mu\nu}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=-Dg_{\mu\nu}=-(\mathrm{d}g_{\mu\nu}-\omega^{\rho}_{\mu}g_{\rho\nu}-\omega^{\rho}_{\nu}g_{\mu\rho})=Q_{\mu\nu\rho}\theta^{\rho}\,,\vphantom{\frac{1}{2}}\end{aligned}$
(1)

respectively. The curvature $R\indices{{}^{\mu}_{\nu}{}_{\rho}{}_{\sigma}}$ is
the Riemann tensor, while $T\indices{{}^{\mu}_{\rho}{}_{\sigma}}$ is the
torsion and $Q_{\mu\nu\rho}$ the non-metricity tensor. All three of the fields
strengths satisfy a corresponding Bianchi identity

$\displaystyle\begin{split}D\Omega\indices{{}^{\mu}_{\nu}}&=0\,,\\\
DT\indices{{}^{\mu}_{\hphantom{\nu}}}&=\Omega\indices{{}^{\mu}_{\nu}}\wedge\theta^{\nu}\,,\\\
DQ_{\mu\nu}&=\Omega_{\mu\nu}+\Omega_{\nu\mu}\,.\end{split}$ (2)

We introduce a boundary $\partial\mathcal{M}$ in $\mathcal{M}$ and a geometry
on $\partial\mathcal{M}$ via two sets of vector fields. First, we define
$\partial\mathcal{M}$ through the normal vector $n^{\mu}$, normalized as
$n_{\mu}n^{\mu}=\varepsilon=-1$ for spacelike and
$n_{\mu}n^{\mu}=\varepsilon=+1$ for timelike boundaries. Second, we introduce
the vielbein $e^{\mu}_{a}$ as a basis of tangent vectors on
$\partial\mathcal{M}$, $n_{\mu}e^{\mu}_{a}=0$, where Latin indices take values
in $\left\\{0,\dots,n-2\right\\}$. The vielbein and its dual $e^{a}_{\mu}$
induce a geometry on $\partial\mathcal{M}$ via pullback from $\mathcal{M}$.
For example, the boundary coframe basis $\phi^{a}$ is obtained from the
manifold coframe basis $\theta^{\mu}$ as $e^{a}_{\mu}\theta^{\mu}=\phi^{a}$,
while the induced metric on $\partial\mathcal{M}$ is
$\gamma_{ab}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=e^{\mu}_{a\vphantom{b}}e^{\nu\vphantom{\mu}}_{b}g_{\mu\nu}$.
While the metric and coframe project to the boundary via pullback, the
connection is related to its boundary value $\omega^{a}_{b}$ in a more
involved way. In particular, we assume that $\omega^{\mu}_{\nu}$ projects to
$\omega^{a}_{b}$ via the vielbein postulate

$\displaystyle\omega^{b}_{a}=e^{b}_{\mu}\left(\mathrm{d}e^{\mu}_{a}+\omega^{\mu}_{\nu}e^{\nu}_{a}\right)\,.$
(3)

We prove the transformation property (3) within the MAG framework [23, 40] in
section 4.3. To our knowledge this is the most general theory that includes
curvature, torsion and non-metricity. It is thus natural to assume that (3)
holds for a large range of theories featuring fewer fields or more symmetries.

Apart from the curvature, torsion and non-metricity associated to the
projected connection, coframe and metric on the boundary, the embedding of
$\partial\mathcal{M}$ in $\mathcal{M}$ allows for additional quantities that
describe the geometry of $\partial\mathcal{M}$ relative to that of
$\mathcal{M}$. For example, in the following we use extensively the extrinsic
curvatures

$\displaystyle
K^{a}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=e^{a}_{\mu}Dn^{\mu}\qquad\text{and}\qquad\tilde{K}_{a}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=e^{\mu}_{a}Dn_{\mu}=K_{a}-e^{\mu}_{a}n^{\nu}Q_{\mu\nu}\,.$
(4)

The two definitions of the extrinsic curvature in (4) differ only by a
deformation in the $n-e_{a}$ plane due to non-metricity. Thus, for metric
compatible connections satisfying $Dg_{\mu\nu}=0$ it suffices to consider only
one of the extrinsic curvature definitions. The tensor components of $K^{a}$
tangent to the boundary give the extrinsic curvature familiar from the
literature, $K\indices{{}^{a}_{b}}=e^{a}_{\mu}e_{b}^{\nu}\nabla_{\nu}n^{\mu}$,
where $\nabla_{\nu}$ is the covariant derivative on $\mathcal{M}$ [41, 36,
42].

This concludes our brief review of the geometry on $\mathcal{M}$ and the
induced geometry on $\partial\mathcal{M}$. For investigating the dynamics of
this geometry, we consider the most general action possible,

$\displaystyle
S_{\mathrm{orig}}[g_{\mu\nu},\omega^{\mu}_{\nu},\theta^{\mu}]=\int_{\mathcal{M}}\mathcal{L}(\Omega\indices{{}^{\mu}_{\nu}},T^{\mu},Q_{\mu\nu})\,,$
(5)

where the Lagrangian $n$-form
$\mathcal{L}(\Omega\indices{{}^{\mu}_{\nu}},T^{\mu},Q_{\mu\nu})$ is an
arbitrary function of the curvature, torsion and non-metricity defined in (1).
Not fixing a particular form for $\mathcal{L}$ ensures that our results hold
for any action of interest and may be applied to a specific system by choosing
$\mathcal{L}$ appropriately. Note that we have restricted ourselves to actions
which are polynomial in the field strengths but do not involve their
derivatives. This is not a severe restriction since the Bianchi identities (2)
tell us that derivatives of curvature vanish, while those of torsion and non-
metricity can be reduced to polynomial terms in curvature.

Our goal is to make the variation of the action (5) well-defined by adding the
appropriate boundary terms to it. To find these terms we vary $S_{\rm orig}$
and isolate the terms on the boundary that do not vanish after enforcing the
boundary conditions on the fields. To be precise, the boundary conditions we
consider are $\left.\delta
g_{\mu\nu}\right|_{\mathrm{\partial\mathcal{M}}}=0$,
$\left.\delta\theta^{\mu}\right|_{\mathrm{\partial\mathcal{M}}}=0$ and
$\left.\delta\omega^{\mu}_{\nu}\right|_{\mathrm{\partial\mathcal{M}}}=0.$
However, according to [36] it suffices to demand that the Dirichlet boundary
conditions

$\displaystyle\delta\gamma_{ab}=0\,,\leavevmode\nobreak\ \leavevmode\nobreak\
\delta\phi^{a}=0\,,\leavevmode\nobreak\ \leavevmode\nobreak\
\delta\omega^{a}_{b}=0$ (6)

hold since the original conditions may be reinstated by gauge transformations
on $\partial\mathcal{M}$.

In principle, we may now obtain the boundary terms from a direct variation of
the action $S_{\rm orig}$. However, general covariance requires that we write
the boundary terms in terms of geometric quantities on $\partial\mathcal{M}$.
To achieve this we need to perform a $3+1$ decomposition222While we work in
general $n$ dimensions and perform a full $(n-1)+1$ decomposition, we use the
term $3+1$ decomposition for compactness throughout. of $S_{\rm orig}$ which
is impossible to write down if $\mathcal{L}$ is not specified. We circumvent
this issue by the method of Lagrange multipliers expounded in [43] for the
case of $\mathcal{L}(\Omega\indices{{}^{\mu}_{\nu}})$ gravity. This method
essentially makes the action linear in the field strengths without losing any
information of the dynamics induced by $\mathcal{L}$. In order to isolate the
boundary terms, we introduce the Lagrange multipliers
$\varphi\indices{{}_{\mu}^{\nu}}$, $t_{\mu}$ and $q^{\mu\nu}$ and turn the
Langrangian into a linear function of $\Omega\indices{{}^{\mu}_{\nu}}$,
$T^{\mu}$ and $Q_{\mu\nu}$. In particular, we consider the gravitational
action

$\displaystyle\begin{split}S[g_{\mu\nu},\omega^{\mu}_{\nu},\theta^{\mu},\varphi\indices{{}^{\mu}_{\nu}},\varrho\indices{{}^{\mu}_{\nu}},t_{\mu},\tau^{\mu},q^{\mu\nu},&\sigma_{\mu\nu}]\\\
=\int_{\mathcal{M}}&\left[\mathcal{L}(\varrho\indices{{}^{\mu}_{\nu}},\tau^{\mu},\sigma_{\mu\nu})+*\varphi\indices{{}_{\mu}^{\nu}}\wedge(\Omega\indices{{}^{\mu}_{\nu}}-\varrho\indices{{}^{\mu}_{\nu}})\right.\\\
&\left.\vphantom{*\varphi\indices{{}_{\mu}^{\nu}}}+*t_{\mu}\wedge\left(T^{\mu}-\tau^{\mu}\right)+*q^{\mu\nu}\wedge\left(Q_{\mu\nu}-\sigma_{\mu\nu}\right)\right]\,,\end{split}$
(7)

where $\varrho\indices{{}^{\mu}_{\nu}}$, $\tau^{\mu}$ and $\sigma_{\mu\nu}$
are auxiliary fields and $*$ is the Hodge duality. Expressing $S$ as in (7)
allows us to directly access $\Omega\indices{{}^{\mu}_{\nu}}$, $T^{\mu}$ and
$Q_{\mu\nu}$ regardless of the explicit form of $\mathcal{L}$. We choose the
Lagrange multipliers to be independent of each other and of the fields
$\omega^{\mu}_{\nu}$, $\theta^{\mu}$ and $g_{\mu\nu}$. In this way we ensure
that the equations of motion for $\Omega\indices{{}^{\mu}_{\nu}}$, $T^{\mu}$
and $Q_{\mu\nu}$ are kept unchanged if we first impose the equations of motion

$\displaystyle\Omega\indices{{}^{\mu}_{\nu}}=\varrho\indices{{}^{\mu}_{\nu}}\,,\qquad
T^{\mu}=\tau^{\mu}\,,\qquad Q_{\mu\nu}=\sigma_{\mu\nu}$ (8)

for the Lagrange multipliers $\varphi\indices{{}_{\mu}^{\nu}}$, $t_{\mu}$ and
$q^{\mu\nu}$, respectively. To that end, we additionally demand these Lagrange
multipliers to have the exact same symmetries as the corresponding field
strengths.

In order to express the boundary terms in terms of a diffeomorphic invariant
Lagrangian on the boundary, we consider projections of the Lagrange
multipliers on $\partial\mathcal{M}$. In particular, we project each index of
the Lagrange multipliers either to the boundary by contraction with the
vielbein $e^{a}_{\mu}$ or normal to the boundary by contraction with
$n^{\mu}$. We abbreviate these contractions as
$\varphi_{a\mathbf{n}}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=e_{a}^{\mu}n^{\nu}\varphi_{\mu\nu}$
for instance. These projections may be regarded as an extension of the $3+1$
decomposition in general relativity and can be derived by expanding every
contraction of Greek indices in (7) using

$\displaystyle\delta^{\mu}_{\nu}=e^{\mu}_{a}e^{a}_{\nu}+\varepsilon
n^{\mu}n_{\nu}\,.$ (9)

For example $A^{\mu}\wedge B_{\mu}=A^{a}\wedge B_{a}+\varepsilon
A_{\mathbf{n}}\wedge B_{\mathbf{n}}$ for generic differential forms $A^{\mu},$
$B_{\mu}$. The explicit form of the action $S$ in terms of the $3+1$
decomposition is rather lengthy and not that enlightening. So we leave (7)
unchanged and keep in mind that for further calculations any of the Greek
indices is $3+1$ decomposed as just described.

While the equations of motion (8) of the Lagrange multipliers
$\varphi\indices{{}_{\mu}^{\nu}}$, $t_{\mu}$ and $q^{\mu\nu}$ ensure the
equivalence of $S$ and $S_{\rm orig}$, the equations of motion of the
additional fields $\varrho\indices{{}^{\mu}_{\nu}}$, $\tau^{\mu}$ and
$\sigma_{\mu\nu}$ yield constraints which enable us to determine the explicit
forms of $\varphi\indices{{}_{\mu}^{\nu}},t_{\mu}$ and $q^{\mu\nu}$ in terms
of $\mathcal{L}$. Among these constraints the ones which yield
$*\varphi^{\mathbf{n}a}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=n^{\alpha}e_{\beta}^{a}*\varphi\indices{{}_{\alpha}^{\beta}}$,
$*\varphi_{a\mathbf{n}}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=e^{\alpha}_{a}n_{\beta}*\varphi\indices{{}_{\alpha}^{\beta}}$
and
$*\varphi_{\mathbf{n}\mathbf{n}}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=n^{\alpha}n_{\beta}*\varphi\indices{{}_{\alpha}^{\beta}}$
are particularly important for us. These are

$\displaystyle\begin{split}*\varphi^{\mathbf{n}a}\wedge\delta\varrho_{\mathbf{n}a}&=\varepsilon\delta_{\varrho_{\mathbf{n}a}}\mathcal{L}(\varrho_{\mathbf{n}a},\varrho^{a\mathbf{n}},\varrho_{\mathbf{n}\mathbf{n}},\dots)\,,\\\
*\varphi_{a\mathbf{n}}\wedge\delta\varrho^{a\mathbf{n}}&=\varepsilon\delta_{\varrho^{a\mathbf{n}}}\mathcal{L}(\varrho_{\mathbf{n}a},\varrho^{a\mathbf{n}},\varrho_{\mathbf{n}\mathbf{n}},\dots)\,,\\\
*\varphi^{\mathbf{n}\mathbf{n}}\wedge\delta\varrho_{\mathbf{n}\mathbf{n}}&=\hphantom{\varepsilon}\delta_{\varrho_{\mathbf{n}\mathbf{n}}}\mathcal{L}(\varrho_{\mathbf{n}a},\varrho^{a\mathbf{n}},\varrho_{\mathbf{n}\mathbf{n}},\dots)\,,\end{split}$
(10)

obtained from varying (7) with respect to $\varrho_{\mathbf{n}a}$,
$\varrho_{a\mathbf{n}}$ and $\varrho_{\mathbf{n}\mathbf{n}}$.333Only these
variations are relevant, since the boundary conditions (6) do not fix
$\omega^{a}_{\mathbf{n}}$, $\omega^{\mathbf{n}}_{a}$ and
$\omega^{\mathbf{n}}_{\mathbf{n}}$. To put it simple,
$\varphi\indices{{}^{\mu}_{\nu}}$ is the Hodge dual of the equations of motion
for $\Omega\indices{{}^{\mu}_{\nu}}$.444Note that these are not Einstein’s
equations which result from a variation with respect to the connection.

In addition to the $3+1$ decomposition of the Lagrange multipliers and
auxiliary fields we also need to introduce the $3+1$ decomposition of the
curvature, torsion and non-metricity into the action $S$ in (7). The full
details of the derivation are give in section 4. Here we mention the $3+1$
decomposition of curvature which plays an important role in the examples we
consider in section 3. We have

$\begin{aligned}
\vphantom{\frac{1}{2}}e_{\mu}^{a}e^{\nu}_{b}\Omega\indices{{}^{\mu}_{\nu}}&=\Omega\indices{{}^{a}_{b}}-\varepsilon
K^{a}\wedge\tilde{K}_{b}\,,\\\
e^{a}_{\mu}n^{\nu}\Omega\indices{{}^{\mu}_{\nu}}&=DK^{a}+\frac{\varepsilon}{2}K^{a}\wedge
Q_{\mathbf{n}\mathbf{n}}\,,\end{aligned}\qquad\begin{aligned}
n_{\mu}e^{\nu}_{a}\Omega\indices{{}^{\mu}_{\nu}}&=D\tilde{K}_{a}+\frac{\varepsilon}{2}\tilde{K}_{a}\wedge
Q_{\mathbf{n}\mathbf{n}}\,,\\\
n_{\mu}n^{\nu}\Omega\indices{{}^{\mu}_{\nu}}&=\frac{1}{2}DQ_{\mathbf{n}\mathbf{n}}+K^{a}\wedge\tilde{K}_{a}\,,\end{aligned}$
(11)

where we define
$\Omega\indices{{}^{a}_{b}}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\mathrm{d}\omega^{a}_{b}+\omega^{a}_{c}\wedge\omega^{c}_{b}$
for the curvature on the boundary.555Note the components of
$\Omega\indices{{}^{a}_{b}}$ also include normal contributions in general. The
purely hypersurface curvature, $\Omega\indices{{}^{(n-1)a}_{b}}$, is found by
means of projecting the components of $\Omega\indices{{}^{a}_{b}}$ to the
hypersurface, that is
$\Omega\indices{{}^{(n-1)a}_{b}}=\Omega\indices{{}^{a}_{b}}(\varphi_{c},\varphi_{d})\phi^{c}\wedge\phi^{d}$,
with $\varphi_{a}$ the dual of $\phi^{a}$ (see (44)). The result (11) may be
considered a generalized version of the Gauß-Codazzi equations.

Finally, let us mention the strategy we use to obtain the GHY term for the
generic action (5) after the $3+1$ decomposition has been made. Expressing the
action in a $3+1$ form results in two types of terms appearing in the
Lagrangian; one containing the fields $\omega^{\mu}_{\nu}$, $\theta^{\mu}$,
$g_{\mu\nu}$ and their projection to the boundary and another one containing
their derivatives. Due to the Dirichlet boundary conditions (6) the first type
does not contribute at the boundary and, thus, only the second type of terms
can lead to boundary contributions under variation. To calculate them, we
employ Stokes’ theorem to rewrite these terms as a pure boundary integral. We
then vary the boundary integral and isolate the terms that do not vanish after
variation. These are the terms which have to be subtracted from $S$ by means
of the GHY term. Carrying out this calculation in section 4 we find the
explicit form of the GHY term for a MAG theory with Lagrangian $\mathcal{L}$
to be

$\displaystyle\begin{split}S_{\mathrm{GHY}}&=-\int_{\partial\mathcal{M}}\left.\left(-\varepsilon\tilde{K}_{a}\wedge*\varphi^{\mathbf{n}a}+\varepsilon
K^{a}\wedge*\varphi_{a\mathbf{n}}+\frac{1}{2}Q_{\mathbf{n}\mathbf{n}}\wedge*\varphi_{\mathbf{n}\mathbf{n}}\right)\right|_{\mathrm{\partial\mathcal{M}}}\\\
&=-\int_{\partial\mathcal{M}}\left.\left(\varepsilon
K^{a}\wedge*\left(\varphi_{a\mathbf{n}}-\varphi_{\mathbf{n}a}\right)+\varepsilon
Q_{\mathbf{n}a}\wedge*\varphi^{\mathbf{n}a}+\frac{1}{2}Q_{\mathbf{n}\mathbf{n}}\wedge*\varphi_{\mathbf{n}\mathbf{n}}\right)\right|_{\mathrm{\partial\mathcal{M}}}\,,\end{split}$
(12)

where $\varphi_{a\mathbf{n}},$ $\varphi_{\mathbf{n}a}$ and
$\varphi_{\mathbf{n}\mathbf{n}}$ are implicitly expressed in terms of
$\mathcal{L}$ through the constraints (10).

A couple of comments about $S_{\rm GHY}$ are in order. First, we note that our
result should be expected. All terms in (12) depend on the first derivatives
of the metric as in typical general relativity. That torsion does not appear
explicitly in the GHY terms should also be expected. Torsion may contribute
boundary terms only via the derivative of the frame field it contains, see
(1). This derivative measures the non-holonomicity of the frame and is not a
true geometric invariant of the theory. Therefore, in a generally covariant
theory, vanishing non-holonomicity is locally enforceable and as a result the
torsion two-form is only a polynomial in $\omega^{\mu}_{\nu}$ and
$\theta^{\mu}$. This is why the torsion two-form cannot appear explicitly in
the GHY terms. Of course, this does not forbid turning the GHY term from a
function of the curvature to a function of torsion if additional constraints
between the fields strengths are taken into account as in teleparallel
theories of gravity for example [44].

Second, in the important case of vanishing non-metricity, $Q_{\mu\nu}=0$, the
GHY term (12) simplifies considerably to

$\displaystyle
S_{\mathrm{GHY}}^{Q=0}=2\int_{\mathrm{\partial\mathcal{M}}}\left.\varepsilon
K^{a}\wedge*\varphi_{\mathbf{n}a}\right|_{\mathrm{\partial\mathcal{M}}}\,.$
(13)

That is, the GHY term is a direct generalization of the GHY term of general
relativity which is proportional to the extrinsic curvature of the boundary.

Finally, some technical notes are in place. First, for the calculation we
considered $\varrho_{\mathbf{n}a}$ and $\varrho_{a\mathbf{n}}$ as independent
although curvature fulfills $\Omega_{\mu\nu}+\Omega_{\nu\mu}=DQ_{\mu\nu}$
according to (2) and $\varrho_{\mu\nu}$ has the same symmetry as
$\Omega_{\mu\nu}$. This simplifies our calculation considerably without
altering the final result. Alternatively, it is possible to invoke the
symmetry relation satisfied by $\varrho_{\mu\nu}$ prior to the variation. In
appendix B we explicitly show that this yields the same resulting GHY term
(77). Second, we worked entirely in the first order formalism where all fields
are independent. If one wants to consider a second order one, the equations of
motion must be used to express all fields in terms of the independent ones. In
traditional general relativity, this involves solving for $\omega^{\mu}_{\nu}$
in terms $g_{\mu\nu}$. Substituting the solution into $\mathcal{L}$ yields a
new Lagrangian $\tilde{\mathcal{L}}$ for the independent fields. If
$\tilde{\mathcal{L}}$ is a function of the remaining independent field
strengths, say
$\tilde{\mathcal{L}}=\tilde{\mathcal{L}}(\Omega\indices{{}^{\mu}_{\nu}})$, our
algorithm may still be applied mutatis mutandis to obtain the GHY term for
$\tilde{\mathcal{L}}$. We have implicitly used this observation when writing
down (13) for the GHY terms of metric compatible theories in the following
section.

To recap, the calculation of the GHY term for any MAG theory follows these
steps: First, choose the Lagrangian $\mathcal{L}$ of the MAG of interest.
Second, express $\mathcal{L}$ in a $3+1$ decomposed form (see section 4).
Third, derive the Lagrange multiplier $\varphi_{\mu\nu}$ through the
constraint equations (10). Finally, substitute the result into the general
form (12) of the GHY term. In the following section, we apply this algorithm
to several particular theories of gravity as a consistency check of our
results.

##  3 Examples for Gibbons-Hawking-York terms

In the current section we apply the algorithm explained in the previous
section to calculate the GHY terms for several theories of gravity. The
examples we consider involve the Einstein-Hilbert action in general
dimensions, the action of 4d Chern-Simons modified gravity and the 5d
Lovelock-Chern-Simons action. We also calculate the GHY term for the most
general MAG Lagrangian quadratic in the field strengths. Since both the
calculation and the result for the MAG GHY terms are quite lengthy, but do not
add any additional insight regarding the formalism, we relegate them to
appendix A.

###  3.1 Einstein-Hilbert gravity $\mathcal{L}\propto R$

As a very basic consistency check of our algorithm, we first consider the
Einstein-Hilbert action

$\displaystyle
S^{\mathrm{EH}}=\frac{1}{2\kappa}\int\mathrm{d}^{4}x\sqrt{-g}R=\frac{1}{2\kappa}\int\eta^{\mu\nu}\wedge\Omega_{\mu\nu}\,,$
(14)

where
$\eta^{\mu\nu}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=*\left(\theta^{\mu}\wedge\theta^{\nu}\right)$.
The Lagrangian for this action is
$\mathcal{L}=\mathcal{L}(\Omega\indices{{}^{\mu}_{\nu}})=\eta^{\mu\nu}\wedge\Omega_{\mu\nu}/2\kappa$.
Hence, the general form of the extended action (7) including the auxiliary
fields and Lagrange multipliers reads

$\displaystyle S^{\rm EH}_{\rm
aux}=\int\left(\frac{1}{2\kappa}\eta^{\mu\nu}\wedge\varrho_{\mu\nu}+*\varphi\indices{{}_{\mu}^{\nu}}\wedge(\Omega\indices{{}^{\mu}_{\nu}}-\varrho\indices{{}^{\mu}_{\nu}})\right)$
(15)

in the case of Einstein-Hilbert gravity. The relevant terms in the $3+1$
decomposition of $\mathcal{L}(\varrho_{\mu\nu})$ for our calculation of the
GHY term are

$\displaystyle\mathcal{L}(\varrho_{\mu\nu})=\frac{1}{2\kappa}\eta^{\mu\nu}\wedge\varrho_{\mu\nu}\simeq\frac{1}{2\kappa}\varepsilon\eta_{\mathbf{n}a}\wedge\varrho^{\mathbf{n}a}\,,$
(16)

where $\simeq$ omits irrelevant terms of the decomposition. According to (10),
the equations of motion for $\varrho^{\mathbf{n}a}$ read

$\displaystyle*\varphi_{\mathbf{n}a}\wedge\delta\varrho^{\mathbf{n}a}=\varepsilon\delta_{\varrho^{\mathbf{n}a}}\mathcal{L}(\varrho)=\frac{1}{2\kappa}\eta_{\mathbf{n}a}\wedge\delta\varrho^{\mathbf{n}a}$
(17)

which fix the form of $\varphi_{\mathbf{n}a}$ to

$\displaystyle*\varphi_{\mathbf{n}a}=\frac{1}{2\kappa}\eta_{\mathbf{n}a}\,.$
(18)

Substituting (18) into our general result (13) for the GHY term, we obtain

$\displaystyle
S_{\mathrm{GHY}}^{\mathrm{EH}}=2\int_{\mathrm{\partial\mathcal{M}}}\left.\varepsilon
K^{a}\wedge*\varphi_{\mathbf{n}a}\right|_{\mathrm{\partial\mathcal{M}}}=\frac{\varepsilon}{\kappa}\int_{\partial\mathcal{M}}\mathrm{d}^{3}x\sqrt{|\gamma|}K\indices{{}^{a}_{a}}$
(19)

which is the well-known result for the GHY term in general relativity [34,
35]. Note that we used the $4$d Einstein-Hilbert action, but since the
differential geometric formulation of the action (14) holds for an arbitrary
number of dimensions, our result is actually more general. In particular, the
GHY term

$\displaystyle
S_{\mathrm{GHY}}^{\mathrm{EH}}=\frac{\varepsilon}{\kappa}\int_{\partial\mathcal{M}}\mathrm{d}\text{Vol}_{\partial\mathcal{M}}\sqrt{|\gamma|}K\indices{{}^{a}_{a}}$
(20)

solves the Dirichlet problem for Einstein-Hilbert gravity on any $n$-manifold
$\mathcal{M}$. The abbreviation
$\mathrm{d}\text{Vol}_{\partial\mathcal{M}}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\phi^{0}\wedge\dots\wedge\phi^{n-2}$
is equal to the standard integral measure $\mathrm{d}^{n-1}y$ in holonomic
coordinates $\phi^{a}=\mathrm{d}y^{a}$.

###  3.2 4d Chern-Simons modified gravity $\mathcal{L}\propto\Omega^{2}$

As a next check of our formalism we turn to the four-dimensional Chern-Simons
action as used in Chern-Simons modified gravity [45]. This provides a non-
trivial check since the action is quadratic in curvature and the GHY term was
already derived in [46].

The Chern-Simons part of the full action is666In the differential geometric
version of (21) we could alternatively use a notation which involves the Hodge
dual of one of the curvature two-forms. We refrain from that since the
variation of Hodge duals is non-trivial. For details on the variation of Hodge
duals as well as interior products and their inclusion into the framework of
this paper, see appendix A.1.

$\displaystyle
S^{\mathrm{CS}}=\frac{\kappa}{4}\int\mathrm{d}^{4}x\sqrt{-g}\,\theta\,*RR=\frac{\kappa}{2}(-1)^{\mathrm{ind}g}\int\theta\,\Omega\indices{{}^{\mu}_{\nu}}\wedge\Omega\indices{{}^{\nu}_{\mu}}\,,$
(21)

where the background scalar field $\theta$ must not be confused with the
coframe basis $\theta^{\mu}$. We read off
$\mathcal{L}(\Omega\indices{{}^{\mu}_{\nu}})=\frac{\kappa}{2}(-1)^{\mathrm{ind}g}\theta\,\Omega\indices{{}^{\mu}_{\nu}}\wedge\Omega\indices{{}^{\nu}_{\mu}}$
from the action (21) so that the equation of motion (10) for
$\varrho^{\mathbf{n}a}$ yields

$\displaystyle*\varphi_{\mathbf{n}a}\wedge\delta\varrho^{\mathbf{n}a}=\varepsilon\delta_{\varrho^{\mathbf{n}a}}\mathcal{L}=(-1)^{\mathrm{ind}g}\kappa\theta\varrho_{a\mathbf{n}}\wedge\delta\varrho^{\mathbf{n}a}$
(22)

which leads to
$*\varphi^{\mathbf{n}a}=(-1)^{\mathrm{ind}g}\kappa\theta\varrho^{a\mathbf{n}}$.

We observe that $\varphi^{\mathbf{n}a}$ depends explicitly on
$\varrho^{a\mathbf{n}}$, unlike the Einstein-Hilbert action. To proceed, we
use the equations of motion (8) of $\varphi\indices{{}^{\mu}{}^{\nu}}$ to fix
$\varrho_{\mu\nu}=\Omega_{\mu\nu}$. Subsequently, we employ the $3+1$
decomposition of curvature which is given by the Gauß-Codazzi equations (11)
to express $\varphi^{\mathbf{n}a}$ in terms of geometric quantities on
$\partial\mathcal{M}$. We find

$\displaystyle\left.\vphantom{(-1)^{\mathrm{ind}g}}*\varphi^{\mathbf{n}a}\right|_{\varrho_{\mu\nu}=\Omega_{\mu\nu}}=\left.(-1)^{\mathrm{ind}g}\kappa\theta\varrho^{a\mathbf{n}}\right|_{\varrho_{\mu\nu}=\Omega_{\mu\nu}}=(-1)^{\mathrm{ind}g}\kappa\theta
DK^{a}$ (23)

which evaluated on the boundary reads

$\displaystyle\left.*\varphi^{\mathbf{n}a}\right|_{\partial\mathcal{M}}=(-1)^{\mathrm{ind}g}\kappa\theta\nabla_{a}K\indices{{}^{c}_{b}}\phi^{a}\wedge\phi^{b}\,.$
(24)

Here we define the boundary covariant derivative as
$\nabla_{a}K\indices{{}^{c}_{b}}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\partial_{a}K\indices{{}^{c}_{b}}+\Gamma^{c}_{ad}K\indices{{}^{d}_{b}}-\Gamma^{d}_{ab}K\indices{{}^{c}_{d}}$,
where the connection coefficients are related to the boundary connection (3)
as $\omega^{a}_{b}=\Gamma^{a}_{cb}\phi^{c}$. To match the conventions of [46]
we choose $|g|=-g$, $|\gamma|=\gamma$ as well as torsion freedom. This, in
conjunction with (13), leads to the GHY term

$\displaystyle
S_{\mathrm{GHY}}^{\mathrm{CS}}=2\kappa\int_{\mathcal{M}_{3}}\mathrm{d}^{3}x\sqrt{\gamma}\theta\epsilon^{ijk}K\indices{{}_{i}^{l}}\nabla_{j}K_{kl}\,,$
(25)

where $\epsilon^{ijk}$ denotes the totally antisymmetric tensor which differs
from the $\varepsilon^{ijk}$-symbol by a factor of $1/\sqrt{\gamma}$. The
result (25) coincides with the result of [46] up to a factor of 2. However, a
calculation of this GHY term via the method of [43] supports our result (25)
including the factor of 2. We now turn to the derivation of the GHY term for
Lovelock-Chern-Simons gravity which is not known in literature as far as we
are aware.

###  3.3 Lovelock-Chern-Simons gravity

For our final example, we consider Lovelock-Chern-Simons gravity in a $5$d
spacetime with boundary. This example involves both a non-trivial polynomial
Lagrangian for curvature and non-vanishing torsion. In particular, we
investigate the action

$\displaystyle
S^{\mathrm{LCS}}=\kappa\int_{\mathcal{M}_{5}}\epsilon_{\mu\nu\alpha\beta\sigma}\left(\Omega^{\mu\nu}\wedge\Omega^{\alpha\beta}+\frac{2}{3}\Omega^{\mu\nu}\wedge\theta^{\alpha}\wedge\theta^{\beta}+\frac{1}{5}\theta^{\mu}\wedge\theta^{\nu}\wedge\theta^{\alpha}\wedge\theta^{\beta}\right)\wedge\theta^{\sigma}$
(26)

employed in [47, 16]. As in the previous examples we identify the integrand in
(26) as $\mathcal{L}$ and use the equations of motion (8), (10) for
$\varphi\indices{{}^{\mu}_{\nu}}$ and $\varrho\indices{{}^{\mu}_{\nu}}$ to
obtain

$\displaystyle\left.\vphantom{(-1)^{\mathrm{ind}g}}*\varphi_{\mathbf{n}a}\right|_{\varrho_{\mu\nu}=\Omega_{\mu\nu}}=2\varepsilon\kappa\epsilon_{\mu\nu\alpha\beta\sigma}n^{\mu}e^{\nu}_{a}\left(\Omega^{\alpha\beta}+\frac{1}{3}\theta^{\alpha}\wedge\theta^{\beta}\right)\wedge\theta^{\sigma}\,.$
(27)

Note that since $\epsilon_{\mu\nu\alpha\beta\sigma}$ in (27) is contracted
with $n^{\mu}$, the $\alpha,\beta$ and $\sigma$ indices are all on
$\partial\mathcal{M}$ which means that they need to be contracted with
$e^{\mu}_{a}$ when evaluating the $3+1$ decomposition of
$\varphi_{\mathbf{n}a}$. Thus, we may write (27) as

$\displaystyle\left.\vphantom{(-1)^{\mathrm{ind}g}}*\varphi_{\mathbf{n}a}\right|_{\varrho_{\mu\nu}=\Omega_{\mu\nu}}=2\varepsilon\kappa\epsilon_{abcd}\left(e^{b}_{\alpha}e^{c}_{\beta}\Omega^{\alpha\beta}+\frac{1}{3}\phi^{b}\wedge\phi^{c}\right)\wedge\phi^{d}\,.$
(28)

Employing the Gauß-Codazzi equations (11) for vanishing non-metricity, we
evaluate $\varphi_{\mathbf{n}a}$ in terms of geometric quantities on the
boundary as

$\displaystyle\left.\vphantom{(-1)^{\mathrm{ind}g}}*\varphi_{\mathbf{n}a}\right|_{\mathcal{M}_{4}}=2\varepsilon\kappa\epsilon_{abcd}\left.\left(\Omega^{bc}-\varepsilon
K^{b}\wedge
K^{c}+\frac{1}{3}\phi^{b}\wedge\phi^{c}\right)\wedge\phi^{d}\right|_{\mathcal{M}_{4}}\,,$
(29)

where
$\Omega\indices{{}^{a}_{b}}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\mathrm{d}\omega^{a}_{b}+\omega^{a}_{c}\wedge\omega^{c}_{b}$
was defined in (11). By virtue of (13) the GHY term of Lovelock-Chern-Simons
gravity is then

$\displaystyle
S_{\mathrm{GHY}}^{\mathrm{LCS}}=-4\kappa\int_{\mathcal{M}_{4}}\left.\epsilon_{abcd}\left(\Omega^{ab}-\varepsilon
K^{a}\wedge K^{b}+\frac{1}{3}\phi^{a}\wedge\phi^{b}\right)\wedge\phi^{c}\wedge
K^{d}\right|_{\mathcal{M}_{4}}\,.$ (30)

In components (30) takes the form

$\displaystyle
S_{\mathrm{GHY}}^{\mathrm{LCS}}=-4\kappa\int_{\mathcal{M}_{4}}\mathrm{d}\mathrm{Vol}_{\mathcal{M}_{4}}\sqrt{-\gamma}\left[3!\left(\frac{1}{2}R\indices{{}^{a}{}^{b}_{[ab}}-\varepsilon
K\indices{{}^{a}_{[a}}K\indices{{}^{b}_{b}}\right)K\indices{{}^{c}_{c]}}+2K\indices{{}^{a}_{a}}\right]\,.$
(31)

General considerations regarding (30) show it is consistent with expectations.
To see this, consider the Lovelock-Chern-Simons action (26) in more detail.
This action consists of three terms. The first of them which is a quadratic
curvature term in five dimensions is new to us so far. Because of its
quadratic curvature form we expect it to yield a GHY term of a new form which
is of higher order in the extrinsic curvature just as we observed it in the
example of $4$d Chern-Simons modified gravity in section 3.2. We immediately
observe this behaviour from the generalized Gauß-Codazzi equations (11) which
give the curvature $3+1$ decomposition. The third term in (26) has no
curvature contribution and is not expected to yield a GHY contribution since
these contributions all relate to variations of curvature, see (12). The
remaining second term we already know. In fact, we rewrite it as

$\displaystyle S^{\mathrm{LCS,\leavevmode\nobreak\ 2nd\leavevmode\nobreak\
term}}=\kappa\int_{\mathcal{M}_{5}}\epsilon_{\mu\nu\alpha\beta\sigma}\frac{2}{3}\Omega^{\mu\nu}\wedge\theta^{\alpha}\wedge\theta^{\beta}\wedge\theta^{\sigma}=8\kappa\int_{\mathcal{M}_{5}}\sqrt{-g}\,\eta^{\mu\nu}\wedge\Omega_{\mu\nu}\,,$
(32)

where
$\eta^{\mu\nu}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=*\left(\theta^{\mu}\wedge\theta^{\nu}\right)$
was defined in section 3.1. Note that this is nothing but a 5-dimensional
Einstein-Hilbert action. In section 3.1 we already noticed that actions of
this form always have the same GHY term regardless of their dimension. Hence,
we expect a contribution to the total Lovelock-Chern-Simons GHY term of the
type

$\displaystyle S_{\mathrm{GHY}}^{\mathrm{LCS,\leavevmode\nobreak\
2nd\leavevmode\nobreak\
term}}\propto\int_{\partial\mathcal{M}}\mathrm{d}\text{Vol}_{\partial\mathcal{M}}\sqrt{|\gamma|}K\indices{{}^{a}_{a}}\,.$
(33)

We only write a proportionality instead of an equality in the latter equation
since there could be additional contributions from the quadratic curvature
term which we omit in this consistency consideration. We observe all of these
expected contributions in our result (31).

Some results regarding the GHY term for Lovelock-Chern-Simons theory already
exist in the literature. Namely, the action of Lovelock-Chern-Simons (26) was
renormalized in [47]. To achieve this, the authors of [47] considered a finite
Fefferman-Graham expansion and compensated the divergent terms by addition of
counterterms to the action. The non-divergent terms in this procedure were not
given explicitly in [47] but calculated in [16], where they were interpreted
as the GHY term of Lovelock-Chern-Simons gravity. This term constitutes an on-
shell result whereas our result (30) is the universal GHY term for the
Lovelock-Chern-Simons action (26). From this point of view, our result (30)
may be considered as a generalization of the Lovelock-Chern-Simons GHY term in
[16].

Our results for the GHY term of Lovelock-Chern-Simons constitute a new result
in the literature. An additional novel result can be found in appendix A,
where we derive the GHY term for the most general MAG Lagrangian quadratic in
the fields. In the same appendix we include details on how to treat interior
products and Hodge duals under the variations necessary to apply our
algorithm. These allow to work directly in the differential form version of
the Lagrangian, which usually provides the most economical expression for the
Lagrangian. Before proceeding to appendix A, we suggest reading section 4.
This section contains the explicit derivation of the general formula for the
GHY term (12) along with some useful formulae regarding the $3+1$
decomposition of the field strengths.

##  4 Derivation of the Gibbons-Hawking-York term

In this section we present the explicit calculation leading to the GHY term
(12) given in section 2. The essence of the calculation of the GHY term
involves the $3+1$ decomposition of the fundamental fields,
$\omega^{\mu}_{\nu}$, $g_{\mu\nu}$ and $\theta^{\mu}$, and their field
strengths, $\Omega\indices{{}^{\mu}_{\nu}}$, $Q_{\mu\nu}$ and $T^{\mu}$
respectively. In order to derive such a decomposition, we first consider the
geometric setup for spacetime foliations introduced in section 2 in more
detail. This involves an investigation of how the frame fields $\theta^{\mu}$,
$n^{\mu}$ and $e^{a}_{\mu}$ decompose with respect to the spacetime foliation.

###  4.1 Frame decomposition

In this subsection, we first introduce frame decompositions on manifolds in
the vielbein formalism. These frame decompositions constitute the basis for
the subsequent investigation of foliations in the $3+1$ formalism [48, 42,
41], in which the manifold is described as a stack of hypersurfaces. We assume
that neither torsion nor non-metricity vanishes and use differential forms
throughout to simplify the analysis (see e.g. [40, 23, 49]).

We consider an $n$-dimensional manifold $\mathcal{M}$ equipped with a generic
coframe $\theta^{\mu}$, $\mu\in\\{0,\dots,n-1\\}$. Locally, a coframe is a
basis of the cotangent space at each point $p$ of $\mathcal{M}$. In order to
describe how the boundary is embedded in $\mathcal{M}$, we aim at decomposing
all geometric quantities on $\mathcal{M}$ into contributions tangent and
normal to the boundary. For this reason, we assume that $\mathcal{M}$ is
foliated by a family of hypersurfaces $\\{\Sigma_{\lambda}\\}_{\lambda}$,
where $\lambda=$const defines each hypersurface $\Sigma_{\lambda}$. In order
to be able to describe the embedding of the boundary as well as the boundary
components of the manifold’s geometric quantities torsion, curvature, and non-
metricity, we choose this foliation such that the asymptotic boundary
$\partial\mathcal{M}$ is one of the hypersurfaces. The most basic quantity to
decompose in this foliation is the coframe $\theta^{\mu}$. We associate a
frame field $\vartheta_{\mu}$ to this coframe by means of
$\theta^{\mu}(\vartheta_{\nu})=\delta^{\mu}_{\nu}=\vartheta_{\nu}(\theta^{\mu})$,
and decompose both according to

$\displaystyle\theta^{\mu}$
$\displaystyle=e^{\mu}_{a}\tilde{\phi}^{a}+t^{\mu}\phi\,,$ (34a)
$\displaystyle\vartheta_{\mu}$
$\displaystyle=E^{a}_{\mu}\varphi_{a}+\frac{\varepsilon}{N}n_{\mu}\tilde{\varphi}\,,$
(34b)

where $a\in\\{0,\dots,n-2\\}$. The new tensors introduced in this
decomposition are given by

$\displaystyle
e^{\mu}_{a}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\theta^{\mu}(\varphi_{a})\,,\qquad
E^{a}_{\mu}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\vartheta_{\mu}(\tilde{\phi}^{a})\,,\qquad
t^{\mu}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\theta^{\mu}(\tilde{\varphi})\,.$
(35)

Using the induced metric on $\Sigma_{\lambda}$,
$\gamma_{ab}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=e^{\mu}_{a}e^{\nu}_{b}g_{\mu\nu}$,
we define

$\displaystyle\varepsilon\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\mathrm{sgn}\left(\frac{\det
g_{\mu\nu}}{\det\gamma_{ab}}\right)\,,\qquad
N\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=1/\sqrt{|g^{\mu\nu}\vartheta_{\mu}(\phi)\vartheta_{\nu}(\phi)|}\,,\qquad
n_{\mu}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\varepsilon
N\vartheta_{\mu}(\phi)\,.$ (36)

The lapse function $N$ is introduced to normalize $n_{\mu}$,
$n_{\mu}n^{\mu}=\varepsilon$. Moreover, the sign $\varepsilon$ in $n_{\mu}$ is
a convenient choice for the direction of $n_{\mu}$ (see [41]), such that
$\varepsilon^{2}=1$. We impose orthogonality for the frame decomposition (34)
by virtue of

$\displaystyle\tilde{\phi}^{a}(\varphi_{b})=\delta^{a}_{b}=\varphi_{b}(\tilde{\phi}^{a})\,,\qquad\phi(\tilde{\varphi})=1=\tilde{\varphi}(\phi)\,,$
(37)

with all other pairings between the $\phi$ and $\varphi$ frames vanishing, for
instance $\phi(\varphi_{a})=0$. These pairings imply that the tensors
$e^{\mu}_{a}$, $E^{a}_{\mu}$, $t^{\mu}$ and $n_{\mu}$ are not independent of
each other. To see this, we insert the decompositions of frame and coframe
(34) into their defining relation
$\theta^{\mu}(\vartheta_{\nu})=\delta^{\mu}_{\nu}$ to obtain

$\displaystyle\delta^{\mu}_{\nu}=e^{\mu}_{a}E^{a}_{\nu}+\frac{\varepsilon}{N}t^{\mu}n_{\nu}\,.$
(38)

Next, we contract this equation with $n^{\mu}$ as well as
$e_{\nu}^{a}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\gamma^{ab}g_{\mu\nu}e^{\mu}_{b}$
to find the relations

$\displaystyle t^{\mu}=Nn^{\mu}+N^{a}e^{\mu}_{a}\quad\text{ and }\quad
E^{a}_{\mu}=e^{a}_{\mu}-\frac{\varepsilon}{N}n_{\mu}t^{\nu}e^{a}_{\nu}=e^{a}_{\mu}-\frac{\varepsilon}{N}n_{\mu}N^{a}-\varepsilon
n_{\mu}n^{\nu}e^{a}_{\nu}\,,$ (39)

where we define the shift vector
$N^{a}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=-Nn^{\mu}E^{a}_{\mu}$.
Note that the expressions (39) relate the fields $e^{\mu}_{a}$, $E^{a}_{\mu}$,
$t^{\mu}$ and $n_{\mu}$ with each other. Nevertheless, their relation may be
examined further by means of777Equation (40) can be proven by expressing the
completeness relation (38) in terms of $e^{\mu}_{a}$, $n^{\mu}$ via (39) to
find $\delta^{\mu}_{\nu}=e^{\mu}_{a}e^{a}_{\nu}+\varepsilon
n^{\mu}n_{\nu}-\varepsilon n^{\rho}e_{\rho}^{a}e_{a}^{\mu}n_{\nu}$. Using
this, we can write
$n_{\mu}e^{\mu}_{a}=n_{\mu}\delta^{\mu}_{\nu}e^{\nu}_{a}=n_{\mu}e^{\mu}_{a}(\varepsilon
n^{\rho}e_{\rho}^{b}n_{\nu}e^{\nu}_{b})$. This implies $n_{\mu}e^{\mu}_{a}=0$,
since $1=\varepsilon n^{\rho}e^{a}_{\rho}n_{\nu}e^{\nu}_{a}$ leads to $1=0$
when we take the trace of the completeness relation.

$\displaystyle n_{\mu}e^{\mu}_{a}=0$ (40)

which implies that $t^{\mu}E_{\mu}^{a}=0$ holds, as well. Using the
orthogonality conditions (37) and (40), we may revisit the relations (38) and
(39) and rewrite them as

$\displaystyle\delta^{\mu}_{\nu}$
$\displaystyle=e^{\mu}_{a}e^{a}_{\nu}+\varepsilon n^{\mu}n_{\nu}\,,$ (41a)
$\displaystyle t^{\mu}$ $\displaystyle=Nn^{\mu}+N^{a}e^{\mu}_{a}\,,$ (41b)
$\displaystyle E^{a}_{\mu}$
$\displaystyle=e^{a}_{\mu}-\frac{\varepsilon}{N}n_{\mu}N^{a}\,.$ (41c)

In the following section we use these equations to find a version of the frame
decomposition (34) which is adapted to the foliation of the manifold
$\mathcal{M}$ by means of hypersurfaces.

###  4.2 Adapted frame decomposition

In the frame decomposition (34) we have introduced four a priori independent
tensors $e^{\mu}_{a}$, $E^{a}_{\mu}$, $t^{\mu}$ and $n_{\mu}$. We imposed
orthogonality conditions in (37) and saw that they allow to express
$E^{a}_{\mu}$ and $t^{\mu}$ in terms of $e^{\mu}_{a}$ and $n_{\mu}$ as in
(41). Thus, we use the relations (41) to eliminate $E^{a}_{\mu}$ and $t^{\mu}$
in the original frame decomposition (34), to obtain

$\displaystyle\theta^{\mu}$
$\displaystyle=e^{\mu}_{a}\phi^{a}+Nn^{\mu}\phi\,,$ (42a)
$\displaystyle\vartheta_{\mu}$
$\displaystyle=e^{a}_{\mu}\varphi_{a}+\frac{\varepsilon}{N}n_{\mu}\varphi\,,$
(42b)

where we introduced the adapted frame

$\displaystyle\phi^{a}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\tilde{\phi}^{a}+N^{a}\phi\,,\qquad\varphi\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\tilde{\varphi}-N^{a}\varphi_{a}\,.$
(43)

The pairings (37) of the $(\phi^{a},\phi,\varphi_{a},\varphi)$ frames are
invariant under this transformation and read

$\displaystyle\phi^{a}(\varphi_{b})=\delta^{a}_{b}=\varphi_{b}(\phi^{a})\,,\qquad\phi(\varphi)=1=\varphi(\phi)$
(44)

with all remaining pairings vanishing. Hence, the new frame fields
$(\phi^{a},\phi,\varphi_{a},\varphi)$ are orthogonal as well. To gain an
intuition for what these fields are, consider a generic one-form
$A=A_{\mu}\theta^{\mu}$ and a vector $B=B^{\mu}\vartheta_{\mu}$. We use (42)
to expand $A$ and $B$ as

$\displaystyle A$
$\displaystyle=A_{\mu}e^{\mu}_{a}\phi^{a}+NA_{\mu}n^{\mu}\phi\,,$ (45a)
$\displaystyle B$
$\displaystyle=B^{\mu}e^{a}_{\mu}\varphi_{a}+\frac{\varepsilon}{N}B^{\mu}n_{\mu}\varphi\,.$
(45b)

Recall that a tensor or differential form is called tangent (normal) to a
hypersurface if it vanishes when any of its indices is contracted with
$n_{\mu}$ ($e^{\mu}_{a}$). We apply these definition to (45) and find that
tangent one-forms and vectors take the form

$\displaystyle A=A_{\mu}e^{\mu}_{a}\phi^{a}\,,\qquad
B=B^{\mu}e^{a}_{\mu}\varphi_{a}$ (46)

while normal ones are expanded as

$\displaystyle A=NA_{\mu}n^{\mu}\phi\,,\qquad
B=\frac{\varepsilon}{N}B^{\mu}n_{\mu}\varphi\,.$ (47)

Combining these notions of being tangent and normal with the orthogonality
conditions (44), we may interpret $\varphi_{a}$ as a frame basis on
$\Sigma_{\lambda}$ with associated coframe basis $\phi^{a}$. Hence, the frame
decomposition (42) is adapted to the foliation and we proceed to work with it
in the following. Having set up the frame decomposition, we proceed by
investigating how fields transform.

###  4.3 Decomposition of metric and connection

In the previous subsections we have investigated the decomposition of the
frame fields in the foliation of the manifold $\mathcal{M}$ by means of
hypersurfaces. We use this decomposition next to investigate how the basic
geometric objects on $\mathcal{M}$ decompose in that foliation. In particular,
apart from the frame field, we consider the metric tensor $g_{\mu\nu}$ as well
as the connection one-form $\omega^{\mu}_{\nu}$ as independent dynamical
fields. We start by investigating the decomposition of $g_{\mu\nu}$ by
applying the adapted frame decomposition (42). Contracting (42) with
$e^{\mu}_{a}$ and $n_{\mu}$, we exploit the normality condition (40),
$n_{\mu}e^{\mu}_{a}=0$, to identify
$\phi=\frac{\varepsilon}{N}n_{\mu}\theta^{\mu}$ and
$\phi^{a}=e^{a}_{\mu}\theta^{\mu}$. We use the manifold metric $g_{\mu\nu}$
and the hypersurface metric $\gamma_{ab}=e^{\mu}_{a}e^{\nu}_{b}g_{\mu\nu}$ to
raise and lower indices in $e^{a}_{\mu}=\gamma^{ab}g_{\mu\nu}e^{\nu}_{b}$.
Using these relations, we rewrite the metric tensor by means of (42) as

$\displaystyle\mathrm{d}s^{2}=g_{\mu\nu}\theta^{\mu}\otimes\theta^{\nu}=\left(\gamma_{ab}e^{a}_{\mu}e^{b}_{\nu}+\varepsilon
n_{\mu}n_{\nu}\right)\theta^{\mu}\otimes\theta^{\nu}\,.$ (48)

From (48) we can read of the $3+1$ decomposition of the metric tensor
$g_{\mu\nu}$ as

$\displaystyle g_{\mu\nu}=\gamma_{ab}e^{a}_{\mu}e^{b}_{\nu}+\varepsilon
n_{\mu}n_{\nu}$ (49)

in hypersurface tangent and normal contributions. The decomposition (49) is
used heavily in the following discussion.

For deriving the transformation of the connection one-form
$\omega^{\mu}_{\nu}$ to the hypersurfaces, we use the transformation law of
connections with respect to the metric-affine group. For
$\Lambda\indices{{}^{\mu}_{\nu}}\in$ GL(n,$\mathbb{R}$), this transformation
is given by [23]

$\displaystyle\omega^{\mu}_{\nu}\rightarrow{\omega^{\prime}}^{\mu}_{\nu}=\Lambda\indices{{}^{-1}{}^{\mu}_{\rho}}\omega^{\rho}_{\sigma}\Lambda\indices{{}^{\sigma}_{\nu}}+\Lambda\indices{{}^{-1}{}^{\mu}_{\rho}}\mathrm{d}\Lambda\indices{{}^{\rho}_{\nu}}\,.$
(50)

Motivated by the frame decomposition (42) we choose the transformation

$\displaystyle\begin{split}\Lambda\indices{{}^{\mu}_{\nu}}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=&e^{\mu}_{a}\delta^{a}_{\nu}+Nn^{\mu}\delta_{\nu}^{n-1}\,,\\\
\Lambda\indices{{}^{-1}{}^{\mu}_{\nu}}=&e^{a}_{\nu}\delta^{\mu}_{a}+\frac{\varepsilon}{N}n_{\nu}\delta^{\mu}_{n-1}\,.\end{split}$
(51)

After inserting this transformation into (50), we only consider the components
$\mu,\nu=(0,\dots,n-2)$ to find

$\displaystyle\omega^{b}_{a}=e^{b}_{\mu}\left(\mathrm{d}e^{\mu}_{a}+\omega^{\mu}_{\nu}e^{\nu}_{a}\right)\qquad\Leftrightarrow\qquad
e^{b}_{\mu}De^{\mu}_{a}=0\,,$ (52)

where we suppressed the prime on the left hand side since the indices
immediately clarify which connection is meant here. This is the transformation
of the connection on $\mathcal{M}$ to the connection on $\Sigma_{\lambda}$.
Essentially, the above argument is a proof of the vielbein postulate for the
frame field on the hypersurface for MAG, along the lines of a similar proof in
[23]. We interpret the hypersurface connection coefficients by means of (52)
as the contribution of $De^{\mu}_{a}$ which is tangent to the hypersurfaces.
It is, thus, natural to consider its normal contributions next to obtain an
expression for $De^{\mu}_{a}$, which constitutes a differential form version
of the Gauß-Weingarten equation.

###  4.4 Foundations of the 3+1 formalism in general frames

In this section we derive the fundamental equations of the $3+1$ formalism in
terms of differential forms. This involves the decomposition of $De^{\mu}_{a}$
into contributions normal and tangent to the hypersurfaces in the foliation of
the manifold $\mathcal{M}$, which can be considered as a generalized version
of the Gauß-Weingarten equations. We furthermore derive a similar equation for
the normal vector $n^{\mu}$, and take exterior covariant derivatives of these
relations to derive generalized versions of the Ricci identities. This set of
equations is the foundation of the $3+1$ formalism in differential forms.

We start our calculation by investigating $De^{\mu}_{a}$. While we found the
tangent contributions of this one-form in (52), we still need to calculate its
hypersurface normal parts. To that end we define a one-form corresponding to
the extrinsic curvature tensor
$K\indices{{}^{a}_{b}}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=e^{a}_{\mu}e^{\nu}_{b}\nabla_{\nu}n^{\mu}$.
Due to non-vanishing non-metricity, we furthermore define extrinsic curvature
in a slightly different manner by
$\tilde{K}_{ab}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=e_{a}^{\mu}e^{\nu}_{b}\nabla_{\nu}n_{\mu}$,
and we work with both definitions interchangeably in the following. Thus, we
also define two different extrinsic curvature one-forms by

$\displaystyle\begin{split}K^{a}&\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=e^{a}_{\mu}Dn^{\mu}\quad\text{and}\\\
\tilde{K}_{a}&\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=e^{\mu}_{a}Dn_{\mu}\,.\end{split}$
(53)

We use the frame decomposition (42) to calculate the components of these
quantities in our foliation,

$\displaystyle\begin{split}K^{a}&=K\indices{{}^{a}_{b}}\phi^{b}+Ne^{a}_{\mu}a^{\mu}\phi\,,\\\
\tilde{K}_{a}&=\tilde{K}_{ab}\phi^{b}+Ne^{\mu}_{a}\tilde{a}_{\mu}\phi\,,\end{split}$
(54)

where we defined
$a^{\mu}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\nabla_{\mathbf{n}}n^{\mu}$
and
$\tilde{a}_{\mu}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\nabla_{\mathbf{n}}n_{\mu}$
analogous to the different notions of extrinsic curvature.

With the extrinsic curvature one-form at hand, we now investigate the
covariant derivative of generic forms $A^{\mu}=e_{a}^{\mu}A^{a}$ tangent to
$\Sigma_{\lambda}$. We use the $3+1$ decomposition of the metric (49) and the
vielbein postulate (52) to evaluate

$\displaystyle
DA^{\mu}=D\left(g^{\mu}_{\nu}A^{\nu}\right)=\left(e^{\mu}_{a}e^{a}_{\nu}+\varepsilon
n^{\mu}n_{\nu}\right)DA^{\nu}=e^{\mu}_{a}DA^{a}-\varepsilon
n^{\mu}e^{\nu}_{a}Dn_{\nu}\wedge A^{a}\,.$ (55)

At the same time $A^{\mu}$ being tangent to the hypersurface implies

$\displaystyle DA^{\mu}=D\left(e_{a}^{\mu}A^{a}\right)=De^{\mu}_{a}\wedge
A^{a}+e^{\mu}_{a}DA^{a}\,.$ (56)

We compare both equations and noting they hold for arbitrary $A^{a}$, we find

$\displaystyle De^{\mu}_{a}=-\varepsilon n^{\mu}\tilde{K}_{a}\,.$ (57)

This is the expression of the Gauß-Weingarten equation in terms of
differential forms.

We can derive a similar expression like the Gauß-Weingarten equation for the
normal vector $n^{\mu}$. To that end, we first need to define the non-
metricity one-form,

$\displaystyle
Q_{\mu\nu}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=-Dg_{\mu\nu}=Q_{\mu\nu\rho}\theta^{\rho}\,,$
(58)

with its tensor components given by $Q_{\mu\nu\rho}=-\nabla_{\rho}g_{\mu\nu}$.
Abbreviating
$Q_{\mathbf{n}\mathbf{n}}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=n^{\mu}n^{\nu}Q_{\mu\nu}$,
we investigate $0=D\varepsilon$ to obtain

$\displaystyle n_{\mu}Dn^{\mu}=\frac{1}{2}Q_{\mathbf{n}\mathbf{n}}\,.$ (59)

We use this result for the calculation of $Dn^{\mu}$ and follow the same steps
as described in the case of $De^{\mu}_{a}$ above to find

$\displaystyle
Dn^{\mu}=e^{\mu}_{a}K^{a}+\frac{\varepsilon}{2}n^{\mu}Q_{\mathbf{n}\mathbf{n}}\,.$
(60)

This is the hypersurface normal Gauß-Weingarten equation.

The last missing fundamental equations of the differential geometric $3+1$
decomposition are the Ricci identities for $e^{\mu}_{a}$ and $n^{\mu}$.
Straightforward evaluation of the second covariant exterior derivatives of
$e^{\mu}_{a}$ and $n^{\mu}$ yields

$\displaystyle D^{2}e^{\mu}_{a}$
$\displaystyle=e^{\nu}_{a}\Omega\indices{{}^{\mu}_{\nu}}-e^{\mu}_{b}\Omega\indices{{}^{b}_{a}}\,,$
(61a) $\displaystyle D^{2}n^{\mu}$
$\displaystyle=\Omega\indices{{}^{\mu}_{\nu}}n^{\nu}\,,$ (61b)

where the curvature two-form on $\mathcal{M}$ is given by

$\displaystyle\Omega\indices{{}^{\mu}_{\nu}}$
$\displaystyle\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\mathrm{d}\omega^{\mu}_{\nu}+\omega^{\mu}_{\rho}\wedge\omega^{\rho}_{\nu}=\frac{1}{2}R\indices{{}^{\mu}_{\nu}{}_{\rho}{}_{\sigma}}\theta^{\rho}\wedge\theta^{\sigma}\,,$
(62)

and we defined
$\Omega\indices{{}^{a}_{b}}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\mathrm{d}\omega^{a}_{b}+\omega^{a}_{c}\wedge\omega^{c}_{b}$,
the boundary curvature, analogously. The tensor components
$R\indices{{}^{\mu}_{\nu}{}_{\rho}{}_{\sigma}}$ on the right hand side of (62)
are those of the Riemann curvature tensor.

The equations we have derived in this section are sufficient to derive the
Gauß-Codazzi equations, which are at the heart of the $3+1$ formalism. They
describe how curvature decomposes into contributions normal and tangent to the
hypersurfaces of the foliation. We perform the derivation of these equations
and their analogues for torsion and non-metricity in the following subsection.
We begin by investigating the $3+1$ decomposition of non-metricity.

###  4.5 Decomposition of the field strengths

In order to examine which boundary terms the non-metricity one-form
$Q_{\mu\nu}$ admits, we decompose this differential form into contributions
normal and tangent to the hypersurfaces given by the foliation of
$\mathcal{M}$. For this derivation we make extensive use of the $3+1$
decompositions derived in the previous subsections, among which the
generalized Gauß-Weingarten equations (57), (60) are particularly important.
We start our investigation by using (41a) to decompose uncontracted manifold
indices of a differential form $A_{\mu}$ as

$\displaystyle
A_{\mu}=\delta^{\nu}_{\mu}A_{\nu}=e_{\mu}^{a}e^{\nu}_{a}A_{\nu}+\varepsilon
n_{\mu}n^{\nu}A_{\nu}\,.$ (63)

Analogous decompositions hold for contravariant indices. We apply this index
decomposition to the non-metricity one-form $Q_{\mu\nu}$ to obtain888We use
symmetrization as
$A_{(\mu_{1}\dots\mu_{p})}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\frac{1}{p!}\sum_{\sigma\in\mathcal{S}}A_{\sigma(\mu_{1})\dots\sigma(\mu_{p})}$
over all permutations $\sigma$ in the permutation group $\mathcal{S}$.
Likewise, we define
$A_{[\mu_{1}\dots\mu_{p}]}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\frac{1}{p!}\sum_{\sigma\in\mathcal{S}}\mathrm{sgn}(\sigma)A_{\sigma(\mu_{1})\dots\sigma(\mu_{p})}$
as antisymmetrization, where $\mathrm{sgn}(\sigma)$ equals $+1$ if $\sigma$
consists of an even number of transpositions and $-1$ else.

$\displaystyle
Q_{\mu\nu}=e_{\mu}^{a}e_{\nu}^{b}\left(e_{a}^{\alpha}e_{b}^{\beta}Q_{\alpha\beta}\right)+2\varepsilon
e_{(\mu}^{a}n_{\nu)}^{\vphantom{a}}\left(e_{a}^{\alpha}n^{\beta}Q_{\alpha\beta}\right)+n_{\mu}n_{\nu}\left(n^{\alpha}n^{\beta}Q_{\alpha\beta}\right)\,.$
(64)

To make the decomposition (64) useful, we recall the definition of non-
metricity (58) and use the $3+1$ decomposition of the metric (49) along with
with repeated use of the generalized Gauß-Weingarten equation (57) and its
normal counterpart (60). We obtain

$\displaystyle Q_{\mu\nu}=e^{a}_{\mu}e^{b}_{\nu}(-D\gamma_{ab})+2\varepsilon
e_{(\mu}^{a}n_{\nu)}^{\vphantom{a}}\left(K_{a}-\tilde{K}_{a}\right)+2n_{\mu}n_{\nu}n_{\alpha}Dn^{\alpha}\,.$
(65)

This result decomposes the indices of $Q_{\mu\nu}$ into contributions normal
and tangent to the hypersurfaces in the foliation of $\mathcal{M}$. In
contrast to (64), (65) expresses these contributions solely in terms of
fundamental fields on the hypersurface. Hence, the comparison of (65) and (64)
yields the $3+1$ decomposition of the non-metricity one-form that reads

$\displaystyle e^{\mu}_{a}e^{\nu}_{b}Q_{\mu\nu}=-D\gamma_{ab}\,,\qquad
e^{\mu}_{a}n^{\nu}Q_{\mu\nu}=K_{a}-\tilde{K}_{a}\,,\qquad
n^{\mu}n^{\nu}Q_{\mu\nu}=2n_{\mu}Dn^{\mu}\,.$ (66)

This is the decomposition of non-metricity we aim for. The projections on the
left hand sides of (66) are expressed through boundary data on the right hand
sides. Note that $n_{\mu}Dn^{\mu}$ is a manifold scalar and does not have to
be express through boundary data any further.

Next we derive the corresponding $3+1$ decompositions for torsion and
curvature. Torsion is defined as the field strength of the coframe field as

$\displaystyle
T^{\mu}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=D\theta^{\mu}=\frac{1}{2}T\indices{{}^{\mu}_{\rho}{}_{\sigma}}\theta^{\rho}\wedge\theta^{\sigma}\,.$
(67)

As in the case of non-metricity, we need to decompose the torsion two-form
into contributions which are normal and tangent to the hypersurfaces w.r.t.
the foliation of the manifold. The calculation proceeds identically to that of
the non-metricity decomposition above. Its result is that the torsion two-form
is decomposed as

$\displaystyle e^{a}_{\mu}T^{\mu}$
$\displaystyle=D\phi^{a}+NK^{a}\wedge\phi=T^{a}+NK^{a}\wedge\phi\,,$ (68a)
$\displaystyle n_{\mu}T^{\mu}$
$\displaystyle=-\tilde{K}_{a}\wedge\phi^{a}+\varepsilon
D(N\phi)+\frac{N}{2}Q_{\mathbf{n}\mathbf{n}}\wedge\phi\,,$ (68b)

with $T^{a}$ the torsion 2-form on the hypersurfaces.

The final $3+1$ decomposition of the field strengths to consider is the one of
the curvature two-form defined in (62). In addition to the methods used for
non-metricity and torsion we exploit the Ricci identities (61) for
$e^{\mu}_{a}$ and $n^{\mu}$ in the derivation of the curvature decomposition.
Apart from that, the analysis proceeds similar to the one of non-metricity and
yields

$\begin{aligned}
\vphantom{\frac{1}{2}}e_{\mu}^{a}e^{\nu}_{b}\Omega\indices{{}^{\mu}_{\nu}}&=\Omega\indices{{}^{a}_{b}}-\varepsilon
K^{a}\wedge\tilde{K}_{b}\,,\\\
e^{a}_{\mu}n^{\nu}\Omega\indices{{}^{\mu}_{\nu}}&=DK^{a}+\frac{\varepsilon}{2}K^{a}\wedge
Q_{\mathbf{n}\mathbf{n}}\,,\end{aligned}\qquad\begin{aligned}
n_{\mu}e^{\nu}_{a}\Omega\indices{{}^{\mu}_{\nu}}&=-D\tilde{K}_{a}+\frac{\varepsilon}{2}\tilde{K}_{a}\wedge
Q_{\mathbf{n}\mathbf{n}}\,,\\\
n_{\mu}n^{\nu}\Omega\indices{{}^{\mu}_{\nu}}&=\frac{1}{2}DQ_{\mathbf{n}\mathbf{n}}+K^{a}\wedge\tilde{K}_{a}\,.\end{aligned}$
(69)

These equations are generalizations of the Gauß-Codazzi equations. As we have
seen in section 3, they are particularly relevant for the calculation of GHY
terms if a specific Lagrangian $\mathcal{L}$ is considered. In particular, the
calculation of GHY terms which are of second or higher order in curvature
always involves terms like $e^{a}_{\mu}n^{\nu}\Omega\indices{{}^{\mu}_{\nu}}$
which are evaluated by means of (69). Moreover, as we show next, the $3+1$
decompositions of curvature, torsion and non-metricity which we have just
derived are sufficient for calculating the GHY term for generic Lagrangians
which are polynomials or Taylor series of these field strengths.

###  4.6 Post-Riemannian Gibbons-Hawking-York term

This section follows [43] to derive a generalized GHY term from the foliations
of curvature, torsion and non-metricity. For the sake of generality we
consider an action of the form

$\displaystyle
S_{\mathrm{orig}}[g_{\mu\nu},\omega^{\mu}_{\nu},\theta^{\mu}]=\int_{\mathcal{M}}\mathcal{L}(\Omega\indices{{}^{\mu}_{\nu}},T^{\mu},Q_{\mu\nu})\,,$
(70)

where the Lagrangian $\mathcal{L}$ is a polynomial in curvature
$\Omega\indices{{}^{\mu}_{\nu}}$, torsion $T^{\mu}$ and non-metricity
$Q_{\mu\nu}$.999Derivatives of $\Omega\indices{{}^{\mu}_{\nu}}$, $T^{\mu}$ or
$Q_{\mu\nu}$ may be reduced to polynomials by means of the Bianchi identities
(2). We introduce Lagrange multipliers $\varphi\indices{{}_{\mu}^{\nu}}$,
$t_{\mu}$ and $q^{\mu\nu}$ in order to linearize $\cal L$ and derive the GHY
term more easily.

$\displaystyle\begin{split}S[g_{\mu\nu},\omega^{\mu}_{\nu},\theta^{\mu},\varphi\indices{{}^{\mu}_{\nu}},\varrho\indices{{}^{\mu}_{\nu}},t_{\mu},\tau^{\mu},q^{\mu\nu},&\sigma_{\mu\nu}]\\\
=\int_{\mathcal{M}}&\left[\mathcal{L}(\varrho\indices{{}^{\mu}_{\nu}},\tau^{\mu},\sigma_{\mu\nu})+*\varphi\indices{{}_{\mu}^{\nu}}\wedge(\Omega\indices{{}^{\mu}_{\nu}}-\varrho\indices{{}^{\mu}_{\nu}})\right.\\\
&\left.\vphantom{*\varphi\indices{{}_{\mu}^{\nu}}}+*t_{\mu}\wedge\left(T^{\mu}-\tau^{\mu}\right)+*q^{\mu\nu}\wedge\left(Q_{\mu\nu}-\sigma_{\mu\nu}\right)\right]\,.\end{split}$
(71)

Integrating out the multipliers shows that the action (71) yields the same
equations of motion as (70). The remaining fields
$\varrho\indices{{}^{\mu}_{\nu}}$, $\tau^{\mu}$ and $\sigma_{\mu\nu}$ in (71)
introduced are auxiliary fields that we demand to have the same symmetries as
the corresponding field strength. This symmetry identification is also
required for the Lagrange multipliers $\varphi\indices{{}_{\mu}^{\nu}}$,
$t_{\mu}$ and $q^{\mu\nu}$ in order to allow equivalence of the equations of
motion of (70) and (71) [43].

To proceed we substitute the $3+1$ decompositions of
$\Omega\indices{{}^{\mu}_{\nu}}$, $T^{\mu}$ and $Q_{\mu\nu}$ in the action
(71) in order to decompose $S$ into boundary tangent and normal contributions
and, subsequently, isolate the boundary terms. For this, we write by means of
(41a) for the linearized terms in $S$

$\displaystyle\begin{split}*\varphi\indices{{}_{\mu}^{\nu}}\wedge\Omega\indices{{}^{\mu}_{\nu}}=&\left(e^{\alpha}_{a}e_{\beta}^{b}*\varphi\indices{{}_{\alpha}^{\beta}}\right)\wedge\left(e_{\mu}^{a}e^{\nu}_{b}\Omega\indices{{}^{\mu}_{\nu}}\right)+\varepsilon\left(e^{\alpha}_{a}n_{\beta}*\varphi\indices{{}_{\alpha}^{\beta}}\right)\wedge\left(e_{\mu}^{a}n^{\nu}\Omega\indices{{}^{\mu}_{\nu}}\right)\\\
&+\varepsilon\left(n^{\alpha}e_{\beta}^{a}*\varphi\indices{{}_{\alpha}^{\beta}}\right)\wedge\left(n_{\mu}e^{\nu}_{a}\Omega\indices{{}^{\mu}_{\nu}}\right)+\left(n^{\alpha}n_{\beta}*\varphi\indices{{}_{\alpha}^{\beta}}\right)\wedge\left(n_{\mu}n^{\nu}\Omega\indices{{}^{\mu}_{\nu}}\right)\,,\end{split}$
(72a) $\displaystyle\begin{split}*t_{\mu}\wedge
T^{\mu}=&\left(e^{\nu}_{a}*t_{\nu}\right)\wedge\left(e_{\mu}^{a}T^{\mu}\right)+\varepsilon\left(n^{\nu}*t_{\nu}\right)\wedge\left(n_{\mu}T^{\mu}\right)\,,\end{split}$
(72b) $\displaystyle\begin{split}*q^{\mu\nu}\wedge
Q_{\mu\nu}=&\left(e_{\alpha}^{a}e_{\beta}^{b}*q^{\alpha\beta}\right)\wedge\left(e^{\mu}_{a}e^{\nu}_{b}Q_{\mu\nu}\right)+\varepsilon\left(e_{\alpha}^{a}n_{\beta}*q^{\alpha\beta}\right)\wedge\left(e_{a}^{\mu}n^{\nu}Q_{\mu\nu}\right)\\\
&+\varepsilon\left(n_{\alpha}e^{a}_{\beta}*q^{\alpha\beta}\right)\wedge\left(n^{\mu}e^{\nu}_{a}Q_{\mu\nu}\right)+\left(n_{\alpha}n_{\beta}*q^{\alpha\beta}\right)\wedge\left(n^{\mu}n^{\nu}Q_{\mu\nu}\right)\,.\end{split}$
(72c)

The right hand side of each of these equations can be written in terms of the
$3+1$ decompositions of $Q_{\mu\nu}$, $T^{\mu}$ and
$\Omega\indices{{}^{\mu}_{\nu}}$ shown in (66), (68) and (69). For example,
(72a) takes the form

$\displaystyle\begin{split}*\varphi\indices{{}_{\mu}^{\nu}}&\wedge\Omega\indices{{}^{\mu}_{\nu}}\\\
&=\left(e^{\alpha}_{a}e_{\beta}^{b}*\varphi\indices{{}_{\alpha}^{\beta}}\right)\wedge\left(\Omega\indices{{}^{a}_{b}}-\varepsilon
K^{a}\wedge\tilde{K}_{b}\right)+\varepsilon\left(e^{\alpha}_{a}n_{\beta}*\varphi\indices{{}_{\alpha}^{\beta}}\right)\wedge\left(DK^{a}+\frac{\varepsilon}{2}K^{a}\wedge
Q_{\mathbf{n}\mathbf{n}}\right)\\\
&+\varepsilon\left(n^{\alpha}e_{\beta}^{a}*\varphi\indices{{}_{\alpha}^{\beta}}\right)\wedge\left(D\tilde{K}_{a}+\frac{\varepsilon}{2}\tilde{K}_{a}\wedge
Q_{\mathbf{n}\mathbf{n}}\right)+\left(n^{\alpha}n_{\beta}*\varphi\indices{{}_{\alpha}^{\beta}}\right)\wedge\left(\frac{1}{2}DQ_{\mathbf{n}\mathbf{n}}+K^{a}\wedge\tilde{K}_{a}\right)\,.\end{split}$
(73)

We insert this decomposition and the analogues for torsion and non-metricity
into the action (71). The derivative terms yield by means of Stokes’ theorem
[50, 51] a boundary action, which reads

$\displaystyle\begin{split}S_{\mathrm{bdy}}=&\int_{\partial\mathcal{M}}\left[\omega^{a}_{b}\wedge\left(e^{\alpha}_{a}e_{\beta}^{b}*\varphi\indices{{}_{\alpha}^{\beta}}\right)+\phi^{a}\wedge\left(e^{\nu}_{a}*t_{\nu}\right)-\gamma_{ab}\left(e_{\alpha}^{a}e_{\beta}^{b}*q^{\alpha\beta}\right)\vphantom{\frac{1}{2}}\right.\\\
&\left.\left.\vphantom{\frac{1}{2}}-\varepsilon\tilde{K}_{a}\wedge\left(n^{\alpha}e_{\beta}^{a}*\varphi\indices{{}_{\alpha}^{\beta}}\right)+\varepsilon
K^{a}\wedge\left(e^{\alpha}_{a}n_{\beta}*\varphi\indices{{}_{\alpha}^{\beta}}\right)+\frac{1}{2}Q_{\mathbf{n}\mathbf{n}}\wedge\left(n^{\alpha}n_{\beta}*\varphi\indices{{}_{\alpha}^{\beta}}\right)\right]\right|_{\partial\mathcal{M}}\,.\end{split}$
(74)

Note that we use the restriction of the integrand in (74) to
$\partial\mathcal{M}$ as abbreviation for its pullback to
$\partial\mathcal{M}$.

We finally construct the GHY term which cancels the boundary contributions to
the action (71). Only some of the terms in (74) actually contribute boundary
terms when we consider variations of the action. In particular, the
variational principle assumes that $\left.\delta
g_{\mu\nu}\right|_{\mathrm{\partial\mathcal{M}}}=0$,
$\left.\delta\theta^{\mu}\right|_{\mathrm{\partial\mathcal{M}}}=0$ and
$\left.\delta\omega^{\mu}_{\nu}\right|_{\mathrm{\partial\mathcal{M}}}=0$.
According to [36] it equivalently suffices to demand that the Dirichlet
boundary conditions $\delta\gamma_{ab}=0$, $\delta\phi^{a}=0$ and
$\delta\omega^{a}_{b}=0$ hold since the original conditions may be reinstated
by gauge transformations on $\partial\mathcal{M}$.101010By the same argument
the residual terms of the curvature, torsion and non-metricity foliations
which are not exact forms do not contribute to the GHY term. We have, hence,
disregarded them in the consideration above. This implies that the GHY term of
(70) only has to cancel the contributions from the second line of (74) and,
thus, takes the form

$\displaystyle
S_{\mathrm{GHY}}=-\int_{\partial\mathcal{M}}\left.\left(-\varepsilon\tilde{K}_{a}\wedge*\varphi^{\mathbf{n}a}+\varepsilon
K^{a}\wedge*\varphi_{a\mathbf{n}}+Q_{\mathbf{n}\mathbf{n}}\wedge\frac{1}{2}*\varphi_{\mathbf{n}\mathbf{n}}\right)\right|_{\mathrm{\partial\mathcal{M}}}\,,$
(75)

where we abbreviated $*\varphi^{\mathbf{n}a}\equiv
n^{\alpha}e_{\beta}^{a}*\varphi\indices{{}_{\alpha}^{\beta}}$,
$*\varphi_{a\mathbf{n}}\equiv
e^{\alpha}_{a}n_{\beta}*\varphi\indices{{}_{\alpha}^{\beta}}$ and
$*\varphi_{\mathbf{n}\mathbf{n}}\equiv
n^{\alpha}n_{\beta}*\varphi\indices{{}_{\alpha}^{\beta}}$. The result (75) is
the general form of the GHY term in theories with torsion and non-metricity we
aim for. It is the main result of this paper. In particular, (75) is the GHY
term for any theory which is constituted by a Lagrangian $\mathcal{L}$ that
is, possibly an infinite, polynomial in curvature, torsion and non-metricity.

To apply our result to a specific Lagrangian $\mathcal{L}$ in (70), we need to
know the Lagrange multipliers $\varphi^{\mathbf{n}a}$, $\varphi_{a\mathbf{n}}$
and $\varphi_{\mathbf{n}\mathbf{n}}$. To obtain these, we write the action
(71) in a $3+1$ form in terms of $(\varphi^{\mathbf{n}a},\leavevmode\nobreak\
\varphi_{a\mathbf{n}},\leavevmode\nobreak\
\varphi_{\mathbf{n}\mathbf{n}},\dots)$ instead of
$\varphi\indices{{}_{\mu}^{\nu}}$, using (72). Then we can read the required
components of $\varphi$ from the equations of motion of $\varrho_{\mu\nu}$

$\displaystyle\begin{split}*\varphi^{\mathbf{n}a}\wedge\delta\varrho_{\mathbf{n}a}&=\varepsilon\delta_{\varrho_{\mathbf{n}a}}\mathcal{L}(\varrho_{\mathbf{n}a},\varrho^{a\mathbf{n}},\varrho_{\mathbf{n}\mathbf{n}},\dots)\,,\\\
*\varphi_{a\mathbf{n}}\wedge\delta\varrho^{a\mathbf{n}}&=\varepsilon\delta_{\varrho^{a\mathbf{n}}}\mathcal{L}(\varrho_{\mathbf{n}a},\varrho^{a\mathbf{n}},\varrho_{\mathbf{n}\mathbf{n}},\dots)\,,\\\
*\varphi^{\mathbf{n}\mathbf{n}}\wedge\delta\varrho_{\mathbf{n}\mathbf{n}}&=\hphantom{\varepsilon}\delta_{\varrho_{\mathbf{n}\mathbf{n}}}\mathcal{L}(\varrho_{\mathbf{n}a},\varrho^{a\mathbf{n}},\varrho_{\mathbf{n}\mathbf{n}},\dots)\end{split}$
(76)

derived from (71). Through these constraints, we may calculate
$S_{\mathrm{GHY}}$ for any given action polynomial in curvature, torsion and
non-metricity.

We emphasize that (75) and (76) are the only equations necessary to calculate
the GHY term for a specific action. The advantage of our method is, thus, not
only the universality of our results but furthermore the efficiency of the
calculation. We have already observed this efficiency in the examples examined
in section 3.

Finally we note that the GHY term (75) simplifies considerably if non-
metricity is absent as shown below.

####  4.6.1 Gibbons-Hawking-York term for metric compatible theories

Let us simplify the above result for theories which are metric compatible,
that is, in which the non-metricity one-form $Q_{\mu\nu}$ vanishes. For metric
compatible theories, the Bianchi identity of non-metricity (2),
$\Omega_{\mu\nu}+\Omega_{\nu\mu}=DQ_{\mu\nu}$, forces the curvature two-form
to be antisymmetric. Since the corresponding Lagrange multipliers $\varphi$
and $\varrho$ are required to have the same symmetry as
$\Omega\indices{{}^{\mu}_{\nu}}$, it follows that $\varphi$ is antisymmetric
in its two indices. This allows to simplify the Gibbons-Hawking York term as

$\displaystyle
S_{\mathrm{GHY}}^{Q=0}=2\int_{\mathrm{\partial\mathcal{M}}}\left.\varepsilon
K^{a}\wedge*\varphi_{\mathbf{n}a}\right|_{\mathrm{\partial\mathcal{M}}}$ (77)

in the metric-compatible case. Note that due to the antisymmetry of the
curvature two-form, formally we must consider the constraints (76) as not
independent. Nevertheless, we may forego this caveat if we treat
$\delta\varrho_{\mathbf{n}a}$ and $\delta\varrho_{a\mathbf{n}}$ as independent
variations and use the symmetry conditions only after the variational
calculus. In appendix B we show explicitly that both methods yield the same
result. This concludes the explicit derivation of the GHY boundary term for
any action polynomial in curvature, torsion and non-metricity, as anticipated
in section 2. We refer to the interpretation of our results in section 2.

##  5 Summary and discussion

In conclusion, in this paper we have accomplished the first step towards the
full holographic renormalization of metric affine gravity (MAG) theories,
namely the derivation of the Gibbons-Hawking-York term (12) in closed form for
any given polynomial action (5) involving curvature, torsion and non-
metricity.111111This includes non-polynomial Lagrangians that can be Taylor
expanded in the field strengths. In addition, our method for calculating the
GHY term is very efficient in practical terms, since it amounts to evaluating
a single variation, (10). Interpreting the main result (12), we notice that
only explicitly curvature-related terms in the Lagrangian contribute to the
GHY boundary terms. If a Lagrangian depends solely on torsion and non-
metricity, we do not need to introduce GHY terms in order for the Dirichlet
variational problem to be well-defined.121212Of course, boundary terms may
arise when these theories must satisfy additional constraints, such as setting
the curvature two-form $\Omega\indices{{}^{\mu}_{\nu}}=0$ as in teleparallel
theories of gravity [44].

We tested our method explicitly for two theories with known GHY terms,
Einstein-Hilbert gravity and 4d Chern-Simons modified gravity, in sections 3.1
and 3.2 respectively. Moreover, we used our method to derive the GHY term for
Lovelock-Chern-Simons theory in section 3.3 and for metric-affine gravity in
appendix A. Our new result for the Lovelock-Chern-Simons GHY term meets
intuitive expectations: Since the Lovelock-Chern-Simons action has a form
similar to the Einstein-Hilbert and 4d Chern-Simons modified ones, we expect
their GHY terms to also be comparable. This expectation bears out as explained
in more detail in section 3.3. We emphasize that for Lovelock-Chern-Simons
theory, our result generalizes that of [16] to off-shell field configurations.
It would be interesting to evaluate our GHY term on the solution of [16] as a
consistency check.

The next step towards understanding spin and hypermomentum transport by means
of the AdS/CFT correspondence is now to complete the holographic
renormalization procedure for MAG theories.131313Recall spin and hypermomentum
are dual to torsion and non-metricity respectively [23]. Evaluating the MAG
action and the associated GHY term given in appendix A on the torsionful and
non-metric AdS Reissner-Nordström solution [52] will provide a hint on what
kind of divergences may appear in this procedure. In general, it will be
necessary to find the asymptotic expansion for torsion and non-metricity
coupled to an asymptotically AdS metric. This will provide us with the
holographic dictionary for torsion and non-metricity. We then have to follow
e.g. [53] and calculate the regularized on-shell action in order to find all
possible divergences, and to construct appropriate counterterms to cancel
them. This will allow us to derive the thermodynamic properties of black holes
with spin and hypermomentum. It may also enable us to study Hawking-Page type
phase transitions [54] relying entirely on spin and hypermomentum. In fact we
expect such transitions to exist, since the entropy of a black hole
generically decreases when it starts rotating [55]. Holographic
renormalization then sets the stage for applying the fluid/gravity
correspondence [39, 56] to MAG and for deriving the complete hydrodynamic
expansion of strongly-coupled holographic systems with non-trivial spin and
hypermomentum transport. We note that non-trivial spin transport in
hydrodynamics has already been discussed in [15, 17, 18, 19, 16, 20, 21], but
hypermomentum transport is still mostly unexplored. The inclusion of
hypermomentum is particularly interesting, since it contains degrees of
freedom largely left unexplored in the literature.141414See [57] for a
discussion on non-metricity of the Weyl type. These degrees of freedom are
expected to lead to interesting changes in the transport properties of matter,
since their source, non-metricity, is interpreted geometrically as the
modification of the causal structure of spacetime. Since non-metricity is by
definition first order in the hydrodynamic derivative expansion, we expect it
to contribute to hydrodynamic transport only at second order in derivatives or
higher. This makes conformal fluids, where the second order derivative
expansion is well-known [58], the most suitable candidate for carrying out
this procedure.

Apart from the AdS/CFT correspondence, MAG theories are relevant in their own
right. For example, they provide us with extensions of Einstein’s theory of
relativity based on a gauge principle [23]. Thus, they are a promising
standing point for exploring deviations from general relativity in search for
clues and constraints helpful for the construction of consistent quantum
gravity models. Moreover, torsion appears naturally in supergravity and,
hence, top-down string theory constructions [59]. In addition, non-metricity
may play a role for conformal gravity as follows: When the non-metricity is
pure trace, i.e. $Q_{\mu\nu}=\sigma g_{\mu\nu}$, it may be absorbed into the
covariant derivative. This derivative is then metric-compatible and covariant
under both diffeomorphisms and Weyl transformations (see e.g. [57]). We expect
such derivatives to be useful for the construction of non-local actions such
as the Riegert action that gives rise to the Euler term in the four-
dimensional conformal anomaly [60], and for obtaining Weyl covariant
differential operators [61, 62] and curvature scalars [63]. Finally,
restricting ourselves to the realm of classical gravity, our results may be
applied to MAG theories to test cosmological implications of deviations from
Einstein gravity along the lines suggested in [64].

To conclude, we note that our results may also be useful within the context of
studying topological field theories in the presence of torsion and non-
metricity. Topological field theories constitute a convenient way of deriving
non-dissipative transport properties, in this case for spin and hypermomentum,
without the necessity of solving the underlying electronic dynamics (see [7]
for instance). For example, we may follow [65] instead of a more complicated
entropy current analysis. Our results and specifically our hypersurface
decomposition for torsion and non-metricity allows writing down such theories
in a spacetime with a timelike boundary. We expect them to be relevant for
describing systems relevant for condensed matter physics.

##  6 Acknowledgments

We thank Daniel Grumiller, Umut Gürsoy, Stefan Theisen, and Zhou-Yu Xian for
useful discussions. The work in Würzburg is funded by the Deutsche
Forschungsgemeinschaft (DFG, German Research Foundation) through Project-ID
258499086—SFB 1170 ?ToCoTronics? and through the Würzburg-Dresden Cluster of
Excellence on Complexity and Topology in Quantum Matter – ct.qmat Project-ID
390858490—EXC 2147. The research of B.H. is funded by the Bundesministerium
für Bildung und Forschung (BMBF, German Federal Ministry of Education and
Research) through the Cusanuswerk - Bischöfliche Studienförderung. I.M.’s
research is supported by the ?Curiosity Driven Grant 2020? of the University
of Genova and by the INFN Scientific Initiatives SFT: ?Statistical Field
Theory, Low-Dimensional Systems, Integrable Models and Applications?.

##  Appendix A GHY term for metric-affine gravity (MAG)

Metric-affine gravity contains non-vanishing non-metricity in addition to
curvature and torsion. According to [23, 52] the most general Lagrangian
density which is at most quadratic in these fields and parity conserving is

$\displaystyle\begin{split}V_{\mathrm{MAG}}=\frac{1}{2\kappa}\left[\vphantom{\frac{1}{2\kappa}}\right.&-a_{0}R^{\alpha\beta}\wedge\eta_{\alpha\beta}-2\Lambda\eta+T^{\alpha}\wedge*\left(\sum\limits_{I=1}^{3}a_{I}\,{}^{(I)}T_{\alpha}\right)\\\
&+2\left(\sum\limits_{I=2}^{4}c_{I}\,{}^{(I)}Q_{\alpha\beta}\right)\wedge\theta^{\alpha}\wedge*T^{\beta}+Q_{\alpha\beta}\wedge*\left(\sum\limits_{I=1}^{4}b_{I}\,{}^{(I)}Q^{\alpha\beta}\right)\\\
&+b_{5}\left({}^{(3)}Q_{\alpha\gamma}\wedge\theta^{\alpha}\right)\wedge*\left({}^{(4)}Q^{\beta\gamma}\wedge\theta_{\beta}\right)\left.\vphantom{\frac{1}{2\kappa}}\right]\\\
&-\frac{1}{2\rho}R^{\alpha\beta}\wedge*\left[\vphantom{\frac{1}{2\kappa}}\right.\sum\limits_{I=1}^{6}w_{I}\,{}^{(I)}W_{\alpha\beta}+w_{7}\theta_{\alpha}\wedge(e_{\gamma}\rfloor{}^{(5)}W\indices{{}^{\gamma}_{\beta}})\\\
&+\sum\limits_{I=1}^{5}z_{I}\,{}^{(I)}Z_{\alpha\beta}+z_{6}\,\theta_{\gamma}\wedge(e_{\alpha}\rfloor{}^{(2)}Z\indices{{}^{\gamma}_{\beta}})+\sum\limits_{I=7}^{9}z_{I}\,\theta_{\alpha}\wedge(e_{\gamma}\rfloor{}^{(I-4)}Z\indices{{}^{\gamma}_{\beta}})\left.\vphantom{\frac{1}{2\kappa}}\right]\,,\end{split}$
(78)

where
$a_{0},\dots,a_{3},b_{1},\dots,b_{5},c_{2},c_{3},c_{4},w_{1},\dots,w_{7},z_{1},\dots,z_{9}$
are dimensionless constants and the curvature two-form
$R\indices{{}_{\nu}^{\mu}}$ relates to the definition in (62) as
$R\indices{{}_{\nu}^{\mu}}=\Omega\indices{{}^{\mu}_{\nu}}$. The various terms
account for the 4+3+11 irreducible contributions to $Q_{\alpha\beta}$,
$T^{\alpha}$ and $R\indices{{}_{\alpha}^{\beta}}$ under the (pseudo)orthogonal
group, respectively. The explicit form of these irreducible decompositions is
given in appendix B of [23]. $e_{\mu}$ denotes the basis dual to
$\theta^{\mu}$ and $\rfloor$ is the interior product so that
$e_{\mu}\rfloor\theta^{\nu}=\delta^{\nu}_{\mu}$. While $\kappa$ is related to
Newton’s constant $G_{n}$ in $n$ dimensions as $\kappa=8\pi G_{n}$, the
coefficient $\rho$ controls the curvature squared terms and is, thus, called
strong gravity coupling constant.

Let us consider the terms in (78) which are proportional to $(2\kappa)^{-1}$
first. The first one of them, $-a_{0}R^{\alpha\beta}\wedge\eta_{\alpha\beta}$,
is exactly the Einstein-Hilbert Lagrangian which we already discussed in
section 3.1. The remainder of the terms proportional to $(2\kappa)^{-1}$ does
not contain curvature and therefore does not contribute to the GHY term as we
already noticed in sections 2 and 4. Therefore, the interesting part of the
MAG Lagrangian (78) in view of the GHY term is

$\displaystyle\begin{split}V_{\mathrm{MAG},\rho}=&-\frac{1}{2\rho}R^{\alpha\beta}\wedge*\left[\vphantom{\frac{1}{2\kappa}}\right.\sum\limits_{I=1}^{6}w_{I}\,{}^{(I)}W_{\alpha\beta}+w_{7}\theta_{\alpha}\wedge(e_{\gamma}\rfloor{}^{(5)}W\indices{{}^{\gamma}_{\beta}})\\\
&+\sum\limits_{I=1}^{5}z_{I}\,{}^{(I)}Z_{\alpha\beta}+z_{6}\,\theta_{\gamma}\wedge(e_{\alpha}\rfloor{}^{(2)}Z\indices{{}^{\gamma}_{\beta}})+\sum\limits_{I=7}^{9}z_{I}\,\theta_{\alpha}\wedge(e_{\gamma}\rfloor{}^{(I-4)}Z\indices{{}^{\gamma}_{\beta}})\left.\vphantom{\frac{1}{2\kappa}}\right]\,.\end{split}$
(79)

Since the number of terms in this action is large compared to the actions
considered in 2, some preliminary considerations are in place before we
calculate its GHY term.

###  A.1 Variation of Hodge stars and interior products

For deriving the GHY term for the MAG Lagrangian we need to insert the
explicit forms of the Lagrange multipliers $\varphi^{\mathbf{n}a}$,
$*\varphi_{a\mathbf{n}}$ and $\varphi_{\mathbf{n}\mathbf{n}},\dots)$ into the
generic GHY result (75). We derive these explicit expressions using the
constraints (76) which relate $\varphi^{\mu\nu}$ to the variation of the
Lagrangian with respect to $\varrho_{\mu\nu}$. Considering the relevant part
of the MAG Lagrangian (79) it is immediately obvious that its variation
involves variations of Hodge duals as well as interior products. Therefore, we
investigate variations of these objects next.

To that effect, we consider differential forms $A,\leavevmode\nobreak\ B$ of
the same degree $p$ on an $n$-manifold. For evaluating expressions like
$\delta*A\wedge B$ we use the relation [66, 67, 68]

$\displaystyle*A\wedge B=*B\wedge A\,.$ (80)

Furthermore, we employ the result

$\displaystyle\begin{split}(\delta*-*\delta)A=\delta\theta^{\alpha}\wedge(e_{\alpha}\rfloor*A)&-*\left[\delta\theta^{\alpha}\wedge(e_{\alpha}\rfloor
A)\right]\\\ &+\delta
g_{\alpha\beta}\left[\theta^{(\alpha}\wedge(e^{\beta)}\rfloor*A)-\frac{1}{2}g^{\alpha\beta}*A\right]\end{split}$
(81)

of [67] for commuting $\delta$ and $*$. The right hand side of (81) does not
contribute at the boundary in the variational principle. Hence, it comes in
handy to define $\simeq$ as being equivalent at the boundary and omitting
terms which are irrelevant there. (81) therefore implies

$\displaystyle\delta*A\simeq*{}\delta A$ (82)

which we combine with (80) to find

$\displaystyle\delta*A\wedge B\simeq*B\wedge\delta A\,.$ (83)

For the further evaluation of variations in combination with the Hodge duality
and the interior product we use the relations [66, 67, 68]

$\displaystyle e_{\alpha}\rfloor A$
$\displaystyle=(-1)^{n(p-1)+\mathrm{ind}g}*(\theta_{\alpha}\wedge*A)\,,$ (84a)
$\displaystyle*(e_{\alpha}\rfloor A)$
$\displaystyle=(-1)^{p-1}\theta_{\alpha}\wedge*A\,,$ (84b) $\displaystyle
e_{\alpha}\rfloor*A$ $\displaystyle=*(A\wedge\theta_{\alpha})\,,$ (84c)
$\displaystyle**A$ $\displaystyle=(-1)^{p(n-p)+\mathrm{ind}g}A\,,$ (84d)

where $\mathrm{ind}g$ is the number of negative signs in the signature of $g$.
The combination of (82) with (84a) immediately implies

$\displaystyle\delta\left(e_{\mu}\rfloor A\right)\simeq e_{\mu}\rfloor\delta
A$ (85)

for the variation of interior products. These are all the expressions for the
variation of Hodge duals and interior product which we need to calculate the
variations of the terms in the GHY relevant part of the MAG Lagrangian (79).
Therefore, we proceed by considering variations of the involved terms
separately.

###  A.2 Variation of the MAG Lagrangian

As first step of the irreducible decomposition of curvature we follow [23] and
decompose the symmetric and antisymmetric parts of the curvature two-form as

$\displaystyle R_{\mu\nu}=W_{\mu\nu}+Z_{\mu\nu}\,,$ (86a)
$\displaystyle\text{where}\quad
W_{\mu\nu}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=R_{[\mu\nu]},\quad
Z_{\mu\nu}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=R_{(\mu\nu)}\,.$
(86b)

From this decomposition we immediately read off

$\displaystyle\delta R_{\mu\nu}=\delta W_{\mu\nu}+\delta Z_{\mu\nu}$ (87)

which we use to consider the variations of $W$ and $Z$ instead of the $R$
variation.

We start our investigation by considering variations of the terms in (79)
which include the antisymmetric part of curvature, $W_{\mu\nu}=R_{[\mu\nu]}$.
Performing the variations of the irreducible decompositions of $W_{\mu\nu}$
using the methods from the previous subsections we obtain

$\displaystyle R^{\alpha\beta}\wedge\delta*{}^{(I)}W_{\alpha\beta}$
$\displaystyle\simeq\delta W^{\alpha\beta}\wedge*{}^{(I)}W_{\alpha\beta}$ (88)

for all $I\in\\{1,2,\dots,6\\}$, where $\simeq$ omits terms irrelevant for the
GHY term calculation. (88) is a non-trivial result as we observe in the case
of the term with coefficient $w_{7}$. For this term we obtain

$\displaystyle R^{\alpha\beta}\wedge\delta*$
$\displaystyle\left(\theta_{\alpha}\wedge\left(e_{\gamma}\rfloor{}^{(5)}W\indices{{}^{\gamma}_{\beta}}\right)\right)\simeq-\frac{(-1)^{n+\mathrm{ind}g}}{n-2}\delta
W_{\mu\nu}\wedge\Theta\indices{{}^{\nu}{}^{\mu}_{[\beta\gamma]}{}^{\gamma}_{\alpha}}\left(*R^{\alpha\beta}\right)\,.$
(89)

The operator $\Theta$ is introduced such that for any differential form $A$ we
have

$\displaystyle\Theta_{\mu}(A)\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=*\left(\theta_{\mu}\wedge
A\right)\qquad\text{and}\qquad\Theta_{\mu_{p}\dots\mu_{1}}(A)\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=*\theta_{\mu_{p}}\wedge\Theta_{\mu_{p-1}\dots\mu_{1}}(A)\,.$
(90)

This definition fulfills
$\Theta_{\mu_{p}\dots\mu_{1}}\left(\Theta_{\nu_{q}\dots\nu_{1}}(A)\right)=\Theta_{\mu_{p}\dots\mu_{1}\nu_{q}\dots\nu_{1}}(A)$
which we use in the variation of the symmetric curvature contributions that we
consider next.

We calculate the relevant terms of the $Z_{\mu\nu}$ variation like for the
$W_{\mu\nu}$ one. In contrast to the latter each irreducible component of
$Z_{\mu\nu}$ appears in two terms in the Lagrangian (79). Before we give the
results of the variation we introduce the decomposition
$Z_{\alpha\beta}=\cancelto{}{Z}_{\alpha\beta}+\frac{1}{n}g_{\alpha\beta}Z$,
where
$Z\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=Z\indices{{}^{\alpha}_{\alpha}}$
is the curvature trace.

As in the case of $W_{\mu\nu}$ some of the variations simplify significantly
as

$\displaystyle R^{\alpha\beta}\wedge\delta*{}^{(I)}Z_{\alpha\beta}$
$\displaystyle\simeq\delta\cancelto{}{Z}^{\alpha\beta}\wedge*{}^{(I)}Z_{\alpha\beta}$
(91)

for $I\in\\{2,4,5\\}$. Prominently, $I=3$ is missing in this list. For the
according term we obtain

$\displaystyle R^{\alpha\beta}\wedge\delta*{}^{(3)}Z_{\alpha\beta}$
$\displaystyle\simeq\frac{(-1)^{n+\mathrm{ind}g}}{n^{2}-4}\delta\cancelto{}{Z}_{\mu\nu}\wedge\Theta^{\nu\mu}\left(*\left(n(n-2)\Delta-2Z\right)\right)\,.$
(92)

This result does not simplify in the same way as the other terms did since the
combination of terms in ${}^{(3)}Z_{\alpha\beta}$ is not trivial. In
particular, one of the terms involved in ${}^{(3)}Z_{\alpha\beta}$ is
proportional to $g_{\alpha\beta}$ and in the contraction with
$R^{\alpha\beta}$ thus yields the factor of $Z\equiv
Z\indices{{}^{\alpha}_{\alpha}}$ which we observe in the result. But for
casting the result in the way one naively expects from comparison with (91),
(92) may solely depend on $\cancelto{}{Z}_{\alpha\beta}$ but not on $Z$.
Hence, the structure of ${}^{(3)}Z_{\alpha\beta}$ makes it impossible to
rewrite (92) in the same way as the remaining terms.

Accordingly, the variation of ${}^{(1)}Z_{\alpha\beta}$ is non-trivial, too,
and needs to be evaluated according to

$\displaystyle\begin{split}R^{\alpha\beta}\wedge\delta*{}^{(1)}Z_{\alpha\beta}&=R^{\alpha\beta}\wedge\left[\delta*Z_{\alpha\beta}-\sum_{I=2}^{5}\delta*{}^{(I)}Z_{\alpha\beta}\right]\,.\end{split}$
(93)

The rest of the combinations of curvature terms in the MAG action mixes not
only $\cancelto{}{Z}_{\alpha\beta}$ and $Z$ but furthermore $Z_{\alpha\beta}$
and $W_{\alpha\beta}$ terms in general. Hence, there is no simplification like
the ones presented above. The results for these variations are

$\displaystyle\begin{split}R^{\alpha\beta}\wedge\delta*&\left(\theta^{\gamma}\wedge\left(e_{\alpha}\rfloor{}^{(2)}Z_{\gamma\beta}\right)\right)\\\
\simeq&\frac{1}{2}\delta\cancelto{}{Z}_{\mu\nu}\wedge\theta^{\nu}\wedge\left[-(-1)^{\mathrm{ind}g}*\left(\theta_{\alpha}\wedge\theta_{\gamma}\wedge*\left(\theta^{(\gamma|}\wedge*R^{\alpha|\mu)}\right)\right)\right.\\\
&\hskip
62.0pt\left.+\frac{1}{n-2}\Theta\indices{{}^{\mu}_{\beta}}\left(*\left(\theta_{\alpha}\wedge\theta_{\gamma}\wedge*\left(\theta^{(\gamma|}\wedge*R^{\alpha|\beta)}\right)\right)\right)\right]\,,\end{split}$
(94a)
$\displaystyle\begin{split}R^{\alpha\beta}\wedge\delta*&\left(\theta_{\alpha}\wedge\left(e^{\gamma}\rfloor{}^{(3)}Z_{\gamma\beta}\right)\right)\\\
\simeq&\frac{1}{n^{2}-4}\delta\cancelto{}{Z}_{\mu\nu}\wedge\Theta\indices{{}^{\mu}{}^{\nu}_{\beta}}\left[n(-1)^{n+\mathrm{ind}g}\Theta\indices{{}_{\gamma}^{(\gamma|}{}_{\alpha}}\left(*R^{\alpha|\beta)}\right)-2\Theta_{\alpha}\left(*R^{\alpha\beta}\right)\right]\,,\end{split}$
(94b)
$\displaystyle\begin{split}R^{\alpha\beta}\wedge\delta*&\left(\theta_{\alpha}\wedge\left(e^{\gamma}\rfloor{}^{(4)}Z_{\gamma\beta}\right)\right)\simeq\frac{(-1)^{n+\mathrm{ind}g}}{n}\delta
Z\wedge\Theta_{\beta\alpha}\left(*R^{\alpha\beta}\right)\,,\end{split}$ (94c)
$\displaystyle\begin{split}R^{\alpha\beta}\wedge\delta*&\left(\theta_{\alpha}\wedge\left(e^{\gamma}\rfloor{}^{(5)}Z_{\gamma\beta}\right)\right)\\\
\simeq&\frac{(-1)^{n}}{n}\delta\cancelto{}{Z}_{\mu\nu}\wedge\Theta^{\nu}\left[2\Theta\indices{{}_{\gamma}^{(\gamma|}{}_{\alpha}}\left(*R^{\alpha|\mu)}\right)+\Theta\indices{{}^{\mu}_{\beta}{}_{\gamma}^{(\gamma|}{}_{\alpha}}\left(*R^{\alpha|\beta)}\right)\right]\,.\end{split}$
(94d)

For the construction of the full GHY term of MAG we need to proceed as in the
examples in section 3. In particular, with the variations above it is
straightforward to derive the required Lagrange multipliers from (10) and
apply the Gauß-Codazzi equations (11) to simplify the remaining curvature
forms in these Lagrange multipliers. Subsequently, the multipliers simplified
that way need to be inserted into (12) to obtain the GHY term for MAG. We
outline how this calculation is performed using the variations in (88\- 94)
for vanishing non-metricity next.

###  A.3 GHY term for metric compatible MAG

For constructing the GHY term for MAG let us consider the Lagrangian (79)
again. For simplicity we only consider the case of vanishing non-metricity
here, $Q_{\mu\nu}\equiv-Dg_{\mu\nu}=0$. From the relation
$\Omega_{\mu\nu}+\Omega_{\nu\mu}=DQ_{\mu\nu}$ for the curvature two-form
$\Omega\indices{{}^{\mu}_{\nu}}$ we obtain that $\Omega_{(\mu\nu)}=0$ in this
case. Comparison with the curvature decomposition in (86) implies that
$Z_{\mu\nu}=0$. Thus, we are left with the relevant part of the MAG Lagrangian
(79) for the calculation of the metric compatible GHY term to be

$\displaystyle\begin{split}V_{\mathrm{MAG},\rho}^{Q=0}=&-\frac{1}{2\rho}R^{\alpha\beta}\wedge*\left[\vphantom{\frac{1}{2\kappa}}\sum\limits_{I=1}^{6}w_{I}\,{}^{(I)}W_{\alpha\beta}+w_{7}\theta_{\alpha}\wedge(e_{\gamma}\rfloor{}^{(5)}W\indices{{}^{\gamma}_{\beta}})\right]\,.\end{split}$
(95)

This is the part of the MAG Lagrangian (78) in the presence of torsion and
vanishing non-metricity which contributes new terms to the GHY term. Hence, we
use (95) next to determine the GHY term for metric compatible MAG with the
method presented in chapter 2.

This method requires us to calculate the Lagrange multiplier
$*\varphi^{\mathbf{n}a}$ first, since it is the only term which contributes to
the metric compatible GHY term (13). We calculate $*\varphi^{\mathbf{n}a}$ by
means of the constraints (10) to obtain

$\displaystyle\begin{split}\left.*\varphi^{\mathbf{n}a}\right|_{\varrho=R}=-\frac{1}{2\rho}&\left[\vphantom{\frac{(-1)^{n+\mathrm{ind}g}}{n-2}}2\sum\limits_{I=1}^{6}w_{I}\,*{}^{(I)}W_{\alpha\beta}+w_{7}(-1)^{n+\mathrm{ind}g}\Theta_{\mathbf{n}\gamma}\left(*{}^{(5)}W\indices{{}^{\gamma}_{|a]}}\right)\right.\\\
&\hskip
4.0pt\left.-w_{7}\frac{(-1)^{n+\mathrm{ind}g}}{n-2}\Theta\indices{{}_{[a\mathbf{n}][\mu\gamma]}^{\gamma}{}_{\nu}}\left(*R^{\nu\mu}\right)\right]\,,\end{split}$
(96)

where we used the variational results (88) and (89). We already used the
equations of motion for $\varphi^{\mu\nu}$ that are
$\varrho_{\mu\nu}=R_{\mu\nu}$. As next step, we need to $3+1$ decompose the
curvature indices by means of (41a) and impose the Gauß-Codazzi equations to
replace the curvature decomposition terms. We note that
$R_{\mu\nu}=W_{\mu\nu}$ is implicit in the irreducible components
${}^{(I)}W_{\mu\nu}$ of $W_{\mu\nu}$. For the considered case of vanishing
non-metricity, the generalized Gauß-Codazzi equations (11) simplify to

$\displaystyle\begin{split}\vphantom{\frac{1}{2}}e_{\mu}^{a}e^{\nu}_{b}R\indices{{}^{\mu}_{\nu}}&=-\mathrm{d}\omega^{a}_{b}-\omega^{a}_{c}\wedge\omega^{c}_{b}+\varepsilon
K^{a}\wedge K_{b}\,,\\\
n_{\mu}e^{\nu}_{a}R\indices{{}^{\mu}_{\nu}}&=-e_{a}^{\mu}n^{\nu}R\indices{{}_{\mu}{}_{\nu}}=DK^{a}\,,\end{split}$
(97)

where the extrinsic curvature one-form $K^{a}$ was defined in (4).

Using (12) we finally write the GHY term of (95) as

$\displaystyle\begin{split}S_{\mathrm{GHY\leavevmode\nobreak\
MAG},\rho}^{Q=0}=2\varepsilon\int_{\mathrm{\partial\mathcal{M}}}\left.\
K^{a}\wedge\left.*\varphi_{\mathbf{n}a}\right|_{\varrho=R,\,\text{Gauß-
Codazzi}}\right|_{\mathrm{\partial\mathcal{M}}}\,,\end{split}$ (98)

where $\left.*\varphi_{\mathbf{n}a}\right|_{\varrho=R,\,\text{Gauß-Codazzi}}$
is understood as evaluation of (96) using the Gauß-Codazzi equations (97).
This evaluation is straightforward but yet obviously cumbersome so we state
only the implicit result here and leave the full evaluation to computer
algebra systems. The same holds for the generalization to the full MAG
Lagrangian in presence of non-metricity.

##  Appendix B GHY term for metric compatible theories

In section 2 we comment on the calculation of the GHY term in the case of
vanishing non-metricity, $Q_{\mu\nu}\equiv-Dg_{\mu\nu}=0$. The latter
condition enforces the curvature two-form $\Omega\indices{{}^{\mu}_{\nu}}$ to
be antisymmetric which is obtained from the Bianchi identity (2) of non-
metricity, $\Omega_{\mu\nu}+\Omega_{\nu\mu}=DQ_{\mu\nu}$. The Lagrange
multipliers $\varphi_{\mu\nu}$ and $\varrho_{\mu\nu}$ are assumed to have the
same symmetries as $\Omega_{\mu\nu}$, but we recommend to consider
$\varrho_{\mathbf{n}a}$ and $\varrho_{a\mathbf{n}}$ as being independent for
the variational calculation in section 2 nevertheless. In the current section
we examine the differences if one does not follow this method for the
calculation but uses $\varrho_{\mathbf{n}a}=-\varrho_{a\mathbf{n}}$ in the
variational process instead.

As a first step, we need to employ the antisymmetry of $\varrho_{\mu\nu}$ in
the $3+1$ decomposition of $*\varphi^{\mu\nu}\wedge\delta\varrho_{\mu\nu}$. In
contrast to the constraints for the calculation of $\varphi_{\mu\nu}$ we have
found in (10) we now obtain

$\displaystyle*\varphi^{\mathbf{n}a}\wedge\delta\varrho_{\mathbf{n}a}=\frac{\varepsilon}{2}\delta_{\varrho_{\mathbf{n}a}}\mathcal{L}(\varrho_{\mathbf{n}a},\dots)$
(99)

as the only relevant constraint. Note the factor $\frac{1}{2}$ on the right
hand side of (99) in contrast to (10). This factor results from using the
$\varrho_{\mu\nu}$ antisymmetry. By the same argument we obtain a factor of
$2$ from using the antisymmetry of $\varrho_{\mu\nu}$ in the calculation of
$\delta_{\varrho_{\mathbf{n}a}}\mathcal{L}(\varrho_{\mathbf{n}a},\dots)$ for a
specific theory. This factor of $2$ cancels the factor $\frac{1}{2}$ in (99).
For this reason we recommended to do the variational calculus without assuming
the antisymmetry of $\varrho_{\mu\nu}$ in section 2. Note that this assumption
is made only in the variational calculus when calculating
$\varphi^{\mathbf{n}a}$ by means of the constraints (10). It can be considered
as a method which simplifies the calculation.

Of course, for the asymmetry method to be valid, the GHY terms in both methods
need to coincide. Indeed, using the antisymmetry of
$\Omega\indices{{}^{\mu}_{\nu}}$ and $\varphi_{\mu\nu}$ yields

$\displaystyle
S_{\mathrm{GHY}}^{Q=0}=2\int_{\mathrm{\partial\mathcal{M}}}\left.\varepsilon
K^{a}\wedge*\varphi_{\mathbf{n}a}\right|_{\mathrm{\partial\mathcal{M}}}$ (100)

as GHY term in coincidence with the result (13) from the asymmetry assumption.
The discussion in this appendix generalizes straightforwardly if non-metricity
is not vanishing.

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