# Nonmetricity and Poincarè gauge gravity

James T. Wheeler†

###### Abstract

We show that in a spacetime geometry with general connection, there exist
field redefinitions that replace the mixed-symmetry nonmetricity of the
asymmetric connection with the sum of torsion and the field strength of
special conformal transformations of a Lorentzian connection.

In contrast to general relativity, generic action functionals give different
results for metric versus Palatini-style variations. We consider some of the
resulting issues in Poincarè gauge theory, where it is natural to vary both
the solder form and the spin connection, and where the action may differ from
the Einstein-Hilbert case. Independent variation may introduce nonmetricity as
well as torsion. Appending the defining equation of nonmetricity to the
Maurer-Cartan equations of the Poincarè Lie algebra, we show that the mixed
symmetry components of nonmetricity can be absorbed into an altered torsion
tensor, restoring local Lorentz symmetry and metric compatibility while
including quadratic terms in the redefined curvature.

In order to recover the original torsion and nonmetric fields, we replace the
definition of nonmetricity by an additional structure equation. We show that
the maximal Lie algebra compatible with the enlarged set is isomorphic to the
conformal Lie algebra. Therefore, in a Lorentzian conformal geometry the sum
of torsion and the field strength of special conformal transformations is
given by the mixed symmetry nonmetricity of an equivalent asymmetric system.

Keywords: Poincarè gauge theory, torsion, nonmetricity, nonmetric gravity,
general relativity, conformal, biconformal, scale invariant general relativity

†James T Wheeler, Utah State University Department of Physics, 4415 Old Main
Hill, Logan, UT 84322-4415<EMAIL_ADDRESS>
##  1 Introduction

A well-known special property of general relativity is the equivalence between
two distinct variations: (1) varying the metric while assuming the Levi-Civita
connection, and (2) varying the metric and connection independently. Without
this coincidence, independent variation of an action may lead to torsion
and/or nonmetricity. Here we explore this more general outcome.

Torsion arises naturally within Poincarè gauge theory, where it is normal to
vary the metric and connection independently, and where the Einstein-Hilbert
action is only one choice out of many possible theories consistent with the
Einstein equation. This leaves open the question of whether the metric and
connection are compatible. If compatibility is not assumed, the gauge theory
will differ from canonical general relativity by the presence of both torsion
and nonmetricity. Nonmetricity is of particular concern, since a symmetric
part to the connection will break Lorentz invariance.

We show that even if a Lorentzian theory develops nonmetricity, the
nonmetricity may always be used to antisymmetrize the spin connection,
restoring Lorentz symmetry, thereby giving an explicit realization of Theorem
5.8 of [1]. Within the recast theory, field redefinitions merge the
nonmetricity and torsion into a single modified torsion field, eliminating
their independence. We develop sufficient additional structure to recover the
original nonmetricity and torsion. This maximal extension of the nonmetric
structure is equivalent to a gauge theory of the conformal group, and shows a
suprising equivalence between nonmetricity, torsion, and the special conformal
curvature.

Our procedure is as follows. In the next Section, we begin with the Cartan
equations for Poincarè symmetry. We then drop the antisymmetry of the spin
connection and append the definition of nonmetricity. Somewhat oxymoronically,
we call this intermediate structure “nonmetric Poincarè gauge theory” because
we then show that field redefinitions restore the original Poincarè form of
the structure equations. Lorentz covariance is restored while independence of
the original fields is lost.

In Section (3) we develop an additional Lorentzian structure equation so that
the original fields may be recovered. This involves the introduction of a
potential for part of the nonmetricity. We check integrability of the extended
system, and backtrack to find an equivalent extended nonmetric system.

The addition of an additional variable ignores the fact that the new potential
might appear in the original equations. In Section (4) we establish an
underlying Maurer-Cartan system by including such extra terms while setting
curvature, torsion, and nonmetricity to zero. We ensure maximal consistency by
adding all allowed additional terms that are at least linear in the new
potential to the original nonmetric structure equations, then impose
integrability of the full set to determine the coefficients. Integrability
guarantees that the resulting equations are the Maurer-Cartan equations of a
Lie algebra. Finally, we redefine fields to recover a Lorentzian system,
showing that the maximal system is equivalent to the conformal Maurer-Cartan
equations.

The extended set may then be used to define a gravity theory by well-known
methods [2, 3]. This extends the original asymmetric system to a maximal set
that, when gauged to include the field strengths, allows us to determine both
the original torsion and nometricity. Using basis changes we relate the
nonmetric curvature, torsion and nonmetricity to the curvature, torsion, and
special conformal curvature of the conformal theory.

All calculations to this point are based on the structure equations and not on
any particular physical theory. In the final Section, summarize our findings
and briefly comment on two conformal gravity theories. We show that in the
auxiliary conformal gauge gravity there is a class of solutions in which the
field strength of the special conformal gauge field explicitly becomes the
nonmetricity. In the biconformal, or Kähler, gauging the interchangeablility
of the torsion and the mixed symmetry nonmetricity is manifest.

The remaining totally symmetric part of the nonmetricity is likely to vanish
without the–possibly unlikely [4]–inclusion of spin-3 sources. In this case we
may conclude that the study of gauge theories of gravity with general
connection may be recast as the study of conformal gravity theories.

##  2 Poincarè gauge theory from nonmetric Poincarè gauge theory

We begin with the Cartan structure equations of Poincarè gauge theory.

$\displaystyle\mathbf{d}\boldsymbol{\omega}_{\;\;\;b}^{a}$ $\displaystyle=$
$\displaystyle\boldsymbol{\omega}_{\;\;\;b}^{c}\wedge\boldsymbol{\omega}_{\;\;\;c}^{a}+\mathbf{R}_{\;\;\;b}^{a}$
(1) $\displaystyle\mathbf{d}\mathbf{e}^{a}$ $\displaystyle=$
$\displaystyle\mathbf{e}^{b}\wedge\boldsymbol{\omega}_{\;\;\;b}^{a}+\mathbf{T}^{a}$
(2)

These specify the curvature $\mathbf{R}_{\;\;\;b}^{a}$ and torsion
$\mathbf{T}^{a}$ 2-forms in terms of the spin connection
$\boldsymbol{\omega}_{\;\;\;b}^{a}$ and the solder form $\mathbf{e}^{a}$.
Poincaré gauge theory is generated from the symmetry of Minkowski space, so
that the spin connection has Lorentzian antisymmetry
$\boldsymbol{\omega}_{\;\;\;b}^{a}=-\eta^{ac}\eta_{bd}\boldsymbol{\omega}_{\;\;\;c}^{d}$
with $\eta_{ab}=diag(-1,1,1,1)$. This antisymmetry justifies the use of
Lorentzian metrics in the resulting curved spacetimes (1, 2).

Constraining to vanishing torsion and specifying the Einstein-Hilbert action
gives the starting point for general relativity, while leaving torsion
unconstrained gives the Einstein-Cartan-Sciama-Kibble (ECSK) theory of gravity
[5, 6, 7, 8, 9, 10, 11, 12, 13]. Including generic sources in the ECSK theory
drives the curvature through the resulting energy tensor and the torsion
through the spin tensor [14] when the connection and metric are varied
independently. Additional terms in the torsion [15, 16, 17, 18] and properties
of nonmetricity [19, 20, 21, 22, 23, 24, 25] have been considered extensively.

Here we consider the structure equations with independent solder form and spin
connection, and show that the system nonetheless results in local Lorentz
symmetry. Replace the spin connection $\boldsymbol{\omega}_{\;\;\;b}^{a}$ in
Eqs.(1) and (2) with an asymmetric spin connection
$\hat{\boldsymbol{\omega}}_{ab}$, and include the resulting nonzero
nonmetricity $\mathbf{Q}_{ab}\equiv\mathbf{D}g_{ab}$. The system is now
described by three equations.

$\displaystyle\mathbf{d}\hat{\boldsymbol{\omega}}_{\;\;\;b}^{a}$
$\displaystyle=$
$\displaystyle\hat{\boldsymbol{\omega}}_{\;\;\;b}^{c}\wedge\hat{\boldsymbol{\omega}}_{\;\;\;c}^{a}+\hat{\boldsymbol{\mathcal{R}}}_{\;\;\;b}^{a}$
(3) $\displaystyle\mathbf{d}\mathbf{e}^{a}$ $\displaystyle=$
$\displaystyle\mathbf{e}^{b}\wedge\hat{\boldsymbol{\omega}}_{\;\;\;b}^{a}+\hat{\mathbf{T}}^{a}$
(4) $\displaystyle\mathbf{d}g_{ab}$ $\displaystyle=$
$\displaystyle\hat{\boldsymbol{\omega}}_{ab}+\hat{\boldsymbol{\omega}}_{ba}+\hat{\mathbf{Q}}_{ab}$
(5)

where we have expanded the equation for the nonmetricity. The integrability of
Eqs.(3) and (4) does not depend on the antisymmetry of the spin connection,
leading to the usual form of the Bianchi identities with torsion. In an
orthonormal frame, $\mathbf{d}g_{ab}=\mathbf{d}\eta_{ab}=0$, so that
$\hat{\boldsymbol{\omega}}_{\left(ab\right)}=-\frac{1}{2}\mathbf{Q}_{ab}$, and
we may decompose the spin connection as

$\hat{\boldsymbol{\omega}}_{ab}=\boldsymbol{\omega}_{ab}-\frac{1}{2}\hat{\mathbf{Q}}_{ab}$
(6)

The circumflex denotes objects of the original nonmetric system (3)-(5) and
$\boldsymbol{\omega}_{ab}=\boldsymbol{\omega}_{\left[ab\right]}$ is a
Lorentzian spin connection.

Now substitute Eq.(6) into the generalized structure equations. For the
torsion equation

$\displaystyle\mathbf{d}\mathbf{e}^{a}$ $\displaystyle=$
$\displaystyle\mathbf{e}^{b}\wedge\left(\boldsymbol{\omega}_{ab}-\frac{1}{2}\hat{\mathbf{Q}}_{ab}\right)+\hat{\mathbf{T}}^{a}$
$\displaystyle=$
$\displaystyle\mathbf{e}^{b}\wedge\boldsymbol{\omega}_{\;\;\;b}^{a}+\hat{\mathbf{T}}^{a}-\hat{\mathbf{Q}}^{a}$

Here we define the mixed symmetry part of the nonmetricity as a 2-form,

$\displaystyle\hat{\mathbf{Q}}^{a}$ $\displaystyle\equiv$
$\displaystyle\frac{1}{2}\mathbf{e}^{b}\wedge\hat{\mathbf{Q}}_{\;\;\;b}^{a}=\frac{1}{2}\hat{Q}_{\;\;\;\left[bc\right]}^{a}\mathbf{e}^{b}\wedge\mathbf{e}^{c}$

where $\hat{Q}_{a\left[bc\right]}$ and $\hat{Q}_{\left(abc\right)}$ are the
irreducible parts of the nonmetricity. Since $\hat{\mathbf{Q}}^{a}$ is a
vector-valued 2-form it merely alters the torsion to
$\mathbf{T}^{a}\equiv\hat{\mathbf{T}}^{a}-\hat{\mathbf{Q}}^{a}$ as shown in
[26].

Expanding the curvature $\hat{\boldsymbol{\mathcal{R}}}_{\;\;\;b}^{a}$ using
Eq.(6) we have the usual extra terms arising from an addition to a connection,

$\displaystyle\mathbf{d}\boldsymbol{\omega}_{\;\;\;b}^{a}$ $\displaystyle=$
$\displaystyle\boldsymbol{\omega}_{\;\;\;b}^{c}\wedge\boldsymbol{\omega}_{\;\;\;c}^{a}+\hat{\boldsymbol{\boldsymbol{\mathcal{R}}}}_{\;\;\;b}^{a}-\frac{1}{2}\mathbf{D}\hat{\mathbf{Q}}_{\;\;\;b}^{a}+\frac{1}{4}\hat{\mathbf{Q}}_{\;\;\;b}^{c}\wedge\hat{\mathbf{Q}}_{\;\;\;c}^{a}$

so that

$\displaystyle\mathbf{d}\boldsymbol{\omega}_{\;\;\;b}^{a}$ $\displaystyle=$
$\displaystyle\boldsymbol{\omega}_{\;\;\;b}^{c}\wedge\boldsymbol{\omega}_{\;\;\;c}^{a}+\mathbf{R}_{\;\;\;b}^{a}$
$\displaystyle\mathbf{d}\mathbf{e}^{a}$ $\displaystyle=$
$\displaystyle\mathbf{e}^{b}\wedge\boldsymbol{\omega}_{\;\;\;b}^{a}+\mathbf{T}^{a}$
(7)

where

$\displaystyle\mathbf{R}_{\;\;\;b}^{a}$ $\displaystyle=$
$\displaystyle\hat{\boldsymbol{\boldsymbol{\mathcal{R}}}}_{\;\;\;b}^{a}-\frac{1}{2}\mathbf{D}\hat{\mathbf{Q}}_{\;\;\;b}^{a}+\frac{1}{4}\hat{\mathbf{Q}}_{\;\;\;b}^{c}\wedge\hat{\mathbf{Q}}_{\;\;\;c}^{a}$
(8) $\displaystyle\mathbf{Q}_{ab}$ $\displaystyle=$ $\displaystyle 0$ (9)

and $\mathbf{D}$ is the covariant exterior derivative of the Lorentzian
connection $\boldsymbol{\omega}_{\;\;\;b}^{a}$. Note that the curvature
$\mathbf{R}_{\;\;\;b}^{a}$ includes contributions from the contorsion
$\mathbf{C}_{\;\;\;b}^{a}$ of the modified torsion,
$\mathbf{T}^{a}=\hat{\mathbf{T}}^{a}-\hat{\mathbf{Q}}^{a}$. The antisymmetry
of $\boldsymbol{\omega}_{\;\;\;b}^{a}$ restores compatibility with the Lorentz
metric while the original symmetric field $\hat{\mathbf{Q}}_{\;\;\;b}^{a}$
folds into the curvature and vanishes from explicit consideration.

Writing the Einstein-Hilbert form of the action,

$\displaystyle S_{EH}$ $\displaystyle=$
$\displaystyle\int\boldsymbol{\mathcal{R}}^{ab}\wedge\mathbf{e}^{c}\wedge\mathbf{e}^{d}e_{abcd}$
$\displaystyle=$
$\displaystyle\int\left(\mathbf{R}^{ab}+\frac{1}{2}\mathbf{D}\hat{\mathbf{Q}}^{ab}-\frac{1}{4}\hat{\mathbf{Q}}^{eb}\wedge\hat{\mathbf{Q}}_{\;\;\;e}^{a}\right)\wedge\mathbf{e}^{c}\wedge\mathbf{e}^{d}e_{abcd}$
$\displaystyle=$
$\displaystyle\int\left(\mathbf{R}^{ab}+\frac{1}{4}\hat{\mathbf{Q}}_{\;\;\;e}^{a}\wedge\hat{\mathbf{Q}}^{be}\right)\wedge\mathbf{e}^{c}\wedge\mathbf{e}^{d}e_{abcd}$

the derivative term drops out by symmetry,
$\mathbf{D}\hat{\mathbf{Q}}^{ab}e_{abcd}=0$, and we are left with a novel
source term determined by the original nonmetricity. Varying
$\hat{\mathbf{Q}}^{ab}$ independently then leads to
$\mathbf{\hat{\mathbf{Q}}}^{ab}=0$ in the absence of sources. However, even if
we constrain to second order field equations linear in second derivatives, we
are no longer restricted to the Einstein-Hilbert form of the action. We may
add any terms up to quadratic in $\hat{\mathbf{Q}}^{ab},\mathbf{\hat{T}}^{a}$
or $\hat{\mathbf{Q}}^{a}$ as examined in [26]. Without including Spin-3 matter
sources or terms cubic in nonmetricity, it seems likely that the totally
symmetric part of the nonmetricity will vanish. However, the mixed symmetry
piece may be sourced by the same fields that drive torsion. An algebraic
solution for the mixed-symmetry nonmetricity in terms of such fields would
allow replacement of the
$\frac{1}{4}\hat{\mathbf{Q}}_{\;\;\;e}^{a}\wedge\hat{\mathbf{Q}}^{be}$ term in
the action with physical source fields.

We have shown that introducing nonmetricity into Poincarè gauge theory changes
only the apparent matter content while leaving the original Lorentzian form of
the structure equations. It is tempting to conjecture that, depending on the
action, this additional matter content might account for dark matter, dark
energy, or some other field of interest, but the formulation above does not
permit independent determination of
$\mathbf{\hat{T}}^{a},\hat{\mathbf{Q}}^{a}$ or $\hat{\mathbf{Q}}^{ab}$.

We now turn to a systematic approach that allows us to determine the original
fields.

##  3 Restoring independence of the nonmetricity

The form of Eqs.(7) confounds any independent prediction of nonmetricity and
torsion. Only the emergent final forms
$\mathbf{R}_{\;\;\;b}^{a},\mathbf{T}^{a}$ enter the structure equations, and
if the action is built from these, the field equations. Now we develop an
additional equation in the form of a structure equation to describe
nonmetricity. This extends the independent variables from
$\left(\mathbf{e}^{a},\boldsymbol{\omega}_{\;\;\;b}^{a}\right)$ to
$\left(\mathbf{e}^{a},\mathbf{f}_{a},\boldsymbol{\omega}_{\;\;\;b}^{a}\right)$
so that solving for all three connection forms determines the curvature,
torsion _and_ the nonmetricity.

###  3.1 Evaporating nonmetricity with separated Weyl vector

We begin again with Eqs.(3)-(5), this time separating the trace of the
nonmetricity $W_{b}\equiv\frac{1}{2n}\hat{Q}_{\;\;\;ab}^{a}$. This part is the
Weyl vector, which may always be included in the antisymmetric part of the
connection. To check this, solve for the spin connection in the usual way.
Separate the nonmetricity into trace and traceless parts,

$\displaystyle\hat{Q}_{abc}$ $\displaystyle=$ $\displaystyle
2\eta_{ab}W_{c}+\tilde{Q}_{abc}$

and define the altered torsion using the original _traceless_ nonmetricity
$\tilde{\mathbf{Q}}^{a}$ only.

$\displaystyle\mathbf{S}^{a}$ $\displaystyle\equiv$
$\displaystyle\hat{\mathbf{T}}^{a}-\tilde{\mathbf{Q}}^{a}$

Substituting
$\hat{\boldsymbol{\omega}}_{\;\;\;b}^{a}=\boldsymbol{\omega}_{\;\;\;b}^{a}-\frac{1}{2}\left(\tilde{\mathbf{Q}}_{\;\;\;b}^{a}+2\delta_{b}^{a}W_{c}\mathbf{e}^{c}\right)$
into the solder form equation (4) results in a Weyl connection, which then
enters the curvature (3) in the usual way. The structure equations have the
form,

$\displaystyle\mathbf{d}\boldsymbol{\omega}_{\;\;\;b}^{a}$ $\displaystyle=$
$\displaystyle\boldsymbol{\omega}_{\;\;\;b}^{c}\wedge\boldsymbol{\omega}_{\;\;\;c}^{a}+\mathbf{R}_{\;\;\;b}^{a}$
(10) $\displaystyle\mathbf{d}\mathbf{e}^{a}$ $\displaystyle=$
$\displaystyle\mathbf{e}^{b}\wedge\boldsymbol{\omega}_{\;\;\;b}^{a}+\boldsymbol{\omega}\wedge\mathbf{e}^{a}+\mathbf{S}^{a}$
(11) $\displaystyle\mathbf{d}\boldsymbol{\omega}$ $\displaystyle=$
$\displaystyle 0$ (12)

with $\boldsymbol{\omega}=W_{c}\mathbf{e}^{c}$. Here the traceless
nonmetricity vanishes from the formulation. Equations (10)-(12) describe an
integrable Weyl geometry with Weyl vector $\boldsymbol{\omega}$ and torsion
$\mathbf{S}^{a}$. Including the Weyl vector in the structure equation for the
solder form does not change its antisymmetry,
$\boldsymbol{\omega}_{ab}=-\boldsymbol{\omega}_{ba}$, so that we still have a
Lorentzian spacetime. Agreement with local scale invariance (i.e., changes of
units) requires the Weyl vector to be integrable,
$\mathbf{d}\boldsymbol{\omega}=0$.

As in the previous Section, all reference to the original nonmetricity has
vanished and the remaining spin connection is Lorentzian.

###  3.2 Independent nonmetricity and torsion

The Weyl-covariant derivative

$\displaystyle\mathbf{D}\mathbf{e}^{a}$ $\displaystyle\equiv$
$\displaystyle\mathbf{d}\mathbf{e}^{a}-\mathbf{e}^{b}\wedge\boldsymbol{\omega}_{\;\;\;b}^{a}-\boldsymbol{\omega}\wedge\mathbf{e}^{a}=\mathbf{S}^{a}$

now determines only one combination of the original two fields
$\hat{\mathbf{T}}^{a},\tilde{\mathbf{Q}}^{a}$. If we wish to recover both
original fields we need an independent equation. The equation for
$\mathbf{h}_{a}=g_{ab}\mathbf{e}^{b}$ explored in [26] is suggestive, but
results in

$\displaystyle\mathbf{d}\mathbf{h}_{a}$ $\displaystyle=$
$\displaystyle\boldsymbol{\omega}_{\;\;\;a}^{d}\wedge\mathbf{h}_{d}+\mathbf{h}_{a}\wedge\boldsymbol{\omega}+\hat{\mathbf{T}}^{a}-\tilde{\mathbf{Q}}^{a}$

which is just Eq.(11) written in covariant form.

To form an independent equation, we write a similar equation,

$\displaystyle\mathbf{d}\mathbf{f}_{a}$ $\displaystyle=$
$\displaystyle\boldsymbol{\omega}_{\;\;\;a}^{b}\wedge\mathbf{f}_{b}+\mathbf{f}_{a}\wedge\boldsymbol{\omega}+\bar{\mathbf{S}}_{a}$
(13)

where we change the field strength to
$\bar{\mathbf{S}}_{a}\equiv\hat{\mathbf{T}}_{a}+\tilde{\mathbf{Q}}_{a}$. The
1-form $\mathbf{f}_{a}$ is now independent of the solder form so that from an
appropriate action the combined system Eqs.(10)-(13) can determine both
$\hat{\mathbf{T}}_{a}$ and $\tilde{\mathbf{Q}}_{a}$.

The nonmetricity vanishes if $\mathbf{f}_{a}=\mathbf{h}_{a}$, so the
difference
$\mathbf{v}_{a}\equiv\frac{1}{2}\left(\mathbf{f}_{a}-g_{ab}\mathbf{e}^{b}\right)$
gives an equation for $\tilde{\mathbf{Q}}_{a}$ alone. Similarly, setting
$\mathbf{u}^{a}\equiv\frac{1}{2}\left(\mathbf{e}^{a}+g^{ab}\mathbf{f}_{b}\right)$
gives an equation for the original torsion. Simplifying the expressions for
$\mathbf{d}\mathbf{u}^{a}$ and $\mathbf{d}\mathbf{v}_{a}$ and noting that
$\mathbf{d}g_{ab}=\boldsymbol{\omega}_{ab}+\boldsymbol{\omega}_{ba}-2g_{ab}\boldsymbol{\omega}=-2g_{ab}\boldsymbol{\omega}$
we have the extended collection:

$\displaystyle\mathbf{d}\boldsymbol{\omega}_{\;\;\;b}^{a}$ $\displaystyle=$
$\displaystyle\boldsymbol{\omega}_{\;\;\;b}^{c}\wedge\boldsymbol{\omega}_{\;\;\;c}^{a}+\mathbf{R}_{\;\;\;b}^{a}$
$\displaystyle\mathbf{d}\mathbf{u}^{a}$ $\displaystyle=$
$\displaystyle\mathbf{u}^{b}\wedge\boldsymbol{\omega}_{\;\;\;b}^{a}+\boldsymbol{\omega}\wedge\mathbf{u}^{a}+\hat{\mathbf{T}}^{a}$
$\displaystyle\mathbf{d}\mathbf{v}_{a}$ $\displaystyle=$
$\displaystyle\boldsymbol{\omega}_{\;\;\;a}^{b}\wedge\mathbf{v}_{b}+\mathbf{v}_{a}\wedge\boldsymbol{\omega}+\tilde{\mathbf{Q}}_{a}$
$\displaystyle\mathbf{d}\boldsymbol{\omega}$ $\displaystyle=$ $\displaystyle
0$ (14)

where

$\displaystyle\mathbf{R}_{\;\;\;b}^{a}$ $\displaystyle\equiv$
$\displaystyle\boldsymbol{\mathcal{R}}_{\;\;\;b}^{a}-\frac{1}{2}\mathbf{D}\hat{\mathbf{Q}}_{\;\;\;b}^{a}+\frac{1}{4}\hat{\mathbf{Q}}_{\;\;\;b}^{c}\wedge\hat{\mathbf{Q}}_{\;\;\;c}^{a}$

and we have included the integrability of the Weyl vector. The full system now
has field strengths equal to the original torsion and nonmetricity of the
asymmetric connection. When we set $\mathbf{v}_{a}=0$ and gauge the Weyl
vector to zero we return to the original recovered Poincarè theory, Eqs.(7),
(8), and (9).

Retracing the calculation backwards from Eqs.(14), we find that the equations
in the $\left(\mathbf{e}^{a},\mathbf{f}_{a}\right)$ basis written with the
original asymmetric connection take the form

$\displaystyle\mathbf{d}\hat{\boldsymbol{\omega}}_{\;\;\;b}^{a}$
$\displaystyle=$
$\displaystyle\hat{\boldsymbol{\omega}}_{\;\;\;b}^{c}\wedge\hat{\boldsymbol{\omega}}_{\;\;\;c}^{a}+\boldsymbol{\mathcal{R}}_{\;\;\;b}^{a}$
$\displaystyle\mathbf{d}\mathbf{e}^{a}$ $\displaystyle=$
$\displaystyle\mathbf{e}^{b}\wedge\hat{\boldsymbol{\omega}}_{\;\;\;b}^{a}+\hat{\mathbf{T}}^{a}$
$\displaystyle\mathbf{d}\mathbf{f}_{a}$ $\displaystyle=$
$\displaystyle\hat{\boldsymbol{\omega}}_{\;\;\;a}^{b}\wedge\mathbf{f}_{b}+\hat{\mathbf{K}}_{a}$
$\displaystyle\mathbf{d}\boldsymbol{\omega}$ $\displaystyle=$ $\displaystyle
0$ (15)

where

$\hat{\mathbf{K}}_{a}\equiv\hat{\mathbf{T}}_{a}+\frac{1}{2}\left(\mathbf{e}^{b}-g^{bc}\mathbf{f}_{c}\right)\wedge\tilde{\mathbf{Q}}_{ab}$

As expected, the dependence on nonmetricity is proportional to the difference
$\mathbf{e}^{b}-g^{bc}\mathbf{f}_{c}$, so that when
$\mathbf{f}_{a}=g_{ab}\mathbf{e}^{b}$ the nometricity equation simply
replicates the torsion equation. We have then recovered the original nonmetric
system.

Next, we examine consistency of the new set of equations.

###  3.3 Integrability of nonmetric Poincarè gauge theory

Other than the parallel structure, we have no reason to expect that including
Eq.(13) with the original Cartan equations (10)-(12) gives a meaningful set of
structure equations. At the very least, when the curvature, torsion, and
nonmetricity all vanish the resulting vacuum equations should be integrable.
Integrability of Eqs.(10)-(13) will guarantee that the corresponding dual
vectors will satisfy closed commutation relations and the Jacobi identity. If
this is case, the extended equations must be the Maurer-Cartan equations of a
Lie algebra.

We therefore regard the vacuum form of the original equations as those
describing some larger Lie algebra with $\mathbf{f}_{a}$ set to zero. We
establish the existence of that larger algebra and find its form by adding
general $\mathbf{f}_{a}$-dependent terms to each of the original three
equations and adjoining the fourth equation for $\mathbf{f}_{a}$. Then by
demanding integrability we fix the remaining constants, and the collection
must the Maurer-Cartan equations for some Lie algebra. This algebra will
generate nonmetric Poincarè gauge theory.

Before finding this maximal extension, we look at integrability of the reduced
set. Setting the curvatures to zero, Eqs.(10)-(13) become

$\displaystyle\mathbf{d}\boldsymbol{\omega}_{\;\;\;b}^{a}$ $\displaystyle=$
$\displaystyle\boldsymbol{\omega}_{\;\;\;b}^{c}\wedge\boldsymbol{\omega}_{\;\;\;c}^{a}$
(16) $\displaystyle\mathbf{d}\mathbf{e}^{a}$ $\displaystyle=$
$\displaystyle\mathbf{e}^{b}\wedge\boldsymbol{\omega}_{\;\;\;b}^{a}+\boldsymbol{\omega}\wedge\mathbf{e}^{a}$
(17) $\displaystyle\mathbf{d}\mathbf{f}_{a}$ $\displaystyle=$
$\displaystyle\boldsymbol{\omega}_{\;\;\;a}^{b}\wedge\mathbf{f}_{b}+\mathbf{f}_{a}\wedge\boldsymbol{\omega}$
(18) $\displaystyle\mathbf{d}\boldsymbol{\omega}$ $\displaystyle=$
$\displaystyle 0$ (19)

Even though this does not give the maximal form of the Lie algebra, we gain
some insight.

Eqs.(16)-(19) are integrable, whether the spin connection is Lorentzian or
asymmetric. In the Lorentzian case Eqs.(16) and (17) describe the Weyl Lie
algebra, already known to be integrable, and integrability of the Weyl vector
is assumed. For the $\mathbf{f}_{a}$ equation we easily check that
$\mathbf{d}^{2}\mathbf{f}_{a}\equiv 0$ and integrability is established. The
proofs do not depend on the antisymmetry of the spin connection.

Since integrability of the Maurer-Cartan equations is equivalent to the Jacobi
identity, these equations describe a Lie algebra. The corresponding Lie group
is not hard to identify: in addition to Lorentz transformations and
dilatations, we now have two sets of translational gauge fields. As proof,
note that Eq.(16) shows the spin connection to be pure (local Lorentz) gauge,
$\boldsymbol{\omega}_{\;\;\;b}^{a}=-\mathbf{d}\Lambda_{\;\;\;c}^{a}\bar{\Lambda}_{\;\;\;b}^{c}$.
Choosing a cross section with constant $\Lambda_{\;\;\;b}^{a}$ we have
vanishing spin connection, $\boldsymbol{\omega}_{\;\;\;b}^{a}=0$. The same
argument holds for the Weyl vector, so we gauge to $\boldsymbol{\omega}=0$.
The remaining equations are simply

$\displaystyle\mathbf{d}\mathbf{e}^{a}$ $\displaystyle=$ $\displaystyle 0$
$\displaystyle\mathbf{d}\mathbf{f}_{a}$ $\displaystyle=$ $\displaystyle 0$

and we conclude the existence of functions $x^{a},y_{a}$ such that

$\displaystyle\mathbf{e}^{a}$ $\displaystyle=$ $\displaystyle\mathbf{d}x^{a}$
$\displaystyle\mathbf{f}_{a}$ $\displaystyle=$ $\displaystyle\mathbf{d}y_{a}$

We assume that $\mathbf{e}^{a}$ is the usual solder form, and exact
orthonormal frames describe flat spaces. Therefore, with the maximal extension
of the coordinates $x^{a}$, the $\mathbf{d}x^{a}$ span Minkowski space with
$\mathbf{e}^{a}$ dual to the generator of translations. While
$\mathbf{f}_{a}=\mathbf{d}y_{a}$ may be degenerate or not, we consider their
maximal extension in $\mathbb{R}^{4}$. Then the $y_{a}$ coordinates may or may
not be independent of the $x^{a}$, giving two Lorentz covariant possibilities:

  1. 1.

The potential $\mathbf{f}_{a}$ is linearly dependent on the solder form,
$\mathbf{f}_{a}=b_{ab}\left(x^{c}\right)\mathbf{e}^{b}$.

  2. 2.

The $\mathbf{f}_{a}$ are independent of the $\mathbf{e}^{a}$, so that
$\left(\mathbf{e}^{a},\mathbf{f}_{b}\right)$ span a flat, 8-dimensional space
comprised of two copies of Minkowski space.

We describe these two cases in more detail in our discussion Section, but
before considering the system further, we find the maximal extension.

##  4 The maximal Lie algebra

We found equations (16)-(19) by introducing an equation for a new gauge
potential $\mathbf{f}_{a}$. Now we allow the new potential to enter the
original equations. Keeping Eq.(18) for $\mathbf{d}\mathbf{f}_{a}$, but
allowing an asymmetric connection, we seek the maximal form of the Lie algebra
by including $\mathbf{f}_{a}$-dependent terms in the remaining structure
equations. We then limit the coefficients by enforcing integrability.

We begin with the original asymmetric geometry, Eqs.(15), separate the trace
of the nonmetricity to include the Weyl vector, then set the curvatures to
zero to study the original vacuum algebra. Appending the additional terms to
Eqs.(16)-(19) we write

$\displaystyle\mathbf{d}\hat{\boldsymbol{\omega}}_{\;\;\;b}^{a}$
$\displaystyle=$
$\displaystyle\hat{\boldsymbol{\omega}}_{\;\;\;b}^{c}\wedge\hat{\boldsymbol{\omega}}_{\;\;\;c}^{a}+\boldsymbol{\Lambda}_{\;\;\;b}^{a\;\;\quad
c}\land\mathbf{f}_{c}$ $\displaystyle\mathbf{d}\mathbf{e}^{a}$
$\displaystyle=$
$\displaystyle\mathbf{e}^{b}\wedge\hat{\boldsymbol{\omega}}_{\;\;\;b}^{a}+\boldsymbol{\omega}\wedge\mathbf{e}^{a}+\boldsymbol{\Lambda}^{ac}\land\mathbf{f}_{c}$
$\displaystyle\mathbf{d}\mathbf{f}_{a}$ $\displaystyle=$
$\displaystyle\hat{\boldsymbol{\omega}}_{\;\;\;a}^{b}\wedge\mathbf{f}_{b}+\mathbf{f}_{a}\wedge\boldsymbol{\omega}$
$\displaystyle\mathbf{d}\boldsymbol{\omega}$ $\displaystyle=$
$\displaystyle\boldsymbol{\Lambda}^{c}\land\mathbf{f}_{c}$

where the 1-forms $\boldsymbol{\Lambda}_{\;\;\;b}^{a\;\;\quad
c},\boldsymbol{\Lambda}^{ac},\boldsymbol{\Lambda}^{c}$ must be built from the
available tensors, $g_{ab},\delta_{b}^{a},g^{ab},e_{abcd},e^{abcd}$ and gauge
fields
$\left(\boldsymbol{\omega}_{\;\;\;b}^{a},\mathbf{e}^{a},\mathbf{f}_{a},\boldsymbol{\omega}\right)$,
with $\boldsymbol{\Lambda}_{\;\;\;b}^{a\;\;\quad c}$ antisymmetric on $ab$. We
work with a general form of the metric $g_{ab}$. Since we are considering the
homogeneous algebra, we have

$\mathbf{Q}^{ab}=\mathbf{d}g^{ab}+\hat{\boldsymbol{\omega}}^{ab}+\hat{\boldsymbol{\omega}}^{ba}-2\boldsymbol{\omega}g^{ab}=0$
(20)

The possible forms of the 1-forms $\boldsymbol{\Lambda}_{\;\;\;b}^{a\;\;\quad
c},\boldsymbol{\Lambda}^{ac},\boldsymbol{\Lambda}^{c}$ are

$\displaystyle\boldsymbol{\Lambda}_{\;\;\;b}^{a\;\;\quad c}$ $\displaystyle=$
$\displaystyle\alpha e_{\;\;\;b\;\;\quad d}^{a\;\;\quad
c}\mathbf{e}^{d}+\beta\left(\delta_{d}^{a}\delta_{b}^{c}-g^{ac}g_{bd}\right)\mathbf{e}^{d}+\mu\left(g^{ad}\delta_{b}^{c}-g^{ac}\delta_{b}^{d}\right)\mathbf{f}_{d}$
$\displaystyle\boldsymbol{\Lambda}^{ac}$ $\displaystyle=$
$\displaystyle\rho\hat{\boldsymbol{\omega}}^{ac}+\lambda e_{\;\;\quad
bd}^{ac}\hat{\boldsymbol{\omega}}^{bd}+\sigma g^{ac}\boldsymbol{\omega}$
$\displaystyle\boldsymbol{\Lambda}^{c}$ $\displaystyle=$
$\displaystyle\gamma\mathbf{e}^{c}+\nu g^{cd}\mathbf{f}_{d}$

with constants $\alpha,\beta,\mu,\rho,\lambda,\sigma,\gamma,\nu$ so that the
vacuum structure equations become

$\displaystyle\mathbf{d}\hat{\boldsymbol{\omega}}_{\;\;\;b}^{a}$
$\displaystyle=$
$\displaystyle\hat{\boldsymbol{\omega}}_{\;\;\;b}^{c}\wedge\hat{\boldsymbol{\omega}}_{\;\;\;c}^{a}+\alpha
e_{\;\;\;b\;\;\quad d}^{a\;\;\quad
c}\mathbf{e}^{d}\land\mathbf{f}_{c}+\beta\left(\delta_{d}^{a}\delta_{b}^{c}-g^{ac}g_{bd}\right)\mathbf{e}^{d}\land\mathbf{f}_{c}+\mu\left(g^{ad}\delta_{b}^{c}-g^{ac}\delta_{b}^{d}\right)\mathbf{f}_{d}\land\mathbf{f}_{c}$
$\displaystyle\mathbf{d}\mathbf{e}^{a}$ $\displaystyle=$
$\displaystyle\mathbf{e}^{b}\wedge\hat{\boldsymbol{\omega}}_{\;\;\;b}^{a}+\boldsymbol{\omega}\wedge\mathbf{e}^{a}+\rho\hat{\boldsymbol{\omega}}^{ac}\land\mathbf{f}_{c}+\lambda
e_{\;\;\quad bd}^{ac}\hat{\boldsymbol{\omega}}^{bd}\land\mathbf{f}_{c}+\sigma
g^{ac}\boldsymbol{\omega}\land\mathbf{f}_{c}$
$\displaystyle\mathbf{d}\mathbf{f}_{a}$ $\displaystyle=$
$\displaystyle\hat{\boldsymbol{\omega}}_{\;\;\;a}^{b}\wedge\mathbf{f}_{b}+\mathbf{f}_{a}\wedge\boldsymbol{\omega}$
$\displaystyle\mathbf{d}\boldsymbol{\omega}$ $\displaystyle=$
$\displaystyle\gamma\mathbf{e}^{c}\land\mathbf{f}_{c}+\nu
g^{cd}\mathbf{f}_{d}\land\mathbf{f}_{c}=\gamma\mathbf{e}^{c}\land\mathbf{f}_{c}$

Substituting these into their exterior derivatives we find the integrability
conditions eventually reduce to:

$\displaystyle 0\equiv\mathbf{d}^{2}\hat{\boldsymbol{\omega}}_{\;\;\;b}^{a}$
$\displaystyle=$ $\displaystyle 2\beta\sigma\boldsymbol{\omega}\wedge
g^{ac}\mathbf{f}_{c}\land\mathbf{f}_{b}$ $\displaystyle
0\equiv\mathbf{d}^{2}\mathbf{e}^{a}$ $\displaystyle=$ $\displaystyle-\lambda
e^{acbd}\hat{\boldsymbol{\omega}}_{bd}\land\hat{\boldsymbol{\omega}}_{\;\;\;c}^{e}\wedge\mathbf{f}_{e}-\lambda\mathbf{f}_{c}\wedge
e^{bcde}\hat{\boldsymbol{\omega}}_{de}\land\hat{\boldsymbol{\omega}}_{\;\;\;b}^{a}+\lambda
e^{acbd}\hat{\boldsymbol{\omega}}_{\;\;\;d}^{e}\wedge\hat{\boldsymbol{\omega}}_{be}\land\mathbf{f}_{c}$
$\displaystyle-\rho\hat{\boldsymbol{\omega}}^{ac}\land\hat{\boldsymbol{\omega}}_{\;\;\;c}^{b}\wedge\mathbf{f}_{b}+\left(\gamma+\beta\right)\mathbf{e}^{c}\land\mathbf{f}_{c}\wedge\mathbf{e}^{a}+2\beta\lambda
e^{acde}g_{db}\mathbf{e}^{b}\land\mathbf{f}_{e}\land\mathbf{f}_{c}$
$\displaystyle-\left(2\mu+\rho\beta+\gamma\sigma\right)g^{ae}\mathbf{e}^{c}\land\mathbf{f}_{e}\land\mathbf{f}_{c}+2\mu\lambda
e^{acfe}\mathbf{f}_{f}\land\mathbf{f}_{e}\land\mathbf{f}_{c}$
$\displaystyle-\rho\boldsymbol{\omega}\wedge\hat{\boldsymbol{\omega}}^{ac}\land\mathbf{f}_{c}-\lambda
e_{\;\;\quad
bd}^{ac}\boldsymbol{\omega}\wedge\hat{\boldsymbol{\omega}}^{bd}\land\mathbf{f}_{c}-\rho\hat{\boldsymbol{\omega}}^{ac}\land\mathbf{f}_{c}\wedge\boldsymbol{\omega}-\lambda
e^{acbd}\hat{\boldsymbol{\omega}}_{bd}\land\mathbf{f}_{c}\wedge\boldsymbol{\omega}$
$\displaystyle+\sigma\left(\mathbf{d}g^{ac}+\hat{\boldsymbol{\omega}}^{ca}+\hat{\boldsymbol{\omega}}^{ac}-\boldsymbol{\omega}g^{ac}\right)\land\boldsymbol{\omega}\wedge\mathbf{f}_{c}$
$\displaystyle 0\equiv\mathbf{d}^{2}\mathbf{f}_{a}$ $\displaystyle=$
$\displaystyle\alpha e_{\;\;\;a\;\;\quad d}^{b\;\;\quad
c}\mathbf{e}^{d}\land\mathbf{f}_{c}\wedge\mathbf{f}_{b}+\left(\beta+\gamma\right)\mathbf{e}^{b}\land\mathbf{f}_{a}\wedge\mathbf{f}_{b}$
$\displaystyle 0\equiv\mathbf{d}^{2}\boldsymbol{\omega}$ $\displaystyle=$
$\displaystyle\gamma\rho\hat{\boldsymbol{\omega}}^{ec}\land\mathbf{f}_{c}\land\mathbf{f}_{e}+\gamma\lambda
e_{\;\;\quad
bd}^{ec}\hat{\boldsymbol{\omega}}^{bd}\land\mathbf{f}_{c}\land\mathbf{f}_{e}$

The independence of the remaining terms implies:

$\displaystyle\gamma$ $\displaystyle=$ $\displaystyle-\beta$
$\displaystyle\lambda$ $\displaystyle=$
$\displaystyle\rho=\mu=\alpha=\sigma=0$

Finally, the choice of $\beta$ amounts to a rescaling of $\mathbf{f}_{a}$ so
setting $\beta=-1$ we have the maximal integrable system.

$\displaystyle\mathbf{d}\hat{\boldsymbol{\omega}}_{\;\;\;b}^{a}$
$\displaystyle=$
$\displaystyle\hat{\boldsymbol{\omega}}_{\;\;\;b}^{c}\wedge\hat{\boldsymbol{\omega}}_{\;\;\;c}^{a}-\left(\delta_{d}^{a}\delta_{b}^{c}-g^{ac}g_{bd}\right)\mathbf{e}^{d}\land\mathbf{f}_{c}$
$\displaystyle\mathbf{d}\mathbf{e}^{a}$ $\displaystyle=$
$\displaystyle\mathbf{e}^{b}\wedge\hat{\boldsymbol{\omega}}_{\;\;\;b}^{a}+\boldsymbol{\omega}\wedge\mathbf{e}^{a}$
$\displaystyle\mathbf{d}\mathbf{f}_{a}$ $\displaystyle=$
$\displaystyle\hat{\boldsymbol{\omega}}_{\;\;\;a}^{b}\wedge\mathbf{f}_{b}+\mathbf{f}_{a}\wedge\boldsymbol{\omega}$
$\displaystyle\mathbf{d}\boldsymbol{\omega}$ $\displaystyle=$
$\displaystyle\mathbf{e}^{c}\land\mathbf{f}_{c}$ (21)

Surprisingly, the form of the maximal nonmetric Eqs.(21) have the same form as
the Maurer-Cartan equations for the conformal group. Here, however, the spin
connection is asymmetric, so the group will be general linear.

To see that Eqs.(21) actually describes the conformal group, we use (20) to
rewrite the connection as

$\displaystyle\hat{\boldsymbol{\omega}}_{\;\;\;b}^{a}$ $\displaystyle=$
$\displaystyle\boldsymbol{\omega}_{\;\;\;b}^{a}+\frac{1}{2}g^{ac}\left(\hat{\boldsymbol{\omega}}_{bc}+\hat{\boldsymbol{\omega}}_{cb}\right)$
$\displaystyle=$
$\displaystyle\boldsymbol{\omega}_{\;\;\;b}^{a}+\frac{1}{2}g^{ac}\left(\mathbf{d}g_{bc}+2\boldsymbol{\omega}g_{cb}\right)$

Substituting this into the solder form equation and choosing the orthonormal
metric gives metric compatibility

$\displaystyle\mathbf{d}\mathbf{e}^{a}$ $\displaystyle=$
$\displaystyle\mathbf{e}^{b}\wedge\hat{\boldsymbol{\omega}}_{\;\;\;b}^{a}+\boldsymbol{\omega}\wedge\mathbf{e}^{a}$
$\displaystyle=$
$\displaystyle\mathbf{e}^{b}\wedge\boldsymbol{\omega}_{\;\;\;b}^{a}$

in the new basis. For a general metric we then have
$\mathbf{d}g_{bc}+2\boldsymbol{\omega}g_{bc}=0$ and therefore

$\displaystyle\hat{\boldsymbol{\omega}}_{\;\;\;b}^{a}$ $\displaystyle=$
$\displaystyle\boldsymbol{\omega}_{\;\;\;b}^{a}$

so the structure equations have the same form in the Lorentz and asymmetric
cases.

$\displaystyle\mathbf{d}\boldsymbol{\omega}_{\;\;\;b}^{a}$ $\displaystyle=$
$\displaystyle\boldsymbol{\omega}_{\;\;\;b}^{c}\wedge\boldsymbol{\omega}_{\;\;\;c}^{a}+\left(\delta_{d}^{a}\delta_{b}^{c}-g^{ac}g_{bd}\right)\mathbf{f}_{c}\land\mathbf{e}^{d}$
$\displaystyle\mathbf{d}\mathbf{e}^{a}$ $\displaystyle=$
$\displaystyle\mathbf{e}^{b}\wedge\boldsymbol{\omega}_{\;\;\;b}^{a}+\boldsymbol{\omega}\wedge\mathbf{e}^{a}$
$\displaystyle\mathbf{d}\mathbf{f}_{a}$ $\displaystyle=$
$\displaystyle\boldsymbol{\omega}_{\;\;\;a}^{b}\wedge\mathbf{f}_{b}+\mathbf{f}_{a}\wedge\boldsymbol{\omega}$
$\displaystyle\mathbf{d}\boldsymbol{\omega}$ $\displaystyle=$
$\displaystyle\mathbf{e}^{a}\land\mathbf{f}_{a}$ (22)

These are the Maurer-Cartan structure equations of the conformal group. This
makes rigorous the conjecture of [26], that the mixed symmetry nonmetricity is
related to conformal symmetry.

Therefore, _the Maurer-Cartan equations of the maximal nonmetric Poincarè
system may be recast as the Maurer-Cartan equations of conformal symmetry._

This equivalence becomes more striking when we modify to include curvatures.

##  5 Developing the curvatures

To develop the curvatures, we return to the $\mathbf{u}^{a},\mathbf{v}_{a}$
basis. In this basis the torsion and nonmetricity separate.

From Eqs.(22) we substitute

$\displaystyle\mathbf{f}_{a}$ $\displaystyle=$
$\displaystyle\mathbf{v}_{a}+g_{ab}\mathbf{u}^{b}$
$\displaystyle\mathbf{e}^{a}$ $\displaystyle=$
$\displaystyle\mathbf{u}^{a}-g^{ab}\mathbf{v}_{b}$

into the conformal system (22), resulting in

$\displaystyle\mathbf{d}\boldsymbol{\omega}_{\;\;\;b}^{a}$ $\displaystyle=$
$\displaystyle\boldsymbol{\omega}_{\;\;\;b}^{c}\wedge\boldsymbol{\omega}_{\;\;\;c}^{a}+2\left(g^{ae}\mathbf{v}_{e}\land\mathbf{v}_{b}-\mathbf{u}^{a}\land
g_{bc}\mathbf{u}^{c}\right)$ $\displaystyle\mathbf{d}\mathbf{u}^{a}$
$\displaystyle=$
$\displaystyle\mathbf{u}^{b}\wedge\boldsymbol{\omega}_{\;\;\;b}^{a}+\boldsymbol{\omega}\wedge\mathbf{u}^{a}$
$\displaystyle\mathbf{d}\mathbf{v}_{a}$ $\displaystyle=$
$\displaystyle\boldsymbol{\omega}_{\;\;\;a}^{b}\wedge\mathbf{v}_{b}+\mathbf{v}_{a}\wedge\boldsymbol{\omega}$
$\displaystyle\mathbf{d}\boldsymbol{\omega}$ $\displaystyle=$ $\displaystyle
2\mathbf{u}^{a}\land\mathbf{v}_{a}$

The structure equations for $\left(\mathbf{u}^{a},\mathbf{v}_{a}\right)$ are
unchanged from the original separation of torsion and nonmetricity given by
Eqs.(14). Therefore, when we restore curvatures we retain the previous
relations for $\mathbf{d}\mathbf{u}^{a}$ and $\mathbf{d}\mathbf{v}_{a}$,
giving

$\displaystyle\mathbf{d}\boldsymbol{\omega}_{\;\;\;b}^{a}$ $\displaystyle=$
$\displaystyle\boldsymbol{\omega}_{\;\;\;b}^{c}\wedge\boldsymbol{\omega}_{\;\;\;c}^{a}+2\left(\mathbf{u}^{a}\land
g_{bc}\mathbf{u}^{c}-g^{ac}\mathbf{v}_{c}\land\mathbf{v}_{b}\right)+\mathbf{R}_{\;\;\;b}^{a}$
$\displaystyle\mathbf{d}\mathbf{u}^{a}$ $\displaystyle=$
$\displaystyle\mathbf{u}^{b}\wedge\boldsymbol{\omega}_{\;\;\;b}^{a}+\boldsymbol{\omega}\wedge\mathbf{u}^{a}+\hat{\mathbf{T}}^{a}$
$\displaystyle\mathbf{d}\mathbf{v}_{a}$ $\displaystyle=$
$\displaystyle\boldsymbol{\omega}_{\;\;\;a}^{b}\wedge\mathbf{v}_{b}+\mathbf{v}_{a}\wedge\boldsymbol{\omega}+\tilde{\mathbf{Q}}_{a}$
$\displaystyle\mathbf{d}\boldsymbol{\omega}$ $\displaystyle=$ $\displaystyle
2\mathbf{u}^{a}\wedge\mathbf{v}_{a}+\boldsymbol{\Omega}$

To relate the torsion and nonmetricity to conformal curvatures we invert to
transform back to the conformal frame
$\left(\mathbf{e}^{a},\mathbf{f}_{a}\right)$, with
$\mathbf{v}_{a}=\frac{1}{2}\left(\mathbf{f}_{a}+g_{ab}\mathbf{e}^{b}\right)$
and
$\mathbf{u}^{a}=\frac{1}{2}\left(\mathbf{e}^{a}-g^{ab}\mathbf{f}_{b}\right)$
to find

$\displaystyle\mathbf{d}\boldsymbol{\omega}_{\;\;\;b}^{a}$ $\displaystyle=$
$\displaystyle\boldsymbol{\omega}_{\;\;\;b}^{c}\wedge\boldsymbol{\omega}_{\;\;\;c}^{a}+\left(\delta_{c}^{a}\delta_{b}^{d}-g^{ad}g_{bc}\right)\mathbf{f}_{d}\land\mathbf{e}^{c}+\mathbf{R}_{\;\;\;b}^{a}$
$\displaystyle\mathbf{d}\mathbf{e}^{a}$ $\displaystyle=$
$\displaystyle\mathbf{e}^{b}\wedge\boldsymbol{\omega}_{\;\;\;b}^{a}+\boldsymbol{\omega}\wedge\mathbf{e}^{a}+\mathbf{S}^{a}$
$\displaystyle\mathbf{d}\mathbf{f}_{a}$ $\displaystyle=$
$\displaystyle\boldsymbol{\omega}_{\;\;\;a}^{b}\wedge\mathbf{f}_{b}+\mathbf{f}_{a}\wedge\boldsymbol{\omega}+\bar{\mathbf{S}}^{a}$
$\displaystyle\mathbf{d}\boldsymbol{\omega}$ $\displaystyle=$
$\displaystyle\mathbf{e}^{a}\wedge\mathbf{f}_{a}+\boldsymbol{\Omega}$ (23)

with $\mathbf{S}^{a},\bar{\mathbf{S}}^{a}$ as defined in Section (3). This is
precisely the form of the gauged conformal group. The evident parallel
structure of the solder form and special conformal transformations,
$\hat{\mathbf{T}}^{a}\pm\tilde{\mathbf{Q}}_{a}$, emphasizes the understanding
of mixed symmetry nonmetricity $\tilde{\mathbf{Q}}_{a}$ as torsion-like.

Bringing the calculation full circle, we reconstruct the Cartan equations with
the asymmetric connection. Choose the orthonormal metric so that
$\mathbf{d}\eta_{ab}=0$. Then

$\displaystyle\boldsymbol{\omega}_{\;\;\;b}^{a}$ $\displaystyle=$
$\displaystyle\hat{\boldsymbol{\omega}}_{\;\;\;b}^{a}-\frac{1}{2}\eta^{ac}\left(\hat{\boldsymbol{\omega}}_{bc}+\hat{\boldsymbol{\omega}}_{cb}\right)$
$\displaystyle=$
$\displaystyle\hat{\boldsymbol{\omega}}_{\;\;\;b}^{a}+\frac{1}{2}\tilde{\mathbf{Q}}_{\;\;\;b}^{a}+\delta_{b}^{a}\boldsymbol{\omega}$
$\displaystyle=$
$\displaystyle\hat{\boldsymbol{\omega}}_{\;\;\;b}^{a}+\frac{1}{2}\hat{\mathbf{Q}}_{\;\;\;b}^{a}$

Substituting we find the maximal nonmetric system

$\displaystyle\mathbf{d}\hat{\boldsymbol{\omega}}_{\;\;\;b}^{a}$
$\displaystyle=$
$\displaystyle\hat{\boldsymbol{\omega}}_{\;\;\;b}^{c}\wedge\hat{\boldsymbol{\omega}}_{\;\;\;c}^{a}+\left(\delta_{c}^{a}\delta_{b}^{d}-\eta^{ad}\eta_{bc}\right)\mathbf{f}_{d}\land\mathbf{e}^{c}+\boldsymbol{\mathcal{R}}_{\;\;\;b}^{a}$
$\displaystyle\mathbf{d}\mathbf{e}^{a}$ $\displaystyle=$
$\displaystyle\mathbf{e}^{b}\wedge\hat{\boldsymbol{\omega}}_{\;\;\;b}^{a}+\hat{\mathbf{T}}^{a}$
$\displaystyle\mathbf{d}\mathbf{f}_{a}$ $\displaystyle=$
$\displaystyle\hat{\boldsymbol{\omega}}_{\;\;\;a}^{b}\wedge\mathbf{f}_{b}+\hat{\mathbf{T}}^{a}+\frac{1}{2}\left(\mathbf{e}^{b}-\eta^{bc}\mathbf{f}_{c}\right)\wedge\tilde{\mathbf{Q}}_{\;\;\;b}^{a}$
$\displaystyle\mathbf{d}\boldsymbol{\omega}$ $\displaystyle=$
$\displaystyle\mathbf{e}^{a}\wedge\mathbf{f}_{a}+\boldsymbol{\Omega}$ (24)

where
$\boldsymbol{\mathcal{R}}_{\;\;\;b}^{a}=\mathbf{R}_{\;\;\;b}^{a}+\frac{1}{2}\mathbf{D}\hat{\mathbf{Q}}_{\;\;\;b}^{a}-\frac{1}{4}\hat{\mathbf{Q}}_{\;\;\;b}^{c}\wedge\hat{\mathbf{Q}}_{\;\;\;c}^{a}$.
Eqs.(24) reduce to the original system when
$\mathbf{f}_{a}=\mathbf{h}_{a}=\eta_{ab}\mathbf{e}^{b}$.

In the asymmetric frame, the curvature has a symmetric part. Writing
$\mathbf{d}\hat{\boldsymbol{\omega}}^{ab}+\mathbf{d}\hat{\boldsymbol{\omega}}^{ba}$
to symmetrize the equation, then expanding the connection in terms of
antisymmetric and symmetric parts,
$\boldsymbol{\omega}^{ca}+\frac{1}{2}\left(\hat{\boldsymbol{\omega}}^{ac}+\hat{\boldsymbol{\omega}}^{ca}\right)$
we find that
$-\frac{1}{2}\mathbf{D}\hat{\mathbf{Q}}_{\;\;\;b}^{a}+\frac{1}{4}\hat{\mathbf{Q}}_{\;\;\;b}^{c}\wedge\hat{\mathbf{Q}}_{\;\;\;c}^{a}$
cancels identically leaving

$\displaystyle\mathbf{R}^{ab}+\mathbf{R}^{ba}$ $\displaystyle=$ $\displaystyle
0$

This shows that the Lorentzian curvature is properly antisymmetric.

Notice that if the symmetric part of the curvature
$\boldsymbol{\Omega}^{ab}+\boldsymbol{\Omega}^{ba}=\mathbf{D}\mathbf{Q}^{ab}$
vanishes then it follows that

$\displaystyle 0$ $\displaystyle=$
$\displaystyle\eta_{ab}\mathbf{D}\mathbf{Q}^{ab}$ $\displaystyle=$
$\displaystyle\mathbf{D}\left(\eta_{ab}\mathbf{Q}^{ab}\right)-\mathbf{D}\eta_{ab}\wedge\mathbf{Q}^{ab}$
$\displaystyle=$ $\displaystyle
2n\,\mathbf{d}\boldsymbol{\omega}-\mathbf{Q}_{ab}\wedge\mathbf{Q}^{ab}$

and since $\mathbf{Q}_{ab}\wedge\mathbf{Q}^{ab}=0$ the Weyl vector is
integrable, $\mathbf{d}\boldsymbol{\omega}=0$.

##  6 Discussion

Curvature, torsion, and nonmetricity have distinct physical effects. The
geodesic deviation described by curvature is familiar, and it is not hard to
see that torsion can produce anomalous precession of angular momentum beyond
the Lense-Thirring effect. By contrast to these, nonmetricity affects the
parallel transport of angles, so that two idential cubes parallel transported
along different paths become non-identical parallelepipeds when later
compared. It is therefore surprising that there is a direct equivalence
between a substantial part of the nonmetricity and the torsion.

In demonstrating these relationships we considered three systems, equivalent
via field redefinitions. Summarizing each by its independent variables and
corresponding curvatures or field strengths, we have the nonmetric
$\left(NM\right)$, Lorentzian with separated sources $\left(L,sep\right)$ and
the conformal $\left(C\right)$ :

$\left[\begin{array}[]{cc}\hat{\boldsymbol{\omega}}_{\;\;\;b}^{a}&\boldsymbol{\mathcal{R}}_{\;\;\;b}^{a}\\\
\mathbf{e}^{a}&\hat{\mathbf{T}}^{a}\\\
\mathbf{f}_{a}&\tilde{\mathbf{Q}}_{\;\;\;b}^{a}\\\
\boldsymbol{\omega}&\boldsymbol{\Omega}\end{array}\right]_{NM}\cong\left[\begin{array}[]{cc}\boldsymbol{\omega}_{\;\;\;b}^{a}&\mathbf{R}_{\;\;\;b}^{a}\\\
\mathbf{u}^{a}&\hat{\mathbf{T}}^{a}\\\
\mathbf{v}_{a}&\tilde{\mathbf{Q}}_{a}\\\
\boldsymbol{\omega}&\boldsymbol{\Omega}\end{array}\right]_{L,sep}\cong\left[\begin{array}[]{cc}\boldsymbol{\omega}_{\;\;\;b}^{a}&\mathbf{R}_{\;\;\;b}^{a}\\\
\mathbf{e}^{a}&\hat{\mathbf{T}}^{a}-\tilde{\mathbf{Q}}^{a}\\\
\mathbf{f}_{a}&\hat{\mathbf{T}}^{a}+\tilde{\mathbf{Q}}^{a}\\\
\boldsymbol{\omega}&\boldsymbol{\Omega}\end{array}\right]_{C}$ (25)

There are two striking results here. Firstly, symmetric contributions to the
spin connection due to nonmetricity may be completely absorbed by field
redefinitions to restore the antisymmetric spin connection of a Lorentzian
system111This is one of those results that, if you stare at it long enough,
seems obvious–naturally we can always choose an orthonormal basis to reduce a
general linear system to an orthonormal (Lorentzian) one (see Theorem 5.8 in
[1]). But here this correspondence drops effortlessly out of the structure
equations.. Secondly, the field strength $\mathbf{D}\mathbf{f}_{a}$ of the
special conformal gauge field $\mathbf{f}_{a}$ equals the sum of the original
torsion and nonmetricity.

$\left.\mathbf{D}\mathbf{f}_{a}\right|_{Conformal\,basis}=\left.\hat{\mathbf{T}}_{a}+\tilde{\mathbf{Q}}_{a}\right|_{Asymmetric\,basis}$

This is an unexpected property of the special conformal transformations222It
does not matter how long we stare..

The set of Cartan equations (23) describe the usual conformal gauge theory.
This form of $SO\left(4,2\right)$ gauge theory has a long history ([27]-[41])
and multiple interpretations, depending on our choice of fiber bundle. We
remark on the two interpretations of $\mathbf{f}_{a}=\mathbf{d}y_{\alpha}$
identified in Subsection (3.3):

  * •

Auxiliary conformal gravity is based locally on the quotient of the conformal
group by its inhomogeneous Weyl subgroup,
$\mathcal{M}^{4}=\mathcal{C}/\mathcal{IW}$, on a $4$-dimensional ($n$-dim)
manifold. This corresponds to the first case,
$\mathbf{f}_{a}=b_{ab}\left(x^{c}\right)\mathbf{e}^{b}$. When the torsion
$\mathbf{S}^{a}$ vanishes the special conformal gauge field equals the
Schouten tensor,

$\mathbf{f}_{a}=\mathscr{R}_{ab}\mathbf{e}^{b}=\frac{1}{n-2}\left(R_{ab}-\frac{1}{2\left(n-1\right)}\eta_{ab}R\right)\mathbf{e}^{b}$

This solution enforces conformal structure by making
$\boldsymbol{\Omega}_{\;\;\;b}^{a}$ equal to the Weyl curvature
$\mathbf{C}_{\;\;\;b}^{a}$, regardless of the particular action. In an
Einstein space, $R_{ab}-\frac{1}{2}g_{ab}R=\Lambda g_{ab}$ we have

$\displaystyle\mathscr{R}_{ab}$ $\displaystyle=$
$\displaystyle-\frac{1}{6}\Lambda g_{ab}$

and the field strength $\mathbf{D}\boldsymbol{\mathscr{R}}_{ab}$ of the
special conformal gauge field $\mathbf{f}_{a}$ is exactly proportional to the
nonmetricity
$\mathbf{D}\boldsymbol{\mathscr{R}}_{ab}=-\frac{1}{6}\Lambda\mathbf{Q}_{ab}$.

  * •

Biconformal gravity [38] is based on the quotient of the conformal group by
the homogeneous Weyl group, $\mathcal{M}^{2n}=\mathcal{C}/\mathcal{W}$. The
quotient is a Kähler manifold, with $y_{a}$ independent of $x^{a}$. The volume
element is dimensionless and it becomes possible to write a scale invariant
curvature-linear action in any dimension [39]. With vanishing torsion
$\mathbf{S}_{a}$, the resulting gravity theory reduces to scale covariant
(i.e., integrable Weyl) general relativity on the co-tangent bundle [38, 39,
41]. Here the torsion $\mathbf{S}_{a}$ and the co-torsion
$\bar{\mathbf{S}}_{a}$ play equivalent roles as torsions associated with
translations at the antipodes of compactified Minkowski space.

In conclusion, independent variation of the metric and connection may lead, in
generic gravity theories, to torsion and/or nonmetricity. We therefore studied
Poincarè gauge gravity, which already includes torsion, when nonmetricity is
also allowed. We found that the Poincarè-Cartan structure equations may be
modified back to a Lorentzian metric-compatible theory by field redefinitions,
writing the connection as a conformal geometry with torsion
$\mathbf{S}^{a}=\hat{\mathbf{T}}^{a}-\tilde{\mathbf{Q}}^{a}$ and special
conformal field strength
$\bar{\mathbf{S}}^{a}=\hat{\mathbf{T}}^{a}+\tilde{\mathbf{Q}}^{a}$. In the
course of our proof, we found the three equivalent systems (25) of Cartan
equations.

The nonmetricity is comprised of irreducible mixed-symmetry
$\hat{\mathbf{Q}}^{a}$ and totally symmetric $Q_{\left(abc\right)}$ parts,
with the latter driven only by Spin-3 sources or terms higher than quadratic
in $Q_{\left(abc\right)}$ in the action. In the absence of consistent Spin-3
fields and the equivalence of $\hat{\mathbf{Q}}^{a}$ to a combination of
conformal fields, we conclude that the study of gravity theories with general
connections may be recast as a study of conformal gauge theories of gravity.

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11institutetext: Rutgers University, Piscataway, NJ 08854, USA 11email:
<EMAIL_ADDRESS>
22institutetext: SenseBrain Research
33institutetext: Department of Medicine, Division of Cardiology. University of
Arkansas for Medical Sciences, Little Rock, AR, USA
44institutetext: Department of Clinical Physiology, Skane University Hospital
Lund, Lund University, 221 85 Lund, Sweden
55institutetext: Department of Radiology, New York University, New York, NY
10016, USA.

# DeepRecon: Joint 2D Cardiac Segmentation and 3D Volume Reconstruction via A
Structure-Specific Generative Method

Qi Chang 11 Zhennan Yan 22 Mu Zhou 22 Di Liu 11 Khalid Sawalha 33 Meng Ye 11
Qilong Zhangli 11 Mikael Kanski 44 Subhi Al’Aref 33 Leon Axel 55 Dimitris
Metaxas 11

###### Abstract

Joint 2D cardiac segmentation and 3D volume reconstruction are fundamental in
building statistical cardiac anatomy models and understanding functional
mechanisms from motion patterns. However, due to the low through-plane
resolution of cine MR and high inter-subject variance, accurately segmenting
cardiac images and reconstructing the 3D volume are challenging. In this
study, we propose an end-to-end latent-space-based framework, DeepRecon, that
generates multiple clinically essential outcomes, including accurate image
segmentation, synthetic high-resolution 3D image, and 3D reconstructed volume.
Our method identifies the optimal latent representation of the cine image that
contains accurate semantic information for cardiac structures. In particular,
our model jointly generates synthetic images with accurate semantic
information and segmentation of the cardiac structures using the optimal
latent representation. We further explore downstream applications of 3D shape
reconstruction and 4D motion pattern adaptation by the different latent-space
manipulation strategies. The simultaneously generated high-resolution images
present a high interpretable value to assess the cardiac shape and motion.
Experimental results demonstrate the effectiveness of our approach on multiple
fronts including 2D segmentation, 3D reconstruction, downstream 4D motion
pattern adaption performance.

###### Keywords:

3D Reconstruction Cardiac MRI GAN Latent Space

##  1 Introduction

Comprehensive image-based assessment of cardiac structure and motion through
3D heart modeling is essential for early detection, cardiac function
understanding, and treatment planning of cardiovascular diseases (CVD) [2,
13]. As a standard clinical diagnostic tool, cine magnetic resonance imaging
(cMRI) has been used to characterize the complex shape and motion of the
heart. cMRI presents multiple advances, including the high temporal and in-
plane resolution, minimal radiation exposure, and improved soft tissue
definition [20]. Yet conventional practice acquires a stack of 2D short-axis
(SAX) slices with large between-slice spacing, so the accurate heart tissue
segmentation and 3D image-based heart modeling is challenging due to the
missing structure information.

Heart modeling from cMRI typically consists of carefully-designed steps,
including image segmentation [3, 17, 6] and 3D reconstruction [4]. Deep
learning has shown its progress in the segmentation of cardiac structures [3,
5], which addresses the challenges of analyzing the complex and variable shape
of the heart and ill-defined borders in MR images. After obtaining 2D
segmentation, 3D reconstruction can be implemented by subsequent
interpretations between neighboring slices. Conventional approaches [8, 14]
often struggle when significant anatomical changes appear in the consecutive
slices. For instance, deformable models have been proposed for 3D surface
construction of left ventricular (LV) wall and motion tracking [27, 26]. Yet
the parameter initialization is sensitive, making it difficult to generalize
across clinical settings. Despite that 3D high-resolution image acquisition
and generation are gaining momentum to assess disease status [4, 13, 24], the
integrative analysis of MR data enables high-quality MR image segmentation,
reconstruction, and subsequent interpretation has not been explicitly
addressed.

In this study, we propose an end-to-end, latent-space-based framework,
DeepRecon, that generates multiple clinically essential outcomes, including
accurate image segmentation, synthetic high-resolution 3D image, and 3D
reconstructed volume (see Fig. 1). Our method could jointly generate 2D
segmentation and 3D volume (by interpolating latent codes) simultaneously in
the evaluation stage. Thus, we do not require another step to reconstruct the
3D volume from the sparse 2D segmentations. Meanwhile, the simultaneously
generated synthetic images present a high interpretable value to assess the
shape and motion of cardiology. Our study draws inspiration from StyleGANs
[11, 12], where a synthetic image can be generated from a random latent code.
Our findings are built upon the rationale that the latent code can be used to
reconstruct the realistic synthetic image and simultaneously generate accurate
segmentation [15, 21]. Experimental results demonstrate the effectiveness of
our approach on 2D segmentation, 3D reconstruction, and 4D motion pattern
adaptation performance.

##  2 Method

Fig.1 illustrates the end-to-end framework of DeepRecon that learns a latent
space from MR imaging to yield a broad range of outcomes, including generation
of high-quality 2D cine image and corresponding segmentation, high-resolution
3D reconstructed volume, and 4D motion adaptation of the heart. We first
describe the architecture design and the learning process and then elaborate
on latent-space-based 3D reconstruction and motion adaptation.

###  2.1 DeepRecon architecture

Figure 1: (a) illustrates the architecture of DeepRecon. The black arrow
workflow demonstrates the hybrid inversion method to acquire the optimized
latent code. The black dash workflow shows the segmentation training stage,
and the red workflow shows the 3D image and volume reconstruction process. (b)
shows a 3D volume reconstruction (both images and masks) by interpolating the
latent code of adjacent slices $x_{1}$ and $x_{2}$. (c) shows a source case
$x_{S_{1}}$ can adapt the motion pattern extracted from temporal frames
$x_{T_{2}}$ and $x_{T_{1}}$ of a target case, so the source’s synthetic motion
has similar volume changes as the target. Best viewed in color.

Our architecture consists of a latent-code encoder $E$, a MR image generator
$G$ followed by a segmentation network $Seg$, as shown in Fig.1 (a). The $E$
takes an real MR image $x$ as input and outputs its latent code representation
$w^{+}$. The optional motion correction is described in [25, 7]. Then, $G$ can
generate synthetic image $\hat{x}$ from the $w^{+}$, and $Seg$ produces the
segmentation $\hat{y}$. As an intuitive downstream application, we can
interpolate the latent codes of two neighboring MR slices to produce 3D images
and masks by stacking continuous synthetic outputs (spatial domain
synthesize), as shown in Fig.1 (b). Furthermore, we show that the manipulation
of latent codes enables a 4D motion adaptation that can transfer a motion
pattern of one subject to another subject (temporal domain synthesize), as
shown in Fig. 1 (c) and described in section 2.3.

###  2.2 Learning of DeepRecon

The training process of DeepRecon includes three stages: training of the
generator $G$ and the encoder $E$, hybrid optimization for the latent code,
and latent space based segmentation module. The training of the $G$ and $E$
follow the methods in [12] and [21], respectively. Then, we use GAN inversion
to find the optimal latent codes for real cine SAX images. We adopt a hybrid
inversion strategy for efficiency and accuracy. Given a target image $x$, we
predict the latent code $w^{+}=E(x)$. The $w^{+}$ is then optimized by
reconstructing the target image. We formulate the objective function of the
optimization by three losses, including a reconstruction term $L_{smooth}$, a
similarity term $L_{simi}$, and a domain regularization term $L_{domain}$.

$w^{+*}=\operatorname*{argmin}\limits_{w^{+}}\mathcal{L}(x,G(w^{+}))\\\
=\operatorname*{argmin}\limits_{w^{+}}(\lambda_{1}\mathcal{L}_{recon}+\lambda_{2}\mathcal{L}_{simi}+\lambda_{3}\mathcal{L}_{domain})$
(1) $\displaystyle\mathcal{L}_{recon}=$
$\displaystyle\mathcal{L}_{LPIPS}(x,G(w^{+}))+\lVert
ROI(x-G(w^{+}))\rVert_{2}^{2},$ (2) $\displaystyle\mathcal{L}_{simi}=$
$\displaystyle-L_{NCC}(x,G(w^{+})),\mathcal{L}_{domain}=\lVert
w^{+}-E(G(w^{+}))\rVert^{2}_{2}$

where $\lambda_{1},\lambda_{2},\lambda_{3}$ are balancing hyperparameters. For
the reconstruction term, we use the Learned Perceptual Image Patch Similarity
(LPIPS) distance [28] to encourage smooth generator mapping from latent codes.
We also include a ROI-based regularization by a weighted L2 term to encourage
the similarity in the ROI. For the similarity term, we adopt a normalized
local cross-correlation (NCC) metric[1] to ensure robustness for intensity-
variant cine images. The last term regularizes the optimization trajectory to
stay in the training domain and keep the interpolation of latent-space vectors
smoothness [29].

We store the optimized latent codes of training samples for training the $Seg$
(e.g., U-Net [22]). The $Seg$ takes synthetic images $\hat{x}=G(w^{+*})$ as
input and train the network with the Cross-Entropy(CE) and DICE loss[23, 22].
The loss of the segmentation task is defined as
$\mathcal{L}_{seg}=\mathcal{L}_{ce}+\mathcal{L}_{dice}$.

###  2.3 3D reconstruction and 4D motion adaptation

After learning of the DeepRecon, we use the learned model towards 3D shape
reconstruction and 4D motion adaptation that take fully advantage of the
learned latent space for the MR images. Our key motivation is that, by
identifying the optimal latent-space representation, the spatial interpolation
of two neighboring SAX slices can be achieved by generating images from the
interpolation of their corresponding latent codes (see Fig. 1 (b)). In this
way, DeepRecon is able to output a smooth super-resolution cine image volume
and the corresponding 3D heart shape from the generated segmentations.

Beyond the spatial interpolation, we can adapt a motion pattern of a target
subject $x_{T}$ to a source subject $x_{S}$. In this way, we can synthesize
and visualize a healthy-heart motion of a diseased subject and compare with
its real motion pattern in an intuitive manner. The motion adaptation is
demonstrated in Fig. 1 (c). Formally, we define the motion adaptation module
as:

$\hat{x}_{S_{i+1}}=G(w^{+}_{S_{0}}+\sum^{i}_{n=0}\mathcal{R}_{S}\Delta_{T_{n+1,n}})$
(3)

where $\hat{x}_{S_{i+1}}$ is the generated source images in time $i+1$;
$w^{+}_{S_{0}}$ is the latent code for $x_{S_{0}}$;
$\Delta_{T_{n+1,n}}=(w^{+}_{T_{n+1}}-w^{+}_{T_{n}})$ and $w^{+}_{T_{n}}$ is
the latent code for target subject in time $n$. Because the size of heart
varies, $\mathcal{R}_{S}$ represents the resampling function that align the
number of target slices to the number of source slices.

##  3 Experiments

We evaluate multiple outputs of DeepRecon including latent-space-based 2D
segmentation (3.1), 3D volume reconstruction (3.2), and motion adaptation
(3.3).

Dataset: The UK Biobank dataset [19] consists of SAX and LAX cine CMR images
of normal subjects. Cardiac structures, LV cavity (LVC), LV myocardium (LVM),
and right-ventricle cavity (RVC) were manually annotated on SAX images at the
end-diastolic (ED) and end-systolic (ES) cardiac phases [19]. We use 6,846
cases containing 3,569,990 2D SAX MR images as a pre-training subset. We adopt
the first 1,010 annotated subjects and split them into two sets of 810/200 for
training and validation of DeepRecon. We select another 100 cases from the UK
Biobank as the testing set that has no overlap with the pre-training, training
and validation subsets. Besides, two cardiologists annotated and verified the
LAX images of 50 testing cases to evaluate the 3D volume reconstruction task.
Finally, we validate the motion adaptation on a private dataset with diseased
cases to assess the motion patterns among various cardiac functional diseases.

###  3.1 Latent-space-based 2D segmentation

Settings: We first evaluate the generative segmentation method based on the
latent representation of MR images. The purpose of this task is to ensure the
generated 2D segmentation is accurate enough to reconstruct 3D cardiac shape
precisely. We pre-train the MR image generator $G$ on the pre-training subset
based on StyleGAN2 method [12]. The training took 14 days with 4 RTX 8000 GPUs
and achieved Fid50k score of 18.09. We perform the following experiments for
comparison. 1) $DirectSeg$. We train a segmentation network with the same
architecture as the $Seg$ in our method directly on the real images. This
result shows the approximate upper bound of the segmentation task. 2)
$3D$-$Unet$. In this method, we evaluate the 3D-Unet segmentation on the
original low-resolution Cine data. Since there is no high-resolution 3D data
for training the 3D-UNet, it can not be directly applied to the generated
high-resolution 3D images to obtain the 3D segmentation volume. 3)
$SemanticGAN$ [15]. In this method, the architecture of the generator is
extended to output the image and segmentation at the same time (no additional
segmentation network is required). 4) $W^{+}SegNet$. A variant of our method
keeps the same architecture except the $Seg$ network takes latent code as
input to directly predict the segmentation. 5) $DeepRecon_{no}$. The proposed
method without the hybrid latent code optimization. 6) $DeepRecon_{1k}$,
$DeepRecon_{10k}$. These two settings of the proposed method use the latent
code generated by the encoder $E$, followed by 1k and 10k optimization steps,
respectively. We train all the models in 2D and use 3D DICE and 95% quantile
of Hausdorff distance (HD95) for evaluation.

Result: We report the quantitative segmentation results in Table 1, and
illustrates a representative example in Fig. 2. Table 1 shows that DeepRecon
consistently outperforms the other latent-space-based methods and achieves
similar performance as the upper bound. The proposed $DeepRecon_{1k}$ performs
the best overall, especially in HD95, because more accurate image
reconstruction with fine details helps to optimize the segmentation network.
The $DeepRecon_{10k}$ further improves the DICE score of LVM while it drops in
terms of HD95 (may due to overfitting) and takes 10x time to retrieve the
optimized latent code. Thus, the $DeepRecon_{1k}$ presents a good trade-off
between the accuracy and efficiency and is used for evaluation in following
experiments. We also observe that other latent-space-based methods tend to
have significantly worse results for the LVM compared with the LVC and RVC. A
possible reason is that the sub-optimal latent code predicted from the encoder
is challenging to accurately reconstruct such a thin annular shape. Our
$DeepRecon_{1k}$, $DeepRecon_{10k}$ are statistically better than SematicGAN
in all metrics ($p<0.05$) and comparable with DirectSeg ($p>0.05$) in most
metrics. Additionally, we measure the image quality of the ground-truth and
synthetic images using the averaged peak signal-to-noise ratio (PSNR) and
structural similarity index (SSIM) in Fig. 2. The results show the improvement
of the synthesized image quality.

Table 1: Segmentation results by various latent-code-based methods compared with a direct segmentation from real images: mean (standard deviation). Method | LVC | LVM | RVC
---|---|---|---
DICE$\uparrow$ | HD95$\downarrow$ | DICE$\uparrow$ | HD95$\downarrow$ | DICE$\uparrow$ | HD95$\downarrow$
$DirectSeg$ | $0.928(0.04)$ | $2.31(9.8)$ | $0.861(0.03)$ | $1.38(3.5)$ | $0.892(0.04)$ | $4.69(15.8)$
$3D$-$UNet$ | $0.938(0.046)$ | $3.78(11.4)$ | $0.861(0.03)$ | $2.31(1.8)$ | $0.90(0.03)$ | $5.79(3.9)$
$SemanticGAN$ | $0.891(0.05)$ | $5.23(7.1)$ | $0.787(0.04)$ | $3.11(1.9)$ | $0.843(0.05)$ | $4.77(4.2)$
$W^{+}SegNet$ | $0.895(0.05)$ | $5.97(9.5)$ | $0.789(0.06)$ | $3.11(2.7)$ | $0.822(0.06)$ | $6.33(6.2)$
$DeepRecon_{no}$ | $0.895(0.04)$ | $2.37(8.1)$ | $0.792(0.04)$ | $1.87(5.8)$ | $0.847(0.05)$ | $2.33(5.6)$
$DeepRecon_{1k}$ | 0.926(0.04) | 1.25(0.8) | $0.835(0.05)$ | 1.27(1.7) | $0.883(0.04)$ | 1.51(4.4)
$DeepRecon_{10k}$ | $0.925(0.04)$ | $3.15(14.8)$ | 0.858(0.03) | $2.14(9.1)$ | 0.890(0.04) | $2.70(10.2)$
Figure 2: Segmentation results of different latent-space-based models. Each
group shows the synthetic image and corresponding segmentation. The red lines
are the ground truth boundaries, and the green lines are the predictions. We
notice that the sub-optimal latent codes may differ in the cardiac region,
especially thin structures.

###  3.2 3D volume reconstruction

Settings: We evaluate the reconstructed 3D cardiac shape continuity from the
interpolated latent codes. Since 3D MR image and segmentation volume are
unavailable, we utilize the LAX annotations and calculate the overlapping
accuracy (2D DICE) of the intersection areas, i.e., the intersection of the
reconstructed 3D volume and the LAX slice. We correct the misalignment for
each SAX slice based on the method described in [25]. The LAX slices usually
include 2-chamber (2ch), 3-chamber (3ch),and 4-chamber (4ch) views. The
following experiments are performed: 1) Original. The intersections of tiles
of the original SAX annotations and each LAX view. 2) Linear Interpolation. 3)
Coherent Point Drifting (CPD). The CPD algorithm [18] can construct 3D surface
based on a 3D point cloud and the deformable registration. 4)
$DeepRecon_{no}$,$DeepRecon_{1k}$,$DeepRecon_{10k}$ experiments follow the
same configuration described in 3.1. The DICE scores on different views and
their average score are computed.

Result: In Table 2, the DICE scores measure the accuracy for the LVC region
between the LAX annotation and the intersection of 3D reconstructed volume and
the corresponding LAX plane. We also show three examples in Fig. 3 for each
LAX view. Overall, our method achieves improved performance in each view
compared with other approaches. We observe that the SAX slices are not
complete compared to the LAX slices in the basal and apex region, but our
methods still achieve a better performance in 2ch and 4ch views on average.

Table 2: Evaluation of reconstructed 3D volumes in terms of the DICE score of their intersections on each LAX plane. Method | Average DICE | 2ch view DICE | 3ch view DICE | 4ch view DICE
---|---|---|---|---
Original | $0.780\pm{0.111}$ | $0.787\pm{0.091}$ | $0.793\pm{0.105}$ | $0.766\pm{0.128}$
Linear Interp | $0.781\pm{0.080}$ | $0.797\pm{0.051}$ | $0.773\pm{0.070}$ | $0.768\pm{0.102}$
CPD[18] | $0.790\pm{0.099}$ | $0.803\pm{0.084}$ | $\textbf{0.815}\pm{\textbf{0.094}}$ | $0.767\pm{0.109}$
$DeepRecon_{no}$ | $0.806\pm{0.111}$ | $0.830\pm{0.069}$ | $0.799\pm{0.111}$ | $0.787\pm{0.091}$
$DeepRecon_{1k}$ | $\textbf{0.817}\pm{\textbf{0.097}}$ | $\textbf{0.848}\pm{\textbf{0.056}}$ | $0.802\pm{0.141}$ | $\textbf{0.797}\pm{\textbf{0.091}}$
$DeepRecon_{10k}$ | $0.809\pm{0.094}$ | $0.839\pm{0.053}$ | $0.800\pm{0.136}$ | $0.787\pm{0.090}$
Figure 3: Visualization of the cross-sections of reconstructed 3D images and
label volumes by DeepRecon in different LAX views (last column). The first and
third columns show the original low-resolution SAX images and labels. The
second and fourth columns show the high-resolution reconstructions from the
proposed method.

###  3.3 Motion pattern adaptation

Settings: We further perform an exploratory analysis to assess the cardiac 4D
motion adaptation qualitatively. We extract a target case’s motion pattern and
assign such motion pattern to the initial shape of a source heart. First, we
select two random healthy cases from the UK Biobank dataset. Second, we use
one normal case as the target and one diseased case from our private dataset
as the source. The diseased heart has shown severe mitral regurgitation with
significant LV dilation and decreased function.

Results:

Figure 4: Two examples of motion adaptation: normal-to-normal (left) and
diseased-to-normal (right). The motions are plotted by volume changes of LVC
in a cardiac cycle.

The volume changes of the LVC over time within a full cardiac cycle are
presented in Fig. 4. The healthy example (Fig. 4 left) shows the consistency
of cavity volume change. By transferring the cardiac motion pattern from the
target case to the source shape, the motion of the source heart (the blue
Adapted curve) becomes similar to the target case (the green curve) while the
heart scale remains the same. In the second experiment, shown in Fig. 4 right,
we can synthesize a normal-like motion for the source diseased heart (the blue
curve), which shows a more efficient contraction than its real motion (the
orange curve). The ejection fraction of the source case is significantly
smaller than the adapted motion. Such motion adaptation method provides a
unique tool to potentially help cardiologists analyze functional differences
between various cases (see video clips in the supplementary).

##  4 Conclusion

Integrative analysis of cMRI is of great clinical significance in cardiac
function understanding and assessment. This paper proposes an end-to-end
latent-space-based generative method, DeepRecon, that generates multiple
outcomes, including 2D image segmentation, 3D reconstructed volume, and
extended motion pattern adaptation. Our findings show that the learned latent
representation can lead to high-level performance on cMRI feature analysis. In
particular, our approach opens up new perspectives on building scalable 3D/4D
synthetic cardiac models for cardiac functional research. In the future, we
will consider Transformer-based approaches [10, 16, 9] and investigate a
large-scale study to validate our approach. We also plan to examine
pathological identification based on the learned latent space.

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