# Elasticity and Glocality: Initiation of Embryonic Inversion in _Volvox_

Pierre A. Haas  Raymond E. Goldstein  Department of Applied Mathematics and
Theoretical Physics, Centre for Mathematical Sciences, University of
Cambridge, Wilberforce Road, Cambridge CB3 0WA, United Kingdom

###### Abstract

Elastic objects across a wide range of scales deform under local changes of
their intrinsic properties, yet the shapes are _glocal_ , set by a complicated
balance between local properties and global geometric constraints. Here, we
explore this interplay during the inversion process of the green alga _Volvox_
, whose embryos must turn themselves inside out to complete their development.
This process has recently been shown [S. Höhn _et al._ , _Phys. Rev. Lett._
114, 178101, (2015)] to be well described by the deformations of an elastic
shell under local variations of its intrinsic curvatures and stretches,
although the detailed mechanics of the process have remained unclear. Through
a combination of asymptotic analysis and numerical studies of the bifurcation
behavior, we illustrate how appropriate local deformations can overcome global
constraints to initiate inversion.

##  I Introduction

The shape of many a deformable object arises through the competition of
multiple constraints on the object: this competition may be between different
global constraints, such as in Helfrich’s analysis helfrich  of the shape of a
red blood cell (where intrinsic curvature effects coexist with constrained
membrane area and enclosed volume). It may also be the competition between
local and global constraints. Such deformations, which we shall term _glocal_
, arise for example in origami patterns silverberg  (where local folds must be
compatible with the global geometry). They are of considerable interest in the
design of programmable materials bende  at macro- and microscales, where one
asks: can a sequence of local deformations overcome global constraints and
direct the global deformations of an object?

This is a problem that, at the close of their development, the embryos of the
green alga _Volvox_ kirkbook  are faced with in the ponds of this world.
_Volvox_ (Fig. 1a) is a multicellular green alga belonging to a lineage (the
Volvocales) which has been recognized since the time of Weismann Weismann  as
a model organism for the evolution of multicellularity, and which more
recently has emerged as the same for biological fluid dynamics ARFM . The
Volvocales span from unicellular _Chlamydomonas_ , through organisms such
_Gonium_ , consisting of $8$ or $16$ _Chlamydomonas_ -like cells in a quasi-
planar arrangement, to spheroidal species (_Pandorina_ and _Pleodorina_) with
scores or hundreds of cells at the surface of a transparent extracellular
matrix (ECM). The largest members of the Volvocales are the species of
_Volvox_ , which display germ-soma differentiation, having sterile somatic
cells at the surface of the ECM and a small number of germ cells in the
interior which develop to become the daughter colonies.

Figure 1: (color online). _Volvox_ invagination and elastic model. (a) Adult
_Volvox_ , with somatic cells and one embryo labelled. (b)  _Volvox_ embryo at
the start of inversion. (c) Mushroom-shaped invaginated _Volvox_ embryo. (d)
Cell shape changes to wedge shapes and motion of cytoplasmic bridges (CB) bend
the cell sheet. Red line indicates position of cytoplasmic bridges. (e,f)
Cross-sections of the stages shown in panels (b,c). Cell shape changes as in
(d) occur in the marked regions. (g) Geometry of undeformed spherical shell of
radius $R$ and thickness $h$. (h) Geometry of deformed shell. Scale bars: (a)
$50\,\mbox{\textmu m}$, (e,f) $20\,\mbox{\textmu m}$. False color images
obtained from light-sheet microscopy provided by Stephanie Höhn and Aurelia R.
Honerkamp-Smith.

Following a period of substantial growth, the germ cells of _Volvox_ undergo
repeated rounds of cell division, at the end of which each embryo (Fig. 1b,e)
consists of a few thousand cells arrayed to form a thin spherical sheet
kirkbook . These cells are connected to each other by the remnants of
incomplete cell division, thin membrane tubes called _cytoplasmic bridges_
kirk81_1 ; kirk81_2 . The ends of the cells whence emanate the flagella,
however, point into the sphere at this stage, and so the ability to swim is
only acquired once the alga turns itself inside out through an opening at the
top of the cell sheet, called the _phialopore_ viamontes77 ; kirkreview ;
hallmann .

Of particular interest in the present context is the crucial first step of
this process, the formation of a circular invagination in so-called ‘type B’
inversion (Fig. 1c,f) followed by the engulfing of the posterior by the
anterior hemisphere hallmann ; hohn11 . (This scenario is distinct from ‘type
A’ inversion in which the initial steps involve four lips which peel back from
a cross-shaped phialopore.) The invaginations of cell sheets found in type B
inversion are very generic deformations during morphogenetic events such as
gastrulation and neurulation he ; lowery ; eiraku ; sawyer , but, in animal
model organisms, they often arise from an intricate interplay of cell
division, intercalation, migration and cell shape changes. Modelling these
therefore requires cell-based models, as pioneered by Odell _et al._ odell ,
but simpler models of simpler morphogenetic processes are required to
elucidate the underlying mechanics of these problems howard . Inversion in
_Volvox_ is, however, driven by active cell shape changes alone: inversion
starts when cells close to the equator of the shell elongate and become wedge-
shaped hohn11 . Simultaneously, the cytoplasmic bridges migrate to the wedge
ends of the cells, thus splaying the cells locally and causing the cell sheet
to bend hohn11  (Fig. 1d). Additional cell shape changes have been implicated
in the relative contraction of one hemisphere with respect to the other in
order to facilitate invagination hohn14 . After invagination, the bend region
expands, allowing the posterior hemisphere to invert fully.

At a more physical level, it has been shown recently that the inversion
process is simple enough to be amenable to a mathematical description hohn14 :
the deformations of the alga are well reproduced by a simple elastic model in
which the cell shape changes and motion of cytoplasmic bridges impart local
variations of intrinsic curvature and stretches to an elastic shell hohn14 .
The associated mechanics have remained unclear, however. Here, we perform an
asymptotic analysis at small deformations to clarify the geometric distinction
between deformations resulting from intrinsic bending and intrinsic
stretching, respectively. A numerical study of the bifurcation behavior
further serves to illustrate how a sequence of local deformations can achieve
invagination, and how contraction complements bending in this picture.

##  II Elastic Model

Following Höhn _et al._ hohn14 , we inscribe _Volvox_ inversion into the very
general framework of the axisymmetric deformations of a thin elastic spherical
shell of radius $R$ and thickness $h\ll R$ under variations of its intrinsic
curvature and stretches. The undeformed, spherical, configuration of the shell
is characterized by arclength $s$ and the distance of the shell from its axis
of revolution, $r_{0}(s)$ (Fig. 1g). To these correspond arclength $S(s)$ and
distance from the axis of revolution $r(s)$ in the deformed configuration
(Fig. 1h). The undeformed and deformed configurations are related by the
meridional and circumferential stretches,

$f_{s}(s)=\frac{\mathrm{d}S}{\mathrm{d}s}\ \ \ \ \ \ {\rm and}\ \ \ \ \ \
f_{\phi}(s)=\frac{r(s)}{r_{0}(s)}~{}.$ (1)

(These definitions do not require that the undeformed configuration be
spherical, and apply for the deformations of any axisymmetric object.) These
define the strains

$\displaystyle E_{s}=f_{s}-f_{s}^{0},$ $\displaystyle
E_{\phi}=f_{\phi}-f_{\phi}^{0},$ (2)

and curvature strains

$\displaystyle K_{s}=f_{s}\kappa_{s}-f_{s}^{0}\kappa_{s}^{0},$ $\displaystyle
K_{\phi}=f_{\phi}\kappa_{\phi}-f_{\phi}^{0}\kappa_{\phi}^{0},$ (3)

where $\kappa_{s}$ and $\kappa_{\phi}$ denote the meridional and
circumferential curvatures of the deformed shell. The intrinsic curvatures and
stretches introduced by $f_{s}^{0},f_{\phi}^{0}$ and
$\kappa_{s}^{0},\kappa_{\phi}^{0}$ extend Helfrich’s work on membranes
helfrich . The deformed configuration of the shell minimises an energy of the
Hookean form libai ; audolypomeau ; knoche11

$\displaystyle\mathcal{E}$ $\displaystyle=\dfrac{\pi
Eh}{1-\nu^{2}}\int_{0}^{\pi R}{r_{0}\Bigl{(}E_{s}^{2}+E_{\phi}^{2}+2\nu
E_{s}E_{\phi}\Bigr{)}\,\mathrm{d}s}$ $\displaystyle\hskip
14.22636pt+\dfrac{\pi Eh^{3}}{12(1-\nu^{2})}\int_{0}^{\pi
R}{r_{0}\Bigl{(}K_{s}^{2}+K_{\phi}^{2}+2\nu
K_{s}K_{\phi}\Bigr{)}\,\mathrm{d}s}.$ (4)

with material parameters the elastic modulus $E$ and Poisson’s ratio $\nu$. In
computations, we take $h/R=0.15$ and $\nu=1/2$ appropriate for _Volvox_
inversion hohn14 .

In general, deformations of the shell arise from a complex interplay of
intrinsic stretches and curvatures, and the global geometry of the shell. To
clarify these, we begin by considering two simple kinds of deformations, in
which the competition is between two effects only. How these effects conspire
in general we shall explore in the main body of the paper.

####  II.0.1 Simple Deformations: Stretching and Bending

The simplest intrinsic deformation is one of uniform stretching or
contraction, which does not affect the global, spherical geometry of the
shell. This corresponds to $f_{s}^{0}=f_{\phi}^{0}=f$ and
$\kappa_{s}^{0}=\kappa_{\phi}^{0}=1/fR$. With these intrinsic stretches and
curvatures, the original sphere deforms to a sphere of radius $R^{\prime}$.
Then $f_{s}=f_{\phi}=R^{\prime}/R$, and so the strains are
$E_{s}=E_{\phi}=R^{\prime}/R-f$. However,
$\kappa_{s}=\kappa_{\phi}=1/R^{\prime}$. Thus
$f_{s}\kappa_{s}=f_{\phi}\kappa_{\phi}=f_{s}^{0}\kappa_{s}^{0}=f_{\phi}^{0}\kappa_{\phi}^{0}=1/R$,
and so $K_{s}=K_{\phi}=0$. The energy density is therefore proportional to
$(R^{\prime}/R-f)^{2}$ and is minimized for $R^{\prime}=fR$, at which point
$\mathcal{E}=0$ (Fig. 2a). (Indeed, uniform contraction is a homothetic
transformation: the angles between material points are unchanged, and so there
is no bending involved. In other words, the shell is blind to its intrinsic
curvature on this spherical solution branch.)

The intrinsic stretches and curvatures need not be compatible in this way,
however: suppose that $f_{s}^{0}=f_{\phi}^{0}=f$, but
$\kappa_{s}^{0}=\kappa_{\phi}^{0}=1/f^{\prime}R$ with $f\not=f^{\prime}$. The
energy still has spherical minima of radius $R^{\prime}=fR$, but now with
$\mathcal{E}\not=0$ (Fig. 2a). This illustrates that, conversely, even if the
equilibrium shape is spherical, the intrinsic curvatures and stretches cannot
straightforwardly be inferred from the resulting shape.

Figure 2: (color online). Simple intrinsic deformations. (a) A sphere can be
shrunk to smaller spheres of equal radii by both compatible and incompatible
intrinsic deformations. (b) Contraction of a circular region of radius $R$ in
a plane elastic sheet by a factor $f$. The boundary of this region is
contracted to $s=FR$. (c) Numerical result for $F$ (+) agrees with analytical
calculation (13) (solid line).

####  II.0.2 Simple Deformations: Stretching and Geometry

To illustrate how the global geometry affects these deformations, we consider
contraction of a plane elastic sheet, with $f_{s}^{0}=f_{\phi}^{0}=f<1$ for
$s<R$ (Fig. 2b). This does not involve any bending of the sheet, and, upon
non-dimensionalising lengths with $R$, the shell minimises

$\displaystyle\int_{0}^{\infty}{s\Bigl{\\{}\bigl{[}r^{\prime}(s)-f(s)\bigr{]}^{2}+\bigl{[}r(s)/s-f(s)\bigr{]}^{2}}$
$\displaystyle\hskip
42.67912pt+2\nu\bigl{[}r^{\prime}(s)-f(s)\bigr{]}\bigl{[}r(s)/s-f(s)\bigr{]}\Bigr{\\}}\,\mathrm{d}s,$
(5)

where

$\displaystyle f(s)=\left\\{\begin{array}[]{cl}f&\text{if }s<1\\\ 1&\text{if
}s>1.\end{array}\right.$ (8)

The resulting Euler–Lagrange equation is

$\displaystyle\dfrac{\mathrm{d}}{\mathrm{d}s}\left(s\dfrac{\mathrm{d}r}{\mathrm{d}s}\right)-\dfrac{r}{s}=(1+\nu)(1-f)s\,\delta(s-1).$
(9)

This is a homogeneous equation, and the solution satisfying the geometric
conditions $r(0)=0$ and $r(s)\sim s$ as $s\rightarrow\infty$ as well as
continuity of $r$ at $s=1$ is

$\displaystyle r(s)=\left\\{\begin{array}[]{cl}Fs&\text{if }s<1\\\
s+\dfrac{F-1}{s}&\text{if }s>1.\end{array}\right.$ (12)

The constant $F=r(1)$ is determined by the jump condition at $s=1$, or,
physically, by requiring the stress to be continuous across $s=1$. This
finally yields

$\displaystyle F=\dfrac{1}{2}\left[(1-\nu)+(1+\nu)f\right].$ (13)

This simplified problem serves as a test case for numerical solution of the
more general Euler–Lagrange equations associated with (4). These boundary-
value problems can be solved numerically with the solver bvp4c of Matlab (The
Mathworks, Inc.); our numerical setup of the governing equations otherwise
mimicks that of knoche11 . In this particular example, the linear relationship
in (13) is indeed confirmed numerically (Fig. 2c). Notice that the governing
equation (9) is independent of the forcing applied away from $s=1$; the
solution is determined by geometric boundary conditions.

##  III Results

The most drastic cell shape changes at the start of inversion occur when cells
in a narrow region close to the equator become wedge-shaped (Fig. 1d). These
are accompanied by motion of the cytoplasmic bridges to the thin tips of the
cells to splay the cell sheet and drive its inward bending. For this reason,
Höhn  _et al._ hohn14  started by considering a piecewise constant functional
form for the intrinsic curvature, in which this curvature took negative values
in a narrow region close to the equator. It was found, however, that with this
ingredient alone the energy minimizers could not reproduce the mushroom shapes
adopted by the embryos in the early of stages of inversion (Fig. 1c,f),
producing instead a shape cinched in at those points – the so-called ‘purse-
string’ effect. However, analysis of thin sections had previously revealed
that the cells in the posterior hemisphere become thinner at the start of
inversion hohn11 . When the resulting contraction of the posterior hemisphere
was incorporated into the model, it could indeed reproduce, quantitatively,
the shapes of invaginating _Volvox_ embryos.

Höhn _et al._ have thus identified two different types of active deformations
that contribute to the shapes of inverting _Volvox_ at the invagination stage:
first, a localized region of active inward bending (corresponding to negative
intrinsic curvature), and second, relative contraction of one hemisphere with
respect to the other. We shall focus on these two types of deformation in what
follows and clarify the ensuing elastic and geometrical balances.

Figure 3: (color online). Asymptotic analysis of invagination and contraction.
(a) Numerical shape resulting from contracting the posterior to a radius
$r_{\mathrm{p}}<R$. (b) Numerical “hourglass” shape resulting from pure
invagination. (c) Geometry of contraction with posterior radius
$r_{\mathrm{p}}<R$, resulting in upward motion of the posterior by a distance
$d$. (d) Geometry of pure invagination solution. (e) Asymptotic geometry: in
the limit $h\ll R$, deformations are localized to an asymptotic inner layer of
width $\delta$ about $x=X$, where $x=s/R$ is the angle that the undeformed
normal makes with the vertical. In the deformed configuration, this angle has
changed to $\beta(x)$. (f) Asymptotic invagination: upward motion of posterior
by a distance $d$ requires inward deformations scaling as $(\delta d)^{1/2}$
in the inner layer of width $\delta$. (g) Relation between preferred curvature
$k$ and width of invagination $\lambda$ for a given amount of upward posterior
motion $d$, from asymptotic calculations. (h) Inward rotation of midpoint of
invagination with, and without contraction, from asymptotic calculations.

###  III.1 Asymptotic Analysis

We start by seeking equilibrium configurations in the limit of a thin shell,
$h\ll R$. In this limit, the shapes (Fig. 3a,b) corresponding to contraction
or (pure) invagination (by which we mean, here, deformations driven by a
region of high intrinsic curvature only) result from the matching of spherical
shells of different radii or disparate relative positions (Fig. 3c,d).
Deviations from these outer solutions are localized to an asymptotic inner
layer of non-dimensional width $\delta$ about $x=X$, where $x=s/R$ is the
angle that the normal to the undeformed shell makes with the vertical (Fig.
3e). Here, we consider an incipient deformation where the normal angle
$\beta(x)$ to the deformed shell deviates but slightly from its value in the
spherical configuration, viz $\beta(x)=x+b(x)$, with $b\ll 1$.

####  III.1.1 Geometric Considerations

We begin by clarifying the geometric distinction between contraction and
invagination. The radial and vertical displacements obey

$\displaystyle u_{r}^{\prime}$
$\displaystyle=f_{s}\cos{\beta}-\cos{x}=-b\sin{X}+O\bigl{(}\delta
b,b^{2}\bigr{)},$ (14a) $\displaystyle u_{z}^{\prime}$
$\displaystyle=f_{s}\sin{\beta}-\sin{x}=b\cos{X}+O\bigl{(}\delta
b,b^{2}\bigr{)},$ (14b)

where dashes denote differentiation with respect to $x$, and where we have
assumed the scaling $f_{s}=1+O(\delta b)$ which we shall derive presently. Let
$d$ denote the (non-dimensional) distance by which the posterior moves up.
Matching to the outer solutions requires the net displacements $U_{r}$ and
$U_{z}$, obtained by integrating (14) across the inner layer, to obey

$\displaystyle U_{r}^{\text{(c)}}$ $\displaystyle=d\sin{X},$ $\displaystyle
U_{z}^{\text{(c)}}$ $\displaystyle=-d\cos{X},$ (15a) $\displaystyle
U_{r}^{\text{(i)}}$ $\displaystyle=0,$ $\displaystyle U_{z}^{\text{(i)}}$
$\displaystyle=-d,$ (15b)

where the superscripts (c) and (i) refer, respectively, to the solutions
corresponding to contraction and (pure) invagination. In the case of
contraction, (14) and (15a) give the scaling $b^{\text{(c)}}\sim d/\delta$. If
there is no contraction, however, (14) and (15b) imply that the leading-order
solution does not yield any upward motion of the posterior, which is
associated with a higher-order solution only. This suggests that the
appropriate scaling is $b^{\text{(i)}}\sim(d/\delta)^{1/2}$, which we shall
verify presently.

Our assumption $b\ll 1$ thus translates to $d\ll\delta$. Hence, in the
invagination case, upward motion of the posterior requires comparatively large
inward displacements of order $(\delta d)^{1/2}\gg d$ (Fig. 3f). This
asymptotic difference of the deformations corresponding to contraction and
invagination arises purely from geometric effects; it is the origin of the
‘purse-string’ shapes found by Höhn _et al._ in the absence of contraction
hohn14 .

####  III.1.2 Elasto-Geometric Considerations

Here, we discuss the detailed solution for pure invagination. Upon non-
dimensionalising distances with $R$ and stresses with $Eh$, the Euler–Lagrange
equations of (4), derived in the appendix, can be cast into the form

$\displaystyle
f_{s}\mathit{\Sigma}\sin{x}\tan{\beta}-\varepsilon^{2}\cos{\beta}\Bigl{(}1-\nu\beta^{\prime}-\sin{\beta}\operatorname{cosec}{x}\Bigr{)}$
$\displaystyle\hskip
17.07164pt-\varepsilon^{2}\dfrac{\mathrm{d}}{\mathrm{d}x}\Bigl{(}\beta^{\prime}\sin{x}+\nu\bigl{(}\sin{\beta}-\sin{x}\bigr{)}\Bigr{)}=k^{0}(x),$
(16a)
$\displaystyle\dfrac{\mathrm{d}}{\mathrm{d}x}\Bigl{(}\mathit{\Sigma}\sec{\beta}\sin{x}\Bigr{)}-A-\nu\mathit{\Sigma}=0,$
(16b)

with the small parameter

$\displaystyle\varepsilon^{2}=\dfrac{1}{12(1-\nu^{2})}\dfrac{h^{2}}{R^{2}}\ll
1.$ (17)

In these equations, $\mathit{\Sigma}$ is the non-dimensional meridional
stress, and $A=e_{\phi}$ is the dimensionless hoop strain. The contribution
from the intrinsic curvature is

$k^{0}(x)=\varepsilon^{2}\left\\{\nu\kappa_{s}^{0}(x)\cos{\beta}-\dfrac{\mathrm{d}}{\mathrm{d}x}\Bigl{(}\kappa_{s}^{0}\sin{x}\Bigr{)}\right\\}.$
(18)

The equations are closed by the geometric relation

$\displaystyle\dfrac{\mathrm{d}}{\mathrm{d}x}\Bigl{(}A\sin{x}\Bigr{)}=f_{s}\cos{\beta}-\cos{x}.$
(19)

Introducing $\gamma=(d/\delta)^{1/2}$, scaling gives the leading balances
$\mathit{\Sigma}\sim\varepsilon^{2}\gamma/\delta^{2}$,
$\mathit{\Sigma}/\delta\sim A$, and $A/\delta\sim\gamma$ in (16,19). Hence
$\delta\sim\varepsilon^{1/2}$, and we define an inner coordinate $\xi$ via
$x=X+\delta\xi$. We also introduce the expansions

$\displaystyle\beta$ $\displaystyle=X+\gamma\bigl{(}b_{0}+\gamma
b_{1}+\gamma^{2}b_{2}+\cdots\bigr{)},$ (20a) $\displaystyle A$
$\displaystyle=\delta\gamma\bigl{(}a_{0}+\gamma a_{1}+\cdots\bigr{)},$ (20b)
$\displaystyle\mathit{\Sigma}$
$\displaystyle=\delta^{2}\gamma\cot{X}\bigl{(}\sigma_{0}+\gamma\sigma_{1}+\cdots\bigr{)}.$
(20c)

This further proves the scaling $f_{s}=1+O(\delta\gamma)$ that we have assumed
previously.

The pure invagination configuration is forced by intrinsic curvature that
differs from the curvature of the undeformed sphere in a region of width
$\lambda$ about $x=X$, where $\kappa_{s}^{0}=-k$. Writing
$\mathit{\Lambda}=\lambda\big{/}\varepsilon^{1/2}$, we thus have, at leading
order,

$\kappa_{s}^{0}(\xi)=-\dfrac{d^{1/2}}{\varepsilon^{3/4}}K\Bigl{(}\mathrm{H}\bigl{(}\xi+\tfrac{1}{2}\mathit{\Lambda}\bigr{)}-\mathrm{H}\bigl{(}\xi-\tfrac{1}{2}\mathit{\Lambda}\bigr{)}\Bigr{)},$
(21)

where $k=d^{1/2}\varepsilon^{-3/4}K$, and where H denotes the Heaviside
function. Thus

$\displaystyle k^{0}(\xi)=\varepsilon^{3/4}d^{1/2}K^{0}(\xi)\sin{X},$ (22)

where
$K^{0}(\xi)=K\left[\delta\left(\xi+\tfrac{1}{2}\mathit{\Lambda}\right)-\delta\left(\xi-\tfrac{1}{2}\mathit{\Lambda}\right)\right]$.
We note that $\gamma^{3}\gg\gamma\delta$ provided that $d\gg\varepsilon$
(which we shall assume to be the case); thus we may set $x=X$ to the order at
which we are working. Expanding (16,19), we then find

$\displaystyle\sigma_{0}-b_{0}^{\prime\prime}=K^{0}(\xi),$
$\displaystyle\sigma_{0}^{\prime}-a_{0}=0,$ $\displaystyle
a_{0}^{\prime}=-b_{0},$ (23)

at lowest order, where dashes now denote differentiation with respect to
$\xi$. At next order,

$\displaystyle\sigma_{1}+\sigma_{0}b_{0}\sec{X}\operatorname{cosec}{X}-b_{1}^{\prime\prime}=0,$
(24a)
$\displaystyle\sigma_{1}^{\prime}+\dfrac{\mathrm{d}}{\mathrm{d}\xi}\bigl{(}b_{0}\sigma_{0})\sin{X}\sec{X}-a_{1}=0,$
(24b)

with $a_{1}^{\prime}=-b_{1}-\tfrac{1}{2}b_{0}^{2}\cot{X}$. We are left to
determine the matching conditions by expanding (14) to find

$\displaystyle u_{r}^{\prime}$ $\displaystyle=-\delta\gamma
b_{0}\sin{X}-\delta\gamma^{2}\bigl{(}b_{1}\sin{X}+\tfrac{1}{2}b_{0}^{2}\cos{X}\bigr{)}$
$\displaystyle\hskip
17.07164pt-\delta\gamma^{3}\bigl{(}b_{2}\sin{X}+b_{0}b_{1}\cos{X}-\tfrac{1}{6}b_{0}^{3}\sin{X}\bigr{)},$
(25a) $\displaystyle u_{z}^{\prime}$ $\displaystyle=\delta\gamma
b_{0}\cos{X}+\delta\gamma^{2}\bigl{(}b_{1}\cos{X}-\tfrac{1}{2}b_{0}^{2}\sin{X}\bigr{)}$
$\displaystyle\hskip
17.07164pt+\delta\gamma^{3}\bigl{(}b_{2}\cos{X}-b_{0}b_{1}\sin{X}-\tfrac{1}{6}b_{0}^{3}\cos{X}\bigr{)},$
(25b)

up to corrections of order
$O\bigl{(}\delta\gamma^{4},\delta^{2}\gamma\bigr{)}$. Applying (15b), at
lowest order, we find

$\int_{-\infty}^{\infty}{b_{0}\;\mathrm{d}\xi}=0.$ (26)

At next order, (15b) is a system of two linear equations for two integrals,
with solution

$\displaystyle\int_{-\infty}^{\infty}{b_{0}^{2}\;\mathrm{d}\xi}=2\sin{X},$
$\displaystyle\int_{-\infty}^{\infty}{b_{1}\;\mathrm{d}\xi}=-\cos{X}.$ (27)

We note in particular that the resulting condition on the leading-order
solution has only arisen in the second-order expansion of the matching
conditions. Similarly, at order $O\bigl{(}\delta\gamma^{3}\bigr{)}$, we find

$\int_{-\infty}^{\infty}{b_{0}b_{1}\;\mathrm{d}\xi}=0.$ (28)

The leading-order problem is thus

$b_{0}^{\prime\prime\prime\prime}+b_{0}=K\Bigl{(}\delta^{\prime\prime}\bigl{(}\xi+\tfrac{1}{2}\mathit{\Lambda}\bigr{)}-\delta^{\prime\prime}\bigl{(}\xi-\tfrac{1}{2}\mathit{\Lambda}\bigr{)}\Bigr{)},$
(29)

with matching conditions (26) and the first of (27). Symmetry ensures that the
first of (26) is satisfied. After a considerable amount of algebra, the first
of (27) reduces to a relation between $K$ and $\mathit{\Lambda}$,

$K^{2}=\dfrac{8\sqrt{2}\sin{X}}{1+\mathrm{e}^{-\mathit{\Lambda}/\sqrt{2}}\Bigl{(}\bigl{(}\sqrt{2}\mathit{\Lambda}-1\bigr{)}\sin{\frac{\mathit{\Lambda}}{\sqrt{2}}}-\cos{\frac{\mathit{\Lambda}}{\sqrt{2}}}\Bigr{)}}.$
(30)

This function exhibits a global minimum at $\mathit{\Lambda}\approx 2.44$
(Fig. 3g). This is a first indication that narrow invaginations are more
efficient than those resulting from wider regions of high intrinsic curvature,
a statement that we shall make more precise later.

Symmetry also implies that there is no inward rotation of the midpoint of the
invagination at this order. Rather, inward folding is a second-order effect,
for which we need to consider the second-order problem,

$b_{1}^{\prime\prime\prime\prime}+b_{1}=\left\\{\dfrac{\mathrm{d}^{2}}{\mathrm{d}\xi^{2}}\Bigl{(}b_{0}\sigma_{0}\Bigr{)}-\tfrac{1}{2}b_{0}^{2}\right\\}\cot{X},$
(31)

with matching conditions (28) and the second of (27). The rotation of the
midpoint of the invagination is thus

$\mathit{\Delta}\beta^{\text{(i)}}=\Bigl{(}B^{\text{(i)}}(\mathit{\Lambda})\cos{X}\Bigr{)}\dfrac{d}{\varepsilon^{1/2}},$
(32)

where $B^{\text{(i)}}(\mathit{\Lambda})$ is determined by the solution of
(31). The detailed solution reveals that

$B^{\text{(i)}}(\mathit{\Lambda})=\frac{2\sqrt{2}\mathrm{e}^{-\frac{\mathit{\Lambda}}{2\sqrt{2}}}\left[4\mathrm{e}^{\frac{\mathit{\Lambda}}{2\sqrt{2}}}\sin{\frac{\mathit{\Lambda}}{\sqrt{2}}}-\mathrm{e}^{\frac{\mathit{\Lambda}}{\sqrt{2}}}\sin{\frac{\mathit{\Lambda}}{2\sqrt{2}}}-3\sin{\frac{3\mathit{\Lambda}}{2\sqrt{2}}}+\Bigl{(}\mathrm{e}^{\frac{\mathit{\Lambda}}{\sqrt{2}}}-1\Bigr{)}\cos{\frac{\mathit{\Lambda}}{2\sqrt{2}}}\right]}{5\left[\mathrm{e}^{\frac{\mathit{\Lambda}}{\sqrt{2}}}+\bigl{(}\sqrt{2}\mathit{\Lambda}-1\bigr{)}\sin{\frac{\mathit{\Lambda}}{\sqrt{2}}}-\cos{\frac{\mathit{\Lambda}}{\sqrt{2}}}\right]},$
(33)

but the geometric factor in (32) is the main point: this factor resulting from
the global geometry of the shell hampers the inward rotation of the midpoint
of the invagination. (This is as expected: by symmetry, invagination at the
equator, where $\cos{X}=0$, yields no rotation.)

An analogous, though considerably more straightforward, calculation can be
carried out for contraction: non-dimensionally, upward posterior motion by $d$
requires $f_{s}^{0}=f_{\phi}^{0}=1-d$ for $x<X$, and leads to

$\displaystyle\mathit{\Delta}\beta^{\text{(c)}}=\dfrac{1}{2\sqrt{2}}\dfrac{d}{\varepsilon^{1/2}}.$
(34)

At this order, the above solutions for pure invagination and contraction can
be superposed; in particular, the solutions at order
$O\bigl{(}\gamma^{2}\bigr{)}$ have the same symmetry, and so (28) is
satisfied. For contraction, there is thus no geometric obstacle to inward
folding (Fig. 3h). Contraction is thus not only a means of creating the
disparity in the radii of the anterior and posterior hemispheres required to
fit the partly inverted latter into the former, but also drives the inward
folding of the invagination, by breaking its symmetry. In _Volvox_ inversion,
this symmetry breaking is at the origin of the formation of the second passive
bend region highlighted by Höhn _et al._ hohn14  to stress the non-local
character of these deformations.

Figure 4: (color online). Setup for numerical calculations, following hohn14 .
(a) Geometrical setup: the intrinsic curvature $\kappa_{s}^{0}$ of a spherical
shell of undeformed radius $R$ differs from the undeformed curvature in the
range $\lambda_{\max}>s>\lambda_{\max}-\lambda$, where $s$ is arclength.
Posterior contraction is taken into account by a reduced posterior radius
$r_{\mathrm{p}}<R$. These intrinsic curvature and contraction result in
deformations that move up the posterior pole by a distance $d$. (b)
Corresponding functional form of $\kappa_{s}^{0}$; in the bend region,
$\kappa_{s}^{0}=-k<0$. (c) Form of the intrinsic stretches
$f_{s}^{0},f_{\phi}^{0}$ for posterior contraction. (d) Functional form of
$\kappa_{\phi}^{0}$ for posterior contraction.

###  III.2 Bifurcation Behaviour

The asymptotic analysis has shown that the coupling of elasticity and geometry
constrains small invagination-like deformations both locally and globally, but
that contraction can help overcome these global constraints. These ideas carry
over to larger deformations of the shell, which must however be studied
numerically. For this purpose, we extend the setup of hohn14 , motivated by
direct observation of thin sections of fixed embryos: the intrinsic curvature
$\kappa_{s}^{0}$ differs from that of undeformed sphere in the range
$\lambda_{\max}>s>\lambda_{\max}-\lambda$ of arclength along the shell (Fig.
4a). In this region of length $\lambda$, $\kappa_{s}^{0}=-k$, where $k>0$
(Fig. 4b). This imposed intrinsic curvature results in upward motion of the
posterior pole by a distance $d$.

####  III.2.1 Stability Statements

Our first observation is that, at fixed $\lambda_{\max}$, more than one
solution may arise for the same input parameters $(k,\lambda)$. Further
understanding is gained by considering, at fixed $\lambda_{\max}$ and for
different values of $\lambda$, the relation between $k$ and $d$. The typical
behaviour of these branches is plotted in Fig. 5. (The shapes eventually self-
intersect; accordingly, these branches end but we expect them to be joined up
smoothly to configurations with opposite sides of the shell in contact. The
study of such contact configurations typically requires some simplifying
assumptions to be made knoche11 , but we do not pursue this further, here.)

Figure 5: (color online). Bifurcation behaviour of invagination solutions.
Solution space for $\lambda_{\max}=1.1R$ and $r_{\mathrm{p}}=R$: each line
shows the relation between $k$ and $d$ at some constant $\lambda$. A critical
branch (at $\lambda=\lambda_{*}$) separates different types of branches.
Branches with $\lambda>\lambda_{\ast}$ feature two extrema; the resulting
spinodal curve (thick dashed line) defines a critical point. Insets illustrate
representative solution shapes. See text for further explanation. Figure 6:
(color online). Stability of invagination solutions. (a) Possible topologies
of a double fold in the distinguished $(k,\varkappa)$ bifurcation diagram.
Dashed branches are those that the results of maddocks87  imply to be
unstable. (b) For $\lambda>\lambda_{\ast}$, S-shaped folds arise in the
$(k,\varkappa)$ diagram. From general theory maddocks87 , the middle part of
the branch is unstable, while the outer parts are stable. An additional region
of metastability is identified by the Maxwell construction. (c) Resulting
picture: a region of unstable and metastable solutions expands underneath the
critical point.

At the distinguished value $\lambda=\lambda_{\ast}$, a critical branch arises
(Fig. 5). It separates two types of branches: first, those with
$\lambda<\lambda_{\ast}$, on which $d$ varies mononotonically with $k$, and
second, those with $\lambda>\lambda_{\ast}$, where the relation between $d$
and $k$ is more complicated. At large values of $\lambda$, these branches may
have a rather involved topology involving loops. At values of $\lambda$ just
above $\lambda_{\ast}$, however, there is a range of values of $k$ for which
there exist three configurations (Fig. 5). We note that the two outer
configurations have $\partial d/\partial k>0$, while the middle one has
$\partial d/\partial k<0$. The latter behaviour prefigures instability, which
we shall discuss in more detail below. There are thus two points on these
branches where $k$, viewed locally as a function of $d$, reaches an extremum.
The curve joining up these extrema for different values of $\lambda$ we shall
term the ‘spinodal curve’. This curve, in turn, has a maximum at a point on
the critical branch, which we shall call the ‘critical point’ and which is
characterised by $\lambda_{\ast}$ and the critical curvature, $k_{\ast}$.

The stability of the configurations in Fig. 5 can be assessed by means of
general results of bifurcation theory maddocks87 , used recently to discuss
the stability of the buckled equilibrium shapes of a pressurised elastic
spherical shell knoche11 ; knoche14 . If we let
$\varkappa=-\partial\mathcal{E}/\partial k$ denote the conjugate variable to
$k$, the key result of maddocks87  is that stability, at fixed $\lambda$, of
extremizers of the energy $\mathcal{E}$ can be assessed from the folds in the
$(k,\varkappa)$ bifurcation diagram. In particular, stability can only change
at folds in the bifurcation diagram. Expanding the bending part of the energy
functional (4) for $f_{s}^{0}=f_{\phi}^{0}=1$ and $\kappa_{\phi}^{0}=0$, we
find

$\varkappa=-G\int_{\lambda_{\max}-\lambda}^{\lambda_{\max}}{r_{0}\Bigl{(}f_{s}\kappa_{s}+\nu
f_{\phi}\kappa_{\phi}+k\Bigr{)}\,\mathrm{d}s}$ (35)

with $G=\pi Eh^{3}\big{/}6\bigl{(}1-\nu^{2}\bigr{)}$. (The last term in the
integrand is independent of the solution, and may therefore be ignored in what
follows.)

The two folds that arise in the $(k,d)$ diagram for $\lambda>\lambda_{\ast}$
(Fig. 5) are compatible _a priori_ with four fold topologies in the
$(k,\varkappa)$ diagram (Fig. 6a). However, since a single solution exists for
small $k$ (at fixed $\lambda$), the lowest branch must be stable. Further,
since the branches do not self-intersect in the $(k,d)$ diagram, they cannot
self-intersect in the $(k,\varkappa)$ diagram either. The results of
maddocks87  imply that only the first topology in Fig. 6a is compatible with
this, and so the fold is S-shaped and traversed upwards in the $(k,\varkappa)$
diagram. It follows in particular that the middle branch, with $\partial
d/\partial k<0$ is unstable, and that right branch is stable. (Numerically,
one confirms that the branches are indeed S-shaped.) Thus the stability of the
branches in this simple bifurcation diagram could also be inferred from the
$(k,d)$ diagram (though, in general problems, as discussed in maddocks87 ,
different bifurcation diagrams may suggest contradictory stability results).
However, the Maxwell construction of equal areas landaulifshitzsp  can be
applied to the $(k,\varkappa)$ diagram (Fig. 6b) to identify metastable
solutions beyond the unstable branch. These stability considerations may
appear rather technical, but they are in fact very natural: under reflection,
the $(d,k)$ diagram maps to the diagram of isotherms of a classical van-der-
Waals gas, for which the middle branch is well known to be unstable
landaulifshitzsp . Under this analogy, $\mathcal{E}$ corresponds to the Gibbs
free energy of the gas.

This analysis cannot immediately be extended to the more exotic topologies
that arise for $\lambda$ close to $\lambda_{\max}$ (Fig. 5). We note however
that part of these branches must be unstable, too: as above, a single solution
exists for small $d$, and so the corresponding branch must be stable. The
first fold must be traversed upwards, and the first branch with $\partial
d/\partial k<0$ is thus unstable, as above.

An analogous analysis can be carried out for deformations that vary $\lambda$
while keeping $k$ fixed: for $k>k_{\ast}$, the $(d,\lambda)$ diagram is
monotonic, but this ceases to be the case for $k<k_{\ast}$. As above, the
stability can be inferred from the $(\lambda,d)$ diagram, and the middle
branch with $\partial d/\partial\lambda<0$ is unstable, too.

The picture that emerges from this discussion is the following: solutions in a
region of parameter space underneath the critical point bounded by the
spinodal curve are unstable; a band of solutions on either side of this region
and below the critical point are metastable (Fig. 6c), both to perturbations
varying $k$ and to perturbations varying $\lambda$. If invagination, driven by
a localized region of active bending, is to be stable, it must move around the
critical point: if it were to enter the unstable region, the shell would flip
back and forth between the ‘shallow’ and ‘deep’ invagination states on either
side of the unstable region and suffer large strains in the process. (This
makes this kind of instability different from the classical buckling
instability of a rod or a ‘popper’ toy pandey : the latter is directed in
that, once the instability threshold is crossed, the system will snap to the
new preferred configuration and remain there.) The need for a sequence of
stable deformations to move around the critical point rationalises the
timecourse of invagination in _Volvox_ : initially, a narrow band of cells
undergoes cell shape changes, thereby acquiring a high intrinsic curvature.
This region of cells then widens, moving around the critical point, whereupon
the preferred curvature relaxes and posterior inversion can complete.

####  III.2.2 Contraction and Criticality

For different values of $\lambda_{\max}$, the critical point traces out a
trajectory in parameter space, characterised by $k_{\ast}$ and
$\lambda_{\ast}$ (Fig. 7). As $\lambda_{\max}$ increases, $k_{\ast}$
increases, while $\lambda_{\ast}$ decreases. Thus the closer to the equator,
the more difficult invagination is, not only because there is less room to fit
the posterior into the anterior, but also because a stable invagination
requires narrower and narrower invaginations of higher and higher intrinsic
curvature.

Figure 7: (color online). Contraction and the critical point. Trajectories of
critical point in parameter space as $\lambda_{\max}$ is varied, for different
values of $r_{\mathrm{p}}$. Thin dotted lines are curves of constant
$\lambda_{\max}$. At constant $\lambda_{\max}$, increased contraction leads to
decreased $k_{\ast}$ and increased $\lambda_{\ast}$.

We are left to explore how contraction affects the position of the critical
point, and hence the invagination. We introduce a reduced posterior radius
$r_{\mathrm{p}}<R$ as in hohn14  (Fig. 4a), and modify the intrinsic
curvatures and stretches accordingly (Fig. 4b,c,d). Numerically, we observe
that, at constant $\lambda_{\max}$, increasing contraction (that is, reducing
$r_{\mathrm{p}}$) decreases the critical curvature $k_{\ast}$, and increases
$\lambda_{\ast}$ (Fig. 7). Hence contraction aids invagination not only
geometrially, but also mechanically: first, it allows invagination close to
the equator (which would otherwise be prevented by different parts of the
shell touching), and second, it makes stable invagination easier, by reducing
$k_{\ast}$. Thus, again, contraction appears as a mechanical means to overcome
global geometric constraints.

####  III.2.3 Asymptotic Analogy

In the asymptotic analysis in the previous section, we restricted ourselves to
small deviations of the normal angle from the spherical configuration so that
the problem remained analytically tractable. While the leading scaling
balances remain the same for large rotations, the resulting non-linear “deep-
shell equations” cannot be rescaled so that the dependance on $X$ drops out
audolypomeau . Some further insight can, however, be gained in the shallow-
shell limit $X\ll 1$: in terms of the inner coordinate $\xi$, we write

$\displaystyle\beta(\xi)=X\,B(\xi)\ \ \ \ {\rm and}\ \ \ \
\mathit{\Sigma}(\xi)=\varepsilon\,S(\xi).$ (36)

In the absence of forcing by intrinsic curvature or stretches, the leading-
order balance is

$\displaystyle 2S^{\prime\prime}=1-B^{2}\ \ \ \ {\rm and}\ \ \ \
B^{\prime\prime}=SB,$ (37)

where dashes denote, as before, differentiation with respect to $\xi$.

This balance arises also in the study of a spherical shell pushed by a plane
audolypomeau : at large indentations, the shell dimples and the plane remains
in contact with it only in a circular transition region joining up the
undeformed shell to the isometric dimple. With the matching conditions
$B\rightarrow\pm 1$ as $\xi\rightarrow\pm\infty$, (37) describe the leading-
order shape of this transition region audolypomeau . Remarkably, this
deformation is independent of the contact force, which only arises at the next
order in the expansion audolypomeau .

Figure 8: (color online). Examples of “preferred” deformation modes, which are
solutions of (37). Of the modes shown, the middle one has the lowest elastic
energy.

The appropriate boundary conditions for the invagination case are
$B\rightarrow 1$ as $\xi\rightarrow\pm\infty$, and non-constant solutions of
(37) can indeed be found numerically (some solutions are shown in Fig. 8). In
these modes, the deformations are, in a sense, large compared to intrinsic
curvature imposed, making them geometrically ‘preferred’. Their existence lies
at the heart of the bifurcation behaviour discussed above.

##  IV Conclusion

In this paper, we have explored perhaps the simplest intrinsic deformations of
a spherical shell: elastic and geometric effects conspire to constrain
deformations resulting from a localized region of intrinsic bending.
Contraction, a somewhat more global deformation, alleviates these constraints
and thereby facilitates the stable transition from one configuration of the
shell to another. This rich mechanical behaviour makes a mathematically
interesting problem in its own right, yet this analysis has implications for
_Volvox_ inversion and wider material design problems.

Experimental studies of _Volvox_ inversion hohn11 ; hohn14  had revealed the
existence of posterior contraction, and indeed, the simple elastic model that
underlies this paper can only reproduce _in vivo_ shapes once posterior
contraction is included hohn14 . Of course, contraction is an obvious means of
creating a disparity in the anterior and posterior radii required ultimately
to fit one hemisphere into the other, but the present analysis reveals that,
beyond this geometric effect, there is another, more mechanical side to the
coin: if contraction is present, lower intrinsic curvatures, i.e. less drastic
cell shape changes, are required to stably invert the posterior hemisphere.
This ascribes a previously unrecognized additional role to these secondary
cell shape changes (i.e. those occurring away from the main bend region): just
as the shape of the deformed shell arises from a glocal competition between
elastic and geometric effects, a combination of local and more global
intrinsic properties allow inversion to proceed stably. Thus, as we have
pointed out previously, this mechanical analysis rationalises the timecouse of
the observed cell shape changes, thereby lending further support to the
observation of Höhn _et al._ hohn14 , that it is a spatio-temporally well
regulated sequences of cell shape changes that drives inversion. Thus, the
remarkable process of _Volvox_ inversion is mechanically more subtle than it
may initially appear to be.

Intrinsic deformations that allow transitions of an elastic object from one
configuration to another are of inherent interest in the material design
context, and divide into two classes: first, snapping transitions for fast
transitions between states, studied in bende , and second, stable sequences of
intrinsic deformations. The glocal behaviour of the latter is illustrated by
the present analysis: in particular, additional transformations such as
contraction can increase the number of stable parameter paths between
configurations of the elastic object. In this material design context, non-
axisymmetric deformations such as polygonal folds or wrinkles vella11  could
also become important, and may warrant a more detailed analysis.

## Acknowledgements

We thank Stephanie Höhn, Aurelia R. Honerkamp-Smith and Philipp Khuc Trong for
extensive discussions. This work was supported in part by an EPSRC studentship
(PAH), an EPSRC Established Career Fellowship (REG), and a Wellcome Trust
Senior Investigator Award (REG).

## Appendix: Governing Equations

In this appendix, we sketch the derivation of the Euler–Lagrange equations of
the energy functional (4), following knoche11 . The variation takes the form

$\displaystyle\dfrac{\delta\mathcal{E}}{2\pi}$ $\displaystyle=\int_{0}^{\pi
R}{r_{0}\Bigl{(}N_{s}\,\delta E_{s}+N_{\phi}\,\delta
E_{\phi}\Bigr{)}\;\mathrm{d}s}$ $\displaystyle\qquad\qquad+\int_{0}^{\pi
R}{r_{0}\Bigl{(}M_{s}\,\delta K_{s}+M_{\phi}\,\delta
K_{\phi}\Bigr{)}\;\mathrm{d}s},$ (A1)

where we have introduced the stresses and moments

$\displaystyle N_{s}$ $\displaystyle=C\bigl{(}E_{s}+\nu E_{\phi}\bigr{)},$
$\displaystyle N_{\phi}$ $\displaystyle=C\bigl{(}\nu E_{s}+E_{\phi}\bigr{)},$
(A2a) $\displaystyle M_{s}$ $\displaystyle=D\bigl{(}K_{s}+\nu
K_{\phi}\bigr{)},$ $\displaystyle M_{\phi}$ $\displaystyle=D\bigl{(}\nu
K_{s}+K_{\phi}\bigr{)},$ (A2b)

with $C=Eh\big{/}\bigl{(}1-\nu^{2}\bigr{)}$ and $D=Ch^{2}/12$. (These stresses
and moments are expressed here relative to the undeformed configuration.)

The deformed shape of the shell is characterised by the radial and vertical
coordinates $r(s)$ and $z(s)$, as well as the angle $\beta(s)$ that the normal
to the deformed shell makes with the vertical direction. These geometric
quantities obey the equations knoche11

$\displaystyle\frac{\mathrm{d}r}{\mathrm{d}s}=f_{s}\cos{\beta},$
$\displaystyle\frac{\mathrm{d}z}{\mathrm{d}s}=f_{s}\sin{\beta},$
$\displaystyle\frac{\mathrm{d}\beta}{\mathrm{d}s}=f_{s}\kappa_{s}.$ (A3)

We note that one of these is redundant. The variations $\delta E_{s},\delta
E_{\phi},\delta K_{s},\delta K_{\phi}$ are purely geometrical, and one shows
that knoche11

$\displaystyle\delta E_{s}$ $\displaystyle=\sec{\beta}\,\delta
r^{\prime}+f_{s}\tan{\beta}\,\delta\beta,$ $\displaystyle\delta E_{\phi}$
$\displaystyle=\dfrac{\delta r}{r_{0}},$ (A4a) $\displaystyle\delta K_{s}$
$\displaystyle=\delta\beta^{\prime},$ $\displaystyle\delta K_{\phi}$
$\displaystyle=\dfrac{\cos{\beta}}{r_{0}}\delta\beta.$ (A4b)

The variation (A1) thus becomes

$\displaystyle\dfrac{\delta\mathcal{E}}{4\pi}$
$\displaystyle=\Bigl{\llbracket}r_{0}N_{s}\sec{\beta}\,\delta
r+r_{0}M_{s}\,\delta\beta\Bigr{\rrbracket}-\int_{0}^{\pi
R}{\biggl{\\{}\frac{\mathrm{d}}{\mathrm{d}s}\Bigl{(}r_{0}N_{s}\sec{\beta}\Bigr{)}-N_{\phi}\biggr{\\}}\delta
r\;\mathrm{d}s}$ $\displaystyle\hskip 170.71652pt+\int_{0}^{\pi
R}{\biggl{\\{}r_{0}f_{s}N_{s}\tan{\beta}+M_{\phi}\cos{\beta}-\frac{\mathrm{d}}{\mathrm{d}s}\Bigl{(}r_{0}M_{s}\Bigr{)}\biggr{\\}}\delta\beta\;\mathrm{d}s},$
(A5)

upon integration by parts, whence

$\displaystyle
r_{0}f_{s}N_{s}\tan{\beta}+M_{\phi}\cos{\beta}-\frac{\mathrm{d}}{\mathrm{d}s}\Bigl{(}r_{0}M_{s}\Bigr{)}=0,$
(A6a)
$\displaystyle\frac{\mathrm{d}}{\mathrm{d}s}\Bigl{(}r_{0}N_{s}\sec{\beta}\Bigr{)}-N_{\phi}=0.$
(A6b)

These equations, together with two of the geometric relations (A3), describe
the shape of the deformed shell. For numerical purposes, it is convenient to
remove the singularity at $\beta=\pi/2$ by introducing the transverse shear
tension libai ; knoche11 , $Q=-N_{s}\tan{\beta}$, expressed here relative to
the undeformed configuration. Force balance arguments libai ; knoche11  show
that $Q$ obeys

$\displaystyle\frac{\mathrm{d}}{\mathrm{d}s}\Bigl{(}r_{0}Q\Bigr{)}+r_{0}f_{s}\kappa_{s}N_{s}+r_{0}f_{\phi}\kappa_{\phi}N_{\phi}=0.$
(A7)

The solution $Q=-N_{s}\tan{\beta}$ is selected by the boundary condition
$Q(0)=0$. At the poles of the shell, the equations have singular terms in
them, but these singularities are either removable or the appropriate boundary
values are set by symmetry arguments knoche11 . This allows appropriate
boundary conditions and values to be derived.

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# Phase transitions in a $\Phi^{4}$ matrix model on a curved noncommutative
space

Dragan Prekrat111speaker at COST Action CA18108: Workshop on theoretical
aspects of quantum gravity, 1-3 September 2022, Belgrade Faculty of Pharmacy,
University of Belgrade, Vojvode Stepe 450
Belgrade, 11221, Serbia
<EMAIL_ADDRESS>Dragana Ranković  Faculty of Pharmacy, University of
Belgrade, Vojvode Stepe 450
Belgrade, 11221, Serbia
<EMAIL_ADDRESS>Neli Kristina Todorović-Vasović  Faculty
of Pharmacy, University of Belgrade, Vojvode Stepe 450
Belgrade, 11221, Serbia
<EMAIL_ADDRESS>Samuel Kováčik  Department of
Theoretical Physics, Faculty of Mathematics, Physics and Informatics, Comenius
University in Bratislava, Mlynská dolina, 842 48
Bratislava, Slovakia
Department of Theoretical Physics and Astrophysics, Faculty of Science,
Masaryk University, Kotlár̆ská 267/2, Vever̆í
Brno, Czech Republic
<EMAIL_ADDRESS>Juraj Tekel  Department of Theoretical Physics,
Faculty of Mathematics, Physics and Informatics, Comenius University in
Bratislava, Mlynská dolina, 842 48
Bratislava, Slovakia
<EMAIL_ADDRESS>
(Day Month Year; Day Month Year)

###### Abstract

In this contribution, we summarize our recent studies of the phase structure
of the Grosse-Wulkenhaar model and its connection to renormalizability. Its
action contains a special term that couples the field to the curvature of the
noncommutative background space. We first analyze the numerically obtained
phase diagram of the model and its three phases: the ordered, the disordered,
and the noncommutative stripe phase. Afterward, we discuss the analytical
derivation of the effective action and the ordered-to-stripe transition line,
and how the obtained expression successfully explains the curvature-induced
shift of the triple point compared to the model without curvature. This shift
also causes the removal of the stripe phase and makes the model
renormalizable.

###### keywords:

Noncommutative geometry; Matrix models; Phase transitions.

PACS numbers: 02.10.Yn, 03.65.Vf, 05.10.Ln, 04.62.+v, 04.60.-m

##  1 Introduction

One plausible way of bringing the domains of gravitation and quantum theory,
which are reluctant to cooperate, closer together is to modify [1] the short-
distance structure of spacetime, for example by introducing coordinate
noncommutativity (NC). This, unexpectedly, complicates the renormalization of
NC models due to UV/IR mixing [2] which intertwines physics at large and small
scales and interferes with the scale separation.

Grosse-Wulkenhaar (GW) model [3] is a prototype for a renormalizable NC model
without UV/IR mixing problems. It features an additional harmonic-potential
$\Omega$-term that introduces symmetry between the scales through the Langman-
Szabo duality [4]. This term also has a nice geometric interpretation — a
coupling with the curvature of the background NC space of the truncated
Heisenberg algebra [5] $\mathfrak{h}^{\text{tr}}$.

Recently, the renormalizability of the GW model has been connected to the
removal of the NC striped phase from its phase diagram. This was done by
analyzing its matrix field formulation, both analytically [6] and numerically
[7, 8]. This contribution summarizes our results obtained on these fronts in
the previous years.

##  2 GW model

Let us first introduce our main actor and set up the NC stage. The two-
dimensional GW model is given by [3]

$S_{\text{\tiny GW}}=\int
dx^{2}\Bigg{(}\frac{1}{2}\partial^{\mu}\phi\star\partial_{\mu}\phi+\frac{\Omega^{2}}{2}((\theta^{-1})_{\mu\rho}x^{\rho}\phi)\star((\theta^{-1})^{\mu\sigma}x_{\sigma}\phi)+\\\
+\frac{m^{2}}{2}\phi\star\phi+\frac{\lambda}{4!}\phi\star\phi\star\phi\star\phi\Bigg{)},$
(1)

and it lives on the Moyal plane equipped with a $\star$-product and NC
coordinates:

$f\star
g=f\,e^{\nicefrac{{i}}{{2}}\,{\mathchoice{\vbox{\offinterlineskip\halign{#\cr\reflectbox{$\displaystyle\vec{}\mkern
4.0mu$}\cr\kern-3.01389pt\cr$\displaystyle\partial$\cr}}}{\vbox{\offinterlineskip\halign{#\cr\reflectbox{$\textstyle\vec{}\mkern
4.0mu$}\cr\kern-3.01389pt\cr$\textstyle\partial$\cr}}}{\vbox{\offinterlineskip\halign{#\cr\reflectbox{$\scriptstyle\vec{}\mkern
4.0mu$}\cr\kern-2.10971pt\cr$\scriptstyle\partial$\cr}}}{\vbox{\offinterlineskip\halign{#\cr\reflectbox{$\scriptscriptstyle\vec{}\mkern
4.0mu$}\cr\kern-1.50694pt\cr$\scriptscriptstyle\partial$\cr}}}}_{\mu}\theta^{\mu\nu}\vec{\partial}_{\nu}}\,g\quad\Rightarrow\quad\commutator{x^{\mu}}{x^{\nu}}_{\star}=i\theta\epsilon^{\mu\nu}.$
(2)

Thanks to the $\Omega$-term, this model is superrenormalizable [9]; without
it, we are left with the nonrenormalizable $\lambda\phi^{4}_{\star}$ model.

Using the Weyl transform and a slight adjustment of coefficients, (1) can be
turned into a matrix model

$S=N\tr\left(\Phi\mathcal{K}\Phi-
g_{r}R\Phi^{2}-g_{2}\Phi^{2}+g_{4}\Phi^{4}\right),$ (3)

where the field $\phi$ becomes a $N\times N$ Hermitian matrix $\Phi$. In
addition to regularizing the model, the finite matrices are also simulation-
friendly and allow a glimpse into nonperturbative results. The model’s curved
background space $\mathfrak{h}^{\text{tr}}$ is now spanned by rescaled NC
coordinates represented by matrices

$X=\frac{1}{\sqrt{2N}}\begin{pmatrix}&\scriptstyle+\sqrt{1}\\\
\scriptstyle+\sqrt{1}&&\scriptstyle+\sqrt{2}\\\
&\scriptstyle+\sqrt{2}&&\begin{rotate}{-5.0}{$\ddots$}\end{rotate}\\\
&&\begin{rotate}{-5.0}{$\ddots$}\end{rotate}&&\scriptstyle+\sqrt{N-1}\\\
&&&\scriptstyle+\sqrt{N-1}&\end{pmatrix},\quad
Y=\frac{i}{\sqrt{2N}}\begin{pmatrix}&\scriptstyle-\sqrt{1}\\\
\scriptstyle+\sqrt{1}&&\scriptstyle-\sqrt{2}\\\
&\scriptstyle+\sqrt{2}&&\begin{rotate}{-5.0}{$\ddots$}\end{rotate}\\\
&&\begin{rotate}{-5.0}{$\ddots$}\end{rotate}&&\scriptstyle-\sqrt{N-1}\\\
&&&\scriptstyle+\sqrt{N-1}&\end{pmatrix},$ (4)

and its rescaled curvature $R$ hides the energy levels of the $\Omega$-term
harmonic oscillator

$R=\frac{15}{2N}-8\left(X^{2}+Y^{2}\right)\stackrel{{\scriptstyle\scriptscriptstyle
N\to\infty}}{{\approx}}-\frac{16}{N}\operatorname{diag}\left(1,2,\ldots
N\right).$ (5)

In the infinite-size limit, we recover the commutation relations of the Moyal
plane. Finally, the differential calculus is defined [5] in such a way that
coordinates also act as momenta, leading to the kinetic operator given by a
double commutator

$\mathcal{K}\Phi=\commutator{X}{\commutator{X}{\Phi}}+\commutator{Y}{\commutator{Y}{\Phi}}\,.$
(6)

We have absorbed the NC scale $\theta$ into definitions of matrices and
couplings and set it to $\sqrt{\theta}$ = 1, which allows us to work with
dimensionless quantities. It will also be useful to keep an eye on the
unscaled222containing the factor $N$ from the action (3). couplings

$G_{2}=Ng_{2}\,,\qquad G_{4}=Ng_{4}\,,$ (7)

as they appear in the analytical results for renormalization.

Having defined our matrix action, we can now ask questions about the
observables $\mathcal{O}$ of interest and have them answered with the help of
well-defined matrix path integrals:

$\expectationvalue{\mathcal{O}}=\frac{\displaystyle\int[d\Phi]\,\mathcal{O}\,e^{-S}}{\displaystyle\int[d\Phi]\,e^{-S}}\,,\qquad\quad\operatorname{Var}\mathcal{O}=\expectationvalue{\mathcal{O}^{\smash{2}}}-{\expectationvalue{\mathcal{O}}}{{}^{2}}\,.$
(8)

##  3 Phase diagram

Since we are interested in the phase structure of the GW model, it is
instructive to look at the possible variants for its classical vacua. The
saddle point method gives us the classical EOM

$2\mathcal{K}\Phi-g_{r}\left(R\Phi+\Phi
R\right)+\Phi\left(-2g_{2}+4g_{4}\Phi^{2}\right)=0\,,$ (9)

whose solutions corresponding to the kinetic ($\mathcal{K}$), curvature ($R$),
and pure-potential333We are interested in the $g_{2}>0$ regime with the
spontaneous symmetry breaking. ($g_{2},g_{4}$) parts are given, respectively,
by

$\;\;\Phi_{\uparrow\uparrow}=\frac{\tr\Phi}{N}\operatorname{\mathbb{1}}\,,\qquad\quad\Phi_{\updownarrow}=\operatorname{\mathbb{0}}\,,\qquad\quad\Phi^{2}_{\uparrow\downarrow}=\frac{g_{2}}{2g_{4}}\operatorname{\mathbb{1}}\,.$
(10)

The first two are the ordered (${\uparrow\uparrow}$) and the disordered
(${\updownarrow}$) vacuums found also in commutative models; the third is NC-
specific, since nontrivial square roots of the identity matrix can occur in
the case of spontaneous symmetry breaking, giving rise to the so-called stripe
phase (${\uparrow\downarrow}$). The stripe phase breaks the translational
symmetry since it has both positive and negative eigenvalues and thus varies
[10, 11, 12] throughout space.

Figure 1: Phase diagrams for the model with $g_{r}=0$ (left) and $g_{r}=0.2$
(right). The two diagonal bright stripes on each of the plots represent the
transition lines. The ${\updownarrow}$-phase occupies the bottom right, the
${\uparrow\uparrow}$-phase the top left, while the NC
${\uparrow\downarrow}$-phase is sandwiched in-between. The diagrams are
constructed from contour plots of the specific heat for $N=24$ and represent
the updated and modified version of Fig. 1 from Ref. 7.

As shown in Fig. 1, such a phase structure is precisely what we found [6, 7,
8] using numerical Hamiltonian Monte Carlo simulations [13], aided
occasionally by an eigenvalue-flipping algorithm [14]. These phase diagrams
are constructed for both the model with (GW) and without
($\lambda\phi^{4}_{\star}$) curvature effects by scanning the $(g_{2},g_{4})$
parameter space for peaks and edges in profiles of specific heat
$C=\operatorname{Var}S/N$, which are some possible signatures of a phase
transition. Darker regions in the diagrams have lower values and lighter
regions have higher values of $C$. In each of the diagrams, two bright stripes
rising from the common triple point from left to right thus represent the
phase-transition lines separating the three phases indicated by the arrows in
the corners of the diagrams. These lines follow [6] a (shifted) deformed
square-root law, the exact version of which is found in the pure-potential
model [15, 16]. As can be seen, the ${\updownarrow}$-phase occupies the lower
right corner due to the stronger $g_{4}$-interaction, the
${\uparrow\uparrow}$-phase is in the upper left corner where the mass
parameter $g_{2}$ is greater, and the ${\uparrow\downarrow}$-phase is
sandwiched in-between. This diagram structure is also found in matrix models
on other NC spaces such as the fuzzy sphere [17, 18, 19], the fuzzy disc [20],
and the fuzzy complex projective space [21, 22].

The extensive analysis of various other observables, different matrix sizes
and parameter scalings, as well as the details on the numerical method and
algorithms, can be found in Ref. 7 and Ref. 8.

##  4 Renormalization

As advertized in the introduction, the extension of the
${\uparrow\downarrow}$-phase in the phase diagram of the NC model seems to be
related to the renormalizability of the model. We will now clarify the details
of this connection.

There are several important ingredients that tie these properties together.
The first of these is shown in Fig. 1. Namely, if we compare the two phase
diagrams with $g_{r}=0$ and $g_{r}\neq 0$, we see that there is an overall
shift of the latter towards the larger values of $g_{2}$ that is proportional
to the curvature coupling $g_{r}$. More precisely, we have found [6] that the
triple point of the $\lambda\phi^{4}_{\star}$ model is glued to the origin in
the large-$N$ limit while the triple point of the GW model is separated from
the origin by

$\delta g_{2}^{\text{tp}}\propto g_{r}\,.$ (11)

Since we often calculate the renormalization of the model in unscaled
parameters, we can translate this picture from $(g_{2},g_{4})$ to
$(G_{2},G_{4})$. This is our second ingredient. Since this change stretches
the diagram axis, it affects the large-$N$ limits of the diagrams. It turns
out that after this stretch the triple point of $\lambda\phi^{4}_{\star}$
still remains at the origin while the triple point of the GW model now shifts
to infinity and out of the diagram as

$\delta G_{2}^{\,\text{tp}}=N\,\delta g_{2}^{\text{tp}}\propto
Ng_{r}\to\infty,\qquad N\to\infty\,.$ (12)

Thus, with respect to the unscaled parameters, the nonrenormalizable
$\lambda\phi^{4}_{\star}$ model is left with the ${\uparrow\downarrow}$-phase,
while in the renormalizable GW model, the ${\uparrow\downarrow}$-phase is
completely removed.

Furthermore, if we consider the mass renormalization [23] in the GW model

$\delta
m^{2}_{\text{ren}}=\frac{\lambda}{12\pi(1+\Omega^{2})}\log\frac{\Lambda^{2}\theta}{\Omega}\,,$
(13)

using the cutoff [3] $\Lambda^{2}\propto N$, and translating this into the
language of the matrix model parameters, we obtain

$\delta G_{2}^{\text{\,ren}}\sim-\log N,$ (14)

hence the bare $G_{2}$ must increase by $\absolutevalue{\delta
G_{2}^{\,\text{ren}}}$ to compensate for the quantum effects, shifting
accordingly its position in the phase diagram. Since $\absolutevalue{\delta
G_{2}^{\,\text{ren}}}<\delta G_{2}^{\,\text{tp}}$, this means that the bare
mass of the model remains in the ${\updownarrow}$-phase and consequently out
of the ${\uparrow\downarrow}$-phase in the large-$N$ limit.

We know [3, 9] that the GW model can be used to construct the renormalizable
version of the $\lambda\phi^{4}_{\star}$ by turning off the curvature coupling
sufficiently slowly as we increase $N$. Indeed, since it does not renormalize,
$\Omega$ can serve as a label for a series of superrenormalizable models
$\text{GW}_{\Omega}$ whose limit we will denote by
$\lambda\phi^{4}_{\text{\tiny GW}}$:

$\text{GW}_{\Omega}\to\lambda\phi^{4}_{\text{\tiny
GW}},\qquad\Omega\sim\frac{1}{\log N}\to 0\,.$ (15)

The comparison between the $\lambda\phi^{4}_{\star}$ and
$\lambda\phi^{4}_{\text{\tiny GW}}$ phase diagrams will be our final
ingredient in the argument about the connection between the
${\uparrow\downarrow}$-phase and (non)renormalizability. When we change to
matrix-model parameters, $\Omega\sim 1/\log N$ becomes [7] $g_{r}\sim
1/\log^{2}N$, and accordingly

$\delta G_{2}^{\,\text{tp}}\propto
Ng_{r}\sim\frac{N}{\log^{2}N}\to\infty,\qquad N\to\infty\,.$ (16)

Therefore, although formally both $\lambda\phi^{4}_{\star}$ and
$\lambda\phi^{4}_{\text{\tiny GW}}$ lack the curvature term, the
former/nonrenormalizable has the ${\uparrow\downarrow}$-phase tethered to the
phase-diagram origin while the latter/renormalizable is
${\uparrow\downarrow}$-phase-free.

Looking once more at the mass renormalization, we see that the leading
divergence

$g_{r}\sim\frac{1}{\log^{2}N}\quad\Rightarrow\quad\absolutevalue{\delta
G_{2}^{\,\text{ren}}}\sim\log N\,,$ (17)

still grows slower than the triple-point shift (16) and that the bare mass
still remains outside the ${\uparrow\downarrow}$-phase.

##  5 Analytical results

Since the conclusions of the previous section hinge on the numerically
obtained phase-diagram shift, it would be desirable to derive this shift by
analytical means. This is partially achieved in Ref. 6.

As can be seen from the classical vacuum solutions (10), depending on the
shape and depth of the potential well, their eigenvalues can be all around
zero, all around one minimum of the potential well, or simultaneously around
both the positive and negative minimum. This means that the eigenvalue
distributions in the different phases can be classified according to their
symmetry and the connectedness of their support: the ${\updownarrow}$-phase
has a symmetric one-cut distribution, the ${\uparrow\uparrow}$-phase has an
asymmetric one-cut distribution, while the ${\uparrow\downarrow}$-phase is
described by a two-cut distribution.

To obtain these distributions and the conditions for their existence, and to
extract from them the positions of the phase-transition lines, we must first
integrate out the angular degrees of freedom in the partition function $Z$
coming from the unitary matrices $U$ in the field decomposition $\Phi=U\Lambda
U^{\dagger}$, and obtain the effective action that depends only on the
eigenvalues $\Lambda=\operatorname{diag}\lambda_{i}$. Because of the double
commutators in the kinetic operator
$\mathcal{K}=[X,[X,\cdot\,]]+[Y,[Y,\cdot\,]]$, unlike in the potential term,
unitary matrices cannot be canceled out under the trace, leading to the
complicated angular integral. Since numerical simulations [8, 7] imply that
the curvature gives the main contribution to the shift, while the kinetic term
merely turns on the additional ${\uparrow\uparrow}$-phase, we will drop the
kinetic term and focus on the slightly less complicated curvature contribution

$Z=\int[d\Phi]\,e^{-S}=\int[d\Lambda]\,\Delta^{2}(\Lambda)\,e^{-N\tr\left(-g_{2}\Lambda^{2}+g_{4}\Lambda^{4}\right)}\int[dU]\,e^{\,g_{r}N\tr\left(URU^{\dagger}\Lambda^{2}\right)}\,,$
(18)

where $\Delta(\Lambda)$ is a Vandermonde determinant that depends only on the
argument’s eigenvalues

$\Delta(\Lambda)=\prod_{1\leq i<j\leq N}(\lambda_{j}-\lambda_{i})\,,$ (19)

and appears from the Jacobian of the change of variables in the integral. If
we use the method described in Ref. 24 and Ref. 25, where the exact analytical
treatment of the correlation functions for the $\Phi^{3}$ and hybrid
$\Phi^{3}$/$\Phi^{4}$ NC models is given, we can perform [6] the
$[dU]$-integration to obtain

$Z=\int[d\Lambda]\,\frac{\Delta^{2}(\Lambda)}{\Delta(\Lambda^{2})}\,e^{-N\tr(-g_{2}\Lambda^{2}-g_{r}R\Lambda^{2}+g_{4}\Lambda^{4})}\,,$
(20)

however, since $\Delta(\Lambda^{2})$ does not have a definite sign for all
$\Lambda$, we cannot perform the final step and absorb it into the effective
action in the exponent.

It turns out that (18) can be instead tackled perturbatively with the help of
the Harish-Chandra-Itzykson-Zuber integral [26]

$I=\int\limits_{{\rm
U}(N)}\\!\\![dU]\,e^{t\tr(AUBU^{\dagger})}=\frac{c_{N}\,t^{-N(N-1)/2}}{\Delta(A)\Delta(B)}\begin{vmatrix}e^{ta_{1}b_{1}}&e^{ta_{1}b_{2}}&\cdots&e^{ta_{1}b_{N}}\\\
e^{ta_{2}b_{1}}&e^{ta_{2}b_{2}}&\cdots&e^{ta_{2}b_{N}}\\\
\vdots&\vdots&&\vdots\\\
e^{ta_{N}b_{1}}&e^{ta_{N}b_{2}}&\cdots&e^{ta_{N}b_{N}}\end{vmatrix},$ (21)

where $c_{N}$ is a constant that depends on the matrix size. The trick is to
expand the determinant on the right-hand side of (21) in powers of $t$ and
then re-exponentiate it into the effective action $S(\Lambda)$, which contains
only the field eigenvalues. This was done in Ref. 6 up to $O(g_{r}^{4})$,
leading to the multitrace action

$S(\Lambda)=-\left(g_{2}-8g_{r}\right)N\tr\Lambda^{2}+\left(g_{4}-\frac{32}{3}g_{r}^{2}\right)N\tr\Lambda^{4}+\frac{1024}{45}g_{r}^{4}N\tr\Lambda^{8}+\\\
+\frac{32}{3}g_{r}^{2}\tr^{\mathrlap{2}}\,\Lambda^{2}+\frac{1024}{15}g_{r}^{4}\tr^{\mathrlap{2}}\,\Lambda^{4}-\frac{4096}{45}g_{r}^{4}\tr\Lambda^{6}\tr\Lambda^{2}-\log\Delta^{2}(\Lambda)\,.$
(22)

As we can see from the first term in the action, the leading effect of the
curvature is to shift the mass parameter by $8g_{r}$, which is proportional to
the curvature coupling $g_{r}$, as expected, but only by half the amount found
for the triple-point shift in numerical simulations [7]. This is a good place
to notice [8] that if we replace $R$ in (3) by its eigenvalue with the largest
absolute value, we can expect the triple-point shift to be at most $16g_{r}$
in the large-$N$ limit, so the simulations imply that this bound is saturated.

Figure 2: (left) Comparison of the dashed analytical line (25) and numerical
points with errorbars for the ${\uparrow\uparrow}\to{\uparrow\downarrow}$
transition at $N=24$ and $g_{r}=0.1$. The plot is the modified version of Fig.
5 from Ref. 7. (right) Turning points of the transition line at $O(g_{r}^{2})$
and $O(g_{r}^{4})$ where the dotted-line ending near
$(g_{2},g_{4})=(16g_{r},0)$ represents a naive extrapolation to higher orders.

Similarly to other matrix models [15, 16, 21, 27, 28, 29], by using the saddle
point method and taking the continuous limit $N\to\infty$, we arrive at the
equation for the eigenvalue distribution $\rho(\lambda)$,

$-\left(g_{2}-8g_{r}-\frac{64}{3}g_{r}^{2}m_{2}+\frac{4096}{45}g_{r}^{4}m_{6}\right)\lambda+2\left(g_{4}-\frac{32}{3}g_{r}^{2}+\frac{2048}{15}g_{r}^{4}m_{4}\right)\lambda^{3}-\\\
-\frac{4096}{15}g_{r}^{4}m_{2}\lambda^{5}+\frac{4096}{45}g_{r}^{4}\lambda^{7}=\underset{\mathclap{\text{support}}}{\int}d\lambda^{\prime}\,\frac{\rho(\lambda^{\prime})}{\lambda-\lambda^{\prime}}\,,$
(23)

which also depends on the moments of the distribution

$m_{k}=\underset{\mathclap{\text{support}}}{\int}d\lambda^{\prime}\,\rho(\lambda^{\prime})\,\lambda^{\prime\,k}.$
(24)

The equation (23) can be solved [6] order-by-order in $g_{r}$ with appropriate
ansatz, and from the one-cut solution existence condition we have found that
the ${\uparrow\uparrow}\to{\uparrow\downarrow}$ transition line satisfies

$g_{2}=2\sqrt{g_{4}}+8g_{r}+\frac{32}{3}\frac{g_{r}^{2}}{\sqrt{g_{4}}}+\frac{256}{15}\frac{g_{r}^{4}}{g_{4}\sqrt{g_{4}}}\,,$
(25)

up to $O(g_{r}^{4})$. As the left diagram in Fig. 2 shows, (25) cannot be
trusted near $g_{4}=0$, where it deviates starkly from the numerical results,
which means that the would-be-triple-point shift is not fully described by
$\delta g_{2}^{\text{tp}}=8g_{r}$. However, looking at the right plot and the
positions of the transition-line turning points for $O(g_{r}^{2})$ and
$O(g_{r}^{4})$, where the approximation starts to fail, we notice that they
move towards the desired point $(g_{2},g_{4})=(16g_{r},0)$, i.e. the shift
$\delta g_{2}^{\text{tp}}=16g_{r}$, when higher terms are included in the
effective action.

##  6 Conclusions and outlook

In this contribution, we have compiled the results of the recent investigation
of the phase structure of the GW model and its connection to
renormalizability. The phase diagram is found to be divided into three phases,
which are also present in other NC models: the ordered, the disordered, and
the NC stripe phase. We have also sketched the derivation of the ordered-to-
stripe transition line in the strong interaction regime without the kinetic
term and shown that the expression successfully explains the curvature-induced
shift of the triple point. By expressing the results in unscaled parameters,
we have shown that this shift causes the removal of the stripe phase in the
renormalizable GW model and also in the renormalizable
$\lambda\phi^{4}_{\text{\tiny GW}}$ model obtained by slowly turning off the
curvature coupling, but not in the original nonrenormalizable
$\lambda\phi^{4}_{\star}$ without the curvature.

It would be interesting to investigate whether this connection between phase
structure and renormalizability also holds in other NC models. For example,
one could try to simulate the related model [30, 31] on the Snyder-de Sitter
space that naturally develops the GW harmonic oscillator term, either in the
matrix form (if possible) or on a lattice [11, 32]. Another promising option
would be the nonrenormalizable $U(1)$ gauge model [33] on
$\mathfrak{h}^{\text{tr}}$. This model has two possible classical vacua: the
trivial zero-vacuum and the stripe vacuum proportional to the NC coordinates.
Numerical simulations are needed to determine which of these is actually
realized in the model. It would also be nice to study the correlation
functions of the GW model in a similar way as it is done [34] on the fuzzy
sphere.

On the analytical side, we could try adding $O(g_{r}^{6})$ terms to the
effective action (22), as this would be a straightforward, if somewhat more
involved, continuation of the existing approach, and see if the turning point
of the transition line follows the trend from Fig. 2. This would give us the
first nontrivial extrapolation of the starting point of the transition line.
Finally, we would also like to find a way to extend the analytical treatment
of the curvature term to the kinetic term of the GW model.

## Acknowledgments

This research was funded by:

  * •

the Ministry of Education, Science and Technological Development, Republic of
Serbia through:

    * –

Grant Agreement with University of Belgrade – Faculty of Pharmacy No.
451-03-68/2022-14/200161,

    * –

Grant Agreement with University of Belgrade – Faculty of Physics No.
451-03-68/2022-14/200162,

  * •

the Ministry of Science, Technological Development and Innovation, Republic of
Serbia through Grant Agreement with University of Belgrade – Faculty of
Pharmacy No. 451-03-47/2023-01/200161,

  * •

VEGA 1/0703/20 grant _Quantum structure of spacetime_.

  * •

VEGA 1/0025/23 grant

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