Hessian is compact. Using the shape gradient information, a Sobolev gradient-
based descent scheme was formulated in order to solve the problem numerically
via finite element method. Some numerical experiments in two dimensions are
exhibited and are compared with those obtained using the Kohn-Vogelius
approach. Numerical results showed that the new method has some advantages
when compared to the conventional KV approach since (1) it requires less
overall computational-time-per-iteration when utilized in a Sobolev gradient
based algorithm – at least in the case of the conducted experiments, (2) is
less prone to premature convergence under large domain variations, and (3)
provides more accurate approximation of the optimal shape for coarser meshes.
Nonetheless, in general, KVM requires less number of iterations than CCBM for
the present algorithm to converge. Even so, for small step sizes, the optimal
solutions obtained from the two methods coincide. The new method is also
tested in three dimensions, and various test cases were considered to further
illustrate the feasibility and efficiency of the proposed method. Further
application of CCBM in solving free surface problems under shape optimization
settings is the subject of our next investigation. Also, in our future work,
we will further study the issue of convergence of an evolving domain to a
stationary point under the vanishing of the normal speed of the moving
boundary.

Acknowledgements. The author wishes to thank the referees for their helpful
comments, suggestions, and remarks which greatly improve the quality of this
manuscript. He is especially grateful to one of the referees for pointing out
several inaccuracies and for bringing to his attention references [12] and
[34]. Lastly, the author acknowledges the support from JST CREST Grant Number
JPMJCR2014.

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##  Appendix A Appendices

###  A.1 Proof of Lemma 4.2

###### Proof of Lemma 4.2.

The proof proceeds in three classical steps (sketch here) after formulating
the problem onto the fixed domain. First, we prove its weak convergence to the
material derivative followed then by its convergence in strong sense.
Afterwards, we deduce the shape derivative of the state using the identity
$u^{\prime}=\dot{u}-\nabla u\cdot\bm{\mathbf{V}}$.

Step 1. We recall from Lemma 3.3 the transported problem, and subtract from it
the original one to obtain $(1/t)({u^{t}}-{u})=0$ on $\Gamma$, and for all
$\overline{v}\in\bm{\mathbf{\mathsf{V}}}(\Omega)$, the equation

$\displaystyle\int_{\Omega}{A_{t}\left(\frac{\nabla({u^{t}}-{u})}{t}\right)\cdot\nabla\overline{v}}{\,{\operatorname{\mathnormal{d}}}x}+i\int_{\Sigma}{B_{t}\left(\frac{{u^{t}}-{u}}{t}\right)\overline{v}}{\,{\operatorname{\mathnormal{d}}}\sigma}$
(38) $\displaystyle\
=\int_{\Omega}{\left(\frac{\bm{\mathbf{I}}-A_{t}}{t}\right)\nabla{u}\cdot\nabla\overline{v}}{\,{\operatorname{\mathnormal{d}}}x}+i\int_{\Sigma}{\left(\frac{1-B_{t}}{t}\right){u}\overline{v}}{\,{\operatorname{\mathnormal{d}}}\sigma}+\lambda\int_{\Sigma}{\left(\frac{B_{t}-1}{t}\right)\overline{v}}{\,{\operatorname{\mathnormal{d}}}\sigma}.$

By taking $\frac{1}{t}({u^{t}}-{u})\in\bm{\mathbf{\mathsf{V}}}(\Omega)$ as the
test function, and using the continuity of the maps $t\mapsto A_{t}$ and
$t\mapsto B_{t}$ at $t=0$, and the fact that $A_{t}$ and $B_{t}$ are bounded
for sufficiently small $t>0$, it can be verified that

${\left|\kern-1.07639pt\left|\kern-1.07639pt\left|\frac{{u^{t}}-{u}}{t}\right|\kern-1.07639pt\right|\kern-1.07639pt\right|}_{\bm{\mathbf{\mathsf{V}}}(\Omega)}\lesssim\max\left\\{\left(\left\|\frac{A_{t}-\bm{\mathbf{I}}}{t}\right\|_{\infty}+\left\|\frac{B_{t}-1}{t}\right\|_{\infty}\right){\left|\kern-1.07639pt\left|\kern-1.07639pt\left|\nabla{u}\right|\kern-1.07639pt\right|\kern-1.07639pt\right|}_{\bm{\mathbf{\mathsf{Q}}}},|\Sigma|^{1/2}\left\|\frac{B_{t}-1}{t}\right\|_{\infty}\right\\}.$

Thus, $\left\\{\frac{1}{t}({u^{t}}-{u})\right\\}$ is bounded in
$\bm{\mathbf{\mathsf{V}}}(\Omega)$. Therefore, the sequence is weakly
convergent in $\bm{\mathbf{\mathsf{V}}}(\Omega)$ and its weak limit is the
material derivative $\dot{u}$ of ${u}$.

Step 2. By passing to the limit $t\to 0$ in (38), we see that $\dot{u}$ solves
equation (3.1). We use the said equation to prove the strong convergence in
$\bm{\mathbf{\mathsf{V}}}(\Omega)$. Indeed, setting
$v=\frac{1}{t}({u^{t}}-{u})=:w^{t}\in\bm{\mathbf{\mathsf{V}}}(\Omega)$ in
(38), and noting that, on $\Omega$, we have ${u}=1$ on $\Gamma$ and
$\int_{\Omega}{\nabla{u}\cdot\nabla\overline{v}}{\,{\operatorname{\mathnormal{d}}}x}+i\int_{\Sigma}{{u}\overline{v}}{\,{\operatorname{\mathnormal{d}}}\sigma}=\lambda\int_{\Sigma}{\overline{v}}{\,{\operatorname{\mathnormal{d}}}\sigma}$,
for all $\overline{v}\in\bm{\mathbf{\mathsf{V}}}(\Omega)$, we then get the
equation

$\displaystyle\int_{\Omega}{\nabla{w}^{t}\cdot\nabla\overline{w}^{t}}{\,{\operatorname{\mathnormal{d}}}x}+i\int_{\Sigma}{{w}^{t}\overline{w}^{t}}{\,{\operatorname{\mathnormal{d}}}\sigma}$
$\displaystyle\
=\left\\{-\int_{\Omega}{(A_{t}-\bm{\mathbf{I}})\nabla{w}^{t}\cdot\nabla\overline{w}^{t}}{\,{\operatorname{\mathnormal{d}}}x}-i\int_{\Sigma}{(B_{t}-1){w}^{t}\overline{w}^{t}}{\,{\operatorname{\mathnormal{d}}}\sigma}\right\\}$
$\displaystyle\quad+\left\\{\int_{\Omega}{\left(\frac{\bm{\mathbf{I}}-A_{t}}{t}\right)\nabla{u}\cdot\nabla\overline{w}^{t}}{\,{\operatorname{\mathnormal{d}}}x}+i\int_{\Sigma}{\left(\frac{1-B_{t}}{t}\right){u}\overline{w}^{t}}{\,{\operatorname{\mathnormal{d}}}\sigma}+\lambda\int_{\Sigma}{\left(\frac{B_{t}-1}{t}\right)\overline{w}^{t}}{\,{\operatorname{\mathnormal{d}}}\sigma}\right\\}$
$\displaystyle\ =:\mathbb{I}_{1}(t)+\mathbb{I}_{2}(t).$

Using the weak convergence result for the sequence $\\{{w}^{t}\\}$ obtained in
the previous step, we easily deduce that $\lim_{t\to 0}\mathbb{I}_{1}(t)=0$
and $\lim_{t\to
0}\mathbb{I}_{2}(t)=-\int_{\Omega}{A\nabla{u}\cdot\nabla\overline{\dot{u}}}{\,{\operatorname{\mathnormal{d}}}x}-i\int_{\Sigma}{({\operatorname{div}}_{\Sigma}\bm{\mathbf{V}}){u}\overline{\dot{u}}}{\,{\operatorname{\mathnormal{d}}}\sigma}+\lambda\int_{\Sigma}{({\operatorname{div}}_{\Sigma}\bm{\mathbf{V}})\overline{\dot{u}}}{\,{\operatorname{\mathnormal{d}}}\sigma}$.
Now, from (3.1) in Proposition 3.1, we conclude that
$\mathbb{I}_{2}(t)\to\int_{\Omega}{\nabla\dot{u}\cdot\nabla\overline{\dot{u}}}{\,{\operatorname{\mathnormal{d}}}x}+i\int_{\Sigma}{\dot{u}\overline{\dot{u}}}{\,{\operatorname{\mathnormal{d}}}\sigma}$
as $t\to 0$. This shows the strong convergence of $\nabla{w}^{t}$ to
$\nabla{\dot{u}}$ in $\bm{\mathbf{\mathsf{V}}}(\Omega)$. The equivalence of
norms between the $\bm{\mathbf{\mathsf{V}}}(\Omega)$-norm and the usual
$\bm{\mathbf{\mathsf{H}}}^{1}(\Omega)$-norm implies the strong convergence of
${w}^{t}$ to ${\dot{u}}$ in $\bm{\mathbf{\mathsf{V}}}(\Omega)$.

Step 3. In the last step, we deduce the equations satisfied by the shape
derivative of $u$ using the identity
$u^{\prime}=\dot{u}-\bm{\mathbf{V}}\cdot\nabla u$. Let us first note that
$\dot{u}\in\bm{\mathbf{\mathsf{V}}}(\Omega)$. So,
$u^{\prime}=\dot{u}-\bm{\mathbf{V}}\cdot\nabla u=0$ on $\Gamma$ because
$\bm{\mathbf{V}}=0$ on $\Gamma$. Next, we let the trace of the normal
derivative of the (complex conjugate of the) test function ${v}$ on $\Sigma$
in equation (3.1) be zero. Then, by expansion (17), and integration by parts,
we get
$-\int_{\Omega}{\nabla\dot{u}\cdot\nabla\overline{v}}{\,{\operatorname{\mathnormal{d}}}x}=\int_{\Omega}{\Delta{u}(\bm{\mathbf{V}}\cdot\nabla\overline{v})}{\,{\operatorname{\mathnormal{d}}}x}+\int_{\Omega}{\Delta\overline{v}(\bm{\mathbf{V}}\cdot\nabla{u})}{\,{\operatorname{\mathnormal{d}}}x}=-\int_{\Omega}{\nabla(\bm{\mathbf{V}}\cdot\nabla{u})\cdot\nabla\overline{v}}{\,{\operatorname{\mathnormal{d}}}x}$,
for any smooth function ${v}$ with compact support on $\Omega$ and such that
$\partial_{\bm{\mathbf{n}}}{\overline{v}}=0$ on $\Sigma$. Since $-\Delta{u}=0$
in $\Omega$, we easily find that
$\int_{\Omega}{\nabla(\dot{u}-\bm{\mathbf{V}}\cdot\nabla{u})\cdot\nabla\overline{v}}{\,{\operatorname{\mathnormal{d}}}x}=-\int_{\Omega}{(\Delta{u}^{\prime})\overline{v}}{\,{\operatorname{\mathnormal{d}}}x}=0$.
Varying ${v}$, we obtain $\Delta{u}^{\prime}=0$ in $\Omega$. Let us now choose
${v}\in\bm{\mathbf{\mathsf{H}}}^{2}(\Omega)\cap\bm{\mathbf{\mathsf{V}}}(\Omega)$
such that $\partial_{\bm{\mathbf{n}}}{\overline{v}}=0$ on $\Sigma$. Applying
Green’s theorem and the tangential Green’s formula (22) to equation (3.1), and
noting that $\partial_{\bm{\mathbf{n}}}{u}+i{u}=\lambda$ on $\Sigma$, we
obtain
$\int_{\Sigma}{(\partial_{\bm{\mathbf{n}}}{{u}^{\prime}}+i{u}^{\prime})\overline{v}}{\,{\operatorname{\mathnormal{d}}}\sigma}=-\int_{\Sigma}{(\nabla{\overline{v}}\cdot\nabla{u})V_{n}}{\,{\operatorname{\mathnormal{d}}}\sigma}-\int_{\Sigma}{\overline{v}\left[i(\partial_{\bm{\mathbf{n}}}{u}+\kappa{u})-\lambda\kappa\right]V_{n}}{\,{\operatorname{\mathnormal{d}}}\sigma}$.
Since $\partial_{\bm{\mathbf{n}}}{\overline{v}}=0$ on $\Sigma$, then we can
write
$\int_{\Sigma}{(\nabla{\overline{v}}\cdot\nabla{u})V_{n}}{\,{\operatorname{\mathnormal{d}}}\sigma}=\int_{\Sigma}{(\nabla_{\Sigma}{\overline{v}}\cdot\nabla_{\Sigma}{u})V_{n}}{\,{\operatorname{\mathnormal{d}}}\sigma}=-\int_{\Sigma}{\overline{v}\operatorname{div}_{\Sigma}(V_{n}\nabla_{\Sigma}{u})}{\,{\operatorname{\mathnormal{d}}}\sigma}$,
where the second equality follows again from the tangential Green’s formula
(22) together with the fact that
$(V_{n}\nabla_{\Sigma}{u})\cdot\bm{\mathbf{n}}=0$ on $\Sigma$. Putting the
computed identity to the previous equation above leads to

$\int_{\Sigma}{(\partial_{\bm{\mathbf{n}}}{{u}^{\prime}}+i{u}^{\prime})\overline{v}}{\,{\operatorname{\mathnormal{d}}}\sigma}=\int_{\Sigma}{{\overline{v}}\left[\operatorname{div}_{\Sigma}(V_{n}\nabla_{\Sigma}{u})-i(\partial_{\bm{\mathbf{n}}}{u}+\kappa{u})V_{n}+\lambda\kappa
V_{n}\right]}{\,{\operatorname{\mathnormal{d}}}\sigma}.$

Varying ${v}$ yields the equation
$\partial_{\bm{\mathbf{n}}}{{u}^{\prime}}+i{u}^{\prime}=\operatorname{div}_{\Sigma}(V_{n}\nabla_{\Sigma}{u})-i(\partial_{\bm{\mathbf{n}}}{u}+\kappa{u})V_{n}+\lambda\kappa
V_{n}$ on $\Sigma$. After collecting all equations for the shape derivative
${u}^{\prime}$, we finally obtain (27). ∎

###  A.2 Alternative computation of the shape gradient

We give below the computation of the shape gradient (13) under a
$\mathcal{C}^{2,1}$ regularity assumption on the domain. With the given
regularity, the existence of the shape derivative of the state is guaranteed
and the shape gradient of the cost is easily obtained using Hadamard’s domain
differentiation formula (25) – assuming the perturbation of $\Omega$ preserves
its regularity.

######  Proposition A.1.

Let $\Omega\in\mathcal{C}^{2,1}$ and
$\bm{\mathbf{V}}\in\bm{\mathbf{\Theta}}^{2}$. Then, the shape derivative of
$J$ at $\Omega$ along $\bm{\mathbf{V}}$ is given by
$dJ(\Omega)[\bm{\mathbf{V}}]=\int_{\Sigma}{G{V_{n}}}{\,{\operatorname{\mathnormal{d}}}\sigma}$,
where $G$ is the expression in (13).

###### Proof.

Let us assume that $\Omega$ is of class $\mathcal{C}^{2,1}$ and
$\bm{\mathbf{V}}\in\bm{\mathbf{\Theta}}^{2}$. By classical regularity theory,
$u_{1}$, $u_{2}\in H^{3}(\Omega)$ and so, we can apply formula (25) to obtain
– noting that $\bm{\mathbf{V}}\big{|}_{\Gamma}=0$ – the derivative
${\operatorname{\mathnormal{d}}}J(\Omega)[\bm{\mathbf{V}}]=\int_{\Omega}{u_{2}u_{2}^{\prime}}{\,{\operatorname{\mathnormal{d}}}x}+\frac{1}{2}\int_{\Sigma}{|u_{2}|^{2}V_{n}}{\,{\operatorname{\mathnormal{d}}}\sigma}=:\mathbb{I}_{1}+\mathbb{I}_{2}$.
We focus on rewriting $\mathbb{I}_{1}$ through the adjoint method. To this
end, we consider the adjoint problem (14), multiply it by
$u^{\prime}\in\bm{\mathbf{\mathsf{V}}}(\Omega)$, and then apply integration by
parts to obtain $\int_{\Omega}{\nabla\overline{p}\cdot\nabla
u^{\prime}}{\,{\operatorname{\mathnormal{d}}}x}+i\int_{\Sigma}{\overline{p}u^{\prime}}{\,{\operatorname{\mathnormal{d}}}\sigma}=\int_{\Omega}{u_{2}u^{\prime}}{\,{\operatorname{\mathnormal{d}}}x}$.
We do the same on (27) with the multiplier
$\overline{p}\in\bm{\mathbf{\mathsf{V}}}(\Omega)$ to obtain
$\int_{\Omega}{\nabla
u^{\prime}\cdot\nabla\overline{p}}{\,{\operatorname{\mathnormal{d}}}x}+i\int_{\Sigma}{u^{\prime}\overline{p}}{\,{\operatorname{\mathnormal{d}}}\sigma}=\int_{\Sigma}{\overline{p}\Upsilon(u)[V_{n}]}{\,{\operatorname{\mathnormal{d}}}\sigma}$.
The last two equations lead us to
$\int_{\Omega}{u_{2}u^{\prime}}{\,{\operatorname{\mathnormal{d}}}x}=\int_{\Sigma}{\overline{p}\Upsilon(u)[V_{n}]}{\,{\operatorname{\mathnormal{d}}}\sigma}$.
Comparing the respective real and imaginary parts on both sides of this
equation will give us the form for $\mathbb{I}_{1}$, which, upon adding to
$\mathbb{I}_{2}$ finally yield the desired expression. ∎


(or in the zeroth $G_{{\rm SM}_{p}}$-cobordism class) — luckily Ref. Garcia-
Etxebarria and Montero (2019); Davighi  _et al._ (2020); Wan and Wang (2020);
Wang (2020) checked all local or global anomalies of SM vanish in $G_{{\rm
SM}_{p}}$. We had done the same analysis in Sec. III.2. Just like Sec. III.2,
we can use the modification of Razamat and Tong (2021); Tong (2021)’s model to
deform a gapless mirror SM to a featureless fully gapped mirror sector.

    1. (1).

_15n or 16n Weyl-fermion SM and mirror SM_ : We could consider the three-
family $\mathfrak{su}(3)\times\mathfrak{su}(2)\times\mathfrak{u}(1)_{Y}$ SM
with the 15 or 16 left-handed Weyl fermions (in eq. (1))

$(\bar{d}_{R}\oplus{l}_{L}\oplus
q_{L}\oplus\bar{u}_{R}\oplus\bar{e}_{R})\oplus n_{{\nu}_{R}}\bar{\nu}_{R}$

per family on one domain wall, including the right-handed neutrino or not,
$n_{{\nu_{j}}_{R}}\in\\{0,1\\}$. We like to gap the mirror SM with the right-
handed Weyl fermion

$(\bar{d}^{\prime}_{L}\oplus{l}^{\prime}_{R}\oplus
q^{\prime}_{R}\oplus\bar{u}^{\prime}_{L}\oplus\bar{e}^{\prime}_{L})\oplus
n_{{\nu}_{R}}\bar{\nu}^{\prime}_{L}$

but in the same
$\mathfrak{su}(3)\times\mathfrak{su}(2)\times\mathfrak{u}(1)_{Y}$
representation on the other domain wall. Note that chiral SM and mirror SM are
only mapped onto each other under ${\mathbb{Z}_{2}^{\rm PR}}$ symmetry
transformation; each particle has its own anti-particle under the charge
conjugation C.

We follow the Razamat-Tong model Razamat and Tong (2021) to embed the 15 (or
16) Weyl-fermion SM into the 27 Weyl-fermion with the following fermion matter
content (44) by introducing additional vector-like theory (3 right-handed and
3 left-handed Weyl fermions, in the second and third rows of (44)).

Following the Tong model Tong (2021), we can find an enlarged symmetry (45)
$G_{\text{SM-LR}_{p,p^{\prime}}}\equiv\frac{G_{{\rm SM}_{q}}\times{\rm
SU}(2)_{\mathrm{R}}\times{\rm
U}(1)_{\mathrm{R}}}{\mathbb{Z}_{p^{\prime}}}\equiv\frac{{\rm SU}(3)\times{\rm
SU}(2)_{\mathrm{L}}\times{\rm SU}(2)_{\mathrm{R}}\times{\rm
U}(1)_{\mathrm{L}}\times{\rm
U}(1)_{\mathrm{R}}}{\mathbb{Z}_{p}\times\mathbb{Z}_{p^{\prime}}}$ with Lie
algebra
$\mathfrak{su}(3)\times\mathfrak{su}(2)\times\mathfrak{u}(1)_{Y}\times\mathfrak{su}(2)_{\rm
R}\times\mathfrak{u}(1)_{\rm R}$ $\equiv$ $\mathfrak{su}(3)\times$
$\mathfrak{su}(2)_{\rm L}\times\mathfrak{su}(2)_{\rm R}$ $\times$
$\mathfrak{u}(1)_{\rm L}\times\mathfrak{u}(1)_{\rm R}$ such that the
27-fermion representation in (44) becomes

$\displaystyle\text{chiral SM $\psi$ }\big{(}\bar{d}_{R}\oplus{l}_{L}\oplus
q_{L}\oplus\bar{u}_{R}\oplus\bar{e}_{R}\oplus\bar{\nu}_{R}\big{)}\oplus{\bar{d}^{\prime}}_{R}\oplus
l^{\prime}_{L}\text{ or}$ (83) $\displaystyle\text{mirror SM
$\psi^{\prime}_{M}$ }\big{(}\bar{d}^{\prime}_{L}\oplus{l}^{\prime}_{R}\oplus
q^{\prime}_{R}\oplus\bar{u}^{\prime}_{L}\oplus\bar{e}^{\prime}_{L}\oplus\bar{\nu}^{\prime}_{L}\big{)}\oplus{\bar{d}^{\prime\prime}}_{L}\oplus
l^{\prime\prime}_{R}:$ (84) $\displaystyle\sim\big{(}(\overline{\bf 3},{\bf
1},{\bf 2})_{2,-1}\oplus({\bf 1},{\bf 2},{\bf 2})_{-3,3}\oplus({\bf 3},{\bf
2},{\bf 1})_{1,-2}\oplus(\overline{\bf 3},{\bf 1},{\bf 1})_{-4,2}\oplus({\bf
1},{\bf 1},{\bf 1})_{6,-6}\oplus{({\bf 1},{\bf 1},{\bf
2})_{0,-3}}\big{)}\oplus({\bf 3},{\bf 1},{\bf 1})_{-2,4}\oplus({\bf 1},{\bf
2},{\bf 1})_{3,0}.$ (85)

The chiral fermion multiplet is $\psi$, while the mirror fermion multiplet is
$\psi^{\prime}_{\rm M}$. (In the first line of (83), the prime ′ field is
meant to indicate the extra addition to the original SM content (1). While in
the second line of (83), the last prime is meant to indicate the field’s
mirror doubling partner.) See footnote 7, for the left and right notations, we
use the italic font $L$ and $R$ to denote that of spacetime symmetry, while
use the text font L and R for that of internal symmetry.

    2. (2).

_Gapping 15n vs 16n Weyl-fermion mirror SM_ :
$\bullet$ If we only preserve $G_{{\rm SM}_{p}}$, then without or with the
sterile neutrino $\bar{\nu}_{R}={({\bf 1},{\bf 1})_{0}}$ (namely, 15 or 16
Weyl fermions per family) can lead to a short-range entangled SMG phase
(without any low energy TQFT).
$\bullet$ However, if we preserve not only $G_{{\rm SM}_{p}}$ but also some
${{\bf B}-{\bf L}}$ symmetry, either we need a 16n Weyl-fermion model to
achieve SMG, or we need a 15n Weyl-fermion model plus SGTO with a low energy
TQFT Wang (2020, 2021) to achieve a symmetric gapped phase.

  5. 5.

Mean-field vs non-mean-field SMG gapping:

_Mean-field gapping_ : Notice both Razamat-Tong model Razamat and Tong (2021)
and Tong model Tong (2021) propose certain SMG gapping terms, however there
are either mean-field condensates Tong (2021) or generic complex coupling
coefficients Razamat and Tong (2021) — those make solving the Strong CP
problem by Razamat-Tong or Tong’s SMG deformation Razamat and Tong (2021);
Tong (2021) either difficult or impossible.121212We had explained in Sec.
III.2 why the mean-field condensates $\langle\phi\rangle\neq 0$ fails to solve
the Strong CP problem.
Here we explain why the smooth confinement deformation of ${\cal N}=1$
supersymmetrized SM’s model in Razamat and Tong (2021) is also difficult to
solve the Strong CP problem. First, one introduces three copies of Razamat and
Tong (2021); Tong (2021)’s model with family indices ${\rm I},{\rm
J}\in\\{1,2,3\\}$, and a new ${\rm SU}(2)^{\prime}\times{\rm
SU}(2)^{\prime\prime}\times{\rm SU}(2)^{\prime\prime\prime}$ gauge theory.
Precisely, for the supersymmetrized ${\cal N}=1$ chiral multiplets of (83)
denoted in their superfield forms ${\rm D},{\rm L},{\rm Q},{\rm U},{\rm
E},{\rm D}^{\prime},{\rm L}^{\prime},{\rm N}$, their supersymmetric ${\cal W}$
potential has three family mixing:  $\displaystyle{\cal W}_{\rm
UV}=\lambda_{{\rm I}{\rm J}}^{\rm U}{\rm D}^{\rm I}{\rm D}^{\rm I}{\rm U}^{\rm
J}+\lambda_{{\rm I}{\rm J}}^{\rm Q}{\rm L}^{\rm I}{\rm D}^{\rm I}{\rm Q}^{\rm
J}+\lambda_{{\rm I}{\rm J}}^{\rm D}{\rm D}^{\rm I}{\rm N}^{\rm I}{\rm
D^{\prime}}^{\rm J}+\dots,$ (87) $\displaystyle{\cal W}_{\rm IR}=\lambda_{{\rm
I}{\rm J}}^{\rm U}\tilde{\rm U}^{\rm I}{\rm U}^{\rm J}+\lambda_{{\rm I}{\rm
J}}^{\rm Q}\tilde{\rm Q}^{\rm I}{\rm Q}^{\rm J}+\lambda_{{\rm I}{\rm J}}^{\rm
D}\tilde{\rm D}^{\rm I}{\rm D^{\prime}}^{\rm J}+\dots.\quad$ (88) Each ${\rm
SU}(2)$ couples to one of three families of an extended internal
$\mathfrak{su}(2)_{{\rm R}_{1}}\times\mathfrak{su}(2)_{{\rm
R}_{2}}\times\mathfrak{su}(2)_{{\rm R}_{3}}$ symmetry of the
$\mathfrak{su}(3)\times\mathfrak{su}(2)_{\rm L}\times\mathfrak{u}(1)_{Y}$ SM,
which guarantees the mirror SM to reach the fully gapped SMG phase by the
s-confinement once $\mathfrak{su}(2)_{{\rm R}_{1}}\times\mathfrak{su}(2)_{{\rm
R}_{2}}\times\mathfrak{su}(2)_{{\rm R}_{3}}$ are dynamically gauged. Any
${\theta}$ of $\mathfrak{su}(2)_{{\rm R}_{1}}\times\mathfrak{su}(2)_{{\rm
R}_{2}}\times\mathfrak{su}(2)_{{\rm R}_{3}}$ can be also removed by the chiral
and mirror sectors imposed by the $\mathbb{Z}_{2}^{\rm R}$ symmetry, so
${\theta}=-{\theta}^{\prime}=0$. But there is another worry that any complex
coefficient $\lambda_{{\rm I}{\rm J}}$ etc. in the superpotential can affect
the ${\bar{\theta}^{\prime}_{3}}$. Even if any $\lambda_{{\rm I}{\rm
J}}\in{\mathbb{C}}$ is rotated away via $\lambda_{{\rm I}{\rm J}}\equiv
U_{{{\rm I}{\rm I}^{\prime}}}^{\dagger}\lambda_{{\rm I^{\prime}}{\rm
J}^{\prime}}^{{\rm diag}}U_{{\rm J}^{\prime}{\rm J}}^{\dagger}$, we need to
further ask whether this basis redefinition by $U$ rotation results in any
analogous CKM matrix to the quarks coupling to $\mathfrak{su}(2)_{\rm R}$ or
$\mathfrak{su}(2)_{\rm L}$. In general, there are too many complex
coefficients in the Razamat and Tong (2021)’s model. Generally, the complex
coefficients in Razamat and Tong (2021)’s model violate the CP and T symmetry,
thus the Strong CP problem cannot be solved directly within Razamat and Tong
(2021).

Figure 5: A schematic plot on the Higgs-induced contribution to SM’s quark and
lepton masses in Table 1(a) (shown in the red color) while the SMG-induced
mirror SM’s quark and lepton masses entirely (shown in the blue color). The
SMG-induced mirror SM’s mass gap the fermion doubling completely hidden at UV,
so only the original chiral SM is observed at low energy.

_Non-mean-field gapping_ : In contrast, in Sec. III.2, we already propose the
non-mean-field multi-fermion interactions in (51) and (52) that can gap one
generation of quarks and leptons. Then here we just need to apply the same
non-mean-field multi-fermion interactions to gap the three generations of the
mirror fermions. This mirror fermion gapping implies the mass spectrum like
Fig. 5.

  6. 6.

Solve the Strong CP problem ${\bar{\theta}_{3}=\bar{\theta}^{\prime}_{3}}=0$
from UV to IR:

Earlier we had explained that the imposed ${\mathbb{Z}_{2}^{\rm PR}}$ symmetry
makes the UV theory (82) with no Strong CP angle because
${\bar{\theta}_{3}=\bar{\theta}^{\prime}_{3}}=0$ at that $M_{\rm
P}>E>\Lambda_{\rm SMG}$ scale.

However,
(1) at the lower energy $E<\Lambda_{\rm SMG}$, the dominant SMG interaction
only occurs at the mirror fermion sector;
(2) while even at a further lower energy $E<\Lambda_{\rm EW}$, the dominant
Higgs condensation only occurs at the original chiral fermion sector.
— both cases mean to “ _spontaneously_ ” breaks the ${\mathbb{Z}_{2}^{\rm
PR}}$ symmetry at IR (see footnote 11). Once ${\mathbb{Z}_{2}^{\rm PR}}$ is
broken at low energy, one may be worried that both ${\bar{\theta}_{3}}$ and
${\bar{\theta}^{\prime}_{3}}$ can be deviated from zeros. We need to argue why
${\bar{\theta}_{3}}$ and/or ${\bar{\theta}^{\prime}_{3}}$ stays close to zero
at IR.

Similar to the argument of the UV parent theory, now at low energy IR, we
include the generalized mass matrix $M_{\rm total}$ that

$\displaystyle\langle M_{\rm total}\rangle=\begin{pmatrix}\langle M+g_{\rm
SMG}{\cal O}_{\rm SMG}\rangle&0\\\ 0&\langle
M^{\prime\dagger}+g^{\prime*}_{\rm SMG}{{\cal O}^{\prime}}_{\rm
SMG}^{\dagger}\rangle\end{pmatrix}=\begin{pmatrix}\langle M\rangle&0\\\
0&\langle M^{\prime\dagger}\rangle\end{pmatrix}$ (89)

that includes the mean-field mass matrix $M=\langle M\rangle$ as well as the
multi-fermion interaction type of SMG that offers no mean-field mass
$\langle{\cal O}_{\rm SMG}\rangle=0$. The computation of the
${\bar{\theta}}+{\bar{\theta}^{\prime}}$ for the full theory with the total
fermion mass $M_{\rm total}$ contribution of the whole system is:

$\displaystyle{\bar{\theta}}+{\bar{\theta}^{\prime}}$ $\displaystyle\equiv$
$\displaystyle{\theta}+{\theta^{\prime}}+\arg(\det\langle M_{\rm
total}\rangle)$ (90) $\displaystyle\equiv$
$\displaystyle{\theta}+{\theta^{\prime}}+\arg(\det\langle M+g_{\rm SMG}{\cal
O}_{\rm SMG}\rangle)+\arg(\det\langle M^{\prime\dagger}+g^{\prime*}_{\rm
SMG}{\cal O}^{\prime\dagger}_{\rm SMG}\rangle)$ (91) $\displaystyle=$
$\displaystyle{\theta}+{\theta^{\prime}}+\arg(\det\langle M+g_{\rm SMG}{\cal
O}_{\rm SMG}\rangle\cdot\det\langle M^{\prime\dagger}+g^{\prime*}_{\rm
SMG}{\cal O}^{\prime\dagger}_{\rm SMG}\rangle)$ (92) $\displaystyle=$
$\displaystyle{\theta}+{\theta^{\prime}}+\arg(\det(\langle M+g_{\rm SMG}{\cal
O}_{\rm SMG}\rangle\cdot\langle M^{\prime\dagger}+g^{\prime*}_{\rm SMG}{\cal
O}^{\prime\dagger}_{\rm SMG}\rangle))$ (93) $\displaystyle=$
$\displaystyle{\theta}+{\theta^{\prime}}+\arg(\det(\langle M\rangle\langle
M^{\prime\dagger}\rangle+g_{\rm SMG}\langle{\cal O}_{\rm SMG}\rangle\langle
M^{\prime\dagger}\rangle+g^{\prime*}_{\rm SMG}\langle{\cal
O}^{\prime\dagger}_{\rm SMG}\rangle\langle M\rangle+g_{\rm
SMG}g^{\prime*}_{\rm SMG}\langle{\cal O}_{\rm SMG}\rangle\langle{\cal
O}^{\prime\dagger}_{\rm SMG}\rangle)).$ (94)

Here are some comments concerning the ${\bar{\theta}}+{\bar{\theta}^{\prime}}$
of the whole system at different energy scales $E$:

    1. (1).

$M_{\rm P}>E>\Lambda_{\rm SMG}$:

$\bullet$ _Without even assuming the mean-field interpretation_ of the theta
angle in (15), the imposed ${\mathbb{Z}_{2}^{\rm PR}}$ symmetry demands the UV
theory (82) satisfying ${\theta}=-{\theta^{\prime}}$ and $\arg(\det M+g_{\rm
SMG}{\cal O}_{\rm SMG})=-\arg(\det M^{\prime\dagger}+g^{\prime*}_{\rm
SMG}{\cal O}^{\prime\dagger}_{\rm SMG})$ in general.

$\bullet$ Moreover, at this high energy scale $M_{\rm P}>E>\Lambda_{\rm SMG}$
at UV, there is no mean-field condensation yet (neither SM Higgs nor the SMG’s
$\xi_{\rm I}{\cal O}_{\rm SMG,IJ}\psi_{\rm J}$ condenses yet. So $\langle
M\rangle=\langle M^{\prime}\rangle=0$ within $M_{\rm P}>E>\Lambda_{\rm SMG}$.
In fact $\langle\xi_{\rm I}\psi_{\rm J}\rangle=\langle{\cal O}_{\rm
SMG,IJ}\rangle=0$ is always strictly zero).

$\bullet$ _Assuming the mean-field interpretation_ of the theta angle in (15),
then $\arg(\det\langle M+g_{\rm SMG}{\cal O}_{\rm
SMG}\rangle)=-\arg(\det\langle M^{\prime\dagger}+g^{\prime*}_{\rm SMG}{\cal
O}^{\prime\dagger}_{\rm SMG}\rangle)=0$ is strictly zero within $M_{\rm
P}>E>\Lambda_{\rm SMG}$. An appropriate chiral transformation $\upalpha$ like
(23) allows us to choose both $\bar{\theta}=0$ and $\bar{\theta}^{\prime}=0$

    2. (2).

$\Lambda_{\rm SMG}>E>\Lambda_{\rm EW}$: Below and around the SMG scale, the
${\mathbb{Z}_{2}^{\rm PR}}$ symmetry is “ _spontaneously_ ” broken due to the
SMG interaction $\langle\xi^{\prime}_{\rm I}{\cal O}^{\prime}_{\rm
SMG,IJ}\psi^{\prime}_{\rm J}\rangle\neq 0$ on the mirror fermion sector, but
$\langle\xi_{\rm I}{\cal O}_{\rm SMG,IJ}\psi_{\rm J}\rangle=0$ on the original
SM sector (also strictly $\langle\xi_{\rm I}\psi_{\rm J}\rangle=\langle{\cal
O}_{\rm SMG,IJ}\rangle=\langle\xi^{\prime}_{\rm I}\psi^{\prime}_{\rm
J}\rangle=\langle{\cal O}^{\prime}_{\rm SMG,IJ}\rangle=0$ always for non-mean-
field SMG).

$\bullet$ _Without even assuming the mean-field interpretation_ of the theta
angle in (15), we can still read that
${\bar{\theta}}+{\bar{\theta}^{\prime}}={\theta}+{\theta^{\prime}}+\arg(\det(MM^{\prime\dagger}+g_{\rm
SMG}{\cal O}_{\rm SMG}M^{\prime\dagger}+g^{\prime*}_{\rm SMG}{\cal
O}^{\prime\dagger}_{\rm SMG}M+g_{\rm SMG}g^{\prime*}_{\rm SMG}{\cal O}_{\rm
SMG}{\cal O}^{\prime\dagger}_{\rm SMG}))$. Ref. [Wang, 2022a] argues that in
terms of the configuration in Fig. 3 (b), the most dominant mass term in (90)
is from $g^{\prime*}_{\rm SMG}{\cal O}^{\prime\dagger}_{\rm SMG}M$.

$\bullet$ _Assuming the mean-field interpretation_ of the theta angle in (15),
${\bar{\theta}}+{\bar{\theta}^{\prime}}\equiv{\theta}+{\theta^{\prime}}+\arg(\det(\langle
M\rangle\langle M^{\prime\dagger}\rangle+g_{\rm SMG}\langle{\cal O}_{\rm
SMG}\rangle\langle M^{\prime\dagger}\rangle+g^{\prime*}_{\rm SMG}\langle{\cal
O}^{\prime\dagger}_{\rm SMG}\rangle\langle M\rangle+g_{\rm
SMG}g^{\prime*}_{\rm SMG}\langle{\cal O}_{\rm SMG}\rangle\langle{\cal
O}^{\prime\dagger}_{\rm SMG}\rangle)).$ Because $E>\Lambda_{\rm EW}$, Higgs
does not yet condense in the SM sector, so $\langle M\rangle=0$, also $\langle
M^{\prime\dagger}\rangle=\langle{\cal O}_{\rm SMG}\rangle=\langle{\cal
O}^{\prime\dagger}_{\rm SMG}\rangle=0$.

    3. (3).

$E<\Lambda_{\rm EW}$: Below and around $\Lambda_{\rm EW}$, the
${\mathbb{Z}_{2}^{\rm PR}}$ symmetry is further “ _spontaneously_ ” broken,
because the Higgs induces $\langle M\rangle\neq 0$ on the SM sector.

$\bullet$ _Without even assuming the mean-field interpretation_ of the theta
angle in (15), Ref. [Wang, 2022a] argues that in terms of the configuration in
Fig. 3 (b), the most dominant mass term in (90) is from $g^{\prime*}_{\rm
SMG}{\cal O}^{\prime\dagger}_{\rm SMG}M$. The Higgs-induced mass $M$ is in the
chiral fermion sector, while the SMG-induced mass with operator
$g^{\prime*}_{\rm SMG}{\cal O}^{\prime\dagger}_{\rm SMG}$ is in the mirror
fermion sector. Different mass-generating mechanisms on the chiral and mirror
sector may give a generic complex nonzero phase.

However, because the SMG _multi-fermion interaction_ and disorder scalar
interaction are _highly irrelevant operators_ at IR. The lower the energy, the
weaker effects are these interactions on the IR correction of $\bar{\theta}$.
There gives another reason that the IR correction to our UV solution
$\bar{\theta}=0$ is extremely small.

$\bullet$ _Assuming the mean-field interpretation_ of the theta angle in (15)
which implies (90). Then, Higgs condense $\langle M\rangle\neq 0$ in the SM
sector, but others have no mean-field values $\langle
M^{\prime\dagger}\rangle=\langle{\cal O}_{\rm SMG}\rangle=\langle{\cal
O}^{\prime\dagger}_{\rm SMG}\rangle=0$. Thus (90) only contains
${\bar{\theta}}+{\bar{\theta}^{\prime}}\equiv{\theta}+{\theta^{\prime}}$ while
the $(\langle M\rangle\langle M^{\prime\dagger}\rangle+g_{\rm SMG}\langle{\cal
O}_{\rm SMG}\rangle\langle M^{\prime\dagger}\rangle+g^{\prime*}_{\rm
SMG}\langle{\cal O}^{\prime\dagger}_{\rm SMG}\rangle\langle M\rangle+g_{\rm
SMG}g^{\prime*}_{\rm SMG}\langle{\cal O}_{\rm SMG}\rangle\langle{\cal
O}^{\prime\dagger}_{\rm SMG}\rangle)=0$.

As long as
${\bar{\theta}}+{\bar{\theta}^{\prime}}\equiv{\theta}+{\theta^{\prime}}=0$ at
UV as (81) set by ${\mathbb{Z}_{2}^{\rm PR}}$, our solution has a smaller or
zero ${\bar{\theta}}+{\bar{\theta}^{\prime}}\approx 0$ at IR, even better than
the higher-loop calculation arguments given in Parity Solution Babu and
Mohapatra (1989, 1990); Barr  _et al._ (1991). Because Ref. [Babu and
Mohapatra, 1989, 1990; Barr  _et al._ , 1991] requires only the Higgs
mechanism, thus Higgs mixing on both the chiral and mirror sectors can
generate complex phases at the higher-order quantum corrections beyond the
tree-level semiclassical analysis. Here instead we have two mechanisms in our
solution: Higgs mechanism dominates on the chiral sector and the SMG dominates
on the mirror sector, and there is _no mean-field mass matrix mixing_ to
generate quantum corrections to $\bar{\theta}+{\bar{\theta}^{\prime}}\approx
0$.

We can take any fermion (here any mirror fermion) that receives its entire
mass from SMG and do a chiral transformation with an appropriate $\upalpha$
like (23) allows us to choose both ${\bar{\theta}}={\theta}-\upalpha\approx 0$
and ${\bar{\theta}^{\prime}}={\theta^{\prime}}+\upalpha\approx 0$ even
including the IR correction. Theoretically this Scenario in Sec. IV provides a
candidate Strong CP solution.

##  V Conclusion and Comparison to Other Strong CP Solutions

To conclude, let us summarize and compare the two SMG-based Strong CP
solutions, of Sec. III and Sec. IV, with other well-known Strong CP solutions.

First Solution: Symmetric Mass Gap within the Chiral Fermion:

  1. 1.

The first solution in Sec. III is related to the SMG modification of ’t Hooft
massless $u$ quark solution ’t Hooft (1976a). In ’t Hooft solution ’t Hooft
(1976a), if any of the quark $\psi$ (say $u$ quark) has no mean-field nor
Higgs mass, then we can do the chiral transformation on this quark $\psi$
alone to rotate the $\theta$ away but without gaining a complex phase in the
mean-field mass matrix (since there is no mean-field mass for $\psi$). This
sets $\bar{\theta}=0$. Our SMG version in Sec. III modifies the ’t Hooft
solution by allowing the SMG-induced non-mean-field mass on the $u$ quark and
a set of $G$-symmetry anomaly-free fermions. For the SM example, we can take
$G=G_{{\rm SM}_{p}}\equiv({{\rm SU}(3)}\times{{\rm SU}(2)}\times{{\rm
U}(1)_{Y}})/{\mathbb{Z}_{p}}$, $p=1,2,3,6$ with the Lie algebra
$\mathfrak{su}(3)\times\mathfrak{su}(2)\times\mathfrak{u}(1)_{Y}$.

  2. 2.

The first solution in Sec. III is also related to the SMG modification of
Peccei-Quinn axion solution Peccei and Quinn (1977a, b); Weinberg (1978);
Wilczek (1978). In the weakly gauge limit or global symmetry limit of $G$, the
Peccei-Quinn solution can be regarded as an “approximate symmetry breaking”
solution (see footnote 6) that uses the mean-field Yukawa-Higgs symmetry
breaking mass term. The arbitrariness of $\bar{\theta}\in[0,2\pi)$ can be
relaxed by the symmetry-breaking dynamics to $\bar{\theta}\simeq 0$. The
fluctuation around a fixed $\bar{\theta}=0$ gives a low-energy Goldstone mode
or an axion. The transition from the mean-field mass to the interacting SMG
mass in Sec. III is analogous to the phase transition between the _ordered_
phase and the _disordered_ phase — for example, the famous superfluid-to-
insulator type of phase transition in the condensed matter Fisher  _et al._
(1989).

Second Solution: Symmetric Gapped Mirror Fermion:

We have combined four physics together to build our second model in Sec. IV:
(i) _Nielsen-Ninomiya_ _fermion-doubling_ Nielsen and Ninomiya (1981) imposes
a discrete spatial parity-reflection $\mathbb{Z}_{2}^{\rm PR}$ symmetry
between the chiral SM and mirror SM.
(ii) _Strong CP problem_ is solved by respecting the $\mathbb{Z}_{2}^{\rm PR}$
symmetry so ${\bar{\theta}_{3}}={\bar{\theta}_{3}}^{\prime}=0$ for the chiral
and mirror SM.
(iii) SMG gaps the mirror SM fermion doubler to leave only a chiral SM at low
energy, which also solves the Nielsen-Ninomiya fermion-doubling and chiral
fermion problems altogether.
(iv) _Parity-reflection $\mathbb{Z}_{2}^{\rm PR}$ symmetry is maximally
broken_ in the weak force due to gapping out the mirror fermion, which also
gives a reason for parity violation in the weak force Lee and Yang (1956).

We compare our second model with other previously proposed Strong CP
solutions. One class of solutions is based on imposing discrete CP Nelson
(1984); Barr (1984) or P symmetries Babu and Mohapatra (1989, 1990); Barr  _et
al._ (1991); Hook (2015):

  1. 1.

Nelson-Barr Nelson (1984); Barr (1984) starts with a CP invariant theory then
spontaneously breaks the CP.

  2. 2.

Barr-Chang-Senjanovic Barr  _et al._ (1991) implements the left-right model
Senjanovic and Mohapatra (1975) or two copies of $\mathfrak{su}(2)_{\rm
L}\times\mathfrak{su}(2)_{\rm R}$ for the weak force, and imposes a diagonal
symmetry mixing between the discrete CP or P and the discrete _internal_
symmetry exchanges the left and right $\mathfrak{su}(2)$. Typically, the gauge
group needs to be doubled at least for the weak force (so
$\mathfrak{su}(2)_{\rm L}\times\mathfrak{su}(2)_{\rm R}$).

  3. 3.

In comparison, our second solution in Sec. IV is still different from all of
the above — we impose the discrete parity-reflection spatial symmetry
$\mathbb{Z}_{2}^{\rm PR}$ only due to Nielsen-Ninomiya fermion doubling
Nielsen and Ninomiya (1981) between the energy $M_{\rm P}>E>\Lambda_{\rm
SMG}$, but without imposing any extra discrete _internal_ symmetry. The gauge
group in principle needs not to be doubled, see for example the analogous 1+1d
toy model does not double the gauge group Wang (2022a). However, it turns out
that the convenient SMG deformation of the SM Razamat and Tong (2021); Tong
(2021) naturally happens to introduce $\mathfrak{su}(2)_{\rm
L}\times\mathfrak{su}(2)_{\rm R}$.

In comparison, our second model is still different from all of the above — we
respect a discrete reflection R symmetry only due to Nielsen-Ninomiya fermion
doubling between the energy $M_{\rm P}>E>\Lambda_{\rm SMG}$, also we keep the
original chiral SM gauge group with _only_ a single $\mathfrak{su}(2)_{\rm
L}$, _not_ the vector-like left-right model, _nor_ need two copies of the
gauge group.

In addition to the SM, our second Strong CP solution also works for other
chiral fermion model (e.g., Georgi-Glashow Georgi and Glashow (1974) or
flipped $\mathfrak{su}(5)$ Barr (1982)) and for the vector-like (e.g., Pati-
Salam $\mathfrak{su}(4)\times\mathfrak{su}(2)_{\rm
L}\times\mathfrak{su}(2)_{\rm R}$ Pati and Salam (1974) or left-right
Senjanovic and Mohapatra (1975)) models, by imposing the $\mathbb{Z}_{2}^{\rm
PR}$ then gapping the mirror fermion sector via SMG.

One predictive signature of both SMG-based models, in Sec. III and Sec. IV, is
that even if we turn off the conventional SM Higgs or SM gauge interactions,
some SM fermions or mirror fermions can still be highly interacting, mediated
through hypothetical direct multi-fermion or disordered mass-field
interactions.

If our _theoretical solutions_ in Sec. III and Sec. IV turn out to be also
favored as valid _phenomenological solutions_ to the Strong CP problem, then
this implies that the fermions in our Nature can indeed be _more interacting_
among fermions themselves than what we used to think of in the SM.

##  VI Acknowledgments

JW especially thanks Yuta Hamada for the precious collaboration on Hamada and
Wang (2022), for several crucial discussions at the initial stage, and for
various conversations during the development of this project. JW also thanks
William Detmold, Daniel Jafferis, Dmitri Kharzeev, Zohar Komargodski, Matthew
Reece, Matthew Strassler, Edward Witten, and Yi-Zhuang You for very helpful
conversations. JW appreciates the generous feedback from many participants of
Harvard CMSA Phase Transitions and Topological Defects in the Early Universe
workshop (August 2-5, 2022) and Generalized Global Symmetries, Quantum Field
Theory, and Geometry (September 19-23, 2022). JW is supported by Harvard
University CMSA.

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