In this section, we first show that the convergence holds for all parameters.
We then show that the limit functions are a consistent collection of multivariate cumulative distribution functions.
We further show that they are continuous in all variables.

Let $\tasep(n,t)$ be the height function for the TASEP on the discrete ring of size $2a$ as in Section \ref{sec:pKPZfp}.
For $T>0$, let
\beqq
\widetilde{\tasep}_T(\gamma, \tau):= \frac{\tasep(\gamma T^{2/3}, 2\tau T) - \tau T}{-T^{1/3}},
\qquad (\gamma, \tau) \in \realR \times \realR_+
\eeqq
where the ring size is set as $(2a)^{3/2}=T$.\footnote{To be precise, we set $a= [T^{2/3}]/2$ since $a$ is a half-integer.}
\beqq
\realR_{+, \le}^m = \{ \tau= (\tau_1, \cdots, \tau_m)\in (0, \infty)^m : 0<\tau_1\le\cdots\le \tau_m\}.
\eeqq
For $\tau \in \realR_{+, \le}^m$, define
\beqq
\Omega_+^m(\tau) = \{ \hite=(\hite_1, \cdots, \hite_m)\in \realR^m : \text{$\hite_i< \hite_{i+1}$ if $\tau_i=\tau_{i+1}$}  \}.
\eeqq
It was shown in [2] that for every $\gamma\in \realR^m$ and $\tau\in \realR^m_{+, \le}$, the limit
\begin{equation}
\label{eq:taseplimit_app}
\lim_{T\to \infty} \prob\bigg(
\bigcap_{i=1}^m
\big\{ \widetilde{\tasep}_T(\gamma_i, \tau_i) \le \hite_i  \big\}   \bigg)
= \tasepF_m(\hite; \gamma, \tau)  \text{ converges if $\hite\in \Omega^m_+(\tau)$,}
\end{equation}
as mentioned in \eqref{eq:proplusminus}.

When $m=1$, it was already shown in [1] that the one-point distribution $\tasepF_1$ is a distribution function and is continuous.
We do not need the explicit form of $\tasepF_m$ for the first two results below.

\medskip

We first show that the limit \eqref{eq:taseplimit_app} convergences for every $\hite\in \realR^m$.
For $\tau \in \realR_{+, \le}^m$, define the set
\beqq
\Omega^m(\tau) := \overline{\Omega_+^m(\tau) } = \{ \hite=(\hite_1, \cdots, \hite_m)\in \realR^m : \text{$\hite_i\le  \hite_{i+1}$ if $\tau_i=\tau_{i+1}$}  \}.
\eeqq

\begin{lemma} \label{prop:tasepF_extension}
Let $\gamma\in \realR^m$ and $\tau\in \realR^m_{+, \le}$.
For every $\hat \hite\in \Omega^m(\tau)$, the limit
\begin{equation} \label{eq:tasepF_aux02}
\lim_{\Omega^m(\tau)\ni  \hite \to  \hat \hite  } \tasepF_m(\hite; \gamma, \tau)
\end{equation}
exists. Furthermore, if we denote the limit as $\tasepF_m(\hat \hite; \gamma, \tau)$, then
\begin{equation}
\label{eq:tasepF_extension_prob}
\lim_{T \to \infty} \prob\bigg(
\bigcap_{i=1}^m
\big\{  \widetilde{\tasep}_T(\gamma_i, \tau_i) \le \hat \hite_i  \big\}  \bigg)
= \tasepF_m(\hat\hite, \gamma, \tau ).
\end{equation}
\end{lemma}

\begin{proof}
For $\epsilon>0$, let
\beqq
\hite'_i
=\hat\hite_i - (2-\frac{i}{m+1} ) \epsilon
\quad \text{and} \quad
\hite''_i
=\hat\hite_i + (1+\frac{i}{m+1}) \epsilon
\eeqq
for $i=1, \cdots, m$.
Then, $\hite', \hite'' \in \Omega^m_+(\tau)$.
Furthermore, $(\hite'_1, \cdots, \hite_k', \hite_{k+1}'', \cdots, \hite_m)\in \Omega^m_+(\tau)$ for every $k$.

Let $\beta\in \Omega^m_+(\tau)$ be an arbitrary number satisfying $\sum_{i=1}^m|\hite_i-\hat\hite_i|<\epsilon$.
Note that $\hite'_i \le \hite_i\le \hite''_i$. Thus,
\beqq
\tasepF_m(\hite'; \gamma, \tau) \le \tasepF_m(\hite ; \gamma, \tau) \le \tasepF_m(\hite'' ; \gamma, \tau)
\eeqq
where we used the fact that being a limit of a distribution function, $\tasepF_m(\hite; \gamma, \tau) $
is a weakly increasing function of $\hite\in \Omega_+^m(\tau)$.
From the monotonicity property again, as $\epsilon\downarrow 0$, $\tasepF_m(\hite'; \gamma, \tau)$  increases weakly and $\tasepF_m(\hite'' ; \gamma, \tau)$  decreases weakly.
Therefore, the limit~\eqref{eq:tasepF_aux02} converges if we show that
\begin{equation}
\label{eq:tasepF_bounds_diff}
\lim_{\epsilon \downarrow 0} \tasepF_m(\hite'' ; \gamma, \tau)  - \tasepF_m(\hite' ; \gamma, \tau)  = 0.
\end{equation}

For every $j$, from \eqref{eq:taseplimit_app},
\beqq
\begin{split}
&\tasepF_m(\hite'_1,\cdots,\hite'_{j-1},\hite''_{j},\hite''_{j+1},\cdots,\hite''_m; \gamma, \tau )
-\tasepF_m(\hite'_1,\cdots,\hite'_{j-1},\hite'_{j},\hite''_{j+1},\cdots,\hite''_m; \gamma, \tau )\\
&= \lim_{T\to \infty} \prob \bigg(
\big\{
\hite'_{j}<
\widetilde{\tasep}_T(\gamma_{j} , \tau_j) \le  \hite''_{j}   \big\}
\bigcap_{i=1}^{j-1}
\big\{
\widetilde{\tasep}_T(\gamma_{j} , \tau_j) \le \hite'_i
\big\} \bigcap_{i=j+1}^m
\big\{
\widetilde{\tasep}_T(\gamma_{j} , \tau_j)\le \hite''_{i}
\big\}
\bigg)\\
&\le \lim_{T \to \infty} \prob\big(
\hite'_{j}<
\widetilde{\tasep}_T(\gamma_{j} , \tau_j) \le \hite''_{j}  \big)
=\tasepF_1(\hite''_{j}; \gamma_{j},\tau_{j}) -\tasepF_1(\hite'_{j}; \gamma_{j},\tau_{j}).
\end{split}
\eeqq
Summing over $j$, we obtain
\beqq
\tasepF_m(\hite''_1,\cdots,\hite''_m; \gamma, \tau)
-\tasepF_m(\hite'_1,\cdots,\hite'_m; \gamma, \tau) \le \sum_{j=1}^{m} \left(\tasepF_1(\hite''_{j}; \gamma_{j},\tau_{j}) -\tasepF_1(\hite'_{j} \gamma_{j},\tau_{j})\right).
\eeqq
Since the one-point distribution $\tasepF_1$ is continuous (see [1]), the right side converges to zero as $\epsilon\to 0$. The left-hand side is also nonnegative due to the monotonicity property of $\tasepF_m$. Thus we obtain \eqref{eq:tasepF_bounds_diff}, which implies the convergence of \eqref{eq:tasepF_aux02}.

With the same notations as above, from the monotonicity of probabilities which holds for the parameters without any restrictions,
\beqq
\prob\bigg(
\bigcap_{i=1}^m
\big\{  \widetilde{\tasep}_T(\gamma_i, \tau_i) \le  \hite_i'  \big\}  \bigg)
\le \prob\bigg(
\bigcap_{i=1}^m
\big\{  \widetilde{\tasep}_T(\gamma_i, \tau_i) \le \hat \hite_i  \big\}  \bigg)
\le
\prob\bigg(
\bigcap_{i=1}^m
\big\{  \widetilde{\tasep}_T(\gamma_i, \tau_i) \le  \hite_i''  \big\}  \bigg)
\eeqq
As $T\to \infty$, the lower bound tends to $\tasepF_m(\hite'; \gamma, \tau)$ and the upper bound tends to $\tasepF_m(\hite''; \gamma, \tau)$.
If we let $\epsilon \downarrow 0$, then both of them converge to $\tasepF_m(\hat \hite; \gamma, \tau)$. This shows \eqref{eq:tasepF_extension_prob}.
\end{proof}

\begin{cor}
For every $\gamma\in \realR^m$,  $\tau\in \realR_+^m$, and $\hite\in \realR^m$, the limit
\begin{equation}
\label{eq:tasepF_general_def}
\tasepF_m(\hite; \gamma, \tau) :=
\lim_{T \to \infty} \prob\bigg( \bigcap_{i=1}^m \big\{  \widetilde{\tasep}_T(\gamma_i, \tau_i) \le \hite_i  \big\}  \bigg)
\end{equation}
converges. The function $\tasepF_m(\hite; \gamma, \tau)$ is invariant under the permutations of the triples $(\hite_i,\gamma_i,\tau_i)$, $i=1, \cdots, m$.
\end{cor}

\begin{proof}
The probability $\prob( \cap_{i=1}^m \big\{  \widetilde{\tasep}_T(\gamma_i, \tau_i) \le \hite_i  \}  )$ is defined for every $(\gamma, \tau, \hite)\in \realR^m\times \realR^m_+\times \realR^m$, and is invariant under the permutations of the triples $(\hite_i,\gamma_i,\tau_i)$.
For a permutation $\sigma \in S_m$, let $\gamma^\sigma, \tau^\sigma, \hite^\sigma$ be the parameters obtained from $\gamma, \tau, \hite$ applying the permutation $\sigma$ to the index.
Let $\sigma$ be a permutation so that $\tau^\sigma\in \realR^m_{+, \le}$ and $\hite^\sigma\in \Omega^m_0(\tau^\sigma)$.
There is at least one such permutation.
By the last lemma,
$\prob( \cap_{i=1}^m \big\{  \widetilde{\tasep}_T(\gamma_i, \tau_i) \le \hite_i  \}  ) = \prob( \cap_{i=1}^m \big\{  \widetilde{\tasep}_T(\gamma_i^\sigma, \tau_i^\sigma) \le \hite_i^\sigma  \}  )$ converges.
If there are more than one permutation with the same property, then it is easy to check that they results in the same limit.
The invariance under permutations follow easily.
\end{proof}

Therefore, the convergence \eqref{eq:taseplimit} holds for all parameters, and we use the same notation $\tasepF_m(\hite; \gamma, \tau)$ for the limit.
We now show that $\tasepF_m(\hite; \gamma, \tau)$ are a collection of consistent multivariate cumulative distribution functions.
For the restricted parameters, this fact was proved in  \cite[Section 7]{Baik-Liu19}. Here, we prove it for all parameters.

\begin{prop}\label{prop:tasepF_consistence}
\begin{enumerate}[(a)]
\item For every $m$ and $(\gamma, \tau)\in \realR^m\times \realR^m_+$, $\hite\mapsto \tasepF_m(\hite; \gamma, \tau)$ is a multivariate cumulative distribution function.
\item
Let $(\gamma, \tau, \hite)\in \realR^m\times \realR_+^m \times \realR^m$.
For each $j=1, \cdots, m$, let $\gamma^{(j)}, \tau^{(j)}, \hite^{(j)}$ be the points in $\realR^{m-1}$ obtained from $\gamma, \tau, \hite$ by removing $\gamma_j, \tau_j, \hite_j$, respectively. Then,
\beqq
\lim_{\hite_j\to\infty} \tasepF_m(\hite; \gamma, \tau)
= \tasepF_{m-1}(\hite^{(j)}; \gamma^{(j)}, \tau^{(j)})  .
\eeqq
\end{enumerate}
\end{prop}

\begin{proof}
(a) The equation \eqref{eq:tasepF_general_def} implies the monotone non-decreasing property.
It also implies that $\tasepF_m(\hite; \gamma, \tau)\le \tasepF_1(\hite_j; \gamma_j,\tau_j)$ and $1-\tasepF_m(\hite; \gamma, \tau) \le \sum_{j=1}^{m} (1- \tasepF_1(\hite_j; \gamma_j,\tau_j))$. We thus find the correct limit properties as $\hite$ becomes small or large.

(b) From \eqref{eq:tasepF_general_def} again,
\beqq \begin{split}
\tasepF_{m-1}(\hite^{(j)}; \gamma^{(j)}, \tau^{(j)}) -  \tasepF_m(\hite; \gamma, \tau)
&= \lim_{T\to \infty} \prob\bigg(
\big\{ \widetilde{\tasep}_T(\gamma_j , \tau_j ) > \hite_j \big\}
\bigcap_{\substack{1\le i\le m\\ i\ne j}}
\big\{  \widetilde{\tasep}_T(\gamma_j , \tau_j ) \le \hite_i   \big\}  \bigg) \\
&\le  \lim_{T \to \infty} \prob \big(
\widetilde{\tasep}_T(\gamma_j , \tau_j ) > \hite_j \big)
= 1-\tasepF_1(\hite_j; \gamma_j,\tau_j) .
\end{split} \eeqq
The upper bound tends to $0$ as $\hite_j\to +\infty$ since $\tasepF_1$ is a distribution function [1].
\end{proof}

\medskip

The final result of this Section is the continuity of $\tasepF_m(\hite; \gamma, \tau)$.
When $\hite\in \Omega^m_+(\tau)$, the limit $\tasepF_m(\hite; \gamma, \tau)$ for \eqref{eq:taseplimit_app} is given by the formula
\begin{equation}
\label{eq:def_tasepF}
\begin{split}
&\tasepF_m(\hite; \gamma, \tau)
=\frac{1}{(2\pi \ii)^m } \oint \cdots \oint C(\bfz)D(\bfz) \prod_{i=1}^m \frac{\dd z_i}{z_i}
\end{split}
\end{equation}
where the integrand is same as that of the formula \eqref{eq:jointprointe} (with $p=1$) but the radii of the contour circles satisfy the reverse inequalities $0<|z_m|<\cdots<|z_1|<1$.
From the formula of $C(\bfz)$ and $D(\bfz) $ in Section \ref{sec:formdf} (with $p=1$) and Lemma~\ref{prop:tasepF_extension},
$\tasepF_m(\hite; \gamma, \tau)$ is jointly continuous for $\gamma\in\realR^m$, $\tau \in \realR_{+, \le}^m$, and $\hite\in\Omega^m(\tau) $.
Due to the invariance under permutations of the triples of the parameters, it is continuous on the set
\beqq
\realR^m\times \realR^m_+\times \realR^m
\setminus \{(\gamma, \tau, \hite): \hite_i=\hite_j \text { and } \tau_i=\tau_j \text{ for some } 1\le i<j\le m\}.
\eeqq
The next result shows that it is continuous in all of $\realR^m\times \realR^m_+\times \realR^m$.

\begin{prop} \label{prop:tasepF_continuity}
The function $\tasepF_m(\hite; \gamma, \tau)$ is jointly continuous in $(\gamma, \tau, \hite)\in \realR^m\times \realR^m_+\times \realR^m$.
\end{prop}

\begin{proof}
Let $(\gamma, \tau, \hite)$ be a point in $\realR^m\times \realR^m_+\times \realR^m$.
For $(\gamma', \tau', \hite')\in \realR^m\times \realR^m_+\times \realR^m$ and $1\le j\le m$, let
\beqq
\gamma'_{(j)} = (\cdots, \gamma'_{j-1}, \tau'_j, \gamma_{j+1}, \cdots),
\qquad
\tau'_{(j)}= (\cdots, \tau'_{j-1}, \tau'_j, \tau_{j+1}, \cdots),
\qquad
\hite'_{(j)} = (\cdots, \hite'_{j-1}, \hite'_j, \hite_{j+1}, \cdots).
\eeqq
Fom \eqref{eq:tasepF_general_def}, for every $j$,
\beqq
\begin{split}
&  | \tasepF_m( \hite'_{(j)};  \gamma'_{(j)},  \tau'_{(j)}) - \tasepF_m( \hite'_{(j-1)};  \gamma'_{(j-1)},  \tau'_{(j-1)})  | \\
&\le \lim_{T \to \infty} \prob \big( \widetilde{\tasep}_T(\gamma_j' , \tau_j' ) \le \hite'_j  ,  \,  \widetilde{\tasep}_T(\gamma_j , \tau_j ) > \hite_j \big)
+ \prob \big(\widetilde{\tasep}_T(\gamma_j , \tau_j ) \le \hite_j  , \,  \widetilde{\tasep}_T(\gamma_j' , \tau_j') > \hite'_j  \big) \\
&= \lim_{T \to \infty} \prob \big(    \widetilde{\tasep}_T(\gamma_j' , \tau_j' ) \le \hite'_j  \big)
+ \prob \big(  \widetilde{\tasep}_T(\gamma_j , \tau_j ) \le \hite_j  \big)
- 2   \prob \big( \widetilde{\tasep}_T(\gamma_j' , \tau_j' )\le \hite'_j, \,  \widetilde{\tasep}_T(\gamma_j , \tau_j ) \le \hite_j  \big) .
\end{split}
\eeqq
\beqq
\begin{split}
| \tasepF_m( \hite'_{(j)};  \gamma'_{(j)},  \tau'_{(j)}) - \tasepF_m( \hite'_{(j-1)};  \gamma'_{(j-1)},  \tau'_{(j-1)})  |
&=\tasepF_1(\hite'_j; \gamma'_j,\tau'_j) + \tasepF_1(\hite_j; \gamma_j,\tau_j)
-2\tasepF_2(\hite'_j,\hite_j; \gamma'_j, \gamma_j,\tau'_j, \tau_j) \\
&\le \tasepF_1(\hite'_j; \gamma'_j,\tau'_j) + \tasepF_1(\hite_j; \gamma_j,\tau_j)
-2\tasepF_2(\hite'_j,\hite_j - \epsilon; \gamma'_j, \gamma_j,\tau'_j, \tau_j)
\end{split}
\eeqq
for every $\epsilon>0$.
If $(\gamma'_j, \tau'_j, \hite'_j)$ is close enough to $(\gamma_j, \tau_j, \hite_j)$, then $(\gamma_j', \gamma_j, \tau_j', \tau_j, \hite_j', \hite_j-\epsilon)\in U^2_+$.
Thus, the continuity of $\tasepF_2$ on $U^2_+$ implies that
\beqq
\begin{split}
&\limsup_{(\gamma', \tau', \hite')\to (\gamma, \tau, \hite)} | \tasepF_m( \hite'_{(j)};  \gamma'_{(j)},  \tau'_{(j)}) - \tasepF_m( \hite'_{(j-1)};  \gamma'_{(j-1)},  \tau'_{(j-1)})  |\\
&\quad \le 2\tasepF_1(\hite_j; \gamma_j,\tau_j)
-2\tasepF_2(\hite_j,\hite_j - \epsilon; \gamma_j, \gamma_j,\tau_j, \tau_j)
= 2\tasepF_1(\hite_j; \gamma_j,\tau_j) - 2\tasepF_1(\hite_j-\epsilon; \gamma_j,\tau_j)
\end{split}
\eeqq
where we used the fact that $\tasepF_2(a,b; \gamma,\gamma, \tau, \tau)=\tasepF_1(a; \gamma,\tau)$ if $a<b$, which follows from the definition of~\eqref{eq:tasepF_general_def}.
Since the inequality holds for every $\epsilon>0$ and the one-point distribution function $\tasepF_1$ is continuous, we find that
\beqq
\begin{split}
&\limsup_{(\gamma', \tau', \hite')\to (\gamma, \tau, \hite)} | \tasepF_m( \hite'_{(j)};  \gamma'_{(j)},  \tau'_{(j)}) - \tasepF_m( \hite'_{(j-1)};  \gamma'_{(j-1)},  \tau'_{(j-1)})  |
= 0 .
\end{split}
\eeqq
Summing over $j$, we conclude that
\beqq
\limsup_{(\gamma', \tau', \hite')\to (\gamma, \tau, \hite)} \left|\tasepF_m(\hite'; \gamma', \tau') - \tasepF_m(\hite; \gamma, \tau)  \right|
= 0,
\eeqq
proving the desired continuity.
\end{proof}

\section{Formula of $D_{\bfn}(z)$} \label{sec:Dnformula}

We state the formula of $D(\bfz)$ given in \cite[Lemma 2.10]{Baik-Liu19} and show that it can be written as the form \eqref{eq:def_Dbfz} in Subsection \ref{sec:formdf}.
It is enough to check it  when $p=1$ since the general $p$ case follows from the property~\eqref{eq:scalingprop}.

For complex vectors $W=(w_1, \cdots, w_n)$ and $W'=(w_1',\cdots, w_{n'}')$, we denote
\beqq
\Delta(W)= \prod_{1\le i<j\le n} (w_j-w_i) \quad \text{and} \quad \Delta(W;W')= \prod_{i=1}^n \prod_{j=1}^{n'} (w_i-w_j') .
\eeqq
We also use the notation that for a function $g$ of a single variable and a vector $W=(w_1, \cdots, w_n)$,
\beqq
g(W) := \prod_{i=1}^n g(w_i) .
\eeqq

For $0<|z|<1$, the sets $\rmL_z$ and $\rmR_z$ are the discrete sets in the complex plane defined as
\beqq
\rmL_z=\{w: e^{-w^2/2}=z, \, \Re(w)<0\} \quad \text{and} \quad \rmR_z=\{w: e^{-w^2/2}=z, \, \Re(w)>0\}.
\eeqq
The series formula of $D(\bfz)$ given in \cite[Lemma 2.10]{Baik-Liu19} is
\beqq
D(\bfz) = \sum_{\bfn \in \{0, 1, \cdots\}^m } \frac{1}{(\bfn !)^2} D_{\bfn}(\bfz)
\eeqq
where for $\bfn=(n_1, \cdots, n_m)$ and $0<|z_1|<\cdots<|z_m|<1$,
\beq \label{eq:D_bfzorig} \begin{split}
= \left(1-\frac{z_{i-1}}{z_i} \right)^{n_i}
&\left(1-\frac{z_{i}}{z_{i-1}} \right)^{n_{i-1}}
\sum_{\substack{U^{(i)}\in \rmL_{z_i}^{n_i}\\ V^{(i)}\in \rmR_{z_i}^{n_i} \\ i=1,\cdots,m}}
\prod_{i=1}^m \frac{\Delta(U^{(i)})^2 \Delta(V^{(i)})^2}{\Delta(U^{(i)};V^{(i)})^2} \hat f_i (U^{(i)}) \hat f_i (V^{(i)})\\
&\qquad \times \prod_{i=2}^m \frac{\Delta(U^{(i)};V^{(i-1)})\Delta(V^{(i)};U^{(i-1)})}{\Delta(U^{(i)};U^{(i-1)})\Delta(V^{(i)};V^{(i-1)})}
\frac{e^{-h(V^{(i)},z_{i-1})-h(V^{(i-1)},z_{i})}}{e^{h(U^{(i)},z_{i-1})+h(U^{(i-1)},z_{i})}}
\end{split} \eeq
\beqq \begin{split}
&h(w,z) =\begin{dcases}
-\frac{1}{\sqrt{2\pi}} \int_{-\infty}^w \Li_{1/2}(ze^{(w^2-y^2)/2})\rmd y, \qquad &\Re(w)<0,\\
-\frac{1}{\sqrt{2\pi}} \int_{-\infty}^{-w} \Li_{1/2}(ze^{(w^2-y^2)/2})\rmd y, \qquad &\Re(w)>0,
\end{dcases}
\end{split}\eeqq
\beqq \label{eq:ffuntion}
\begin{split}
e^{-\frac{1}{3}(\tau_i-\tau_{i-1})w^3 +\frac{1}{2}(\gamma_i-\gamma_{i-1})w^2 + (\hite_i-\hite_{i-1})w},
\qquad & \Re(w)<0,\\
e^{\frac{1}{3}(\tau_i-\tau_{i-1})w^3 -\frac{1}{2}(\gamma_i-\gamma_{i-1})w^2 - (\hite_i-\hite_{i-1})w},
\qquad & \Re(w)>0,
\end{dcases}
\end{split}\eeqq
\beqq
\hat f_i(w) = \frac{1}{w} f_i(w) e^{2h(w,z_i)}.
\eeqq

Note that $w\in \rmR_z$ if and only if $-w\in \rmL_z$. Thus, setting $\hat U^{(i)}= - V^{(i)}$, the sum in \eqref{eq:D_bfzorig} can be written as
\beqq \begin{split}
\sum_{\substack{U^{(i)}, \hat U^{(i)} \in \rmL_{z_i}^{n_i} \\ i=1,\cdots,m}}
\prod_{i=1}^m \frac{\Delta(U^{(i)})^2 \Delta(- \hat U^{(i)})^2}{\Delta(U^{(i)};-\hat U^{(i)})^2} \hat f_i (U^{(i)}) \hat f_i (-\hat U^{(i)})\\
&\qquad\qquad \times  \prod_{i=2}^m \frac{\Delta(U^{(i)}; -\hat U^{(i-1)})\Delta( -\hat U^{(i)};U^{(i-1)})}{\Delta(U^{(i)};U^{(i-1)})\Delta( -\hat U^{(i)}; -\hat U^{(i-1)})}
\frac{e^{-h(-\hat U^{(i)},z_{i-1})-h(-\hat U^{(i-1)},z_{i})}}{e^{h(U^{(i)},z_{i-1})+h(U^{(i-1)},z_{i})}} .
\end{split} \eeqq
Since $h(-w,z)=h(w,z)$, after inserting the formula $\hat f_i(w) = \frac{1}{w} f_i(w) e^{2h(w,z_i)}$ and using the notation $E^{i, \pm}$ of \eqref{eq:Edeffff} instead of $f_i$, we can express the above sum as
\beqq \begin{split}
\sum_{\substack{U^{(i)}, \hat U^{(i)} \in \rmL_{z_i}^{n_i} \\ i=1,\cdots,m}}
\prod_{i=1}^m
\frac{e^{2h(\hat U^{(i)},z_{i}) +2h(\hat U^{(i)},z_{i})}}{e^{h(U^{(i-1)},z_{i})+h(U^{(i+1)},z_{i})+ h(\hat U^{(i-1)},z_{i})+h(\hat U^{(i+1)},z_{i})}}
\prod_{i=1}^m \prod_{j=1}^{n_i} E^{i, +}(u^{(i)}_j) E^{i, -}( \hat u^{(i)}_j) \\
&\qquad \qquad \times
\prod_{i=1}^m \prod_{j=1}^{n_i} \frac{-1}{u^{(i)}_j \hat u^{(i)}_j}
\prod_{i=1}^m \frac{\Delta(U^{(i)})^2 \Delta(- \hat U^{(i)})^2}{\Delta(U^{(i)};-\hat U^{(i)})^2}
\prod_{i=2}^m \frac{\Delta(U^{(i)}; -\hat U^{(i-1)})\Delta( -\hat U^{(i)};U^{(i-1)})}{\Delta(U^{(i)};U^{(i-1)})\Delta( -\hat U^{(i)}; -\hat U^{(i-1)})}
\end{split} \eeqq
where we set $U^{(0)}= \hat U^{(0)}= U^{(m+1)} = \hat U^{(m+1)}=\emptyset$ so that $e^{h(U^{(0)}; z_1)}=1$, and so on.
The product involving the function $h$ is $H_{\bfn}(U, \hat U)$ of \eqref{eq:Hdefff}
and the next product involving $E^{i, \pm}$ is $E_{\bfn}(U, \hat U)$ of \eqref{eq:Edeffff}.
Finally, again with the convention $U^{(0)}= \hat U^{(0)}= U^{(m+1)} = \hat U^{(m+1)}=\emptyset$, we have
\beqq \begin{split}
&\prod_{i=1}^m \frac{\Delta(U^{(i)})^2 \Delta(- \hat U^{(i)})^2}{\Delta(U^{(i)};-\hat U^{(i)})^2}
\prod_{i=2}^m \frac{\Delta(U^{(i)}; -\hat U^{(i-1)})\Delta( -\hat U^{(i)};U^{(i-1)})}{\Delta(U^{(i)};U^{(i-1)})\Delta( -\hat U^{(i)}; -\hat U^{(i-1)})} \\
\prod_{i=1}^{m+1} \frac{\Delta(U^{(i-1)}) \Delta(- \hat U^{(i-1)}) \Delta(U^{(i)}; -\hat U^{(i-1)})\Delta( -\hat U^{(i)};U^{(i-1)}) \Delta(U^{(i)}) \Delta(- \hat U^{(i)})}{ \Delta(U^{(i-1)};-\hat U^{(i-1)}) \Delta(U^{(i)};U^{(i-1)})\Delta( -\hat U^{(i)}; -\hat U^{(i-1)}) \Delta(U^{(i)};-\hat U^{(i)}) } \\
(-1)^{n_1+\cdots+n_m}\prod_{i=1}^{m+1} \Cd( U^{(i-1)}, -\hat U^{(i)} ; \hat U^{(i-1)}, - U^{(i)})
\end{split} \eeqq
in terms of the Cauchy determinant \eqref{eq:Caudedef}.
This is a factor of $R_{\bfn}(U, \hat U)$ of \eqref{eq:Rdetfom}, and we thus find that $D_{\bfn}(\bfz)$ is equal to the form \eqref{eq:def_Dbfz}
in Subsection \ref{sec:formdf}.

\def\cydot{\leavevmode\raise.4ex\hbox{.}}
\begin{thebibliography}{10}

[1]
J.~Baik and Z.~Liu.
\newblock Fluctuations of {TASEP} on a ring in relaxation time scale.
\newblock {\em Comm. Pure Appl. Math.}, 71(4):747--813, 2018.

[2]
J.~Baik and Z.~Liu.
\newblock Multipoint distribution of periodic {TASEP}.
\newblock {\em J. Amer. Math. Soc.}, 32(3):609--674, 2019.

[3]
J.~Baik and Z.~Liu.
\newblock Periodic {TASEP} with general initial conditions.
\newblock {\em Probab. Theory Related Fields}, 179(3-4):1047--1144, 2021.

[4]
J.~Baik, Z.~Liu, and G.~L.~F. Silva.
\newblock Limiting one-point distribution of periodic {TASEP}.
\newblock {\em Ann. Inst. Henri Poincar\'{e} Probab. Stat.}, 58(1):248--302,

[5]
J.~Baik, A.~Prokhorov, and G.~L.~F. Silva.
\newblock Differential equations for the {KPZ} and periodic {KPZ} fixed points.
\newblock {\em Comm. Math. Phys.}, 401(2):1753--1806, 2023.

[6]
R.~Basu and S.~Ganguly.
\newblock Time correlation exponents in last passage percolation.
\newblock In {\em In and out of equilibrium 3. {C}elebrating {V}ladas
{S}idoravicius}, volume~77 of {\em Progr. Probab.}, pages 101--123.
Birkh\"{a}user/Springer, Cham, 2021.

[7]
P.~L. Ferrari and A.~Occelli.
\newblock {Universality of the GOE Tracy-Widom distribution for TASEP with
arbitrary particle density}.
\newblock {\em Electronic Journal of Probability}, 23:1--24, 2018.

[8]
S.~Ganguly and M.~Hegde.
\newblock Sharp upper tail estimates and limit shapes for the {KPZ} equation
via the tangent method, 2022.
\newblock arXiv:2208.08922.

[9]
S.~Ganguly, M.~Hegde, and L.~Zhang.
\newblock Brownian bridge limit of path measures in the upper tail of {KPZ}
models, 2023.
\newblock arXiv:2311.12009.

[10]
P.~Y. Gaudreau~Lamarre, Y.~Lin, and L.-C. Tsai.
\newblock K{PZ} equation with a small noise, deep upper tail and limit shape.
\newblock {\em Probab. Theory Related Fields}, 185(3-4):885--920, 2023.

[11]
\newblock Long and short time asymptotics of the two-time distribution in local
random growth.
\newblock {\em Math. Phys. Anal. Geom.}, 23(4):Paper No. 43, 34, 2020.

[12]
J.-H. Li and A.~Saenz.
\newblock Contour integral formulas for {PushASEP} on the ring, 2023.
\newblock arXiv:2308.05372.

[13]
\newblock Multi-point distribution of discrete time periodic {TASEP}.
\newblock {\em Probab. Theory Related Fields}, 182(3-4):1053--1131, 2022.

[14]
\newblock One-point distribution of the geodesic in directed last passage
\newblock {\em Probab. Theory Related Fields}, 2022.

[15]
\newblock When the geodesic becomes rigid in the directed landscape.
\newblock {\em Electronic Communications in Probability}, 27:1--13,

[16]
Z.~Liu and Y.~Wang.
\newblock A conditional scaling limit of the {KPZ} fixed point with height
tending to infinity at one location, 2022.
\newblock arXiv:2208.12215.

[17]
K.~Matetski, J.~Quastel, and D.~Remenik.
\newblock The {KPZ} fixed point.
\newblock {\em Acta Math.}, 227(1):115--203, 2021.

[18]
R.~Nissim and R.~Zhang.
\newblock Asymptotics of the one-point distribution of the {KPZ} fixed point
conditioned on a large height at an earlier point, 2022.
\newblock arXiv:2210.04999.

[19]
J.~Quastel and D.~Remenik.
\newblock K{P} governs random growth off a 1-dimensional substrate.
\newblock {\em Forum Math. Pi}, 10:Paper No. e10, 26, 2022.

\end{thebibliography}

\end{document}


expected, as QCD(F) has no center symmetry in the $R^{3}$-directions.

Summary of 7.2:

Here, we first outlined the QCD-like IR physics on $R^{4}$ and the nonabelian
chiral-symmetry breaking pattern (167). Then, we studied the small-$L$ IR
physics with “CFC” twisted boundary conditions. We showed that the dual photon
shifts under the $U(1)_{X}$ chiral symmetry and is the Goldstone boson. The
order parameter for the chiral-symmetry breaking is the expectation value of
the monopole operator. The chiral-symmetry breaking leads to constituent mass
of the A$+$P fundamental fermions, through the $M$ and $KK$ ’t Hooft vertices.

###  7.3 Continuity between the large-$L$ and small-$L$ symmetry realization.

Finally, we explain the remarkable feature of the CFC-symmetric
compactification noted in the beginning of this Section: the
small-$L$/large-$L$ continuity. Consider the low-energy limit of the $R^{4}$
theory, described by the $SU(4)/USp(4)$ chiral Lagrangian of the Goldstone
bosons. Compactifying on an $S^{1}$ of size $L\gg\Lambda^{-1}$, we have to
remember that the different boundary conditions for the A and P fermions can
be rephrased as the turning on background Wilson lines for the vector global
symmetry, or equivalently, imposing $S^{1}$-boundary conditions twisted by
$\Omega$. In the low-energy chiral lagrangian, valid at $L\gg\Lambda^{-1}$,
these same $\Omega$-twisted boundary conditions on the $S^{1}$ have to be
imposed on the fields in the chiral Lagrangian of the Goldstone bosons.

Now, we recall our earlier symmetry considerations, showing that the CFC
boundary conditions leave the $SU(2)_{P}\times SU(2)_{A}\times U(1)_{X}\in
SU(4)$ part of the broken global symmetry intact, but that only the $U(1)_{X}$
is part of the broken symmetry, i.e. one under which the goldstones transform
nonlinearly. In other words, our CFC boundary conditions explicitly break all
the spontaneously broken (by (167)) symmetries but $U(1)_{X}$. This implies
that of the $5$ $SU(4)/USp(4)$ Goldstone bosons, only the $U(1)_{X}$-Goldstone
will remain exactly massless at finite $L$, while the others should obtain
mass of order $1/L$ from the CFC boundary conditions.121121121This can be
worked out explicitly if needed, but it requires introducing the chiral
Lagrangian for $SU(4)/USp(4)$. For a more familiar example in an $SU(3)$
QCD(F), using the familiar QCD $(SU(3)_{L}\times SU(3)_{R})/SU(3)_{V}$ pion
Lagrangian, see [146]. This single massless Goldstone boson becomes, in the
small-$L$ limit, the dual photon. The upshot is that the massless spectrum of
our P$+$A theory is the same at $L\ll\Lambda^{-1}$ and $L\gg\Lambda^{-1}$.
This provides evidence for the continuous connection between the small-$L$ and
large-$L$ limit, i.e. for “adiabatic continuity” in QCD(F) with CFC boundary
conditions. Notice, however, that the $L\sim\Lambda^{-1}$ regime is one where
neither the $R^{4}$ nor the small-$L$ descriptions used above applies, so we
do not have proof of the continuity.

The reader may feel a certain amount of unhappiness after understanding the
main points of this Section. This may arise from the realization that only the
breaking of the $U(1)_{X}$ subgroup of the nonabelian $SU(4)$ chiral symmetry
can be seen in the calculable small-$L$ regime. Indeed, weak coupling on small
$R^{3}\times S^{1}$ is designed so that the physics abelianizes. The massless
dual photons (which are Goldstone bosons also for the $N>2$ generalizations
[146] of our CFC story) comprise only a subset of the $R^{4}$-pions which
transform, linearly or nonlinearly, under nonabelian global symmetries. The
dual photon(s), on the other hand, only transform under abelian global
symmetries. This may, indeed, be the best that is possible in a calculable
semiclassical regime.

At this point, we leave the judgment and possible applications of this
construction for the future, but should not fail to notice its many
interesting and thought-provoking features, notably the role monopole-
instantons play in the constituent mass generation. We now rest our case and
refer the interested reader to recent work on this continuity for $N>2$,
discussing many aspects not touched upon here [152].

Summary of 7.3:

Here, we showed that considering “CFC” twisted boundary conditions allows one
to exhibit a continuous connection between the IR spectra of QCD(F) at small
and large $L$: the continuous chiral symmetry realization is the same at large
$L\gg\Lambda^{-1}$ and at small $L\ll\Lambda^{-1}$.

##  8 A quick guide to the literature on other theories.

###  8.1 dYM, QCD(adj)/SYM, QCD(F) with gauge group $SU(N)$ and $N>2$.

Having familiarized ourselves with the detailed working of the various
$R^{3}\times S^{1}$-compactified $SU(2)$ theories, it is now natural to ask
about larger numbers of colours. The only reason we stayed away from
discussing $SU(N)$ was to not have to introduce more group theory notation.
This would require venturing into Lie-algebraic territory: using roots,
weights, etc., which are all required for the study of the circle compactified
theories for general $N$ and for other gauge groups. A pedagogical approach
would have required us to introduce these notions, which are not standard fare
in QFT classes. This would have made the lectures even longer.

We believe, however, that the $SU(2)$ case illustrated much of the small-$L$
physics quite nicely. Briefly, the general feature, abelianization on
$R^{3}\times S^{1}$, holds also in dYM, QCD(adj), and SYM for $N>2$ and
appears as spontaneous breaking $SU(N)\rightarrow U(1)^{N-1}$ at a scale
$m_{W}=2\pi/(NL)$. Thus, calculability requires $\Lambda NL\ll 2\pi$. There
are $N-1$ dual photons as well as $N$ different “M and KK” monopole
instantons; these are now most appropriately labeled by the $N-1$ simple roots
and the lowest, or affine, root of $SU(N)$. Their ’t Hooft vertices (and
molecules made thereof) generate the appropriate potentials for the dual
photons. Magnetic bions and neutral bions appear, as in $SU(2)$, but have a
more diverse nature. Overall, the small-$L$ physics of dYM, QCD(adj) and SYM
is quite similar to the $SU(2)$ case. As alluded to in the last Section,
QCD(F) differs because of the fact that it is only $SU(2)$ where the
fundamental and antifundamental quarks are equivalent (for QCD(F) with $N>2$,
see [152]).

Our $SU(2)$ considerations do not allow us to discuss the small-$L$, so-called
“abelian large-$N$” limit. This consists in taking $N\rightarrow\infty$ and
$L\rightarrow 0$ but keeping $NL\Lambda\ll 2\pi$ fixed, in order to have weak
coupling calculability [18]. For $SU(N)$ one can study the properties of
confining strings of various $N$-alities [40, 161, 49]. The most unusual
feature of $SU(N)$ theories in this limit is the appearance of an emergent
dimension: even though one takes $L\rightarrow 0$, thus expecting the long-
distance theory to be 3d, a latticized dimension of $N$ sites and lattice
spacing $m_{W}^{-1}e^{S_{0}}$ (for SYM) emerges from the Cartan space of the
gauge group. For more discussion of this phenomenon, somewhat similar to
T-duality in string theory, see [23].

For the reader who wishes to understand the details of the generalizations to
$SU(N)$ (for other gauge groups, see further below) we offer a brief guide
through the literature. The original papers on dYM [13] and QCD(adj) [11] all
considered $SU(N)$ for arbitrary $N$, assuming familiarity with the Lie-
algebraic notation. This can be learned by studying one of the many group-
theory books, for example Ramond’s [162]. There are also several papers that
introduce aspects of this technology adapted to the $R^{3}\times S^{1}$
compactification. We especially recommend the voluminous appendix of [19],
where the generalization of the Weyl chamber, shown on Figure 10 for $SU(2)$,
for all gauge groups is explicitly worked out; another paper whose appendix
introduces some of the relevant technology is [43].

An aspect we didn’t discuss even for $SU(2)$ was the matching of various ’t
Hooft anomalies in the small-$L$ theory. These include both the traditional
$0$-form symmetry anomalies, which are standard material, reviewed in [27], as
well as the more recent so-called generalized ’t Hooft anomalies involving
higher-form global symmetries [15, 16, 17]. This is a big subject that we can
not possibly do justice to here; unfortunately, an introductory set of notes
does not appear to have been written yet. We already mentioned that the
liberation of quarks on domain walls (exhibited explicitly in our
semiclassical domain, see Sections 4.2.2, 5.3.2, and 6.2) is a manifestation
of the generalized ’t Hooft anomaly [80, 48, 163, 164]. The matching on
$R^{3}\times S^{1}$ is discussed, in various ways, in [69, 165, 105, 166], as
well as in the recent [76]. The appendix of the latter reference has a
physicist-friendly discussion of the gauging of the $1$-form center symmetry
for all gauge groups on the four-torus, relevant for the ’t Hooft anomaly
between center symmetry and parity or discrete chiral symmetry. Further
related recent work is in [130, 125].

###  8.2 Other matter representations with $SU(N)$ gauge group.

The study of $SU(N)$ gauge theories with fermions in other representations
began soon after the work [11, 13] appeared, in the framework which was called
“deformation theory.” This consists of adding the center stabilizing potential
(87) by hand, to ensure abelianization, and then studying the “M and KK”
vertices, taking into account the zero-modes of the various fermions via the
index theorem [102]. The presence or absence of dual photon potentials can be
then deduced from the relevant symmetries [167, 14, 106, 149]. In this manner,
the small-$L$ phases of theories with fermions in varying numbers of
vectorlike fundamental, symmetric, antisymmetric, etc., representations were
studied, as well as some chiral theories [168] (see [169] for interesting
remarks on the supersymmetric version of the latter).

An uncomfortable aspect of this approach is the fact that the nonlocal term
(87) is added by hand, leading to a nonrenormalizable UV-incomplete theory. It
is desirable if the theory abelianizes as a result of the dynamics, as we saw
happens naturally in QCD(adj). This was seen to occur in some cases with
mixed-representation fermions [106]. A general study was performed by Anber et
al. in [170]. They studied general $SU(N)$ asymptotically-free and anomaly-
free theories with fermions in general representations. When taken on
$R^{3}\times S^{1}$, all fermions were taken periodic and their “GPY”
potentials were studied. The conclusion was that theories with only a single-
type of representation do not abelianize on $S^{1}$ and are thus generally
strongly-coupled in the IR. However in a large class of mixed-representation
theories, abelianization occurs. A complete list of mixed-representation
theories was given, see Table 8 of [170]. Some of these theories were studied
further in [171, 172] and were also used to suggest novel possible phases on
$R^{4}$.

###  8.3 Other gauge groups.

A natural question that arises is whether the abelianization—and the resulting
calculability of the IR dynamics on $R^{3}\times S^{1}$—in $SU(N)$ with
$n_{f}$ adjoint fermions also occurs for other gauge groups with adjoints.
Many of the simple Lie groups ($Sp(N),Spin(N),E_{6},E_{7}$) have nontrivial
centers. In all cases, theories with only adjoint fermions respect the
$1$-form center symmetry. A big difference compared to $SU(N)$, however, is
that the order of the center of the gauge group is usually much smaller than
the rank of the group. Let us compare $SU(N)$, where the order of $Z_{N}$ is
$N$ and the rank is $N-1$ with, for example, $Sp(N)$, also known as $USp(2N)$,
whose center is $Z_{2}$ while the rank is $N$. In each case the dimension of
the Weyl chamber is equal to the rank, as was the case for $SU(2)$. The center
symmetry maps the Weyl chamber to itself and the center symmetric vevs are the
fixed points of this map. If the theory is center-symmetric, this fixed-point
set is bound to be an extremum. For $SU(N)$, as for $N=2$, there is a unique
center-symmetric point on the Weyl chamber, where complete abelianization
occurs. As we showed, this point is the minimum of the GPY potential for
$n_{f}>1$.

In contrast, for $Sp(N)$122122122The situation for all other groups with
centers (the maximal order of center is $4$) is similar [19]. there is an
$\lfloor N/2\rfloor$ dimensional fixed-point set which also passes through the
edges of the Weyl chamber, where there is unbroken nonabelian gauge symmetry.
A priori, it is not obvious which point on this fixed-point set is the actual
minimum of the perturbative “GPY” potential in the theory with $n_{f}>1$
adjoints. The question about the behaviour of the one-loop “GPY” potentials in
such theories was studied in the already-mentioned ref. [19]. It turns out
that only QCD(adj) with $n_{f}>1$ and gauge groups $SU(N)$ and $Sp(N)$
abelianizes at small $S^{1}$ (the small-$L$ IR physics of $Sp(N)$ QCD(adj) was
studied in [19, 173]). For all other gauge groups, there is an unbroken
nonabelian gauge group, leading to a strongly-coupled IR physics even at small
$S^{1}$.

The most remarkable feature, however, arises for the $n_{f}=1$ case of SYM
with general gauge group $G$ [122]. There, as for our $SU(2)$ case, due to
supersymmetry, the “GPY” potential vanishes to all orders of perturbation
theory. The nonperturbative physics determines where the holonomy vev lies. It
turns out that for SYM theories, abelianization occurs for arbitrary gauge
group: $G\rightarrow U(1)^{r}$, due to the “M + KK” nonperturbative potentials
on the Weyl chamber (here $r$ is the rank of $G$). The minimum occurs away
from the boundaries of the Weyl chamber at a particular point described in
[122]. In each case, the small-$L$ IR theory is the magnetic dual of an
$U(1)^{r}$ 3d gauge theory.

In the case where $G$ has a center, this point on the Weyl chamber preserves
center symmetry and the abelianization allows one, upon adding a gaugino mass,
to investigate the continuity conjecture to the thermal-YM theory, as in
Section 6.4, see also [134, 43, 144]. This allows also a study of other
nonperturbative properties, not accessible to the usual holomorphy tools, such
as the nature of the various domain walls and of the confining strings.

Summary of 8:

The small-$L$ dynamics of all dYM, QCD(adj)/SYM, QCD(F) theories studied here
proceeds along lines similar to $SU(2)$, for all $SU(N)$ gauge groups.
However, in the nonsupersymmetric case of QCD(adj) only $Sp(N)$ abelianizes at
small $L$, while the other groups retain some unbroken nonabelian subgroup. In
the case of SYM, abelianization occurs for all gauge groups, with or without a
center. Regarding other matter representations, abelianization holds in many
cases of mixed-representation periodic fermions, as per the classification of
[170].

Acknowledgments:

I am grateful to Andreas Wipf for the invitation to lecture at the 2016
“Saalburg” Summer School and to Simon Catterall and Simon Hands for inviting
me to contribute to the special issue of the journal “Symmetry” on “New
Applications of Symmetry in Lattice Field Theory.” Special thanks are due to
Mohamed Anber, John Lai, and F. David Wandler for their comments and
suggestions on the manuscript. This work is supported by an NSERC Discovery
Grant.

##  Appendix A Notations and conventions.

Here, we summarize our conventions and warn about possible confusions caused
by our notation.

We use $\mu=1,2,3$ to label the $R^{3}$ Euclidean spacetime coordinates. In
$R^{4}$ we use instead $M=1,2,3,4$ to label the Euclidean coordinates. We use
the same labeling on $R^{3}\times S^{1}$ with $x^{4}\equiv x^{4}+L$ being the
compact $S^{1}$. In each case, the Euclidean metric is positive definite
$(+,+,...)$. On several occasions in the text, we fail to explicitly show
lower/raised repeated indices that are summed over. The 4d Levi-Cevita tensor
is $\epsilon^{1234}=+1$.

We use $a=1,2,3$ to label $SU(2)$ Lie algebra indices. The generators of the
gauge group are $T^{a}$, obeying Tr($T^{a}T^{b})={1\over 2}\delta^{ab}$ and
can be taken to be one-half the Pauli matrices, $T^{a}={\sigma^{a}/2}$. The
gauge fields and adjoint scalar transform as usual under $SU(2)$ gauge
transformations $g$: $A_{M}=A_{M}^{a}T^{a}\rightarrow
g(A_{M}-i\partial_{M})g^{-1}$. An adjoint field transforms as
$\Phi=\Phi^{a}T^{a}\rightarrow g\Phi g^{-1}$. The field strength tensor is
$F_{MN}=\partial_{M}A_{N}-\partial_{N}A_{M}+i\left[A_{M},A_{N}\right]$ and the
adjoint covariant derivative is $D_{M}\Phi=\partial_{M}\Phi+i[A_{M},\Phi]$. A
fundamental field $\Psi$ transforms as $\Psi\rightarrow g\Psi$ and has
covariant derivative $D_{M}\Psi=\partial_{M}\Psi+iA_{M}\Psi$.

On several occasions, we use $R^{1,3}$ or $R^{1,2}\times S^{1}$ Minkowski
space. We use the same labels $M,N$ (and $\mu,\nu$) for the coordinates, but
with $(+,-,-,..)$ metric. In addition, their ranges are now $M,N=0,1,2,3$ (and
$\mu,\nu=0,1,2$), where $x^{0}$ is the time direction while the $S^{1}$
direction is $x^{3}\equiv x^{3}+L$.

For fermions, we use the “God given” two-component $SL(2,C)$ spinor notation
used in [27]; for the most detailed introduction, see [174]. Lagrangians
involving fermions are written in Minkowski metric. The minimal amount of
information on the two component $SL(2,C)$ spinor notation we need is as
follows. The Weyl spinor is an anticommuting complex object $\lambda_{\alpha}$
($\alpha=1,2$), with $\bar{\lambda}_{\dot{\alpha}}=(\lambda_{\alpha})^{*}$,
$\epsilon^{12}=-\epsilon^{21}=\epsilon^{\dot{1}\dot{2}}=-\epsilon^{\dot{2}\dot{1}}=+1$.
Instead of the usual gamma matrices, one uses the two-by-two matrices
$\bar{\sigma}^{0}=\sigma^{0}=\bf{1}_{2}$, $\bar{\sigma}^{i}=-\sigma^{i}$,
$i=1,2,3$, where $\sigma^{i}$ are the Pauli matrices (when using matrix
notation there is no reason to explicitly show the indices
$\bar{\sigma}^{\dot{\alpha}\alpha}$, as we do in e.g. (3.4.1)). For a massless
Weyl fermion, the kinetic term is
${i}\bar{\lambda}_{\dot{\alpha}}\;\bar{\sigma}^{M\dot{\alpha}\alpha}(\partial_{M}\lambda_{\alpha})$
while a Majorana mass term is ${m\over
2}\lambda_{\alpha}\lambda_{\beta}\epsilon^{\beta\alpha}+{m^{*}\over
2}\bar{\lambda}_{\dot{\alpha}}\epsilon^{\dot{\alpha}\dot{\beta}}\bar{\lambda}_{\dot{\beta}}$.
A Dirac mass term requires two Weyl fermions $\psi_{1,2}$ with the same
kinetic terms and mass term
$m\psi_{1\;\alpha}\psi_{2\;\beta}\epsilon^{\beta\alpha}+\text{c.c.}$. The
Dirac mass term is symmetric w.r.t. interchanging $1\leftrightarrow 2$ due to
anticommutativity of the spinors and the antisymmetry of $\epsilon$. In each
case, gauge invariance is imposed as required by the fermions’ representation.

Possible notational pitfalls:

  * •

The reader should be aware of the relabeling of the $S^{1}$ coordinate
$x^{4}\rightarrow x^{3}$ upon transition from Euclidean to Minkowski space.

  * •

Throughout, we use $Z_{2}^{(1)}$ to denote the 1-form $Z_{2}$ (center)
symmetry along the $S^{1}$ coordinate. Explicitly, this symmetry appears first
in eqn. (58) as acting on the fundamental Wilson line wrapping the $S^{1}$.
This symmetry is often called “0-form center symmetry,” a name justified from
the point of view of the compactified theory. We avoid this terminology and,
in the few occasions where we need to mention the 1-form symmetry in the
noncompact directions, this is explicitly stated.

##  Appendix B The massless adjoint fermion contribution to “GPY” potential.

Here we compute the fermion determinant needed to obtain the finite
$vL$-dependent part of (80). Introducing dimensionless variables ($\vec{q}$
and $x=vL$), we define the quantity $F_{\lambda}[x]$, proportional to our
formal expression for the determinant over the non-Cartan fermions
$\lambda^{\pm}$:

$\displaystyle F_{\lambda}[x]$ $\displaystyle\equiv$
$\displaystyle\sum\limits_{p\in
Z}\int{d^{3}q\over(2\pi)^{3}}\ln\left(\vec{q}^{\;2}+({2\pi
p}-x)^{2}\right)\leavevmode\nobreak\ .$ (172)

We next define the $\zeta$-function of the operator123123123In addition to
[59], zeta-function regularization is also discussed in the textbooks by
Birrell and Davies [175] and Schwartz [176] (in the latter, in the related
context of flat-space Casimir-energy-like problems). The interested reader can
also look up the original GPY calculation which did not use zeta-function
[52]. whose eigenvalues appear inside the logarithm above:

$\displaystyle\zeta_{\lambda}[s,x]$ $\displaystyle\equiv$
$\displaystyle\sum\limits_{p\in
Z}\int{d^{3}q\over(2\pi)^{3}}{1\over\left(\vec{q}^{\;2}+({2\pi
p}-x)^{2}\right)^{s}}\leavevmode\nobreak\ ,$ (173)

and note that the r.h.s. converges for $Re[s]$ large enough. For other values
of $s$, the function $\zeta_{\lambda}[s,x]$ is defined by analytic
continuation. In terms of $\zeta_{\lambda}[s]$ thus defined, we have

$F_{\lambda}[x]=-\lim\limits_{s\rightarrow 0}{d\over
ds}\zeta_{\lambda}[s,x]\leavevmode\nobreak\ .$ (174)

We compute $\zeta_{\lambda}[s,x]$ by first integrating over $d^{3}q$ to obtain
(below, $\lfloor a\rfloor$ denotes the largest integer smaller than $a$)

$\displaystyle\zeta_{\lambda}[s,x]={\Gamma[s-{3\over 2}]\over
8\pi^{3/2}\;\;\Gamma[s]\;(2\pi)^{2s-3}}\;\sum\limits_{p\in
Z}{1\over|p+\hat{x}|^{2s-3}}\leavevmode\nobreak\ ,\leavevmode\nobreak\
\text{where}\leavevmode\nobreak\ \leavevmode\nobreak\ \hat{x}\equiv{x\over
2\pi}-\bigg{\lfloor}{x\over 2\pi}\bigg{\rfloor}\in(0,1)\leavevmode\nobreak\ .$
(175)

Then, we separate the sums over $p\geq 0$ and $p<0$, shifting variables in the
latter sum, to obtain an expression for $\zeta_{\lambda}[s,x]$ in terms of the
incomplete $\zeta$ function:

$\displaystyle\zeta_{\lambda}[s,x]$ $\displaystyle=$
$\displaystyle{\Gamma[s-{3\over 2}]\over
8\pi^{3/2}\;\;\Gamma[s]\;(2\pi)^{2s-3}}\left(\sum\limits_{p\geq
0}{1\over(p+\hat{x})^{2s-3}}+\sum\limits_{p\geq
0}{1\over(p+1-\hat{x})^{2s-3}}\leavevmode\nobreak\ ,\right)$ (176)
$\displaystyle=$ $\displaystyle{\Gamma[s-{3\over 2}]\over
8\pi^{3/2}\;\Gamma[s]\;(2\pi)^{2s-3}}\left(\zeta(2s-3,\hat{x})+\zeta(2s-3,1-\hat{x})\right)$

Next, we realize that to obtain a nonzero $F_{\lambda}[x]$, the derivative in
(174) should act on $1/\Gamma[s]$ (otherwise the contribution vanishes due to
the pole of $\Gamma(s)$ at $s=0$). Then, using $\lim\limits_{s\rightarrow
0}{d\over ds}{1\over\Gamma[s]}=1$ gives the only nonvanishing contribution to
$F_{\lambda}[x]$, recalling that $[vL]\equiv vL\;(\text{mod}2\pi)$,

$\displaystyle F_{\lambda}[x]$ $\displaystyle=$ $\displaystyle-{4\pi^{2}\over
3}\left(\zeta(-3,\hat{x})+\zeta(-3,1-\hat{x})\right)=-{\pi^{2}\over
45}+{1\over 24\pi^{2}}[vL]^{2}(2\pi-[vL])^{2}\leavevmode\nobreak\ ,$ (177)

where to obtain the last equality we recalled that $x=vL$, the definition of
$\hat{x}$ in (175), and used the relation between the incomplete zeta-function
and the Bernoulli polynomials, $\zeta(-3,a)=-({d\over da}B_{5}(a))/20$, where
$B_{5}(a)=a^{5}-5a^{4}/2+5a^{3}/3-a/6$.

##  Appendix C The massive adjoint fermion contribution to “GPY” potential.

Here, in order to compute (85), we consider the massive generalization of
(172). We define, in terms of the dimensionless $x=vL$ and $\mu=mL$,

$F_{\lambda}[x,\mu]=\sum\limits_{p\in
Z}\int{d^{3}q\over(2\pi)^{3}}\ln\left(\vec{q}^{\;2}+({2\pi
p}-x)^{2}+\mu^{2}\right)\leavevmode\nobreak\ =-\lim\limits_{s\rightarrow
0}{d\over ds}\zeta_{\lambda}[s,x,\mu]\leavevmode\nobreak\ ,$ (178)

where we proceeded in complete analogy with (173) to define a zeta-function

$\displaystyle\zeta_{\lambda}[s,x,\mu]$ $\displaystyle\equiv$
$\displaystyle\sum\limits_{p\in
Z}\int{d^{3}q\over(2\pi)^{3}}{1\over\left(\vec{q}^{\;2}+({2\pi
p}-x)^{2}+\mu^{2}\right)^{s}}\leavevmode\nobreak\ $ (179) $\displaystyle=$
$\displaystyle{\Gamma(s-{3\over 2})\over
8\pi^{3/2}\Gamma(s)(2\pi)^{2s-3}}\sum\limits_{p\in
Z}{1\over((p-\hat{x})^{2}+({\mu\over 2\pi})^{2})^{s-3/2}}\leavevmode\nobreak\
.$

The sum over $p$ is now more complicated and is performed using the
identity124124124The derivation can be found in [177] (for easier access, note
that it is also reproduced in the Appendix of [178]).

$f[t;a,c]=\sum\limits_{n\in
Z}{1\over[(n+a)^{2}+c^{2}]^{t}}={\sqrt{\pi}|c|^{1-2t}\over\Gamma(t)}\left[\Gamma(t-{1\over
2})+4\sum\limits_{p=1}^{\infty}(\pi p|c|)^{t-{1\over 2}}\cos(2\pi
pa)K_{t-{1\over 2}}(2\pi p|c|)\right]\leavevmode\nobreak\ .$

The result is then substituted in (178) to obtain, after careful manipulations
similar to those in Appendix B, the desired form (85).

##  Appendix D A flavour of the $R^{3}\times S^{1}$ index theorem: $SU(2)$
spin-$j$ representations.

Here, we provide the reader with a flavour of the index theorem, without
considering the details of its derivation (see [102] and the Appendix of [43]
where Lie-algebraic notation valid for general gauge groups is used). In the
text, we avoided making explicit use of the index theorem and instead relied
on symmetry arguments to write terms involving fermions in the $\mu\ll 1/L$
EFT. Our main point here is simply to illustrate the fact that these arguments
are backed up by the index theorem—unfortunately too technical a subject to
present in detail.

The $R^{3}\times S^{1}$ index theorem for monopole-instantons in a vacuum with
a nonzero holonomy at infinity in $R^{3}$ (i.e. one which completely
abelianizies the gauge group, as everywhere in these notes) is a
generalization of the $R^{3}$ Callias index theorem [103] and has been derived
in the mathematical literature [101]. Using terms accessible to physicists,
the Callias index theorem was derived by E. Weinberg [179, 32]. Ref. [102]
generalized his derivation to $R^{3}\times S^{1}$ to obtain the result of
[101] in a physicist-friendly manner. Below, we only outline the main ideas.
Begin by considering the quantity

$I_{j}=\lim\limits_{\mu^{2}\rightarrow 0}{\text{tr}}{\mu^{2}\over
D^{\dagger}D+\mu^{2}}-{\text{tr}}{\mu^{2}\over
DD^{\dagger}+\mu^{2}}\leavevmode\nobreak\ ,$ (180)

where $D$ is the Weyl operator on $R^{3}\times S^{1}$ in the appropriate
“spin-$j$” representation of $SU(2)$. We explicitly showed this operator for
the $j=1/2$ fundamental in (159) and $j=1$ adjoint representations in (75).
The trace is taken over the space of two-component spinor functions the
operators $D^{\dagger}D$ and $DD^{\dagger}$ act on. Naturally, the Weyl
operator is taken in the monopole-instanton gauge background of interest.

The most relevant observation regarding (180) is that nonzero discrete
eigenvalues do not contribute to $I_{j}$. To see this, note that if $\Psi$ is
a normalizable eigenfunction of $D^{\dagger}D$ with nonzero eigenvalue,
$D^{\dagger}D\Psi=\lambda\Psi$, then $D\Psi$, also normalizable, obeys
$DD^{\dagger}(D\Psi)=\lambda(D\Psi)$, i.e. is an eigenfunction with the same
eigenvalue. Thus, the $\Psi$ and $D\Psi$ respective contributions to the first
and second term in (180) cancel out already for finite $\mu^{2}$. The detailed
analysis of [179], which also holds on $R^{3}\times S^{1}$, showed that the
continuous spectrum also does not contribute to (180), provided the vev of the
holonomy is not at the edges of the Weyl chamber—hence the need for
abelianization of the gauge group. Accepting this, the conclusion is that only
zero eigenvalues of $D^{\dagger}D$ and $DD^{\dagger}$ contribute to (180).
Thus, $I_{j}$ counts the number of zero modes of $D^{\dagger}D$ minus the
number of zero modes of $DD^{\dagger}$. For the $M$ and $KK$ backgrounds,
$I_{j}$ counts the number of zero modes of the Weyl equation, the normalizable
solutions of $D\Psi=0$.

The quantity $I_{j}$ can be seen to depend only on the topological properties
of the gauge field background: on the topological charge, the magnetic charge,
and the value of the holonomy at infinity (this follows from the explicit
calculation, requiring some regularization—for which we refer the reader to
the references). For the particular $M$ and $KK$ monopole-instanton
backgrounds of interest to us, the index (180) is found to be, for fermions
taken periodic on the $S^{1}$:

$I_{j}^{M}=\sum\limits_{m=-j}^{j}2m\bigg{\lfloor}-{mvL\over
2\pi}\bigg{\rfloor}\leavevmode\nobreak\ ,\leavevmode\nobreak\
\leavevmode\nobreak\ I_{j}^{KK}={2j(j+1)(2j+1)\over
3}-I_{j}^{M}\leavevmode\nobreak\ .$ (181)

To unpack this formula, notice that for the fundamental and the adjoint, the
two cases studied in this paper, we find $I_{1/2}^{M}$ = $-\lfloor{-{vL\over
4\pi}}\rfloor+\lfloor{{vL\over 4\pi}}\rfloor$ and $I_{1}^{M}$
=$-2\lfloor{-{vL\over 2\pi}}\rfloor+2\lfloor{{vL\over 2\pi}}\rfloor$. As an
example of using (181), let us now take $vL=\pi$, the center symmetric point.
We immediately obtain from the above formulae $I_{1/2}^{M}=1$ (hence
$I_{1/2}^{KK}=0$) and $I_{1}^{M}=2$ (hence $I_{1}^{KK}=2$), exactly as found
in the text by other means. Recall that we used center symmetry of the EFT as
well as the shift of the dual photon under the chiral symmetry.

One can also find expressions for fermions antiperiodic on the $S^{1}$, by
turning on an appropriate $U(1)_{V}$ Wilson line. For the case of the
fundamental representation relevant for Section 7, one finds $I_{1/2}^{M}=0$
(hence $I_{1/2}^{KK}=1$), i.e. upon changing the $U(1)_{V}$ background to dial
the boundary condition from periodic to an antiperiodic, the fundamental
fermion zero mode jumps from the $M$ to the $KK$ monopole instanton, as we
already found in our discussion of QCD(F). A discussion of this jump and other
jumps (seen to occur upon changing $vL$ in (181)) is given in [102].

## References

  * [1] N. Seiberg, The Power of holomorphy: Exact results in 4-D SUSY field theories, in Particles, Strings, and Cosmology (PASCOS 94), 5, 1994\.  hep-th/9408013.
  * [2] J. Greensite, An introduction to the confinement problem, Lect. Notes Phys. 821 (2011) 1–211.
  * [3] O. Aharony, S. S. Gubser, J. M. Maldacena, H. Ooguri, and Y. Oz, Large N field theories, string theory and gravity, Phys. Rept. 323 (2000) 183–386, [hep-th/9905111].
  * [4] N. Seiberg and E. Witten, Electric - magnetic duality, monopole condensation, and confinement in N=2 supersymmetric Yang-Mills theory, Nucl. Phys. B 426 (1994) 19–52, [hep-th/9407087]. [Erratum: Nucl.Phys.B 430, 485–486 (1994)].
  * [5] A. Gorsky, M. Shifman, and A. Yung, N = 1 supersymmetric quantum chromodynamics: How confined non-Abelian monopoles emerge from quark condensation, Phys. Rev. D 75 (2007) 065032, [hep-th/0701040].
  * [6] J. D. Bjorken, Elements of Quantum Chromodynamics, Prog. Math. Phys. 4 (12, 1979) 423–561.
  * [7] M. Luscher, Some Analytic Results Concerning the Mass Spectrum of Yang-Mills Gauge Theories on a Torus, Nucl. Phys. B 219 (1983) 233–261.
  * [8] P. van Baal, QCD in a finite volume, hep-ph/0008206.
  * [9] K.-M. Lee and P. Yi, Monopoles and instantons on partially compactified D-branes, Phys. Rev. D 56 (1997) 3711–3717, [hep-th/9702107].
  * [10] T. C. Kraan and P. van Baal, Periodic instantons with nontrivial holonomy, Nucl. Phys. B 533 (1998) 627–659, [hep-th/9805168].
  * [11] M. Unsal, Magnetic bion condensation: A New mechanism of confinement and mass gap in four dimensions, Phys. Rev. D 80 (2009) 065001, [arXiv:0709.3269].
  * [12] M. Unsal, Abelian duality, confinement, and chiral symmetry breaking in QCD(adj), Phys. Rev. Lett. 100 (2008) 032005, [arXiv:0708.1772].
  * [13] M. Unsal and L. G. Yaffe, Center-stabilized Yang-Mills theory: Confinement and large N volume independence, Phys. Rev. D 78 (2008) 065035, [arXiv:0803.0344].
  * [14] M. Shifman and M. Unsal, QCD-like Theories on R(3) x S(1): A Smooth Journey from Small to Large r(S(1)) with Double-Trace Deformations, Phys. Rev. D 78 (2008) 065004, [arXiv:0802.1232].
  * [15] D. Gaiotto, A. Kapustin, N. Seiberg, and B. Willett, Generalized Global Symmetries, JHEP 02 (2015) 172, [arXiv:1412.5148].
  * [16] D. Gaiotto, A. Kapustin, Z. Komargodski, and N. Seiberg, Theta, Time Reversal, and Temperature, JHEP 05 (2017) 091, [arXiv:1703.00501].
  * [17] D. Gaiotto, Z. Komargodski, and N. Seiberg, Time-reversal breaking in QCD4, walls, and dualities in 2 + 1 dimensions, JHEP 01 (2018) 110, [arXiv:1708.06806].
  * [18] E. Poppitz and M. Unsal, Seiberg-Witten and ’Polyakov-like’ magnetic bion confinements are continuously connected, JHEP 07 (2011) 082, [arXiv:1105.3969].
  * [19] P. C. Argyres and M. Unsal, The semi-classical expansion and resurgence in gauge theories: new perturbative, instanton, bion, and renormalon effects, JHEP 08 (2012) 063, [arXiv:1206.1890].
  * [20] M. Unsal, Theta dependence, sign problems and topological interference, Phys. Rev. D 86 (2012) 105012, [arXiv:1201.6426].
  * [21] E. Poppitz, T. Schäfer, and M. Unsal, Continuity, Deconfinement, and (Super) Yang-Mills Theory, JHEP 10 (2012) 115, [arXiv:1205.0290].
  * [22] G. V. Dunne and M. Ünsal, New Nonperturbative Methods in Quantum Field Theory: From Large-N Orbifold Equivalence to Bions and Resurgence, Ann. Rev. Nucl. Part. Sci. 66 (2016) 245–272, [arXiv:1601.03414].
  * [23] A. Cherman and E. Poppitz, Emergent dimensions and branes from large-$N$ confinement, Phys. Rev. D 94 (2016), no. 12 125008, [arXiv:1606.01902].
  * [24] A. M. Polyakov, Quark Confinement and Topology of Gauge Groups, Nucl. Phys. B 120 (1977) 429–458.
  * [25] A. M. Polyakov, Gauge Fields and Strings, vol. 3.  1987\.
  * [26] P. Deligne, P. Etingof, D. S. Freed, L. C. Jeffrey, D. Kazhdan, J. W. Morgan, D. R. Morrison, and E. Witten, eds., Quantum fields and strings: A course for mathematicians. Vol. 1, 2.  1999\.
  * [27] M. Shifman, Advanced topics in quantum field theory.: A lecture course.  Cambridge Univ. Press, Cambridge, UK, 2, 2012.
  * [28] T. Banks, Modern Quantum Field Theory: A Concise Introduction.  Cambridge University Press, 12, 2014.
  * [29] A. V. Smilga and A. Vainshtein, Background field calculations and nonrenormalization theorems in 4-D supersymmetric gauge theories and their low-dimensional descendants, Nucl. Phys. B 704 (2005) 445–474, [hep-th/0405142].
  * [30] M. M. Anber and T. Sulejmanpasic, The renormalon diagram in gauge theories on ${\mathrm{\mathbb{R}}}3\times{\mathbb{S}}1$, JHEP 01 (2015) 139, [arXiv:1410.0121].
  * [31] J. A. Harvey, Magnetic monopoles, duality and supersymmetry, in ICTP Summer School in High-energy Physics and Cosmology, 3, 1996.  hep-th/9603086.
  * [32] E. J. Weinberg, Classical solutions in quantum field theory: Solitons and Instantons in High Energy Physics.  Cambridge Monographs on Mathematical Physics. Cambridge University Press, 9, 2012.
  * [33] M. M. Anber and E. Poppitz, Microscopic Structure of Magnetic Bions, JHEP 06 (2011) 136, [arXiv:1105.0940].
  * [34] A. I. Vainshtein, V. I. Zakharov, V. A. Novikov, and M. A. Shifman, ABC’s of Instantons, Sov. Phys. Usp. 25 (1982) 195.
  * [35] S. Coleman, Aspects of Symmetry: Selected Erice Lectures.  Cambridge University Press, Cambridge, U.K., 1985.
  * [36] T. Schäfer and E. V. Shuryak, Instantons in QCD, Rev. Mod. Phys. 70 (1998) 323–426, [hep-ph/9610451].
  * [37] C. Turner, Dualities in 2+1 Dimensions, PoS Modave2018 (2019) 001, [arXiv:1905.12656].
  * [38] A. M. Polyakov, Compact Gauge Fields and the Infrared Catastrophe, Phys. Lett. B 59 (1975) 82–84.
  * [39] I. I. Kogan and A. Kovner, Monopoles, vortices and strings: Confinement and deconfinement in (2+1)-dimensions at weak coupling, hep-th/0205026.
  * [40] E. Poppitz and M. E. Shalchian T., String tensions in deformed Yang-Mills theory, JHEP 01 (2018) 029, [arXiv:1708.08821].
  * [41] G. ’t Hooft, Computation of the Quantum Effects Due to a Four-Dimensional Pseudoparticle, Phys. Rev. D 14 (1976) 3432–3450. [Erratum: Phys.Rev.D 18, 2199 (1978)].
  * [42] G. ’t Hooft, Symmetry Breaking Through Bell-Jackiw Anomalies, Phys. Rev. Lett. 37 (1976) 8–11.
  * [43] M. M. Anber, E. Poppitz, and B. Teeple, Deconfinement and continuity between thermal and (super) Yang-Mills theory for all gauge groups, JHEP 09 (2014) 040, [arXiv:1406.1199].
  * [44] C. Pazarbaşı and M. Ünsal, Cluster expansion and resurgence in Polyakov model, arXiv:2110.05612.
  * [45] M. E. Peskin and D. V. Schroeder, An Introduction to quantum field theory.  Addison-Wesley, Reading, USA, 1995.
  * [46] C. Bachas, Convexity of the Quarkonium Potential, Phys. Rev. D 33 (1986) 2723.
  * [47] E. H. Fradkin and S. H. Shenker, Phase Diagrams of Lattice Gauge Theories with Higgs Fields, Phys. Rev. D 19 (1979) 3682–3697.
  * [48] A. A. Cox, E. Poppitz, and S. S. Y. Wong, Domain walls and deconfinement: a semiclassical picture of discrete anomaly inflow, JHEP 12 (2019) 011, [arXiv:1909.10979].
  * [49] M. W. Bub, E. Poppitz, and S. S. Y. Wong, Confinement on $\mathbb{R}^{3}\times\mathbb{S}^{1}$ and double-string collapse, JHEP 01 (2021) 044, [arXiv:2010.04330].
  * [50] M. M. Anber, The abelian confinement mechanism revisited: new aspects of the Georgi-Glashow model, Annals Phys. 341 (2014) 21–55, [arXiv:1308.0027].
  * [51] A. Cherman, T. Jacobson, S. Sen, and L. G. Yaffe, Higgs-confinement phase transitions with fundamental representation matter, Phys. Rev. D 102 (2020), no. 10 105021, [arXiv:2007.08539].
  * [52] D. J. Gross, R. D. Pisarski, and L. G. Yaffe, QCD and Instantons at Finite Temperature, Rev. Mod. Phys. 53 (1981) 43.
  * [53] K.-M. Lee and C.-h. Lu, SU(2) calorons and magnetic monopoles, Phys. Rev. D 58 (1998) 025011, [hep-th/9802108].
  * [54] G. ’t Hooft, Aspects of Quark Confinement, Phys. Scripta 24 (1981) 841–846.
  * [55] J. Lai, to appear, 2021, .
  * [56] M. M. Anber and E. Poppitz, On the global structure of deformed Yang-Mills theory and QCD(adj) on ${\mathrm{\mathbb{R}}}^{3}\times{\mathbb{S}}^{1}$, JHEP 10 (2015) 051, [arXiv:1508.00910].
  * [57] K. Aitken, A. Cherman, E. Poppitz, and L. G. Yaffe, QCD on a small circle, Phys. Rev. D 96 (2017), no. 9 096022, [arXiv:1707.08971].
  * [58] C. P. Korthals Altes, Constrained effective potential in hot QCD, Nucl. Phys. B 420 (1994) 637–668, [hep-th/9310195].
  * [59] S. W. Hawking, Zeta Function Regularization of Path Integrals in Curved Space-Time, Commun. Math. Phys. 55 (1977) 133.
  * [60] S. Weinberg, The Quantum theory of fields. Vol. 1: Foundations.  Cambridge University Press, 6, 2005.
  * [61] J. Zinn-Justin, Quantum field theory and critical phenomena, Int. Ser. Monogr. Phys. 113 (2002) 1–1054.
  * [62] K. Aitken, A. Cherman, and M. Ünsal, Vacuum structure of Yang-Mills theory as a function of $\theta$, JHEP 09 (2018) 030, [arXiv:1804.06848].
  * [63] J. C. Myers and M. C. Ogilvie, New phases of SU(3) and SU(4) at finite temperature, Phys. Rev. D 77 (2008) 125030, [arXiv:0707.1869].
  * [64] J. C. Myers and M. C. Ogilvie, Phase diagrams of SU(N) gauge theories with fermions in various representations, JHEP 07 (2009) 095, [arXiv:0903.4638].
  * [65] C. Bonati, M. Cardinali, and M. D’Elia, $\theta$ dependence in trace deformed $SU(3)$ Yang-Mills theory: a lattice study, Phys. Rev. D 98 (2018), no. 5 054508, [arXiv:1807.06558].
  * [66] C. Bonati, M. Cardinali, M. D’Elia, and F. Mazziotti, $\theta$-dependence and center symmetry in Yang-Mills theories, Phys. Rev. D 101 (2020), no. 3 034508, [arXiv:1912.02662].
  * [67] A. Bhoonah, E. Thomas, and A. R. Zhitnitsky, Metastable vacuum decay and $\theta$ dependence in gauge theory. Deformed QCD as a toy model, Nucl. Phys. B 890 (2014) 30–47, [arXiv:1407.5121].
  * [68] M. M. Anber and A. R. Zhitnitsky, Oblique Confinement at $\theta\neq 0$ in weakly coupled gauge theories with deformations, Phys. Rev. D 96 (2017), no. 7 074022, [arXiv:1708.07520].
  * [69] K. Aitken, A. Cherman, and M. Ünsal, Dihedral symmetry in $SU(N)$ Yang-Mills theory, Phys. Rev. D 100 (2019), no. 8 085004, [arXiv:1804.05845].
  * [70] L. Del Debbio, G. M. Manca, H. Panagopoulos, A. Skouroupathis, and E. Vicari, Theta-dependence of the spectrum of SU(N) gauge theories, JHEP 06 (2006) 005, [hep-th/0603041].
  * [71] G. ’t Hooft, A Planar Diagram Theory for Strong Interactions, Nucl. Phys. B 72 (1974) 461.
  * [72] E. Witten, Current Algebra Theorems for the U(1) Goldstone Boson, Nucl. Phys. B 156 (1979) 269–283.
  * [73] E. Witten, Large N Chiral Dynamics, Annals Phys. 128 (1980) 363.
  * [74] A. Athenodorou, M. Cardinali, and M. D’Elia, Spectrum of Trace Deformed Yang-Mills Theories, arXiv:2010.03618.
  * [75] G. ’t Hooft, A Property of Electric and Magnetic Flux in Nonabelian Gauge Theories, Nucl. Phys. B 153 (1979) 141–160.
  * [76] A. A. Cox, E. Poppitz, and F. D. Wandler, The mixed 0-form/1-form anomaly in Hilbert space: pouring the new wine into old bottles, JHEP 10 (2021) 069, [arXiv:2106.11442].
  * [77] R. Kitano, R. Matsudo, N. Yamada, and M. Yamazaki, Peeking into the $\theta$ vacuum, arXiv:2102.08784.
  * [78] M. M. Anber, E. Poppitz, and T. Sulejmanpasic, Strings from domain walls in supersymmetric Yang-Mills theory and adjoint QCD, Phys. Rev. D 92 (2015), no. 2 021701, [arXiv:1501.06773].
  * [79] C. G. Callan, Jr. and J. A. Harvey, Anomalies and Fermion Zero Modes on Strings and Domain Walls, Nucl. Phys. B 250 (1985) 427–436.
  * [80] Z. Komargodski, T. Sulejmanpasic, and M. Ünsal, Walls, anomalies, and deconfinement in quantum antiferromagnets, Phys. Rev. B 97 (2018), no. 5 054418, [arXiv:1706.05731].
  * [81] P.-S. Hsin, H. T. Lam, and N. Seiberg, Comments on One-Form Global Symmetries and Their Gauging in 3d and 4d, SciPost Phys. 6 (2019), no. 3 039, [arXiv:1812.04716].
  * [82] E. Witten, Branes and the dynamics of QCD, Nucl. Phys. B507 (1997) 658–690, [hep-th/9706109].
  * [83] S.-J. Rey, unpublished, 1997, .
  * [84] A. Armoni and M. Shifman, On k string tensions and domain walls in N=1 gluodynamics, Nucl. Phys. B 664 (2003) 233–246, [hep-th/0304127].
  * [85] G. V. Dunne, I. I. Kogan, A. Kovner, and B. Tekin, Deconfining phase transition in (2+1)-dimensions: The Georgi-Glashow model, JHEP 01 (2001) 032, [hep-th/0010201].
  * [86] Y. V. Kovchegov and D. T. Son, Critical temperature of the deconfining phase transition in (2+1)-d Georgi-Glashow model, JHEP 01 (2003) 050, [hep-th/0212230].
  * [87] D. Simic and M. Unsal, Deconfinement in Yang-Mills theory through toroidal compactification with deformation, Phys. Rev. D 85 (2012) 105027, [arXiv:1010.5515].
  * [88] M. M. Anber, E. Poppitz, and M. Unsal, 2d affine XY-spin model/4d gauge theory duality and deconfinement, JHEP 04 (2012) 040, [arXiv:1112.6389].
  * [89] M. M. Anber, S. Collier, and E. Poppitz, The SU(3)/$\mathbb{Z}_{3}$ QCD(adj) deconfinement transition via the gauge theory/’affine’ XY-model duality, JHEP 01 (2013) 126, [arXiv:1211.2824].
  * [90] M. M. Anber, S. Collier, E. Poppitz, S. Strimas-Mackey, and B. Teeple, Deconfinement in $\mathcal{N}=1$ super Yang-Mills theory on $\mathbb{R}^{3}\times\mathbb{S}^{1}$ via dual-Coulomb gas and ”affine” XY-model, JHEP 11 (2013) 142, [arXiv:1310.3522].
  * [91] B. Svetitsky and L. G. Yaffe, Critical Behavior at Finite Temperature Confinement Transitions, Nucl. Phys. B 210 (1982) 423–447.
  * [92] L. G. Yaffe and B. Svetitsky, First Order Phase Transition in the SU(3) Gauge Theory at Finite Temperature, Phys. Rev. D 26 (1982) 963\.
  * [93] H. Shimizu and K. Yonekura, Anomaly constraints on deconfinement and chiral phase transition, Phys. Rev. D 97 (2018), no. 10 105011, [arXiv:1706.06104].
  * [94] D. R. Nelson, Study of melting in two dimensions, Phys. Rev. B 18 (178) 2318–2338.
  * [95] S. Catterall, J. Giedt, F. Sannino, and J. Schneible, Phase diagram of SU(2) with 2 flavors of dynamical adjoint quarks, JHEP 11 (2008) 009, [arXiv:0807.0792].
  * [96] A. J. Hietanen, J. Rantaharju, K. Rummukainen, and K. Tuominen, Spectrum of SU(2) lattice gauge theory with two adjoint Dirac flavours, JHEP 05 (2009) 025, [arXiv:0812.1467].
  * [97] L. Del Debbio, B. Lucini, A. Patella, C. Pica, and A. Rago, Conformal versus confining scenario in SU(2) with adjoint fermions, Phys. Rev. D 80 (2009) 074507, [arXiv:0907.3896].
  * [98] A. Athenodorou, Bennett, G. Bergner, and B. Lucini, Investigating the conformal behaviour of SU(2) with one adjoint Dirac flavor, arXiv:2103.10485.
  * [99] G. Bergner, S. Piemonte, and M. Ünsal, Adiabatic continuity and confinement in supersymmetric Yang-Mills theory on the lattice, JHEP 11 (2018) 092, [arXiv:1806.10894].
  * [100] P. van Baal, Some Results for SU(N) Gauge Fields on the Hypertorus, Commun. Math. Phys. 85 (1982) 529.
  * [101] T. M. W. Nye and M. A. Singer, An L**2 index theorem for Dirac operators on S**1 x R**3, math/0009144.
  * [102] E. Poppitz and M. Unsal, Index theorem for topological excitations on R**3 x S**1 and Chern-Simons theory, JHEP 03 (2009) 027, [arXiv:0812.2085].
  * [103] C. Callias, Index Theorems on Open Spaces, Commun. Math. Phys. 62 (1978) 213–234.
  * [104] N. M. Davies, T. J. Hollowood, V. V. Khoze, and M. P. Mattis, Gluino condensate and magnetic monopoles in supersymmetric gluodynamics, Nucl. Phys. B 559 (1999) 123–142, [hep-th/9905015].
  * [105] E. Poppitz and F. D. Wandler, Topological terms and anomaly matching in effective field theories on $\mathbb{R}^{3}\times\mathbb{S}^{1}$:. Part I. Abelian symmetries and intermediate scales, JHEP 01 (2021) 091, [arXiv:2009.14667].
  * [106] E. Poppitz and M. Unsal, Conformality or confinement: (IR)relevance of topological excitations, JHEP 09 (2009) 050, [arXiv:0906.5156].
  * [107] C. Córdova and T. T. Dumitrescu, Candidate Phases for SU(2) Adjoint QCD4 with Two Flavors from $\mathcal{N}=2$ Supersymmetric Yang-Mills Theory, arXiv:1806.09592.
  * [108] Y. Nambu and G. Jona-Lasinio, Dynamical Model of Elementary Particles Based on an Analogy with Superconductivity. 1., Phys. Rev. 122 (1961) 345–358.
  * [109] Y. Nambu and G. Jona-Lasinio, Dynamical model of elementary particles based on an analogy with superconductivity, II, Phys. Rev. 124 (1961) 246–254.
  * [110] C. T. Hill and E. H. Simmons, Strong Dynamics and Electroweak Symmetry Breaking, Phys. Rept. 381 (2003) 235–402, [hep-ph/0203079]. [Erratum: Phys.Rept. 390, 553–554 (2004)].
  * [111] M. M. Anber and E. Poppitz, Two-flavor adjoint QCD, Phys. Rev. D 98 (2018), no. 3 034026, [arXiv:1805.12290].
  * [112] E. Poppitz and T. A. Ryttov, Possible new phase for adjoint QCD, Phys. Rev. D 100 (2019), no. 9 091901, [arXiv:1904.11640].
  * [113] C. Córdova and K. Ohmori, Anomaly Obstructions to Symmetry Preserving Gapped Phases, arXiv:1910.04962.
  * [114] C. Córdova and K. Ohmori, Anomaly Constraints on Gapped Phases with Discrete Chiral Symmetry, Phys. Rev. D 102 (2020), no. 2 025011, [arXiv:1912.13069].
  * [115] M. M. Anber and E. Poppitz, On the baryon-color-flavor (BCF) anomaly in vector-like theories, JHEP 11 (2019) 063, [arXiv:1909.09027].
  * [116] M. M. Anber and E. Poppitz, Generalized ’t Hooft anomalies on non-spin manifolds, JHEP 04 (2020) 097, [arXiv:2002.02037].
  * [117] N. S. Manton and P. Sutcliffe, Topological solitons.  Cambridge Monographs on Mathematical Physics. Cambridge University Press, 2004.
  * [118] E. Witten, Constraints on Supersymmetry Breaking, Nucl. Phys. B 202 (1982) 253.
  * [119] K. Hori, S. Katz, A. Klemm, R. Pandharipande, R. Thomas, C. Vafa, R. Vakil, and E. Zaslow, Mirror symmetry, vol. 1 of Clay mathematics monographs.  AMS, Providence, USA, 2003.
  * [120] M. M. Anber and E. Poppitz, New nonperturbative scales and glueballs in confining supersymmetric gauge theories, JHEP 03 (2018) 052, [arXiv:1711.00027].
  * [121] N. Seiberg and E. Witten, Gauge dynamics and compactification to three-dimensions, in Conference on the Mathematical Beauty of Physics (In Memory of C. Itzykson), pp. 333–366, 6, 1996.  hep-th/9607163.
  * [122] N. M. Davies, T. J. Hollowood, and V. V. Khoze, Monopoles, affine algebras and the gluino condensate, J. Math. Phys. 44 (2003) 3640–3656, [hep-th/0006011].
  * [123] E. B. Bogomolny, Calculation of instanton-anti-instanton contributions in quantum mechanics, Phys. Lett. B 91 (1980) 431–435.
  * [124] J. Zinn-Justin, Multi - Instanton Contributions in Quantum Mechanics, Nucl. Phys. B 192 (1981) 125–140.
  * [125] M. Ünsal, TQFT at work for IR-renormalons, resurgence and Lefschetz decomposition, arXiv:2106.14971.
  * [126] A. Behtash, G. V. Dunne, T. Schäfer, T. Sulejmanpasic, and M. Ünsal, Complexified path integrals, exact saddles and supersymmetry, Phys. Rev. Lett. 116 (2016), no. 1 011601, [arXiv:1510.00978].
  * [127] A. Behtash, E. Poppitz, T. Sulejmanpasic, and M. Ünsal, The curious incident of multi-instantons and the necessity of Lefschetz thimbles, JHEP 11 (2015) 175, [arXiv:1507.04063].
  * [128] A. Behtash, G. V. Dunne, T. Schäfer, T. Sulejmanpasic, and M. Ünsal, Toward Picard–Lefschetz theory of path integrals, complex saddles and resurgence, Ann. Math. Sci. Appl. 02 (2017) 95–212, [arXiv:1510.03435].
  * [129] A. Behtash, G. V. Dunne, T. Schaefer, T. Sulejmanpasic, and M. Ünsal, Critical Points at Infinity, Non-Gaussian Saddles, and Bions, JHEP 06 (2018) 068, [arXiv:1803.11533].
  * [130] M. Ünsal, Strongly coupled QFT dynamics via TQFT coupling, arXiv:2007.03880.
  * [131] T. Fujimori, M. Honda, S. Kamata, T. Misumi, N. Sakai, and T. Yoda, Quantum phase transition and Resurgence: Lessons from 3d $\mathcal{N}=4$ SQED, arXiv:2103.13654.
  * [132] L. Di Pietro, M. Mariño, G. Sberveglieri, and M. Serone, Resurgence and $1/N$ Expansion in Integrable Field Theories, arXiv:2108.02647.
  * [133] A. Behtash, T. Sulejmanpasic, T. Schäfer, and M. Ünsal, Hidden topological angles and Lefschetz thimbles, Phys. Rev. Lett. 115 (2015), no. 4 041601, [arXiv:1502.06624].
  * [134] E. Poppitz, T. Schäfer, and M. Ünsal, Universal mechanism of (semi-classical) deconfinement and theta-dependence for all simple groups, JHEP 03 (2013) 087, [arXiv:1212.1238].
  * [135] H. Murayama, Some Exact Results in QCD-like Theories, Phys. Rev. Lett. 126 (2021), no. 25 251601, [arXiv:2104.01179].
  * [136] K. Holland, M. Pepe, and U. J. Wiese, The Deconfinement phase transition of Sp(2) and Sp(3) Yang-Mills theories in (2+1)-dimensions and (3+1)-dimensions, Nucl. Phys. B 694 (2004) 35–58, [hep-lat/0312022].
  * [137] M. Pepe and U. J. Wiese, Exceptional Deconfinement in G(2) Gauge Theory, Nucl. Phys. B 768 (2007) 21–37, [hep-lat/0610076].
  * [138] M. D’Elia and F. Negro, $\theta$ dependence of the deconfinement temperature in Yang-Mills theories, Phys. Rev. Lett. 109 (2012) 072001, [arXiv:1205.0538].
  * [139] M. D’Elia and F. Negro, Phase diagram of Yang-Mills theories in the presence of a $\theta$ term, Phys. Rev. D 88 (2013), no. 3 034503, [arXiv:1306.2919].
  * [140] E. Shuryak and T. Sulejmanpasic, Holonomy potential and confinement from a simple model of the gauge topology, Phys. Lett. B 726 (2013) 257–261, [arXiv:1305.0796].
  * [141] D. DeMartini and E. Shuryak, Chiral Symmetry Breaking and Confinement from an Interacting Ensemble of Instanton-dyons in Two-flavor Massless QCD, arXiv:2108.06353.
  * [142] E. Shuryak, Lectures on nonperturbative QCD ( Nonperturbative Topological Phenomena in QCD and Related Theories), arXiv:1812.01509.
  * [143] E. Shuryak, Nonperturbative Topological Phenomena in QCD and Related Theories, vol. 977 of Lecture Notes in Physics.  3, 2021.
  * [144] S. Chen, K. Fukushima, H. Nishimura, and Y. Tanizaki, Deconfinement and $\mathcal{CP}$ breaking at $\theta=\pi$ in Yang-Mills theories and a novel phase for SU(2), Phys. Rev. D 102 (2020), no. 3 034020, [arXiv:2006.01487].
  * [145] T. Iritani, E. Itou, and T. Misumi, Lattice study on QCD-like theory with exact center symmetry, JHEP 11 (2015) 159, [arXiv:1508.07132].
  * [146] A. Cherman, T. Schäfer, and M. Ünsal, Chiral Lagrangian from Duality and Monopole Operators in Compactified QCD, Phys. Rev. Lett. 117 (2016), no. 8 081601, [arXiv:1604.06108].
  * [147] A. Cherman, S. Sen, M. Unsal, M. L. Wagman, and L. G. Yaffe, Order parameters and color-flavor center symmetry in QCD, Phys. Rev. Lett. 119 (2017), no. 22 222001, [arXiv:1706.05385].
  * [148] I. Affleck, J. A. Harvey, and E. Witten, Instantons and (Super)Symmetry Breaking in (2+1)-Dimensions, Nucl. Phys. B 206 (1982) 413–439.
  * [149] E. Poppitz and M. Unsal, Conformality or confinement (II): One-flavor CFTs and mixed-representation QCD, JHEP 12 (2009) 011, [arXiv:0910.1245].
  * [150] G. V. Dunne, Y. Tanizaki, and M. Ünsal, Quantum Distillation of Hilbert Spaces, Semi-classics and Anomaly Matching, JHEP 08 (2018) 068, [arXiv:1803.02430].
  * [151] T. Kanazawa and M. Ünsal, Quantum distillation in QCD, Phys. Rev. D 102 (2020), no. 3 034013, [arXiv:1909.05222].
  * [152] M. Ünsal, Graded Hilbert spaces, quantum distillation and connecting SQCD to QCD, arXiv:2104.12352.
  * [153] T. Sulejmanpasic, Global Symmetries, Volume Independence, and Continuity in Quantum Field Theories, Phys. Rev. Lett. 118 (2017), no. 1 011601, [arXiv:1610.04009].
  * [154] E. Witten, An SU(2) Anomaly, Phys. Lett. B 117 (1982) 324–328.
  * [155] E. Poppitz and T. Sulejmanpasic, (S)QCD on $\mathbb{R}^{3}\times\mathbb{S}^{1}$: Screening of Polyakov loop by fundamental quarks and the demise of semi-classics, JHEP 09 (2013) 128, [arXiv:1307.1317].
  * [156] V. Drach, T. Janowski, and C. Pica, Update on SU(2) gauge theory with NF = 2 fundamental flavours, EPJ Web Conf. 175 (2018) 08020, [arXiv:1710.07218].
  * [157] J. Davighi and B. Gripaios, Topological terms in Composite Higgs Models, JHEP 11 (2018) 169, [arXiv:1808.04154].
  * [158] A. Kovner, B. Rosenstein, and D. Eliezer, Photon as Goldstone boson in (2+1)-dimensional Higgs model, Mod. Phys. Lett. A 5 (1990) 2733–2740.
  * [159] A. Kovner, B. Rosenstein, and D. Eliezer, Photon as a Goldstone boson in (2+1)-dimensional Abelian gauge theories, Nucl. Phys. B 350 (1991) 325–354.
  * [160] A. Kovner and B. Rosenstein, New look at QED in four-dimensions: The Photon as a Goldstone boson and the topological interpretation of electric charge, Phys. Rev. D 49 (1994) 5571–5581, [hep-th/9210154].
  * [161] M. M. Anber and V. Pellizzani, Representation dependence of k -strings in pure Yang-Mills theory via supersymmetry, Phys. Rev. D 96 (2017), no. 11 114015, [arXiv:1710.06509].
  * [162] P. Ramond, Group theory: A physicist’s survey.  2010\.
  * [163] M. M. Anber and E. Poppitz, Anomaly matching, (axial) Schwinger models, and high-T super Yang-Mills domain walls, JHEP 09 (2018) 076, [arXiv:1807.00093].
  * [164] M. M. Anber and E. Poppitz, Domain walls in high-T SU(N) super Yang-Mills theory and QCD(adj), JHEP 05 (2019) 151, [arXiv:1811.10642].
  * [165] Y. Tanizaki and M. Ünsal, Modified instanton sum in QCD and higher-groups, JHEP 03 (2020) 123, [arXiv:1912.01033].
  * [166] M. M. Anber and E. Poppitz, Deconfinement on axion domain walls, JHEP 03 (2020) 124, [arXiv:2001.03631].
  * [167] M. Shifman and M. Unsal, On Yang-Mills Theories with Chiral Matter at Strong Coupling, Phys. Rev. D 79 (2009) 105010, [arXiv:0808.2485].
  * [168] E. Poppitz and M. Unsal, Chiral gauge dynamics and dynamical supersymmetry breaking, JHEP 07 (2009) 060, [arXiv:0905.0634].
  * [169] A. Arabi Ardehali, L. Cassia, and Y. Lü, From Exact Results to Gauge Dynamics on $\mathbb{R}^{3}\times S^{1}$, JHEP 08 (2020) 053, [arXiv:1912.02732].
  * [170] M. M. Anber and L. Vincent-Genod, Classification of compactified $su(N_{c})$ gauge theories with fermions in all representations, JHEP 12 (2017) 028, [arXiv:1704.08277].
  * [171] M. M. Anber, Self-conjugate QCD, JHEP 10 (2019) 042, [arXiv:1906.10315].
  * [172] M. M. Anber, Condensates and anomaly cascade in vector-like theories, JHEP 03 (2021) 191, [arXiv:2101.04132].
  * [173] S. Golkar, Conformal windows of $SP(2N)$ and $SO(N)$ gauge theories from topological excitations on $R^{3}\times S^{1}$, JHEP 11 (2009) 076, [arXiv:0909.2838].
  * [174] H. K. Dreiner, H. E. Haber, and S. P. Martin, Two-component spinor techniques and Feynman rules for quantum field theory and supersymmetry, Phys. Rept. 494 (2010) 1–196, [arXiv:0812.1594].
  * [175] N. D. Birrell and P. C. W. Davies, Quantum Fields in Curved Space.  Cambridge Monographs on Mathematical Physics. Cambridge Univ. Press, Cambridge, UK, 2, 1984.
  * [176] M. D. Schwartz, Quantum Field Theory and the Standard Model.  Cambridge University Press, 3, 2014.
  * [177] P. Di Francesco, P. Mathieu, and D. Senechal, Conformal Field Theory.  Graduate Texts in Contemporary Physics. Springer-Verlag, New York, 1997\.
  * [178] E. Ponton and E. Poppitz, Casimir energy and radius stabilization in five-dimensional orbifolds and six-dimensional orbifolds, JHEP 06 (2001) 019, [hep-ph/0105021].
  * [179] E. J. Weinberg, Parameter Counting for Multi-Monopole Solutions, Phys. Rev. D 20 (1979) 936–944.