11institutetext: Department of Physics, Kyushu University, 744 Motooka, Nishi-
ku, Fukuoka 819-0395, Japan22institutetext: Department of Physics, Osaka
University, Toyonaka, Osaka 560-0043, Japan33institutetext: Yukawa Institute
for Theoretical Physics, Kyoto University, Kitashirakawa Oiwakecho, Sakyo-ku,
Kyoto 606-8502, Japan

# Topology of $SU(N)$ lattice gauge theories coupled with $\mathbb{Z}_{N}$
$2$-form gauge fields

Motokazu Abe 2 Okuto Morikawa 1 Soma Onoda 1 Hiroshi Suzuki 3 Yuya Tanizaki
<EMAIL_ADDRESS>
###### Abstract

We extend the definition of Lüscher’s lattice topological charge to the case
of $4$d $SU(N)$ gauge fields coupled with $\mathbb{Z}_{N}$ $2$-form gauge
fields. This result is achieved while maintaining the locality, the $SU(N)$
gauge invariance, and $\mathbb{Z}_{N}$ $1$-form gauge invariance, and we find
that the manifest $1$-form gauge invariance plays the central role in our
construction. This result gives the lattice regularized derivation of the
mixed ’t Hooft anomaly in pure $SU(N)$ Yang–Mills theory between its
$\mathbb{Z}_{N}$ $1$-form symmetry and the $\theta$ periodicity.

††preprint: KYUSHU-HET-258, OU-HET-1177, YITP-23-37

##  1 Introduction

Gauge fields in the continuum description enjoy topological classifications,
and those in different topological sectors cannot be continuously deformed
from one another. This fact allows us to introduce a new parameter, called the
$\theta$ angle, in quantum Yang–Mills theories Belavin:1975fg ; Callan:1976je
; Jackiw:1976pf , and the presence of topological sectors makes the dynamics
of Yang–Mills theories quite rich and highly nontrivial. However, we should
note that it is nontrivial if these considerations based on topology are
robust under quantum fluctuations. Quantum field theories (QFTs) are subject
to ultraviolet divergences, and we need to introduce some regularization to
control it. Lattice regularization achieves it in a gauge invariant manner,
but it becomes unclear if the notion of continuous gauge fields survives under
this process.

Lüscher addressed this issue for $SU(2)$ gauge theories and pointed out the
presence of topological sectors on the lattice by introducing the
admissibility condition Luscher:1981zq  (see also Ref. Phillips:1986qd ).
There, the topological charge on the lattice is explicitly defined in a local
and gauge-invariant way, and one can extend the construction to simple and
simply-connected gauge groups such as $SU(N)$ straightforwardly. In this
paper, we consider its extension for non-simply-connected gauge groups, in
particular $SU(N)/\mathbb{Z}_{N}$.

$SU(N)/\mathbb{Z}_{N}$ gauge fields can be thought of the $SU(N)$ gauge
theories coupled to $\mathbb{Z}_{N}$ $2$-form gauge fields. Treating the
$\mathbb{Z}_{N}$ $2$-form gauge fields as the background field, it is
equivalent to the $SU(N)$ gauge theories in the ’t Hooft twisted boundary
condition tHooft:1979rtg . van Baal computed the topological charge for smooth
gauge fields in this setup, and it turns out that the topological charge has a
fractional shift in the unit of $1/N$ vanBaal:1982ag . These observations now
acquire renewed interest from the viewpoint of generalized symmetries in QFTs.
Pure $SU(N)$ Yang–Mills theory enjoys the $\mathbb{Z}_{N}$ $1$-form symmetry
as the global symmetry Gaiotto:2014kfa , and the $\mathbb{Z}_{N}$ $2$-form
gauge field obtains a natural interpretation as the gauge field coupled to the
$1$-form symmetry Kapustin:2013qsa ; Kapustin:2014gua . Studying the response
of background gauge fields provides a concise and systematic way to extract
nontrivial consequences out of global symmetries, and the fractional shift of
the topological charge turns out to have a huge impact on our understanding of
the Yang–Mills vacua Gaiotto:2017yup .

In this paper, we construct the topological charge of lattice
$SU(N)/\mathbb{Z}_{N}$ gauge fields that maintains the locality, $SU(N)$ gauge
invariance, and $\mathbb{Z}_{N}$ $1$-form gauge invariance. Introducing the
admissibility condition for $SU(N)$ gauge fields coupled to flat
$\mathbb{Z}_{N}$ $2$-form gauge fields, we show that the $SU(N)$-valued
transition functions can be locally defined from the lattice
$SU(N)/\mathbb{Z}_{N}$ gauge fields. We show that those transition functions
satisfy the cocycle condition up to the center elements specified locally by
the $2$-form gauge fields. Thus, the principal $SU(N)/\mathbb{Z}_{N}$ bundle
is obtained from the admissible lattice gauge fields, and we can compute their
topological charge as its second Chern class. We note that all of this
procedure is completely parallel to Lüscher’s one in Ref. Luscher:1981zq . It
turns out that we can keep the manifest $1$-form gauge invariance at every
stage of our construction, which is the essential ingredient to circumvent
various complications potentially caused by the presence of higher-form gauge
fields. Using this lattice topological charge, we define the lattice
Yang–Mills theory with the $\theta$ angle coupled to the background
$\mathbb{Z}_{N}$ $2$-form gauge fields, and we obtain the anomalous relation
for the Yang–Mills partition function on the lattice. Our result justifies the
observation of Ref. Gaiotto:2017yup  that had been obtained in the continuum
analysis assuming smoothness of gauge fields, so we rigorously realize the
mixed ’t Hooft anomaly between the $\mathbb{Z}_{N}$ $1$-form symmetry and the
$\theta$ periodicity with the lattice regularization.

We note that this study is a non-Abelian extension of the previous study about
$U(1)$ lattice gauge theory Abe:2022nfq , while the actual construction in
this paper is based on a different idea from that of Ref. Abe:2022nfq . Still,
applying this construction to the $U(1)$ gauge group should reproduce the
result of Ref. Abe:2022nfq  due to the uniqueness of the principal bundle with
given data.

##  2 Lattice $SU(N)$ gauge theories coupled with $\mathbb{Z}_{N}$ $2$-form
gauge fields

Approximating the closed Euclidean spacetime $M$ as a lattice, we introduce
the $SU(N)$-valued link variables $U_{\ell}\in SU(N)$ for each link $\ell$ and
the $\mathbb{Z}_{N}$-valued plaquette variables
$\mathrm{e}^{\frac{2\pi\mathrm{i}}{N}B_{p}}$ for each plaquette $p$. For the
purpose of numerical simulations, one usually takes $M=T^{4}$ as the Euclidean
spacetime and approximates it as the hypercubic lattice
$\Lambda_{L}=(\mathbb{Z}/L\mathbb{Z})^{4}$ since the hypercubic symmetry
restricts possible UV divergences and simplifies renormalization procedures.
In this section, let us consider more generic cases for the discussion of
general properties, although we shall also restrict to the hypercubic case
later for the construction of a lattice topological charge. Following the
nomenclature in Ref. Luscher:1981zq , the $0$-cells are referred to as sites,
the $1$-cells are links, the $2$-cells are plaquettes, the $3$-cells are
faces, and the $4$-cells are just cells.

When $\ell$ and $\ell^{\prime}$ describe the same link with the opposite
orientation, $\ell^{\prime}=-\ell$, we require that $U_{-\ell}=U_{\ell}^{-1}$.
Similarly, $B_{-p}=-B_{p}\bmod N$. Along a plaquette $p$, we define the
$SU(N)$-valued plaquette variable by the path-ordered product of link
variables,

$U_{p}:=\mathcal{P}\prod_{\ell\in\partial p}U_{\ell},$ (1)

where $\ell\in\partial p$ refers to the link in the positive relative
orientation for the plaquette $p$. We require that the $\mathbb{Z}_{N}$
plaquette variables satisfy the flatness condition,

$\sum_{p\in\partial f}B_{p}=0\bmod N,$ (2)

for each face $f$. We define the Wilson plaquette action as

$S_{\mathrm{W}}[U_{\ell},B_{p}]=\sum_{p}\beta\left[\mathrm{tr}\left(\bm{1}-\mathrm{e}^{-\frac{2\pi\mathrm{i}}{N}B_{p}}U_{p}\right)+\mathrm{c.c.}\right],$
(3)

where $\beta$ is the coupling constant for the lattice Yang–Mills theory.

This Wilson action is invariant under the $SU(N)$ gauge transformation: For
the link $\ell$ connecting two sites $x$ and $y$, which we denote as
$\ell=\langle x\to y\rangle$, the $SU(N)$ transformation is given by

$U_{\ell}\mapsto g_{x}^{-1}U_{\ell}g_{y},$ (4)

with $g_{x}\in SU(N)$. Let us denote $n$ as the initial and final point of the
closed loop for $p$, then $U_{p}\mapsto g_{n}^{-1}U_{p}g_{n}$ under the
$SU(N)$ gauge transformation so that $\mathrm{tr}(U_{p})$ is invariant.

The Wilson action with $\mathbb{Z}_{N}$ plaquette fields $B_{p}$ is invariant
also under the $\mathbb{Z}_{N}$ $1$-form gauge transformation, defined by

$\displaystyle U_{\ell}$
$\displaystyle\mapsto\mathrm{e}^{\frac{2\pi\mathrm{i}}{N}\lambda_{\ell}}U_{\ell},$
$\displaystyle B_{p}$ $\displaystyle\mapsto
B_{p}+(\mathrm{d}\lambda)_{p}\quad\bmod N,$ (5)

where $\lambda_{\ell}\in\mathbb{Z}_{N}$ and
$(\mathrm{d}\lambda)_{p}=\sum_{\ell\in\partial p}\lambda_{\ell}$. We note that
the flatness condition (2), or $(\mathrm{d}B)_{f}=0\bmod N$, is invariant
under the $\mathbb{Z}_{N}$ $1$-form gauge transformations. When $M$ has no
torsion, we can think of $B_{p}$ as a closed surface on the dual lattice using
the Poincaré duality, and the $\mathbb{Z}_{N}$ $1$-form gauge invariance
requires the invariance under its continuous deformation and the
addition/subtraction of contractible closed surfaces.

In order to introduce the notion of topological sectors for the lattice gauge
fields, the admissibility condition Luscher:1981zq  plays the key role. We say
that the $SU(N)$ gauge field $\\{U_{\ell}\\}$ is admissible, if and only if,
for a given $0<\varepsilon\ll 2$, all the plaquettes $U_{p}$ satisfy

$\left\|\bm{1}-\mathrm{e}^{-\frac{2\pi\mathrm{i}}{N}B_{p}}U_{p}\right\|<\varepsilon.$
(6)

Here, $\|\cdot\|$ refers the matrix norm. After diagonalization
$\mathrm{e}^{-\frac{2\pi\mathrm{i}}{N}B_{p}}U_{p}=V\mathrm{e}^{\operatorname{diag}(\mathrm{i}\theta_{1},\dotsc,\mathrm{i}\theta_{N})}V^{-1}$
with $\theta_{1}+\dotsb+\theta_{N}=0$, we find that
$\|\bm{1}-\mathrm{e}^{-\frac{2\pi\mathrm{i}}{N}B_{p}}U_{p}\|=\max_{k}\left|2\sin(\theta_{k}/2)\right|<\varepsilon$.
We denote the set of admissible gauge fields as

$\mathfrak{A}_{\varepsilon}[B_{p}]=\left\\{\left\\{U_{\ell}\right\\}\,\Bigm{|}\text{$\left\|\bm{1}-\mathrm{e}^{-\frac{2\pi\mathrm{i}}{N}B_{p}}U_{p}\right\|<\varepsilon$
for all $p$}\right\\},$ (7)

and $\mathfrak{A}_{\varepsilon}[B_{p}]$ is $SU(N)$ gauge invariant and
$\mathbb{Z}_{N}$ $1$-form gauge covariant. The functional integral is
performed over $\mathfrak{A}_{\varepsilon}[B_{p}]$ so that the partition
function is given as111Here, we introduce the sharp cutoff for the integration
domain. One can instead introduce the bump function to make everything smooth
and it would ensure better analytic properties of the partition function.

$\mathcal{Z}[B_{p}]=\int_{\mathfrak{A}_{\varepsilon}[B_{p}]}{\mathcal{D}}U_{\ell}\,\exp\left(-S_{\mathrm{W}}[U_{\ell},B_{p}]+\dotsb\right),$
(8)

where ${\mathcal{D}}U_{\ell}$ is the product of the $SU(N)$ Haar measure for
all the link variables. We note that $\varepsilon$ shall be determined
independently of the lattice size, coupling constant $\beta$, and the
plaquette gauge fields $B_{p}$. It will be discussed in Appendix C and we find
$\varepsilon\lesssim 0.074$ is sufficiently small.

##  3 Transition functions from lattice gauge fields

In this section, we construct the transition function for admissible lattice
gauge fields coupled to the $\mathbb{Z}_{N}$ $2$-form gauge fields $B_{p}$
extending the seminal work by Lüscher Luscher:1981zq . We work on the
$4$-torus $M=T^{4}$ and its hypercubic lattice discretization
$\Lambda_{L}=(\mathbb{Z}/L\mathbb{Z})^{4}$, i.e., each direction is cut into
$L$ pieces. As illustrated in Fig. 1, we define some variables on the
hypercubic structure as follows: The unit cell is denoted as

$c(n)=\left\\{x\in T^{4}\bigm{|}\text{$0\leq x_{\mu}-n_{\mu}\leq 1$ for
$\mu=1$, \ldots, $4$}\right\\}$ (9)

for $n\in\Lambda_{L}$, and we would like to define the transition function on
each face

$f(n,\mu)=c(n)\cap c(n-\hat{\mu}),$ (10)

where $\hat{\mu}$ is the unit vector along the $\mu$th direction.

For the link $\ell=\langle n\to n+\hat{\mu}\rangle$, we write the link
variable as $U(n,\mu)=U_{\ell}$. For the plaquette including $n$,
$n+\hat{\mu}$, $n+\hat{\nu}$, and $n+\hat{\mu}+\hat{\nu}$, we have the $SU(N)$
plaquette variable

$U_{\mu\nu}(n):=U(n,\mu)U(n+\hat{\mu},\nu)U(n+\hat{\nu},\mu)^{-1}U(n,\nu)^{-1},$
(11)

and $\mathbb{Z}_{N}$ $2$-form gauge field $B_{\mu\nu}(n)=B_{p}$. The $1$-form
gauge invariant plaquette is then given by

$\tilde{U}_{\mu\nu}(n):=\mathrm{e}^{-\frac{2\pi\mathrm{i}}{N}B_{\mu\nu}(n)}U_{\mu\nu}(n),$
(12)

and the admissibility condition (6) requires
$\|\bm{1}-\tilde{U}_{\mu\nu}(n)\|<\varepsilon$ for all the plaquettes.

###  3.1 Complete axial gauge and transition functions at the corner

We first define the complete axial gauge for each cell $c(n)$ and introduce
the transition function at the corner of the cells as the connection formula
of the corresponding link variables. Let $x\in c(n)$ be a corner, i.e.,
$x=n+\sum_{\mu}z_{\mu}\hat{\mu}$ with $z_{\mu}\in\\{0,1\\}$, and we define the
standard parallel transporter from $n$ to $x$ as222We adopt a somewhat
different definition of the standard transporter from Lüscher’s one in Ref.
Luscher:1981zq . Most part of the discussion goes through in the same manner
for both conventions, except for the explicit formulas of $\tilde{u}_{xy}^{m}$
and $\tilde{v}_{n,\mu}(x)$.

$w^{n}(x):=U(n,4)^{z_{4}}U(n+z_{4}\hat{4},3)^{z_{3}}U(n+z_{4}\hat{4}+z_{3}\hat{3},2)^{z_{2}}U(n+z_{4}\hat{4}+z_{3}\hat{3}+z_{2}\hat{2},1)^{z_{1}}.$
(13)

We then set, for two adjacent corners $x$ and $y$ of $c(n)$,

$\displaystyle u_{xy}^{n}$ $\displaystyle:=w^{n}(x)U(x,\mu)w^{n}(y)^{-1},$
$\displaystyle\text{if $y=x+\hat{\mu}$},$ $\displaystyle u_{xy}^{n}$
$\displaystyle:=\left(u_{yx}^{n}\right)^{-1},$ $\displaystyle\text{if
$y=x-\hat{\mu}$}.$ (14)

This can be thought of the link variable in the complete axial gauge of $c(n)$
for $\ell=\langle x\to y\rangle$, since all the link variables along the
standard parallel transporter are set to be $\bm{1}$. When the link
$\ell=\langle x\to y\rangle$ is shared by two cells $c(n)$ and
$c(n-\hat{\mu})$, the link variables $u_{xy}^{n}$ and $u_{xy}^{n-\hat{\mu}}$
take different values due to the different choice of the gauge between these
cells, and we introduce the transition functions at the corner $v_{n,\mu}(x)$
to make the connection between them:

$u^{n-\hat{\mu}}_{xy}=v_{n,\mu}(x)u^{n}_{xy}v_{n,\mu}(y)^{-1}.$ (15)

$f(n,\mu)$$n$$p(n,\mu,\nu)$$U(n,\mu)$$\tilde{v}_{n,\mu}(x)$$c(n-\hat{\mu})$$c(n)$$x_{\mu}$$x_{\nu}$$x_{\rho}$
Figure 1: Illustration for the definition of variables. The transition
functions $\tilde{v}_{n,\mu}(x)$ ($1$-form covariant counterpart of
$v_{n,\mu}(x)$) are defined on the face $f(n,\mu)=c(n)\cap c(n-\hat{\mu})$,
and they are subject to the cocycle condition on $p(n,\mu,\nu)=c(n)\cap
c(n-\hat{\mu})\cap c(n-\hat{\mu})\cap c(n-\hat{\mu}-\hat{\nu})$. At this
moment, $\tilde{v}_{n,\mu}(x)$ is defined only at the corner of $f(n,\mu)$,
and we later interpolate it.

The explicit formula for $v_{n,\mu}(x)$ is given by

$v_{n,\mu}(x)=w^{n-\hat{\mu}}(x)w^{n}(x)^{-1},$ (16)

when $x$ is at the corner of $f(n,\mu)=c(n)\cap c(n-\hat{\mu})$. For later
purpose, we denote the corner of $f(n,\mu)$ as $s_{i}$ ($i=0$, $1$, …, $7$) as
illustrated in Fig. 2, and we introduce the coordinate on $f(n,\mu)$ as
$x=n+y_{\alpha}\hat{\alpha}+y_{\beta}\hat{\beta}+y_{\gamma}\hat{\gamma}$ with
$\alpha$, $\beta$, $\gamma\in\\{1,2,3,4\\}\setminus\\{\mu\\}$ and
$\alpha<\beta<\gamma$.

$s_{0}$$s_{1}$$s_{6}$$s_{2}$$s_{5}$$s_{4}$$s_{7}$$s_{3}$$x_{\alpha}$$x_{\beta}$$x_{\gamma}$
Figure 2: We introduce the coordinate for the face $f(n,\mu)$ as
$x=n+y_{\alpha}\hat{\alpha}+y_{\beta}\hat{\beta}+y_{\gamma}\hat{\gamma}$ with
$0\leq y_{\alpha,\beta,\gamma}\leq 1$ and $\alpha<\beta<\gamma$. We label the
corners of $f(n,\mu)$ (i.e., $y_{\alpha,\beta,\gamma}\in\\{0,1\\}$) as $s_{i}$
($i=0$, $1$, …, $7$) as shown in this figure.

The link in the complete axial gauge $u_{xy}^{m}$ is the product of link
variables along a closed loop that starts and ends at the site $m$. Therefore,
we can make them invariant under the $1$-form gauge transformation by
multiplying appropriate factors of
$\mathrm{e}^{-\frac{2\pi\mathrm{i}}{N}B_{\mu\nu}(n)}$. Due to the flatness
condition (2), there is a unique local way to achieve the $\mathbb{Z}_{N}$
$1$-form gauge invariance, and such $1$-form invariant $u_{xy}^{m}$ are
denoted with the tilde, $\tilde{u}_{xy}^{m}$. This construction is illustrated
in Fig. 3 by taking an example of $\tilde{u}_{s_{7}s_{2}}^{m=n-\hat{3}}$,
where $s_{7}$ and $s_{2}$ denote corners of $f(n,3)$, and it shows that

$\tilde{u}_{s_{7}s_{2}}^{n-\hat{3}}=\mathrm{e}^{\frac{2\pi\mathrm{i}}{N}B_{34}(n-\hat{3})}\mathrm{e}^{\frac{2\pi\mathrm{i}}{N}B_{24}(n)}u_{s_{7}s_{2}}^{n-\hat{3}}.$
(17)

The list of the concrete expressions for $\tilde{u}_{xy}^{m}$ is given in
Appendix B.1.

$w^{n-\hat{3}}(s_{2})^{-1}$$w^{n-\hat{3}}(s_{7})$$B_{24}(n)$$B_{34}(n-\hat{3})$$U(s_{7},-4)$$n-\hat{3}$$s_{0}=n$$s_{1}$$s_{6}$$s_{2}$$s_{5}$$s_{4}$$s_{7}$$s_{3}$$x_{1}$$x_{2}$$x_{4}$Construction
of $u_{s_{7}s_{2}}^{n-\hat{3}}$Construction of
$\tilde{u}_{s_{7}s_{2}}^{n-\hat{3}}$ Figure 3:  Illustration for the
construction of $u_{xy}^{m}$ (left) and its $1$-form invariant counterpart
$\tilde{u}_{xy}^{m}$ (right). We take an example with $m=n-\hat{3}$,
$x=s_{7}=n+\hat{2}+\hat{4}$, and $y=s_{2}=n+\hat{2}$, so we draw two cubes for
faces $f(n,3)$ and $f(n-\hat{3},3)$. (Left) Each building block of
$u_{s_{7}s_{2}}^{n-\hat{3}}=w^{n-\hat{3}}(s_{7})U(s_{7},-4)w^{n-\hat{3}}(s_{2})^{-1}$
is shown in different colors, in which the links in the $\hat{3}$ direction
are drawn with curved lines. (Right) As $u_{xy}^{m}$ forms a closed loop
starting and ending at the site $m$, we attach the $\mathbb{Z}_{N}$ plaquette
fields $\mathrm{e}^{\frac{2\pi\mathrm{i}}{N}B_{\mu\nu}(n)}$ when the loop
surrounds the corresponding plaquettes.

We further want to define $\tilde{v}_{n,\mu}(x)$ for a corner $x$ of
$f(n,\mu)$ such that

$\tilde{u}_{xy}^{n-\hat{\mu}}=\tilde{v}_{n,\mu}(x)\tilde{u}_{xy}^{n}\tilde{v}_{n,\mu}(y)^{-1},$
(18)

where $x$ and $y$ are the corners of $f(n,\mu)$. Also, we impose that
$\tilde{v}_{n,\mu}(x)$ for the corner $x$ behaves covariantly under the
$1$-form gauge transformation,

$\tilde{v}_{n,\mu}(x)\mapsto\mathrm{e}^{\frac{2\pi\mathrm{i}}{N}\lambda_{\mu}(n-\hat{\mu})}\tilde{v}_{n,\mu}(x).$
(19)

This is again achieved by multiplication of appropriate factors of
$\mathrm{e}^{\frac{2\pi\mathrm{i}}{N}B_{\mu\nu}(n)}$, and the concrete
expressions are given in Appendix B.2. We can then confirm Eq. (18) by using
the flatness condition, $(\mathrm{d}B)_{f}=0\bmod N$. From these, we observe
the cocycle condition at the corner $n$ of $p(n,\mu,\nu)=c(n)\cap
c(n-\hat{\mu})\cap c(n-\hat{\nu})\cap c(n-\hat{\mu}-\hat{\nu})$,

$\displaystyle\tilde{v}_{n-\hat{\nu},\mu}(n)\tilde{v}_{n,\nu}(n)\tilde{v}_{n,\mu}(n)^{-1}\tilde{v}_{n-\hat{\mu},\nu}(n)^{-1}=\mathrm{e}^{\frac{2\pi\mathrm{i}}{N}B_{\mu\nu}(n-\hat{\mu}-\hat{\nu})}\bm{1}.$
(20)

We can check it using the relation
$v_{n-\hat{\nu},\mu}(n)v_{n,\nu}(n)=v_{n-\hat{\mu},\nu}(n)v_{n,\mu}(n)$ and
the formulas given in Appendix B.2. This is the desired cocycle condition in
the $SU(N)/\mathbb{Z}_{N}$ gauge theory and actually is expected from the
transformation property under the $1$-form gauge transformation alone: From
the cocycle condition for transition functions without tildes, the right-hand
side must be an element of $\mathbb{Z}_{N}$. This $\mathbb{Z}_{N}$ element
should transform identically as the left-hand side does under the
$\mathbb{Z}_{N}$ $1$-form gauge transformation (19). Then, the unique
possibility is
$\mathrm{e}^{\frac{2\pi\mathrm{i}}{N}B_{\mu\nu}(n-\hat{\mu}-\hat{\nu})}\bm{1}$.

###  3.2 Continuum interpolation of transition functions and cocycle
conditions

So far, we have defined the transition functions $\tilde{v}_{n,\mu}(x)$ only
at the corners of $f(n,\mu)$. In this section, we define the smooth transition
functions

$\tilde{v}_{n,\mu}:f(n,\mu)\to SU(N),$ (21)

by introducing the interpolation functions $\tilde{S}^{m}_{n,\mu}(x)$ so that

$\tilde{v}_{n,\mu}(x)\equiv\tilde{S}_{n,\mu}^{n-\hat{\mu}}(x)^{-1}\tilde{v}_{n,\mu}(n)\tilde{S}_{n,\mu}^{n}(x).$
(22)

In order for this to be an interpolation, the iterative application of Eq.
(18) implies that $\tilde{S}^{m}_{n,\mu}:f(n,\mu)\to SU(N)$ should satisfy

$\tilde{S}_{n,\mu}^{m}(x)=\begin{cases}1,&\text{for $x=s_{0}$, i.e.,
$(y_{\alpha},y_{\beta},y_{\gamma})=(0,0,0)$},\\\
\tilde{u}_{s_{0}s_{1}}^{m},&\text{for $x=s_{1}$, i.e.,
$(y_{\alpha},y_{\beta},y_{\gamma})=(1,0,0)$},\\\
\tilde{u}_{s_{0}s_{2}}^{m},&\text{for $x=s_{2}$, i.e.,
$(y_{\alpha},y_{\beta},y_{\gamma})=(0,1,0)$},\\\
\tilde{u}_{s_{0}s_{3}}^{m},&\text{for $x=s_{3}$, i.e.,
$(y_{\alpha},y_{\beta},y_{\gamma})=(0,0,1)$},\\\
\tilde{u}_{s_{0}s_{3}}^{m}\tilde{u}_{s_{3}s_{7}}^{m}\tilde{u}_{s_{7}s_{4}}^{m},&\text{for
$x=s_{4}$, i.e., $(y_{\alpha},y_{\beta},y_{\gamma})=(1,1,1)$},\\\
\tilde{u}_{s_{0}s_{3}}^{m}\tilde{u}_{s_{3}s_{5}}^{m},&\text{for $x=s_{5}$,
i.e., $(y_{\alpha},y_{\beta},y_{\gamma})=(1,0,1)$},\\\
\tilde{u}_{s_{0}s_{2}}^{m}\tilde{u}_{s_{2}s_{6}}^{m},&\text{for $x=s_{6}$,
i.e., $(y_{\alpha},y_{\beta},y_{\gamma})=(1,1,0)$},\\\
\tilde{u}_{s_{0}s_{3}}^{m}\tilde{u}_{s_{3}s_{7}}^{m},&\text{for $x=s_{7}$,
i.e., $(y_{\alpha},y_{\beta},y_{\gamma})=(0,1,1)$}.\\\ \end{cases}$ (23)

Another important requirement is that this interpolated function should
satisfy the cocycle condition, that is, for $x\in p(n,\mu,\nu)$,

$\tilde{v}_{n-\hat{\nu},\mu}(x)\tilde{v}_{n,\nu}(x)\tilde{v}_{n,\mu}(x)^{-1}\tilde{v}_{n-\hat{\mu},\nu}(x)^{-1}=\mathrm{e}^{\frac{2\pi\mathrm{i}}{N}B_{\mu\nu}(n-\hat{\mu}-\hat{\nu})}\bm{1},$
(24)

which gives a nontrivial condition for $\tilde{S}_{n,\mu}^{m}(x)$.

Such an interpolation $\tilde{S}_{n,\mu}^{m}(x)$ has been constructed
explicitly by Lüscher in Eqs. (22a)–(22f) of Ref. Luscher:1981zq  for the case
of $SU(N)$. We extend the formula to the case of $SU(N)/\mathbb{Z}_{N}$ by
imposing the manifest $1$-form gauge invariance. The explicit form of
$\tilde{S}_{n,\mu}^{m}(x)$ is written as follows:

$\displaystyle\tilde{f}_{n,\mu}^{m}(x_{\gamma})$
$\displaystyle=(\tilde{u}_{s_{3}s_{0}}^{m})^{y_{\gamma}}(\tilde{u}_{s_{0}s_{3}}^{m}\tilde{u}_{s_{3}s_{7}}^{m}\tilde{u}_{s_{7}s_{2}}^{m}\tilde{u}_{s_{2}s_{0}}^{m})^{y_{\gamma}}\tilde{u}_{s_{0}s_{2}}^{m}(\tilde{u}_{s_{2}s_{7}}^{m})^{y_{\gamma}},$
$\displaystyle\tilde{g}_{n,\mu}^{m}(x_{\gamma})$
$\displaystyle=(\tilde{u}_{s_{5}s_{1}}^{m})^{y_{\gamma}}(\tilde{u}_{s_{1}s_{5}}^{m}\tilde{u}_{s_{5}s_{4}}^{m}\tilde{u}_{s_{4}s_{6}}^{m}\tilde{u}_{s_{6}s_{1}}^{m})^{y_{\gamma}}\tilde{u}_{s_{1}s_{6}}^{m}(\tilde{u}_{s_{6}s_{4}}^{m})^{y_{\gamma}},$
$\displaystyle\tilde{h}_{n,\mu}^{m}(x_{\gamma})$
$\displaystyle=(\tilde{u}_{s_{3}s_{0}}^{m})^{y_{\gamma}}(\tilde{u}_{s_{0}s_{3}}^{m}\tilde{u}_{s_{3}s_{5}}^{m}\tilde{u}_{s_{5}s_{1}}^{m}\tilde{u}_{s_{1}s_{0}}^{m})^{y_{\gamma}}\tilde{u}_{s_{0}s_{1}}^{m}(\tilde{u}_{s_{1}s_{5}}^{m})^{y_{\gamma}},$
$\displaystyle\tilde{k}_{n,\mu}^{m}(x_{\gamma})$
$\displaystyle=(\tilde{u}_{s_{7}s_{2}}^{m})^{y_{\gamma}}(\tilde{u}_{s_{2}s_{7}}^{m}\tilde{u}_{s_{7}s_{4}}^{m}\tilde{u}_{s_{4}s_{6}}^{m}\tilde{u}_{s_{6}s_{2}}^{m})^{y_{\gamma}}\tilde{u}_{s_{2}s_{6}}^{m}(\tilde{u}_{s_{6}s_{4}}^{m})^{y_{\gamma}},$
$\displaystyle\tilde{l}_{n,\mu}^{m}(x_{\beta},x_{\gamma})$
$\displaystyle=\left[\tilde{f}_{n,\mu}^{m}(x_{\gamma})^{-1}\right]^{y_{\beta}}\left[\tilde{f}_{n,\mu}^{m}(x_{\gamma})\tilde{k}_{n,\mu}^{m}(x_{\gamma})\tilde{g}_{n,\mu}^{m}(x_{\gamma})^{-1}\tilde{h}_{n,\mu}^{m}(x_{\gamma})^{-1}\right]^{y_{\beta}}$
$\displaystyle\qquad{}\cdot\tilde{h}_{n,\mu}^{m}(x_{\gamma})\left[\tilde{g}_{n,\mu}^{m}(x_{\gamma})\right]^{y_{\beta}},$
$\displaystyle\tilde{S}_{n,\mu}^{m}(x_{\alpha},x_{\beta},x_{\gamma})$
$\displaystyle=(\tilde{u}_{s_{0}s_{3}}^{m})^{y_{\gamma}}\left[\tilde{f}_{n,\mu}^{m}(x_{\gamma})\right]^{y_{\beta}}\left[\tilde{l}_{n,\mu}^{m}(x_{\beta},x_{\gamma})\right]^{y_{\alpha}}.$
(25)

In these formulas, we need to take the fractional power of plaquette variables
as defined in Appendix A, and their well-definedness is ensured by imposing
the admissibility $\\{U_{\ell}\\}\in\mathfrak{A}_{\varepsilon}[B_{p}]$ for
sufficiently small $\varepsilon\lesssim 0.074$. We can confirm that, when $x$
is one of the corners of $f(n,\mu)$, this definition reduces to Eq. (23) as
requested. Moreover, $\tilde{v}_{n,\mu}(x)$ obeys the $1$-form transformation
proeprety (19) since $\tilde{S}^{m}_{n,\mu}$ is $1$-form gauge invariant.

Let us now check if the above $\tilde{v}_{n,\mu}(x)$ satisfies the cocycle
condition (24). It turns out that Lüscher’s proof for the Lemma in Ref.
Luscher:1981zq  works in the same manner by making its every step manifestly
$1$-form gauge invariant. We first note that
$\tilde{S}_{n,\mu}^{m}(x)=\tilde{S}_{n,\lambda}^{m}(x)$ when $x\in
p(n,\mu,\lambda)$ by construction, and we denote it as

$\tilde{P}_{n,\mu\lambda}^{m}(x):=\tilde{S}_{n,\mu}^{m}(x)=\tilde{S}_{n,\lambda}^{m}(x)\qquad\text{for
$x\in p(n,\mu,\lambda)$}.$ (26)

Following Ref. Luscher:1981zq , it is also convenient to introduce

$\tilde{R}_{n,\mu;\lambda}^{m}(x)=\tilde{S}_{n,\mu}^{m}(x)\tilde{P}_{n+\hat{\lambda},\mu\lambda}^{m}(x)^{-1}\qquad\text{for
$x\in p(n+\hat{\lambda},\mu,\lambda)$},$ (27)

and its explicit form is given by

$\displaystyle\tilde{R}_{n,\mu;\alpha}^{m}(x_{\beta},x_{\gamma})$
$\displaystyle=\bigl{[}(\tilde{u}_{s_{0}s_{3}}^{m}\tilde{u}_{s_{3}s_{7}}^{m}\tilde{u}_{s_{7}s_{2}}^{m}\tilde{u}_{s_{2}s_{0}}^{m})^{y_{\gamma}}\tilde{u}_{s_{0}s_{2}}^{m}$
$\displaystyle\qquad{}\cdot(\tilde{u}_{s_{2}s_{7}}^{m}\tilde{u}_{s_{7}s_{4}}^{m}\tilde{u}_{s_{4}s_{6}}^{m}\tilde{u}_{s_{6}s_{2}}^{m})^{y_{\gamma}}\tilde{u}_{s_{2}s_{6}}^{m}\tilde{u}_{s_{6}s_{1}}^{m}(\tilde{u}_{s_{1}s_{6}}^{m}\tilde{u}_{s_{6}s_{4}}^{m}\tilde{u}_{s_{4}s_{5}}^{m}\tilde{u}_{s_{5}s_{1}}^{m})^{y_{\gamma}}$
$\displaystyle\qquad{}\cdot\tilde{u}_{s_{1}s_{0}}^{m}(\tilde{u}_{s_{0}s_{1}}^{m}\tilde{u}_{s_{1}s_{5}}^{m}\tilde{u}_{s_{5}s_{3}}^{m}\tilde{u}_{s_{3}s_{0}}^{m})^{y_{\gamma}}\bigr{]}^{y_{\beta}}(\tilde{u}_{s_{0}s_{3}}^{m}\tilde{u}_{s_{3}s_{5}}^{m}\tilde{u}_{s_{5}s_{1}}^{m}\tilde{u}_{s_{1}s_{0}}^{m})^{y_{\gamma}}\tilde{u}_{s_{0}s_{1}}^{m},$
$\displaystyle\tilde{R}_{n,\mu;\beta}^{m}(x_{\alpha},x_{\gamma})$
$\displaystyle=(\tilde{u}_{s_{0}s_{3}}^{m}\tilde{u}_{s_{3}s_{7}}^{m}\tilde{u}_{s_{7}s_{2}}^{m}\tilde{u}_{s_{2}s_{0}}^{m})^{y_{\gamma}}\tilde{u}_{s_{0}s_{2}}^{m},$
$\displaystyle\tilde{R}_{n,\mu;\gamma}^{m}(x_{\alpha},x_{\beta})$
$\displaystyle=\tilde{u}_{s_{0}s_{3}}^{m}.$ (28)

Using the crucial relation (18) combined with the formula (51), we obtain

$\tilde{R}_{n,\mu;\lambda}^{n-\hat{\mu}}(x)=\tilde{v}_{n,\mu}(n)\tilde{R}_{n,\mu;\lambda}^{n}(x)\tilde{v}_{n,\mu}(n+\hat{\lambda})^{-1}.$
(29)

These show that, for $x\in p(n,\mu,\lambda)$,

$\tilde{v}_{n,\mu}(x)=\tilde{P}_{n,\mu\lambda}^{n-\hat{\mu}}(x)^{-1}\tilde{v}_{n,\mu}(n)\tilde{P}_{n,\mu\lambda}^{n}(x),$
(30)

and, for $x\in p(n+\hat{\lambda},\mu,\lambda)$,

$\displaystyle\tilde{v}_{n,\mu}(x)$
$\displaystyle=\tilde{P}_{n+\hat{\lambda},\mu\lambda}^{n-\hat{\mu}}(x)^{-1}\tilde{R}_{n,\mu;\lambda}^{n-\hat{\mu}}(x)^{-1}\tilde{v}_{n,\mu}(n)\tilde{R}_{n,\mu;\lambda}^{n}(x)\tilde{P}_{n+\hat{\lambda},\mu\lambda}^{n}(x)$
$\displaystyle=\tilde{P}_{n+\hat{\lambda},\mu\lambda}^{n-\hat{\mu}}(x)^{-1}\tilde{v}_{n,\mu}(n+\hat{\lambda})\tilde{P}_{n+\hat{\lambda},\mu\lambda}^{n}(x).$
(31)

Using these, for $x\in p(n,\mu,\nu)$, we find

$\displaystyle\tilde{v}_{n,\mu}(x)$
$\displaystyle=\tilde{P}_{n,\mu\nu}^{n-\hat{\mu}}(x)^{-1}\tilde{v}_{n,\mu}(n)\tilde{P}_{n,\mu\nu}^{n}(x),$
$\displaystyle\tilde{v}_{n,\nu}(x)$
$\displaystyle=\tilde{P}_{n,\mu\nu}^{n-\hat{\nu}}(x)^{-1}\tilde{v}_{n,\nu}(n)\tilde{P}_{n,\mu\nu}^{n}(x),$
$\displaystyle\tilde{v}_{n-\hat{\nu},\mu}(x)$
$\displaystyle=\tilde{P}_{n,\mu\nu}^{n-\hat{\mu}-\hat{\nu}}(x)^{-1}\tilde{v}_{n-\hat{\nu},\mu}(n)\tilde{P}_{n,\mu\nu}^{n-\hat{\nu}}(x),$
$\displaystyle\tilde{v}_{n-\hat{\mu},\nu}(x)$
$\displaystyle=\tilde{P}_{n,\mu\nu}^{n-\hat{\mu}-\hat{\nu}}(x)^{-1}\tilde{v}_{n-\hat{\mu},\nu}(n)\tilde{P}_{n,\mu\nu}^{n-\hat{\mu}}(x),$
(32)

where we have noted $\tilde{P}_{n,\nu\mu}^{m}(x)=\tilde{P}_{n,\mu\nu}^{m}(x)$
by definition. Therefore, the cocycle condition along $p(n,\mu,\nu)$ yields

$\displaystyle\tilde{v}_{n-\hat{\nu},\mu}(x)\tilde{v}_{n,\nu}(x)\tilde{v}_{n,\mu}(x)^{-1}\tilde{v}_{n-\hat{\mu},\nu}(x)^{-1}$
$\displaystyle=\tilde{P}_{n,\mu\nu}^{n-\hat{\mu}-\hat{\nu}}(x)^{-1}\tilde{v}_{n-\hat{\nu},\mu}(n)\tilde{v}_{n,\nu}(n)\tilde{v}_{n,\mu}(n)^{-1}\tilde{v}_{n-\hat{\mu},\nu}(n)^{-1}\tilde{P}_{n,\mu\nu}^{n-\hat{\mu}-\hat{\nu}}(x)$
$\displaystyle=\mathrm{e}^{\frac{2\pi\mathrm{i}}{N}B_{\mu\nu}(n-\hat{\mu}-\hat{\nu})}\bm{1},$
(33)

where we used Eq. (20). This completes the proof of the cocycle condition
(24).

##  4 Fractional topological charge on the lattice and its applications

We now obtain the $SU(N)/\mathbb{Z}_{N}$ bundle on $T^{4}$ characterized by
transition functions $\tilde{v}_{n\mu}(x)$ between the cells $c(n)$ and
$c(n-\hat{\mu})$. The topological charge, or the second Chern class, is given
by Luscher:1981zq ; vanBaal:1982ag

$Q_{\mathrm{top}}[U_{\ell},B_{P}]=\sum_{n\in\Lambda_{L}}q(n),$ (34)

where

$\displaystyle q(n)$
$\displaystyle=-\frac{1}{24\pi^{2}}\sum_{\mu,\nu,\rho,\sigma}\varepsilon_{\mu\nu\rho\sigma}\int_{f(n,\mu)}\mathrm{d}^{3}x\,\mathrm{tr}\left[(\tilde{v}_{n,\mu}^{-1}\partial_{\nu}\tilde{v}_{n,\mu})(\tilde{v}_{n,\mu}^{-1}\partial_{\rho}\tilde{v}_{n,\mu})(\tilde{v}_{n,\mu}^{-1}\partial_{\sigma}\tilde{v}_{n,\mu})\right]$
$\displaystyle\qquad{}-\frac{1}{8\pi^{2}}\sum_{\mu,\nu,\rho,\sigma}\varepsilon_{\mu\nu\rho\sigma}\int_{p(n,\mu,\nu)}\mathrm{d}^{2}x\,\mathrm{tr}\left[(\tilde{v}_{n,\mu}\partial_{\rho}\tilde{v}_{n,\mu}^{-1})(\tilde{v}_{n-\hat{\mu},\nu}^{-1}\partial_{\sigma}\tilde{v}_{n-\hat{\mu},\nu})\right].$
(35)

$Q_{\mathrm{top}}$ introduces the topological classification of admissible
lattice gauge fields $\mathfrak{A}_{\varepsilon}[B_{p}]$, and gauge fields in
different sectors cannot be continuously deformed with each other within
$\mathfrak{A}_{\varepsilon}[B_{p}]$. We note that this topological charge is
manifestly $SU(N)$ and $\mathbb{Z}_{N}$ $1$-form gauge invariant. Moreover,
since $\tilde{v}_{n,\mu}(x)$ is constructed out of the plaquettes
$\tilde{U}_{p}$ in the adjacent cells, the topological charge density is
defined in the local way on the lattice.

###  4.1 Fractional shift of the topological charge and mixed ’t Hooft anomaly

By performing the $1$-form gauge transformations and redefinition of
variables, we find that the $SU(N)/\mathbb{Z}_{N}$ bundle constructed above is
equivalent to the ’t Hooft twisted boundary condition with the ’t Hooft flux
(see, e.g., Appendix A of Ref. Tanizaki:2022ngt  for details)

$z_{\mu\nu}=\sum_{p\in(T^{2})_{\mu\nu}}B_{p}\qquad\bmod N.$ (36)

As shown by van Baal in Ref. vanBaal:1982ag , we then find that

$Q_{\mathrm{top}}\in-\frac{1}{N}\frac{\varepsilon_{\mu\nu\rho\sigma}z_{\mu\nu}z_{\rho\sigma}}{8}+\mathbb{Z},$
(37)

and thus the lattice topological charge is shifted by a fractional value in
the unit of $1/N$. In the above expression, the fractional part may look to be
written using the global data, but it can be written in the local way by using
the cohomological operations:

$Q_{\mathrm{top}}=-\frac{1}{N}\int_{T^{4}}\frac{1}{2}P_{2}(B_{p})\qquad\bmod
1,$ (38)

where the Pontryagin square is defined by

$P_{2}(B_{p})=B_{p}\cup B_{p}+B_{p}\cup_{1}\mathrm{d}B_{p}.$ (39)

For the definition of (higher-)cup products on the hypercubic lattice, see
Ref. Chen:2021ppt . The correction by the $1$-cup product is introduced so
that the $\mathbb{Z}_{N}$ $1$-form gauge invariance is achieved at the cochain
level. We can then confirm the equivalence between Eqs. (37) and (38) by
performing the gauge transformation so that $(\mathrm{d}B)_{f}=0$ in
$\mathbb{Z}$ and the Poincaré dual of $B_{p}$ always has transverse
intersections. Then, $\int\frac{1}{2}P_{2}(B_{p})$ in Eq. (38) counts the
number of intersection points in mod $N$ and it gives the fractional part of
Eq. (37).

As we have defined the local topological charge, we can introduce it as the
$\theta$ term for the Boltzmann weight of the path integral, and we define the
partition function as

$\mathcal{Z}_{\theta}[B_{p}]=\int_{\mathfrak{A}_{\varepsilon}[B_{p}]}{\mathcal{D}}U_{\ell}\,\exp\left(-S_{\mathrm{W}}[U_{\ell},B_{p}]+\mathrm{i}\theta
Q_{\mathrm{top}}[U_{\ell},B_{p}]\right).$ (40)

In order for the well-definedness of the lattice topological charge, we have
to restrict the domain for the path integral to the admissible gauge fields
$\mathfrak{A}_{\varepsilon}[B_{p}]$ with some $\varepsilon\lesssim 0.074$.
Using Eq. (38), we find the following equality for the lattice Yang–Mills
partition function:

$\mathcal{Z}_{\theta+2\pi}[B_{p}]=\exp\left[-\frac{2\pi\mathrm{i}}{N}\int_{T^{4}}\frac{1}{2}P_{2}(B_{p})\right]\mathcal{Z}_{\theta}[B_{p}].$
(41)

In Ref. Gaiotto:2017yup  (see also Refs. Tanizaki:2017bam ;
Komargodski:2017dmc ; Kikuchi:2017pcp ; Tanizaki:2018xto ; Karasik:2019bxn ;
Cordova:2019jnf ), this is interpreted as the ’t Hooft anomaly for the
$\mathbb{Z}_{N}$ $1$-form symmetry and the $\theta$ periodicity of $4$d
$SU(N)$ pure Yang–Mills theory, and one can derive various nontrivial
consequences on the Yang–Mills vacua from this relation. In previous studies,
its derivation is based on the classical analysis of the path-integral measure
for smooth gauge field configurations, and it is assumed that the path
integral over quantum gauge fields does not spoil the relation for classical
action. Here, we have given the complete proof of the anomalous relation (41)
using the lattice regularization that maintains the locality and the gauge
invariance, and thus it is now shown to be a rigorous relation for the quantum
Yang–Mills theory.

###  4.2 Classical continuum limit of the topological charge

Although we obtained the lattice local expression for the topological charge
density (35), it is somewhat unrealistic to use this expression for the
practical computations due to the complicated construction of transition
functions $\tilde{v}_{n,\mu}(x)$. Therefore, it is desirable to find a simpler
expression for the lattice topological charge. Due to the quantization of the
topological charge, we do not even have to perform its exact calculation as
long as we can control the approximation.

Here, we take the classical continuum approximation for computing the lattice
topological charge and make its connection to the naive continuum formula.
Since $Q_{\mathrm{top}}$ is invariant under the continuous deformation of
$\\{U_{\ell}\\}$ as long as the admissibility (6) is satisfied, one can use
the following approximate expression with smeared lattice gauge fields when
one ensures (or monitors) the preservation of the admissibility in the
smearing procedures such as the gradient flow Narayanan:2006rf ;
Luscher:2009eq ; Luscher:2010iy ; Luscher:2011bx ; Bonati:2014tqa .

To take the classical continuum limit, we formally introduce the lattice
constant $a$ and take the $a\to 0$ limit with fixed torus size $aL$. When
$B_{p}=0$ everywhere, we can put the assumption that all the link variables
are close to the identity and then the gauge fields are introduced as their
phase factor. When $B_{p}$ is present, however, this assumption has a self-
contradiction as some link variables necessarily have $O(1)$ deviations from
$\bm{1}$, so we need more careful treatment about the notion of continuum
gauge field. In contrast, we can safely assume the continuity about the
$1$-form gauge-invariant plaquettes. We introduce the $1$-form gauge invariant
field strength $\tilde{F}_{\mu\nu}$ by

$\tilde{U}_{\mu\nu}(n)=\exp\left[\mathrm{i}a^{2}\tilde{F}_{\mu\nu}(an)\right]\simeq\bm{1}+\mathrm{i}a^{2}\tilde{F}_{\mu\nu}(an).$
(42)

For the discussion of continuity, we would like to compare
$\tilde{F}_{\mu\nu}$ at two different lattice points $n$ and $m$. When $m\in
c(n)$, we use the standard parallel transporter $w^{n}(m)$ to bring back
$\tilde{F}_{\mu\nu}(am)$ to the point $n$, and we say that the gauge field is
continuous if

$\left\|\tilde{F}_{\mu\nu}(an)-w^{n}(m)\tilde{F}_{\mu\nu}(am)w^{n}(m)^{-1}\right\|\sim\mathcal{O}(a).$
(43)

Under this assumption of the continuity, we obtain

$\tilde{u}_{xy}^{n}=1-\mathrm{i}\frac{a^{2}}{2}\sum_{\mu<\nu}(x+y-2n)_{\mu}(x-y)_{\nu}\tilde{F}_{\mu\nu}(an)+\mathcal{O}(a^{3}).$
(44)

Then, we substitute this expression to the definition (35), and the second
term on the right-hand-side gives the leading contribution. The rest of
calculations is completely identical to the one for $SU(N)$ theories
Luscher:1981zq , and we can find the result by quoting them as

$q(n)=a^{4}\frac{1}{32\pi^{2}}\varepsilon_{\mu\nu\rho\sigma}\mathrm{tr}\left[\tilde{F}_{\mu\nu}(an)\tilde{F}_{\rho\sigma}(an)\right]+\mathcal{O}(a^{5}).$
(45)

The continuum expression for the topological charge is reproduced at the
leading order under the presence of generic $B_{p}$.

##  5 Summary and outlooks

We have constructed the topological charge on the lattice for $SU(N)$ gauge
fields coupled with $\mathbb{Z}_{N}$ $2$-form gauge fields. Introducing the
admissibility condition for the $SU(N)$ link variables with generic flat
$B_{p}$, we obtain a manifestly local, $SU(N)$ and $\mathbb{Z}_{N}$ $1$-form
gauge-invariant expression for the topological charge density. As an
application of this result, we provide the rigorous derivation of the mixed ’t
Hooft anomaly between the $\mathbb{Z}_{N}$ $1$-form symmetry and the $\theta$
periodicity. We also give the classical continuum limit of the topological
charge density, as the exact expression is so complicated for the practical
use. We introduce the gauge-invariant criterion for the validity of continuum
approximation, so one can use it with smeared gauge field if one ensures that
the admissibility is kept intact under the smearing process such as the
gradient flow.

In this work, we have treated $B_{p}$ as a flat background gauge field, but we
can perform its path integral to obtain the $SU(N)/\mathbb{Z}_{N}$ Yang–Mills
theory. Here, we can add the discrete theta angle as the local counter term,
so there are $N$ distinct $SU(N)/\mathbb{Z}_{N}$ Yang-Mills theory as
clarified in Ref. Aharony:2013hda . In the continuum analysis, it is suggested
that they have dyonic line operators as gauge-invariant line operators, and
their electric charge is specified by the choice of the discrete theta
parameter. In the lattice formulation, we need to violate the flatness
condition for $B_{p}$ along the dyonic line. It would be an interesting future
study to uncover various topological phenomena, such as the Witten effect
Witten:1979ey , on the lattice. As we have defined the admissible gauge fields
only for $SU(N)$ link variables coupled with the flat $2$-form gauge fields in
this paper, we would need further generalizations for studying those
phenomena.

It would also be interesting to extend this work to the case with fundamental
matter fields. In such cases, the flavor symmetry and the gauge redundancy has
the common center, and we may have a ’t Hooft anomaly between the projective
nature of the flavor symmetry and the $\theta$ periodicity Shimizu:2017asf ;
Gaiotto:2017tne ; Tanizaki:2017qhf . To obtain such an anomaly, the background
$2$-form gauge field plays an important role even though the $1$-form symmetry
is not present, so we expect that our result is useful to give the fully
lattice regularized derivation of those ’t Hooft anomalies.

###### Acknowledgements.

The authors appreciate the conference “Lattice and continuum field theories
2022” (YITP-W-22-02) at Yukawa Institute for Theoretical Physics (YITP) in the
last July. This work was partially supported by Japan Society for the
Promotion of Science (JSPS) Grant-in-Aid for Scientific Research Grant Numbers
JP21J30003 (O.M.), JP20H01903 (H.S.), JP22H01218, and JP20K22350 (Y.T.). The
work of Y.T. was supported by Center for Gravitational Physics and Quantum
Information (CGPQI) at YITP.

##  Appendix A Fractional power of special unitary matrices

In construction of transition functions, we need to take a fractional power
$U^{y}$ with $0\leq y\leq 1$ for special unitary matrices $U\in SU(N)$. In
this section, we give the definition including its domain and describe its
properties that are necessary in the discussion in the main text.

Let $\mathfrak{su}(N)$ be the Lie algebra of $SU(N)$, which is given by
$N\times N$ Hermitian traceless matrices. We have the exponential map,

$\exp:\mathfrak{su}(N)\to SU(N),\qquad X\mapsto\exp(\mathrm{i}X),$ (46)

and this map is surjective, which means that any $U\in SU(N)$ can be written
as $U=\exp(\mathrm{i}X)$ for some $X\in\mathfrak{su}(N)$. Thus, we are tempted
to define $U^{y}=\exp(\mathrm{i}yX)$, but this is not well defined as the
exponential map is not injective. To circumvent this issue, we restrict the
domain of the exponential map to the following subset,

$\mathcal{D}=\left\\{X\in\mathfrak{su}(N)\bigm{|}\text{eigenvalues of
$X\in(-\pi,\pi)$}\right\\},$ (47)

that is, when $X$ is diagonalized as
$\operatorname{diag}(\theta_{1},\dotsc,\theta_{N})$ with
$\theta_{1}+\dotsb+\theta_{N}=0$ then $X\in\mathcal{D}$ if and only if the
eigenvalues satisfy $|\theta_{k}|<\pi$ for all $k=1$, …, $N$. The exponential
map is injective on $\mathcal{D}$, and thus $U^{y}$ can be defined as

$U^{y}=\exp(\mathrm{i}yX)$ (48)

for $U=\exp(\mathrm{i}X)\in\exp(\mathrm{i}\mathcal{D})\subset SU(N)$. When
$U\in SU(N)\setminus\exp(\mathrm{i}\mathcal{D})$, we do not define $U^{y}$.

In order to get some insight, let us discuss what kinds of elements in $SU(N)$
are excluded from the definition of $U^{y}$. We can immediately see that the
nontrivial center elements, such as
$\mathrm{e}^{\frac{2\pi\mathrm{i}}{N}}\bm{1}\in SU(N)$, are excluded from the
domain since

$\mathrm{e}^{\frac{2\pi\mathrm{i}}{N}}\bm{1}=\exp\left[\frac{2\pi\mathrm{i}}{N}\operatorname{diag}(1,\dotsc,1,1-N)\right],$
(49)

which shows that $\theta_{N}=\frac{2\pi}{N}(1-N)$ does not fit the condition.
Our definition also excludes

$\operatorname{diag}(-1,-1,1,\dotsc,1)=\exp[\operatorname{diag}(\mathrm{i}\pi,-\mathrm{i}\pi,0,\dotsc,0)],$
(50)

which is a nontrivial center element for a subgroup $SU(2)\subset SU(N)$.
Especially when $N=2$, this is the element treated as the exceptional
configuration in Ref. Luscher:1981zq .

Let $V\in SU(N)$. Since the set of eigenvalues are not affected by the
conjugate operation, $VXV^{-1}\in\mathcal{D}$ if $X\in\mathcal{D}$. This shows
that, for $U\in\exp(\mathrm{i}\mathcal{D})\subset SU(N)$,

$\left(VUV^{-1}\right)^{y}=VU^{y}V^{-1}.$ (51)

This is the key property of $U^{y}$ for the discussions in the main text.

##  Appendix B Explicit formulas

In this appendix, we give the concrete expressions for the $1$-form gauge
invariant links in the complete axial gauge, $\tilde{u}^{n}_{xy}$, and the
$1$-form covariant transition function, $\tilde{v}_{n,\mu}(x)$, at the corner
of $f(n,\mu)$.

###  B.1 Formulas for $\tilde{u}^{n}_{xy}$

#### For $\mu=4$.

$\displaystyle\tilde{u}_{s_{2}s_{7}}^{n,n-\hat{4}}$
$\displaystyle=\mathrm{e}^{-\frac{2\pi\mathrm{i}}{N}B_{23}(n)}u_{s_{2}s_{7}}^{n,n-\hat{4}},$
$\displaystyle\tilde{u}_{s_{1}s_{5}}^{n,n-\hat{4}}$
$\displaystyle=\mathrm{e}^{-\frac{2\pi\mathrm{i}}{N}B_{13}(n)}u_{s_{1}s_{5}}^{n,n-\hat{4}},$
$\displaystyle\tilde{u}_{s_{5}s_{4}}^{n,n-\hat{4}}$
$\displaystyle=\mathrm{e}^{-\frac{2\pi\mathrm{i}}{N}B_{12}(n+\hat{3})}u_{s_{5}s_{4}}^{n,n-\hat{4}},$
$\displaystyle\tilde{u}_{s_{6}s_{4}}^{n,n-\hat{4}}$
$\displaystyle=\mathrm{e}^{-\frac{2\pi\mathrm{i}}{N}B_{23}(n)}\mathrm{e}^{-\frac{2\pi\mathrm{i}}{N}B_{13}(n+\hat{2})}u_{s_{6}s_{4}}^{n,n-\hat{4}},$
$\displaystyle\tilde{u}_{s_{1}s_{6}}^{n,n-\hat{4}}$
$\displaystyle=e^{-\frac{2\pi i}{N}B_{12}(n)}u_{s_{1}s_{6}}^{n,n-\hat{4}}.$
(52)

Others are trivial:
$\tilde{u}_{s_{0}s_{3}}^{n,n-\hat{4}}=\tilde{u}_{s_{3}s_{7}}^{n,n-\hat{4}}=\tilde{u}_{s_{0}s_{2}}^{n,n-\hat{4}}=\tilde{u}_{s_{3}s_{5}}^{n,n-\hat{4}}=\tilde{u}_{s_{0}s_{1}}^{n,n-\hat{4}}=\tilde{u}_{s_{7}s_{4}}^{n,n-\hat{4}}=\tilde{u}_{s_{2}s_{6}}^{n,n-\hat{4}}=\bm{1}$.

#### For $\mu=3$.

$\displaystyle\tilde{u}_{s_{2}s_{7}}^{n}$
$\displaystyle=\mathrm{e}^{-\frac{2\pi\mathrm{i}}{N}B_{24}(n)}u_{s_{2}s_{7}}^{n},$
$\displaystyle\tilde{u}_{s_{1}s_{5}}^{n}$
$\displaystyle=\mathrm{e}^{-\frac{2\pi\mathrm{i}}{N}B_{14}(n)}u_{s_{1}s_{5}}^{n},$
$\displaystyle\tilde{u}_{s_{5}s_{4}}^{n}$
$\displaystyle=\mathrm{e}^{-\frac{2\pi\mathrm{i}}{N}B_{12}(n+\hat{4})}u_{s_{5}s_{4}}^{n},$
$\displaystyle\tilde{u}_{s_{6}s_{4}}^{n}$
$\displaystyle=\mathrm{e}^{-\frac{2\pi\mathrm{i}}{N}B_{24}(n)}\mathrm{e}^{-\frac{2\pi\mathrm{i}}{N}B_{14}(n+\hat{2})}u_{s_{6}s_{4}}^{n},$
$\displaystyle\tilde{u}_{s_{1}s_{6}}^{n}$
$\displaystyle=\mathrm{e}^{-\frac{2\pi\mathrm{i}}{N}B_{12}(n)}u_{s_{1}s_{6}}^{n}.$
(53)

Others are trivial:
$\tilde{u}_{s_{0}s_{3}}^{n}=\tilde{u}_{s_{3}s_{7}}^{n}=\tilde{u}_{s_{0}s_{2}}^{n}=\tilde{u}_{s_{3}s_{5}}^{n}=\tilde{u}_{s_{0}s_{1}}^{n}=\tilde{u}_{s_{7}s_{4}}^{n}=\tilde{u}_{s_{2}s_{6}}^{n}=\bm{1}$.

$\displaystyle\tilde{u}_{s_{0}s_{3}}^{n-\hat{3}}$
$\displaystyle=\mathrm{e}^{-\frac{2\pi\mathrm{i}}{N}B_{34}(n-\hat{3})}u_{s_{0}s_{3}}^{n-\hat{3}},$
$\displaystyle\tilde{u}_{s_{2}s_{7}}^{n-\hat{3}}$
$\displaystyle=\mathrm{e}^{-\frac{2\pi\mathrm{i}}{N}B_{34}(n-\hat{3})}\mathrm{e}^{-\frac{2\pi\mathrm{i}}{N}B_{24}(n)}u_{s_{2}s_{7}}^{n-\hat{3}},$
$\displaystyle\tilde{u}_{s_{1}s_{5}}^{n-\hat{3}}$
$\displaystyle=\mathrm{e}^{-\frac{2\pi\mathrm{i}}{N}B_{34}(n-\hat{3})}\mathrm{e}^{-\frac{2\pi\mathrm{i}}{N}B_{14}(n)}u_{s_{1}s_{5}}^{n-\hat{3}},$
$\displaystyle\tilde{u}_{s_{5}s_{4}}^{n-\hat{3}}$
$\displaystyle=\mathrm{e}^{-\frac{2\pi\mathrm{i}}{N}B_{12}(n+\hat{4})}u_{s_{5}s_{4}}^{n-\hat{3}},$
$\displaystyle\tilde{u}_{s_{6}s_{4}}^{n-\hat{3}}$
$\displaystyle=\mathrm{e}^{-\frac{2\pi\mathrm{i}}{N}B_{34}(n-\hat{3})}\mathrm{e}^{-\frac{2\pi\mathrm{i}}{N}B_{24}(n)}\mathrm{e}^{-\frac{2\pi\mathrm{i}}{N}B_{14}(n+\hat{2})}u_{s_{6}s_{4}}^{n-\hat{3}},$
$\displaystyle\tilde{u}_{s_{1}s_{6}}^{n-\hat{3}}$
$\displaystyle=\mathrm{e}^{-\frac{2\pi\mathrm{i}}{N}B_{12}(n)}u_{s_{1}s_{6}}^{n-\hat{3}}.$
(54)

Others are trivial:
$\tilde{u}_{s_{3}s_{7}}^{n-\hat{3}}=\tilde{u}_{s_{0}s_{2}}^{n-\hat{3}}=\tilde{u}_{s_{3}s_{5}}^{n-\hat{3}}=\tilde{u}_{s_{0}s_{1}}^{n-\hat{3}}=\tilde{u}_{s_{7}s_{4}}^{n-\hat{3}}=\tilde{u}_{s_{2}s_{6}}^{n-\hat{3}}=\bm{1}$.

#### For $\mu=2$.

$\displaystyle\tilde{u}_{s_{2}s_{7}}^{n}$
$\displaystyle=\mathrm{e}^{-\frac{2\pi\mathrm{i}}{N}B_{34}(n)}u_{s_{2}s_{7}}^{n},$
$\displaystyle\tilde{u}_{s_{1}s_{5}}^{n}$
$\displaystyle=\mathrm{e}^{-\frac{2\pi\mathrm{i}}{N}B_{14}(n)}u_{s_{1}s_{5}}^{n},$
$\displaystyle\tilde{u}_{s_{5}s_{4}}^{n}$
$\displaystyle=\mathrm{e}^{-\frac{2\pi\mathrm{i}}{N}B_{13}(n+\hat{4})}u_{s_{5}s_{4}}^{n},$
$\displaystyle\tilde{u}_{s_{6}s_{4}}^{n}$
$\displaystyle=\mathrm{e}^{-\frac{2\pi\mathrm{i}}{N}B_{34}(n)}\mathrm{e}^{-\frac{2\pi\mathrm{i}}{N}B_{14}(n+\hat{3})}u_{s_{6}s_{4}}^{n},$
$\displaystyle\tilde{u}_{s_{1}s_{6}}^{n}$
$\displaystyle=\mathrm{e}^{-\frac{2\pi\mathrm{i}}{N}B_{13}(n)}u_{s_{1}s_{6}}^{n}.$
(55)

Others are trivial:
$\tilde{u}_{s_{0}s_{3}}^{n}=\tilde{u}_{s_{3}s_{7}}^{n}=\tilde{u}_{s_{0}s_{2}}^{n}=\tilde{u}_{s_{3}s_{5}}^{n}=\tilde{u}_{s_{0}s_{1}}^{n}=\tilde{u}_{s_{7}s_{4}}^{n}=\tilde{u}_{s_{2}s_{6}}^{n}=\bm{1}$.

$\displaystyle\tilde{u}_{s_{0}s_{3}}^{n-\hat{2}}$
$\displaystyle=\mathrm{e}^{-\frac{2\pi\mathrm{i}}{N}B_{24}(n-\hat{2})}u_{s_{0}s_{3}}^{n-\hat{2}},$
$\displaystyle\tilde{u}_{s_{3}s_{7}}^{n-\hat{2}}$
$\displaystyle=\mathrm{e}^{-\frac{2\pi\mathrm{i}}{N}B_{23}(n-\hat{2}+\hat{4})}u_{s_{3}s_{7}}^{n-\hat{2}},$
$\displaystyle\tilde{u}_{s_{2}s_{7}}^{n-\hat{2}}$
$\displaystyle=\mathrm{e}^{-\frac{2\pi\mathrm{i}}{N}B_{34}(n-\hat{2})}\mathrm{e}^{-\frac{2\pi\mathrm{i}}{N}B_{24}(n-\hat{2}+\hat{3})}u_{s_{2}s_{7}}^{n-\hat{2}},$
$\displaystyle\tilde{u}_{s_{0}s_{2}}^{n-\hat{2}}$
$\displaystyle=\mathrm{e}^{-\frac{2\pi\mathrm{i}}{N}B_{23}(n-\hat{2})}u_{s_{0}s_{2}}^{n-\hat{2}},$
$\displaystyle\tilde{u}_{s_{1}s_{5}}^{n-\hat{2}}$
$\displaystyle=\mathrm{e}^{-\frac{2\pi\mathrm{i}}{N}B_{24}(n-\hat{2})}\mathrm{e}^{-\frac{2\pi\mathrm{i}}{N}B_{14}(n)}u_{s_{1}s_{5}}^{n-\hat{2}},$
$\displaystyle\tilde{u}_{s_{5}s_{4}}^{n-\hat{2}}$
$\displaystyle=\mathrm{e}^{-\frac{2\pi\mathrm{i}}{N}B_{23}(n-\hat{2}+\hat{4})}\mathrm{e}^{-\frac{2\pi\mathrm{i}}{N}B_{13}(n+\hat{4})}u_{s_{5}s_{4}}^{n-\hat{2}},$
$\displaystyle\tilde{u}_{s_{6}s_{4}}^{n-\hat{2}}$
$\displaystyle=\mathrm{e}^{-\frac{2\pi\mathrm{i}}{N}B_{34}(n-\hat{2})}\mathrm{e}^{-\frac{2\pi\mathrm{i}}{N}B_{24}(n-\hat{2}+\hat{3})}\mathrm{e}^{-\frac{2\pi\mathrm{i}}{N}B_{14}(n+\hat{3})}u_{s_{6}s_{4}}^{n-\hat{2}},$
$\displaystyle\tilde{u}_{s_{1}s_{6}}^{n-\hat{2}}$
$\displaystyle=\mathrm{e}^{-\frac{2\pi\mathrm{i}}{N}B_{23}(n-\hat{2})}e^{-\frac{2\pi
i}{N}B_{13}(n)}u_{s_{1}s_{6}}^{n-\hat{2}}.$ (56)

Others are trivial:
$\tilde{u}_{s_{3}s_{5}}^{n-\hat{2}}=\tilde{u}_{s_{0}s_{1}}^{n-\hat{2}}=\tilde{u}_{s_{7}s_{4}}^{n-\hat{2}}=\tilde{u}_{s_{2}s_{6}}^{n-\hat{2}}=\bm{1}$.

#### For $\mu=1$.

$\displaystyle\tilde{u}_{s_{2}s_{7}}^{n}$
$\displaystyle=\mathrm{e}^{-\frac{2\pi\mathrm{i}}{N}B_{34}(n)}u_{s_{2}s_{7}}^{n},$
$\displaystyle\tilde{u}_{s_{1}s_{5}}^{n}$
$\displaystyle=\mathrm{e}^{-\frac{2\pi\mathrm{i}}{N}B_{24}(n)}u_{s_{1}s_{5}}^{n},$
$\displaystyle\tilde{u}_{s_{5}s_{4}}^{n}$
$\displaystyle=\mathrm{e}^{-\frac{2\pi\mathrm{i}}{N}B_{23}(n+\hat{4})}u_{s_{5}s_{4}}^{n},$
$\displaystyle\tilde{u}_{s_{6}s_{4}}^{n}$
$\displaystyle=\mathrm{e}^{-\frac{2\pi\mathrm{i}}{N}B_{34}(n)}\mathrm{e}^{-\frac{2\pi\mathrm{i}}{N}B_{24}(n+\hat{3})}u_{s_{6}s_{4}}^{n},$
$\displaystyle\tilde{u}_{s_{1}s_{6}}^{n}$
$\displaystyle=\mathrm{e}^{-\frac{2\pi\mathrm{i}}{N}B_{23}(n)}u_{s_{1}s_{6}}^{n}.$
(57)

Others are trivial:
$\tilde{u}_{s_{0}s_{3}}^{n}=\tilde{u}_{s_{3}s_{7}}^{n}=\tilde{u}_{s_{0}s_{2}}^{n}=\tilde{u}_{s_{3}s_{5}}^{n}=\tilde{u}_{s_{0}s_{1}}^{n}=\tilde{u}_{s_{7}s_{4}}^{n}=\tilde{u}_{s_{2}s_{6}}^{n}=\bm{1}$.

$\displaystyle\tilde{u}_{s_{0}s_{3}}^{n-\hat{1}}$
$\displaystyle=\mathrm{e}^{-\frac{2\pi\mathrm{i}}{N}B_{14}(n-\hat{1})}u_{s_{0}s_{3}}^{n-\hat{1}},$
$\displaystyle\tilde{u}_{s_{3}s_{7}}^{n-\hat{1}}$
$\displaystyle=\mathrm{e}^{-\frac{2\pi\mathrm{i}}{N}B_{13}(n-\hat{1}+\hat{4})}u_{s_{3}s_{7}}^{n-\hat{1}},$
$\displaystyle\tilde{u}_{s_{2}s_{7}}^{n-\hat{1}}$
$\displaystyle=\mathrm{e}^{-\frac{2\pi\mathrm{i}}{N}B_{34}(n-\hat{1})}\mathrm{e}^{-\frac{2\pi\mathrm{i}}{N}B_{14}(n-\hat{1}+\hat{3})}u_{s_{2}s_{7}}^{n-\hat{1}},$
$\displaystyle\tilde{u}_{s_{0}s_{2}}^{n-\hat{1}}$
$\displaystyle=\mathrm{e}^{-\frac{2\pi\mathrm{i}}{N}B_{13}(n-\hat{1})}u_{s_{0}s_{2}}^{n-\hat{1}},$
$\displaystyle\tilde{u}_{s_{1}s_{5}}^{n-\hat{1}}$
$\displaystyle=\mathrm{e}^{-\frac{2\pi\mathrm{i}}{N}B_{24}(n-\hat{1})}\mathrm{e}^{-\frac{2\pi\mathrm{i}}{N}B_{14}(n-\hat{1}+\hat{2})}u_{s_{1}s_{5}}^{n-\hat{1}},$
$\displaystyle\tilde{u}_{s_{5}s_{4}}^{n-\hat{1}}$
$\displaystyle=\mathrm{e}^{-\frac{2\pi\mathrm{i}}{N}B_{23}(n-\hat{1}+\hat{4})}\mathrm{e}^{-\frac{2\pi\mathrm{i}}{N}B_{13}(n-\hat{1}+\hat{2}+\hat{4})}u_{s_{5}s_{4}}^{n-\hat{1}},$
$\displaystyle\tilde{u}_{s_{6}s_{4}}^{n-\hat{1}}$
$\displaystyle=\mathrm{e}^{-\frac{2\pi\mathrm{i}}{N}B_{34}(n-\hat{1})}\mathrm{e}^{-\frac{2\pi\mathrm{i}}{N}B_{24}(n-\hat{1}+\hat{3})}\mathrm{e}^{-\frac{2\pi\mathrm{i}}{N}B_{14}(n-\hat{1}+\hat{2}+\hat{3})}u_{s_{6}s_{4}}^{n-\hat{1}},$
$\displaystyle\tilde{u}_{s_{1}s_{6}}^{n-\hat{1}}$
$\displaystyle=\mathrm{e}^{-\frac{2\pi\mathrm{i}}{N}B_{23}(n-\hat{1})}\mathrm{e}^{-\frac{2\pi\mathrm{i}}{N}B_{13}(n-\hat{1}+\hat{2})}u_{s_{1}s_{6}}^{n-\hat{1}},$
$\displaystyle\tilde{u}_{s_{3}s_{5}}^{n-\hat{1}}$
$\displaystyle=\mathrm{e}^{-\frac{2\pi\mathrm{i}}{N}B_{12}(n-\hat{1}+\hat{4})}u_{s_{3}s_{5}}^{n-\hat{1}},$
$\displaystyle\tilde{u}_{s_{0}s_{1}}^{n-\hat{1}}$
$\displaystyle=\mathrm{e}^{-\frac{2\pi\mathrm{i}}{N}B_{12}(n-\hat{1})}u_{s_{0}s_{1}}^{n-\hat{1}},$
$\displaystyle\tilde{u}_{s_{7}s_{4}}^{n-\hat{1}}$
$\displaystyle=\mathrm{e}^{-\frac{2\pi\mathrm{i}}{N}B_{12}(n-\hat{1}+\hat{3}+\hat{4})}u_{s_{7}s_{4}}^{n-\hat{1}},$
$\displaystyle\tilde{u}_{s_{2}s_{6}}^{n-\hat{1}}$
$\displaystyle=\mathrm{e}^{-\frac{2\pi\mathrm{i}}{N}B_{12}(n-\hat{1}+\hat{3})}u_{s_{2}s_{6}}^{n-\hat{1}}.$
(58)

###  B.2 Formulas for $\tilde{v}_{n,\mu}(x)$

We find that the following expressions fulfill the requirements in the main
text:

#### For $\mu=4$

$\tilde{v}_{n,4}(x)=v_{n,4}(x).$ (59)

#### For $\mu=3$

$\tilde{v}_{n,3}(x)=\begin{cases}v_{n,3}(x),&\text{for $x=s_{0}$, $s_{1}$,
$s_{2}$, $s_{6}$},\\\
\mathrm{e}^{\frac{2\pi\mathrm{i}}{N}B_{34}(n-\hat{3})}v_{n,3}(x),&\text{for
$x=s_{3}$, $s_{4}$, $s_{5}$, $s_{7}$}.\\\ \end{cases}$ (60)

#### For $\mu=2$

$\tilde{v}_{n,2}(x)=\begin{cases}v_{n,2}(x),&\text{for $x=s_{0}$, $s_{1}$},\\\
\mathrm{e}^{\frac{2\pi\mathrm{i}}{N}B_{23}(n-\hat{2})}v_{n,2}(x),&\text{for
$x=s_{2}$, $s_{6}$}.\\\
\mathrm{e}^{\frac{2\pi\mathrm{i}}{N}B_{24}(n-\hat{2})}v_{n,2}(x),&\text{for
$x=s_{3}$, $s_{5}$}.\\\
\mathrm{e}^{\frac{2\pi\mathrm{i}}{N}B_{24}(n-\hat{2})}\mathrm{e}^{\frac{2\pi\mathrm{i}}{N}B_{23}(n-\hat{2}+\hat{4})}v_{n,2}(x),&\text{for
$x=s_{4}$, $s_{7}$}.\\\ \end{cases}$ (61)

#### For $\mu=1$

$\tilde{v}_{n,1}(x)=\begin{cases}v_{n,1}(x),&\text{for $x=s_{0}$},\\\
\mathrm{e}^{\frac{2\pi\mathrm{i}}{N}B_{12}(n-\hat{1})}v_{n,1}(x),&\text{for
$x=s_{1}$},\\\
\mathrm{e}^{\frac{2\pi\mathrm{i}}{N}B_{13}(n-\hat{1})}v_{n,1}(x),&\text{for
$x=s_{2}$},\\\
\mathrm{e}^{\frac{2\pi\mathrm{i}}{N}B_{14}(n-\hat{1})}v_{n,1}(x),&\text{for
$x=s_{3}$},\\\
\mathrm{e}^{\frac{2\pi\mathrm{i}}{N}B_{14}(n-\hat{1})}\mathrm{e}^{\frac{2\pi\mathrm{i}}{N}B_{13}(n-\hat{1}+\hat{4})}\mathrm{e}^{\frac{2\pi\mathrm{i}}{N}B_{12}(n-\hat{1}+\hat{3}+\hat{4})}v_{n,1}(x),&\text{for
$x=s_{4}$},\\\
\mathrm{e}^{\frac{2\pi\mathrm{i}}{N}B_{14}(n-\hat{1})}\mathrm{e}^{\frac{2\pi\mathrm{i}}{N}B_{12}(n-\hat{1}+\hat{4})}v_{n,1}(x),&\text{for
$x=s_{5}$},\\\
\mathrm{e}^{\frac{2\pi\mathrm{i}}{N}B_{13}(n-\hat{1})}\mathrm{e}^{\frac{2\pi\mathrm{i}}{N}B_{12}(n-\hat{1}+\hat{3})}v_{n,1}(x),&\text{for
$x=s_{6}$},\\\
\mathrm{e}^{\frac{2\pi\mathrm{i}}{N}B_{14}(n-\hat{1})}\mathrm{e}^{\frac{2\pi\mathrm{i}}{N}B_{13}(n-\hat{1}+\hat{4})}v_{n,1}(x),&\text{for
$x=s_{7}$}.\\\ \end{cases}$ (62)

##  Appendix C Bound on $\varepsilon$ for the admissibility condition

Let $V$ be the vector space with the norm $\|\cdot\|$. For a linear map
$A:V\to V$, its matrix norm is defined by

$\|A\|:=\sup_{\psi\in
V,\|\psi\|=1}\|A\psi\|=\sqrt{\lambda_{\mathrm{max}}(A^{\dagger}A)},$ (63)

where $\lambda_{\mathrm{max}}$ is the maximum eigenvalue. For $U^{y}$ with
$0\leq y\leq 1$ being well-defined, $U\in\exp(\mathrm{i}\mathcal{D})$ as
discussed in Eq. (47). We can characterize $U\in\exp(\mathrm{i}\mathcal{D})$
using the matrix norm in the following way:

$U\in\exp(\mathrm{i}\mathcal{D})\Leftrightarrow\begin{aligned} &\text{There
exists a map $t\in[0,1]\mapsto U(t)\in SU(N)$ such that}\\\
&\text{$U(0)=\bm{1}$, $U(1)=U$, and $\|U(t)-\bm{1}\|<2$ for all
$t\in[0,1]$.}\end{aligned}$ (64)

We can show $\Rightarrow$ by taking $U(t)=U^{t}$. To show $\Leftarrow$, let us
diagonalize $U\in SU(N)\setminus\exp(\mathrm{i}\mathcal{D})$, then their
eigenvalues $\mathrm{e}^{\mathrm{i}\theta_{i}}$ with the choice of the branch
$\theta_{1}+\dotsb+\theta_{N}=0$ should have $\max_{i}|\theta_{i}|\geq\pi$.
Then, there is no way to circumvent $\max_{i}|\theta_{i}(t_{*})|=\pi$ for some
$t=t_{*}$ when we connect $\bm{1}$ and $U$, at which $\|U(t_{*})-\bm{1}\|=2$.

Let $U_{1}$ and $U_{2}$ be unitary matrices. We pick up $\psi\in V$ with
$\|\psi\|=1$, and then

$\displaystyle\|(U_{1}U_{2}-\bm{1})\psi\|$
$\displaystyle=\|(U_{1}-\bm{1})U_{2}\psi+(U_{2}-\bm{1})\psi\|$
$\displaystyle\leq\|(U_{1}-\bm{1})U_{2}\psi\|+\|(U_{2}-\bm{1})\psi\|$
$\displaystyle\leq\|U_{1}-\bm{1}\|+\|U_{2}-\bm{1}\|.$ (65)

By taking $\sup_{\psi}$ of the both sides, we obtain the useful inequality,

$\|U_{1}U_{2}\dotsb U_{n}-\bm{1}\|\leq\sum_{i=1}^{n}\|U_{i}-\bm{1}\|.$ (66)

Combined with Eqs. (64) and (66), we find that for $U_{1}$, …,
$U_{n}\in\exp(\mathrm{i}\mathcal{D})$,

$\|U_{i}-\bm{1}\|<\frac{2}{n}\Rightarrow U_{1}\dotsb
U_{n}\in\exp(\mathrm{i}\mathcal{D}).$ (67)

To prove it, let us set $U(t)=U_{1}^{t}\dotsb U_{n}^{t}$, then $U(0)=\bm{1}$
and $U(1)=U_{1}\dotsb U_{n}$. Since
$\|U(t)-\bm{1}\|\leq\sum_{i}\|U_{i}^{t}-\bm{1}\|\leq\sum_{i}\|U_{i}-\bm{1}\|<\frac{2}{n}\cdot
n=2$, we obtain the result from Eq. (64).

When we have products of $M$ plaquettes $\tilde{U}_{p_{i}}$ ($i=1$, …, $M$),
we can take its fractional power
$(\tilde{U}_{p_{1}}\dotsb\tilde{U}_{p_{M}})^{y}$ if we impose the
admissibility condition $\|\tilde{U}_{p_{i}}-\bm{1}\|<\varepsilon$ with some
$\varepsilon\leq 2/M$ thanks to Eq. (67). The transition functions
$\tilde{v}_{n,\mu}(x)$ are defined via
$\tilde{S}_{n,\mu}^{m}(x_{\alpha},x_{\beta},x_{\gamma})$ given in Eq. (25),
and the most severe bound for $\varepsilon$ is obtained by the number of
plaquettes appearing in $\tilde{l}_{n,\mu}^{m}(x_{\beta},x_{\gamma})$ for
$m=n-\hat{1}$. The number of plaquettes contained in $\tilde{u}_{xy}^{m}$ can
be counted by the number of $B_{\mu\nu}$ given in Appendix B.1. We further
note that $\tilde{f}$, $\tilde{g}$, $\tilde{h}$, $\tilde{k}$ contain the
combinations such as

$\tilde{u}_{s_{0}s_{3}}^{n-\hat{1}}\tilde{u}_{s_{3}s_{7}}^{n-\hat{1}}\tilde{u}_{s_{7}s_{2}}^{n-\hat{1}}\tilde{u}_{s_{2}s_{0}}^{n-\hat{1}},$
(68)

and this is nothing but the plaquette $p=\langle s_{0}\to s_{3}\to s_{7}\to
s_{2}\to s_{0}\rangle$ in the complete axial gauge. Thus, we can simply count
it as one plaquette, and we then find that there are $M=27$ plaquettes in the
$\tilde{l}_{n,\mu}^{n-\hat{1}}$. This shows that we can take the admissibility
condition given with

$\varepsilon<\frac{2}{27}\simeq 0.074.$ (69)

This bound is larger than that for the overlap Dirac operator in Ref.
Hernandez:1998et ,

$\varepsilon_{\mathrm{overlap}}<\frac{1}{30}\simeq 0.033,$ (70)

but smaller than Lüscher’s assertion in Ref. Luscher:1981zq :
$\varepsilon<\sqrt{0.03}\simeq 0.173$. We point out that $\tilde{f}$,
$\tilde{g}$, $\tilde{k}$, $\tilde{l}$, $\tilde{S}$ take the form of
$U_{1}^{-y}(U_{1}U_{2}U_{3}^{-1}U_{4}^{-1})^{y}U_{4}U_{3}^{y}=U_{1}^{1-y}U_{2}(U_{2}^{-1}U_{1}^{-1}U_{4}U_{3})^{1-y}U_{3}^{y-1}$.
We can obtain larger bounds for $\varepsilon$ by estimating their norm as a
function of $y_{\beta}$ and $y_{\gamma}$ as it improves the above discussion,
but we do not pursue it here.

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11institutetext: Korea University 22institutetext: The Catholic University of
Korea, Seoul 33institutetext: MedAI
33email<EMAIL_ADDRESS>
# Exploring 3D U-Net Training Configurations and Post-Processing Strategies
for the MICCAI 2023 Kidney and Tumor Segmentation Challenge

Kwang-Hyun Uhm 11 Hyunjun Cho 11 Zhixin Xu 11 Seohoon Lim 11 Seung-Won Jung 11
Sung-Hoo Hong 22 Sung-Jea Ko ✉ 11 3 3

###### Abstract

In 2023, it is estimated that 81,800 kidney cancer cases will be newly
diagnosed, and 14,890 people will die from this cancer in the United States.
Preoperative dynamic contrast-enhanced abdominal computed tomography (CT) is
often used for detecting lesions. However, there exists inter-observer
variability due to subtle differences in the imaging features of kidney and
kidney tumors. In this paper, we explore various 3D U-Net training
configurations and effective post-processing strategies for accurate
segmentation of kidneys, cysts, and kidney tumors in CT images. We validated
our model on the dataset of the 2023 Kidney and Kidney Tumor Segmentation
(KiTS23) challenge. Our method took the second place in the final ranking of
KiTS23 challenge on unseen test data with an average Dice score of 0.820 and
an average Surface Dice of 0.712.

###### Keywords:

Kidney cancer Medical image segmentation 3D U-Net.

##  1 Introduction

In 2023, it is estimated that 81,800 kidney cancer cases will be newly
diagnosed, and 14,890 people will die from this cancer in the United States
[1]. Kidney cancer is one of the 10 most common cancers, and by far the most
common type of kidney cancer is renal cell carcinoma (RCC), which occurs in 9
out of 10 cases of all kidney cancer [7]. Preoperative dynamic contrast-
enhanced abdominal computed tomography (CT) is often used for the detection
and evaluation of renal tumors [6]. However, there are some overlaps in image-
level features between kidneys, cysts, and renal tumors, which make accurate
segmentation difficult and cause inter-observer variation. These clinical
issues point to the need to develop automatic systems that can reduce
misdiagnosis and inter-observer variation.

In this paper, we explore various 3D U-Net training configurations and
effective post-processing strategies for accurate segmentation of kidneys,
cysts, and kidney tumors in CT images. We investigate a wide variety of
training configurations including training at different scales, cascade
training approaches, and region-based training. We also introduce post-
processing approaches which aim at improving the performance by effectively
combining the predictions from the models trained in different training
configurations. We validated our model on the dataset of 2023 Kidney and
Kidney Tumor Segmentation (KiTS23) challenge, which includes data from
previous challenges, e.g., KiTS19 [3].

##  2 Methods

Figure 1: Overview of our multi-scale prediction fusion framework.

In this paper, we propose to perform segmentation at both low-resolution and
full-resolution and then combine those two predictions based on a task-
specific post-processing scheme, as shown in Fig. 1. Low-resolution 3D CT
volumes are generated by resampling original input CT images. Two independent
3D U-Nets are utilized to produce low-resolution segmentation maps and full-
resolution segmentation maps from low-resolution 3D CT volume and full-
resolution CT volume, respectively. The final segmentation map is produced by
multi-scale post-processing which takes the predictions of low- and full-
resolution segmentation maps and combines them with domain-specific
procedures.

###  2.1 Training and Validation Data

Our submission made use of the official KiTS23 training set alone [4]. We
divide the provided data into a training set and a validation set at a ratio
of 4:1.

###  2.2 Preprocessing

We follow the way in the nnU-Net [5] to preprocess the training data. The
spacing of all official CT images is the same on the x-axis and y-axis, but
different on the z-axis. The original training data have a median voxel
spacing of 0.78 × 0.78 × 3.0 mm3, and the median volume shape of 512 × 512 ×
104 voxels. For the training of low-resolution segmentation network, input
data is resampled to have a spacing of 1.84 × 1.84 × 2.36 mm3, which results
in a median volume of 217 × 217 × 177 mm3 voxels. On the ohter hand, for the
training of full-resolution segmentation network, input data is resampled to
have a spacing of 0.78 × 0.78 × 1.0 mm3, which results in a median volume of
512 × 512 × 417 mm3 voxels.

We clip each case’s intensity values to the 0.5 and 99.5 percentiles of the
intensity values in the foreground regions across the training set, i.e., the
range of [-58, 302]. We subtract the mean value of 103 and then divide it by
the standard deviation of the intensities in the foreground regions, which is
73.3. The foreground class oversampling is used to enforce more than a third
of the samples in a batch contain at least one randomly chosen foreground
class. During training, patches with shape 128 × 128 × 128 are sampled and
input to the network. A variety of data augmentation techniques are applied on
the fly during training: rotations, scaling, mirroring, etc.

###  2.3 Proposed Method

####  2.3.1 Network Architecture

We use 3D U-Net architecture [2] for both low-resolution and full-resolution
segmentation networks. The U-Net consists of an encoder and decoder, where for
all convolutions in the network we use 3 × 3 × 3 convolution kernels. Each
block in the encoder consists of a sequence of two convolutional layers each
of which followed by instance normalization and LeakyReLU activations. In the
decoder, upsampling is done by 3 × 3 × 3 transposed convolutions. At the last
convolutional layer, it outputs the probability distributions for background,
kidney, cyst, and tumor for each voxel. There are a total of 6 stages for each
encoder and decoder. We use cross-entropy loss and dice loss for the training.
The stochastic gradient descent strategy is used for the optimization. We
investigate various network configurations. For example, we increase all the
channel numbers of convolutions in the network by two times. This increases
the model parameter by two times, and also the training time. Another variant
is a residual 3D U-Net which replaces plain convolution blocks in the encoder
with residual blocks. Also, we tested the region-based training, which uses
three sigmoid activations after the final convolutional layer to produce
probabilities for each region, where regions are defined by ”kidney and
masses”, ”masses”, and ”tumor”. For the cascade model, the output of the low-
resolution segmentation network is concatenated to the input image and then
fed to the high-resolution segmentation network.

####  2.3.2 Multi-Scale Post-Processing

Based on our analysis of the results of the validation set, the low-resolution
segmentation network produces well-localized segmentation parts but lacks
sufficient details, while the full-resolution segmentation model provides
finely detailed boundaries of kidneys and masses but generates some false
positives around backgrounds. Therefore, we remove segmented foreground blobs
in full-resolution segmentation maps which do not belong to the foreground
blobs in low-resolution segmentation maps. Moreover, to reduce tumor false
positives (FPs), we first perform the connected component analysis for the
tumor class, and then we treat the tumor regions in each segmentation map as
true positives if they have sufficient overlap with the tumor regions of the
segmentation map from another scale. Specifically, the two tumor regions from
low- and full-resolution segmentation maps should have a Dice coefficient
greater than 0.3 to be determined as true tumor regions. Finally, we join the
regions of predicted segmentation parts from both the full-resolution and low-
resolution segmentation maps to complement each other. This enables us to take
advantage of both the low-resolution and high-resolution segmentation
predictions, boosting the final segmentation performance. In addition, we find
the convex hulls of tumor blobs in the predicted segmentation maps and then
merge the labels inside the detected hulls to reduce noisy prediction results.
We remove foreground blobs that have an area smaller than 10,000 mm3.

Table 1: Results of experiments on the validation set. Method |  Kidney |  Masses |  Tumor |  Average
---|---|---|---|---
Low-resolution |  0.973 |  0.858 |  0.794 |  0.875
Low-resolution - channel$\times$2 |  0.973 |  0.851 |  0.771 |  0.865
Low-resolution - residual |  0.973 |  0.856 |  0.794 |  0.874
Full-resolution |  0.977 |  0.840 |  0.790 |  0.869
Full-resolution - channel$\times$2 |  0.975 |  0.840 |  0.790 |  0.868
Full-resolution - residual |  0.979 |  0.858 |  0.803 |  0.880
Full-resolution - batch 4 |  0.978 |  0.857 |  0.801 |  0.879
Cascade |  0.979 |  0.858 |  0.804 |  0.880
Region-based training |  0.975 |  0.851 |  0.790 |  0.872
Ours (w/ post-processing) |  0.979 |  0.857 |  0.826 |  0.887

##  3 Results

We validated our model on the dataset of the 2023 Kidney and Kidney Tumor
Segmentation (KiTS23) challenge. We report performances of baselines and our
models evaluated on the validation set. We report Dice scores for regions of
”Kidney and Masses”, ”Kidney Masses”, and ”Kidney Tumor”. For brevity, we
denote in the tables in this section the dice scores as ”Kidney”, ”Masses”,
and ”Tumor”.

We summarize the quantitative results in Tab 1. All the results are based on
the validation set, which contains 98 cases. We can see that our method
outperforms other baselines by a large margin in terms of average Dice score.
The average Dice is 0.887, and Dice for kidney, kidney masses, and kidney
tumors are respectively 0.979, 0.857, and 0.826. For the tumor segmentation,
our algorithm performs significantly better than the baseline. These results
demonstrate the effectiveness of our multi-scale post-processing strategy.

Our final submission involves three base networks including “Low-resolution”,
“Low-resolution - residual”, and “Full-resolution - batch 4” which show better
performance than others in several folds. We apply our multi-scale post-
processing strategy to two possible pairs of low- and full-resolution
segmentation results and join the two post-processed results to make the final
segmentation map.

Table 2: KiTS23 leaderboard for final results on test data (top-5). Rank |  Team |  Affiliation |  Dice |  Surface Dice |  Tumor Dice
---|---|---|---|---|---
1 |  Andriy Myronenko et al. |  NVIDIA |  0.835 |  0.723 |  0.756
2 |  Kwang-Hyun Uhm |  Korea University |  0.820 |  0.712 |  0.738
3 |  Yasmeen George |  Monash University |  0.819 |  0.707 |  0.713
4 |  Shuolin Liu |  Independent Researcher |  0.805 |  0.706 |  0.697
5 |  George Stoica et al. |  University of Lasi, SenticLab |  0.807 |  0.691 |  0.713

##  4 Discussion and Conclusion

In this paper, we explore various 3D U-Net training configurations and
effective post-processing strategies for accurate segmentation of kidneys,
cysts, and kidney tumors in CT images. We investigate a wide variety of
training configurations including training at different scales, cascade
training approaches, and region-based training. We also introduce post-
processing approaches that aim at improving performance by effectively
combining the predictions from the models trained in different training
configurations. We validated our model on the dataset of 2023 Kidney and
Kidney Tumor Segmentation (KiTS23) challenge.

Our approach won second place in KiTS23 Challenge. On the test set, our final
model obtained Dice scores of 0.948, 0.776, and 0.738 and Surface Dice (SD)
scores of 0.899, 0.635 and 0.602 for kidney, masses, and tumors, respectively.
It is important to note that by carefully combining the results of 3D U-Nets
trained from different scale settings, we can obtain much better performance
than the results of individual models. Specifically, we achieved a much higher
tumor dice score than lower-ranked teams, which demonstrates that our multi-
scale tumor prediction aggregation strategy is effective for accurate tumor
segmentation. Our method can be applied to other multi-scale approaches in
current literature to improve the segmentation performance for accurate
diagnosis. Our work is only validated on KiTS23 Challenge dataset and lacks
sufficient experimental validation on multiple datasets. In the future, we
will expand the experiment settings of our method to different datasets for
comprehensive evaluation of generality.

## Acknowledgment

We thank the KiTS competition organizers, data providers, and annotators for
their great effort in the challenge. We further thank the creator and
contributors to the nnU-Net framework.

This work was supported by the Korea Medical Device Development Fund grant
funded by the Korea government (the Ministry of Science and ICT, the Ministry
of Trade, Industry and Energy, the Ministry of Health & Welfare, the Ministry
of Food and Drug Safety) (Project Number: 1711195432, RS-2020-KD000096).

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