Fractional topological charge in lattice Abelian gauge theory

We consider a 4D periodic torus $T^{4}$ of size $L$; the Lorentz index is denoted by $\mu$, $\nu$, …, etc. and runs over $1$, $2$, $3$, and $4$:

$$T^{4}\equiv\left\{x\in\mathbb{R}^{4}\mid\text{$0\leq x_{\mu}<L$ for all $\mu$}\right\}.$$

(2.1)

That is, any two points $x$ and $y$ whose coordinates differ by an integer multiple of $L$ are identified, $x\sim y$.

We then consider a 4D lattice $\Lambda$,

$$\Lambda\equiv\left\{n\in\mathbb{Z}^{4}\mid\text{$0\leq n_{\mu}<L$ for all $\mu$}\right\},$$

(2.2)

by dividing $T^{4}$ into hypercubes $c(n)$ specified by the lattice points in Eq. (2.2):

$$c(n)\equiv\left\{x\in\mathbb{R}^{4}\mid\text{$0\leq(x_{\mu}-n_{\mu})\leq 1$ for all $\mu$}\right\}.$$

(2.3)

We assume a $U(1)$ lattice gauge field on $\Lambda$. The link variable

$$U(n,\mu)\in U(1)$$

(2.4)

is residing on the link connecting $n$ and $n+\hat{\mu}$, where $\hat{\mu}$ denotes a unit vector in the positive $\mu$ direction.

Following the idea of Ref. Luscher:1981zq , we define the transition function of the $U(1)/\mathbb{Z}_{q}$ on $T^{4}$ by regarding each hypercube (2.3) as the coordinate patch for $T^{4}$. Thus, the transition function is defined in the intersection between two hypercubes, called the face:

$$f(n,\mu)\equiv\left\{x\in c(n)\mid x_{\mu}=n_{\mu}\right\}=c(n-\hat{\mu})\cap c(n).$$

(2.5)

This is a 3D cube. We then define the transition function at $x\in f(n,\mu)$ by

$$v_{n,\mu}(x)\equiv\omega_{\mu}(x)\check{v}_{n,\mu}(x)\qquad\text{at $x\in f(n,\mu)$}.$$

(2.6)

In this expression, the first factor $\omega_{\mu}(x)$ is given by

$$\omega_{\mu}(x)\equiv\begin{cases}\exp\left(\frac{\pi i}{q}\sum_{\nu\neq\mu}\frac{z_{\mu\nu}x_{\nu}}{L}\right)&\text{for $x_{\mu}=0\bmod L$},\\ 1&\text{otherwise}.\end{cases}$$

(2.7)

Note that this is non-trivial only on the hyperplane $x_{\mu}=0\pmod{L}$. The “twist angles” $z_{\mu\nu}$ are integers and anti-symmetric in indices, $z_{\mu\nu}=-z_{\nu\mu}$. Equation (2.7) is analogous to the loop in $SU(N)/\mathbb{Z}_{N}$ considered in Refs. tHooft:1979rtg ; vanBaal:1982ag . In fact, along two non-trivial intersecting one-cycles on $T^{4}$, the factor $\omega_{\mu}(x)$ defines a loop in $U(1)/\mathbb{Z}_{q}$. The winding of this loop gives rise to a fractional topological charge tHooft:1979rtg ; vanBaal:1982ag . The integers $z_{\mu\nu}$ are the ’t Hooft flux or (the gauge-invariant content of) the $\mathbb{Z}_{q}$ two-form gauge field; see below. $z_{\mu\nu}$ is defined only modulo $q$ and it labels elements of the cohomology group $H^{2}(T^{4},\mathbb{Z}_{q})$. In the present paper, these are fixed numbers and non-dynamical.

The other factor $\check{v}_{n,\mu}(x)$ in Eq. (2.6) is given by Lüscher’s construction of the principal bundle in lattice gauge theory. When the gauge group is $U(1)$, the construction becomes very simple Fujiwara:2000wn and it gives, for $x\in f(n,\mu)$ with $\mu=1$, $2$, $3$, and $4$, respectively,²²2In deriving this, we have adopted the following definition of the standard parallel transporter, which forms the basis of the construction in Ref. Luscher:1981zq (here $x\equiv n+\sum_{\mu=1}^{4}z_{\mu}\hat{\mu}$): $$w^{n}(x)=U(n,4)^{z_{4}}U(n+z_{4}\hat{4},3)^{z_{3}}U(n+z_{4}\hat{4}+z_{3}\hat{3},2)^{z_{2}}U(n+z_{4}\hat{4}+z_{3}\hat{3}+z_{2}\hat{2},1)^{z_{1}}.$$ (2.8)

	$\displaystyle\check{v}_{n,1}(x)$	$\displaystyle=U(n-\hat{1},1)$
		$\displaystyle\qquad{}\times\exp\Bigl{[}iy_{4}\check{F}_{14}(n-\hat{1})+iy_{3}y_{4}\check{F}_{13}(n-\hat{1}+\hat{4})+iy_{3}(1-y_{4})\check{F}_{13}(n-\hat{1})$
		$\displaystyle\qquad\qquad\qquad{}+iy_{2}y_{3}y_{4}\check{F}_{12}(n-\hat{1}+\hat{3}+\hat{4})+iy_{2}y_{3}(1-y_{4})\check{F}_{12}(n-\hat{1}+\hat{3})$
		$\displaystyle\qquad\qquad\qquad{}+iy_{2}(1-y_{3})y_{4}\check{F}_{12}(n-\hat{1}+\hat{4})+iy_{2}(1-y_{3})(1-y_{4})\check{F}_{12}(n-\hat{1})\Bigr{]},$
	$\displaystyle\check{v}_{n,2}(x)$	$\displaystyle=U(n-\hat{2},2)\exp\left[iy_{4}\check{F}_{24}(n-\hat{2})+iy_{3}y_{4}\check{F}_{23}(n-\hat{2}+\hat{4})+iy_{3}(1-y_{4})\check{F}_{23}(n-\hat{2})\right],$
	$\displaystyle\check{v}_{n,3}(x)$	$\displaystyle=U(n-\hat{3},3)\exp\left[iy_{4}\check{F}_{34}(n-\hat{3})\right],$
	$\displaystyle\check{v}_{n,4}(x)$	$\displaystyle=U(n-\hat{4},4),$		(2.9)

where $y_{\mu}\equiv x_{\mu}-n_{\mu}$. The field strength in this expression is defined by

$$\check{F}_{\mu\nu}(n)\equiv\frac{1}{iq}\ln\left[U(n,\mu)U(n+\hat{\mu},\nu)U(n+\hat{\nu},\mu)^{-1}U(n,\nu)^{-1}\right]^{q}\qquad-\pi<\check{F}_{\mu\nu}(n)\leq\pi.$$

(2.10)

Here, the power $q$ is supplemented inside the logarithm so that the field strength is invariant under the $\mathbb{Z}_{q}$ one-form gauge transformation defined below.³³3Since $q$ is a positive integer, Eq. (2.10) is equal to $\ln[U(n,\mu)^{q}U(n+\hat{\mu},\nu)^{q}U(n+\hat{\nu},\mu)^{-q}U(n,\nu)^{-q}]/(iq)$; i.e., this is the logarithm of the plaquette variable in the charge-$q$ representation. We then require the following admissibility condition,

$$\sup_{n,\mu,\nu}\left|\check{F}_{\mu\nu}(n)\right|<\epsilon\qquad 0<\epsilon<\frac{\pi}{3q},$$

(2.11)

for allowed lattice gauge configurations. By applying the argument in Ref. Luscher:1998kn to Eq. (2.10) carefully, one finds that the condition $\epsilon<\pi/(3q)$ ensures that the Bianchi identity for the field strength, i.e., the absence of the monopole current, $\sum_{\nu,\rho,\sigma}\varepsilon_{\mu\nu\rho\sigma}\Delta_{\nu}\check{F}_{\rho\sigma}(n)=0$, holds.⁴⁴4Here and in what follows, the forward difference is defined by $\Delta_{\mu}f(n)\equiv f(n+\hat{\mu})-f(n)$.

Let us examine the cocycle condition associated with the transition function (2.6). It is given by the product of the transition functions in the intersection of four hypercubes, $c(n)$, $c(n-\hat{\mu})$, $c(n-\hat{\nu})$, and $c(n-\hat{\mu}-\hat{\nu})$, i.e.,

$$p(n,\mu,\nu)\equiv\left\{x\in c(n)\mid x_{\mu}=n_{\mu},x_{\nu}=n_{\nu}\right\}\qquad(\mu\neq\nu).$$

(2.12)

This is a 2D plaquette. Note that $p(n,\mu,\nu)$ is extending in directions complementary to $\mu$ and $\nu$ in the present convention Luscher:1981zq . When the gauge group is $U(1)/\mathbb{Z}_{q}$, the cocycle can take a value in $\mathbb{Z}_{q}$. Noting that the global coordinate $x_{\mu}$ on $T^{4}$ possesses the discontinuity at $x_{\mu}=0$, for $x\in p(n,\mu,\nu)$, we find

	$\displaystyle v_{n-\hat{\nu},\mu}(x)v_{n,\nu}(x)v_{n,\mu}(x)^{-1}v_{n-\hat{\mu},\nu}(x)^{-1}$
	$\displaystyle=\begin{cases}\exp\left(\frac{2\pi i}{q}z_{\mu\nu}\right)\in\mathbb{Z}_{q}&\text{for $x_{\mu}=x_{\nu}=0\bmod L$},\\ 1&\text{otherwise},\end{cases}$		(2.13)

where we have used the fact that the transition function (2.9) satisfies the cocycle condition in the $U(1)$ gauge theory Luscher:1981zq , $\check{v}_{n-\hat{\nu},\mu}(x)\check{v}_{n,\nu}(x)\check{v}_{n,\mu}(x)^{-1}\check{v}_{n-\hat{\mu},\nu}(x)^{-1}=1$.⁵⁵5For the expression in terms of the field strength in Eq. (2.9) to fulfill this cocycle condition, one has to use the Bianchi identity Fujiwara:2000wn . Thus the loop factor $\omega_{\mu}(x)$ in Eq. (2.7) gives rise to the “$\mathbb{Z}_{q}$ breaking” of the cocycle condition.

Let us now consider how the transition function transforms under the $\mathbb{Z}_{q}$ one-form transformation. First, we identify the $\mathbb{Z}_{q}$ one-form global transformation with the center transformation on link variables crossing a 3D hypersurface, say $n_{\mu}=0$. For instance, under

$$U(n,\mu)\to\exp\left(\frac{2\pi i}{q}z_{\mu}\right)U(n,\mu)\qquad\text{$n_{\mu}=0$, $z_{\mu}\in\mathbb{Z}$ and $0\leq z_{\mu}<q$},$$

(2.14)

and $U(n,\mu)\to U(n,\mu)$ otherwise, the field strength (2.10) does not change, the transition functions (2.9) are transformed as

$$\check{v}_{n,\mu}(x)\to\begin{cases}\exp\left(\frac{2\pi i}{q}z_{\mu}\right)\check{v}_{n,\mu}(x)&\text{for $x_{\mu}=1$},\\ \check{v}_{n,\mu}(x)&\text{otherwise},\end{cases}$$

(2.15)

and the other transition functions $\check{v}_{n,\nu\neq\mu}(x)$ do not change. Since these are constant multiplications on the transition functions along the hypersurface $x_{\mu}=1$, the one-form global transformation does not affect the cocycle condition, $\check{v}_{n-\hat{\nu},\mu}(x)\check{v}_{n,\nu}(x)\check{v}_{n,\mu}^{-1}(x)\check{v}_{n-\hat{\mu},\nu}^{-1}(x)=1$. Moreover, one can confirm that the cocycle condition is not affected even under a smooth deformation of the hypersurface. We thus conclude that the $\mathbb{Z}_{q}$ one-form global transformation does not induce any further $\mathbb{Z}_{q}$ breaking of the cocycle condition.

On the other hand, if we consider the $\mathbb{Z}_{q}$ one-form local or gauge transformation defined by⁶⁶6Note that the field strength (2.10) and the admissibility (2.11) are invariant under this $\mathbb{Z}_{q}$ one-form gauge transformation.

$$U(n,\mu)\to\exp\left[\frac{2\pi i}{q}z_{\mu}(n)\right]U(n,\mu)\qquad\text{$z_{\mu}(n)\in\mathbb{Z}$, $0\leq z_{\mu}(n)<q$},$$

(2.16)

then the transition functions (2.6) are transformed as

$$v_{n,\mu}(x)\to\exp\left[\frac{2\pi i}{q}z_{\mu}(n-\hat{\mu})\right]v_{n,\mu}(x)\qquad x\in f(n,\mu).$$

(2.17)

The cocycle condition (2.13) is then modified accordingly to

$$v_{n-\hat{\nu},\mu}(x)v_{n,\nu}(x)v_{n,\mu}(x)^{-1}v_{n-\hat{\mu},\nu}(x)^{-1}\equiv\exp\left[\frac{2\pi i}{q}z_{\mu\nu}(n-\hat{\mu}-\hat{\nu})\right],$$

(2.18)

where the $\mathbb{Z}_{q}$ two-form gauge field on the lattice is given by

$$z_{\mu\nu}(n)=z_{\mu\nu}\delta_{n_{\mu},L-1}\delta_{n_{\nu},L-1}+\Delta_{\mu}z_{\nu}(n)-\Delta_{\nu}z_{\mu}(n)+qN_{\mu\nu}(n)\in\mathbb{Z}.$$

(2.19)

Thus the $\mathbb{Z}_{q}$ one-form gauge transformation (2.16) induces an extra factor of a “pure gauge” form in the cocycle condition. Here, we resolve the modulo $q$ ambiguity of $z_{\mu\nu}(n)$ by setting

$$\begin{cases}0\leq z_{\mu\nu}(n)<q&\text{for~{}$\mu<\nu$},\\ z_{\mu\nu}(n)\equiv-z_{\nu\mu}(n)&\text{for~{}$\mu>\nu$}.\end{cases}$$

(2.20)

The last lattice field $N_{\mu\nu}(n)\in\mathbb{Z}$ in Eq. (2.19) is required to restrict the value of $z_{\mu\nu}(n)$ ($\mu<\nu$) in the range (2.20); recall that we have taken $0\leq z_{\mu}(n)<q$ in Eq. (2.16). Considering successive $\mathbb{Z}_{q}$ one-form gauge transformations starting from Eq. (2.19), for a generic $z_{\mu\nu}(n)$, we have

$$z_{\mu\nu}(n)\to z_{\mu\nu}(n)+\Delta_{\mu}z_{\nu}(n)-\Delta_{\nu}z_{\mu}(n)+qN_{\mu\nu}(n).$$

(2.21)

(The fields $z_{\mu}(n)$ and $N_{\mu\nu}(n)$ differ from those in Eq. (2.19).) Our original configuration of the $\mathbb{Z}_{q}$ two-form gauge field in Eq. (2.13), $z_{\mu\nu}(n)=z_{\mu\nu}\delta_{n_{\mu},L-1}\delta_{n_{\nu},L-1}$, is flat, i.e., $(1/2)\sum_{\nu,\rho,\sigma}\varepsilon_{\mu\nu\rho\sigma}\Delta_{\nu}z_{\rho\sigma}(n)=0\bmod q$. This flatness of the $\mathbb{Z}_{q}$ two-form gauge field Kapustin:2014gua ; Tanizaki:2022ngt is obviously preserved under the $\mathbb{Z}_{q}$ one-form gauge transformation on the lattice (2.21) because $N_{\mu\nu}(n)$ are integers and contribute to the flatness condition only by an integer multiple of $q$. In Appendix A, we give a note on the flatness in our present lattice formulation.

Our construction above may be employed to consider, with lattice regularization, the mixed ’t Hooft anomaly between the $\mathbb{Z}_{q}$ one-form symmetry and the time reversal symmetry in the $U(1)$ gauge theory with matter fields of charge $q\in 2\mathbb{Z}$, when the vacuum angle $\theta$ is $\pi$ Honda:2020txe . We can assume a lattice action that is invariant under the $\mathbb{Z}_{q}$ one-form global transformation:⁸⁸8To incorporate the admissibility and the smoothness of the lattice action, a more ingenious construction such as the one in Ref. Luscher:1998du would be desirable; this point is irrelevant in the present discussion concerning symmetries of the action.

$$S\equiv\frac{1}{4g_{0}^{2}}\sum_{n\in\Lambda}\sum_{\mu,\nu}\check{F}_{\mu\nu}(n)\check{F}_{\mu\nu}(n)+S_{\text{matter}}-iq\theta\mathcal{Q},$$

(4.1)

where $g_{0}$ is the bare coupling. This original system does not contain the $\mathbb{Z}_{q}$ two-form gauge field. We also assume that the lattice action except the last topological term is even under the time reversal transformation; this is actually the case for the first pure gauge term in Eq. (4.1).

In the last topological term in Eq. (4.1), $\theta$ is multiplied by $q$ as $iq\theta\mathcal{Q}$ instead of $i\theta\mathcal{Q}$. This is because, with the conventional normalization of $\theta$, the Witten effect Witten:1979ey suggests a $2\pi q$ periodicity of $\theta$ instead of the $2\pi$ periodicity. This rescaling of the periodicity actually occurs at least in the Cardy–Rabinovici model Cardy:1981qy ; Cardy:1981fd as studied in Ref. Honda:2020txe (see also Ref. Hidaka:2019jtv ). That is, the full spectrum of the system including the monopole and dyons is invariant only under a $2\pi q$ shift of the original vacuum angle, instead of a $2\pi$ shift; thus the periodicity of $\theta$ is $2\pi$ only with the combination in Eq. (4.1). Since we do not introduce the monopole and dyons in the present lattice setup, we cannot observe the Witten effect and the associated rescaling of the vacuum angle directly. Nevertheless, it is interesting to see a possible ’t Hooft anomaly in our lattice regularized setup, temporarily accepting the above rescaling of the vacuum angle. This is what we do here.

Now, let us set $\theta=\pi$ in Eq. (4.1). The original partition function is then time reversal invariant because $i\pi q\mathcal{Q}\stackrel{{\scriptstyle\mathcal{T}}}{{\to}}-i\pi q\mathcal{Q}\sim+i\pi q\mathcal{Q}$ because of the postulated $2\pi$ periodicity of $\theta$ (the original $\mathcal{Q}$ is an integer).

We then switch the $\mathbb{Z}_{q}$ two-form gauge field on. From Eqs. (3.4) and (3.5), we have

$$q\mathcal{Q}=\frac{1}{8q}\sum_{\mu,\nu,\rho,\sigma}\varepsilon_{\mu\nu\rho\sigma}z_{\mu\nu}z_{\rho\sigma}+\mathbb{Z}.$$

(4.2)

The factor $e^{i\pi q\mathcal{Q}}$ in the integrand of the functional integral then acquires an extra phase factor under the time reversal, as (recall Eq. (3.8))

	$\displaystyle e^{i\pi q\mathcal{Q}}$	$\displaystyle\stackrel{{\scriptstyle\mathcal{T}}}{{\to}}e^{-i\pi q\mathcal{Q}}=e^{-2\pi iq\mathcal{Q}}\cdot e^{i\pi q\mathcal{Q}}$
		$\displaystyle=\exp\left(-\frac{2\pi i}{8q}\sum_{\mu,\nu,\rho,\sigma}\varepsilon_{\mu\nu\rho\sigma}z_{\mu\nu}z_{\rho\sigma}\right)e^{i\pi q\mathcal{Q}}.$		(4.3)

One should then ask Gaiotto:2017yup whether a certain local gauge-invariant term of the $\mathbb{Z}_{q}$ two-form gauge field $z_{\mu\nu}(n)$ can counter the above breaking of the time reversal symmetry. Using the technique reviewed in Appendix B, it can be seen that a local term that transforms “covariantly” under the lattice $\mathbb{Z}_{q}$ one-form gauge transformation (2.21) is given by

$$\exp\left[\frac{2\pi ik}{4q}\sum_{n\in\Lambda}\sum_{\mu,\nu,\rho,\sigma}\varepsilon_{\mu\nu\rho\sigma}z_{\mu\nu}(n)z_{\rho\sigma}(n+\hat{\mu}+\hat{\nu})\right].$$

(4.4)

It is easy to see that, under the $\mathbb{Z}_{q}$ one-form gauge transformation (2.21), the combination $\sum_{n\in\Lambda}\sum_{\mu,\nu,\rho,\sigma}\varepsilon_{\mu\nu\rho\sigma}z_{\mu\nu}(n)z_{\rho\sigma}(n+\hat{\mu}+\hat{\nu})$ shifts by $4q\mathbb{Z}$; see Appendix B. Therefore, for the phase factor (4.4) to be gauge invariant, the constant $k$ must be an integer, $k\in\mathbb{Z}$.

However, Eq. (4.4), which would be regarded as the wedge product of the elements of $H^{2}(T^{4},\mathbb{Z}_{q})$ on the hypercubic lattice, is not the counterterm with the “finest” coefficient, as known for the corresponding counterterm on the simplicial lattice Kapustin:2013qsa ; Kapustin:2014gua .⁹⁹9We owe the following discussion to Yuya Tanizaki. The counterterm with a “finer” coefficient on $T^{4}$ can be constructed by employing the integral lift of $H^{2}(T^{4},\mathbb{Z}_{q})$ to $H^{2}(T^{4},\mathbb{Z})$ Kapustin:2013qsa ; Kapustin:2014gua . On our periodic hypercubic lattice $\Lambda$, we may construct an analogue of the integral lift, $\bar{z}_{\mu\nu}(n)$, which satisfies the flatness, $(1/2)\sum_{\nu,\rho,\sigma}\varepsilon_{\mu\nu\rho\sigma}\Delta_{\nu}\bar{z}_{\rho\sigma}(n)=0$ (strictly zero not modulo $q$), from $z_{\mu\nu}(n)$ as follows.

First, we define an integer $m_{\mu}(n)\equiv(1/2)\sum_{\nu,\rho,\sigma}\varepsilon_{\mu\nu\rho\sigma}\Delta_{\nu}z_{\rho\sigma}(n)/q$ for each 3D cube, which spans from the site $n$ to directions complementary to $\mu$. $m_{\mu}(n)$ is an integer, because of the modulo $q$ flatness of $z_{\mu\nu}(n)$, $(1/2)\sum_{\nu,\rho,\sigma}\varepsilon_{\mu\nu\rho\sigma}\Delta_{\nu}z_{\rho\sigma}(n)\in q\mathbb{Z}$.

Next, we take a 3D space specified by a fixed $n_{\mu}$ and consider paths connecting centers of cubes in the 3D space. The paths are defined such that $|m_{\mu}(n)|$ paths begin from the cube at $n$ if $m_{\mu}(n)>0$, while $|m_{\mu}(n)|$ paths end at the cube if $m_{\mu}(n)<0$. A certain consistent configuration of paths can be defined in this way, because the total sum of $m_{\mu}(n)$ on the three-dimensional space identically vanishes, $\sum_{n\in\Lambda,\text{$n_{\mu}$ fixed}}m_{\mu}(n)=\sum_{n\in\Lambda,\text{$n_{\mu}$ fixed}}(1/2)\sum_{\nu,\rho,\sigma}\varepsilon_{\mu\nu\rho\sigma}\Delta_{\nu}z_{\rho\sigma}(n)/q=0$. Then, it is obvious that $\bar{z}_{\mu\nu}(n)$ such that $\sum_{\nu,\rho,\sigma}\varepsilon_{\mu\nu\rho\sigma}\Delta_{\nu}\bar{z}_{\rho\sigma}(n)=0$ can be obtained by subtracting $qm_{\mu\nu}(n)$ from $z_{\mu\nu}(n)$. Here, $m_{\mu\nu}(n)$ is the signed number of paths going through the plaquette at which $z_{\mu\nu}(n)$ is residing; the sign of $m_{\mu\nu}(n)$ is defined to be positive if the paths go through the plaquette in the direction of the standard orientation of the plaquette and negative if they go through in the opposite direction. This construction gives the relation $\bar{z}_{\mu\nu}(n)=z_{\mu\nu}(n)-qm_{\mu\nu}(n)$, where $m_{\nu\mu}(n)=-m_{\mu\nu}(n)$.

The integral lift $\bar{z}_{\mu\nu}(n)$ can differ depending on the choice of the configuration of paths but the difference can be expressed in the difference of the field $m_{\mu\nu}(n)$. Since $\bar{z}_{\mu\nu}(n)=z_{\mu\nu}(n)-qm_{\mu\nu}(n)$ has the form of the gauge transformation (2.21), the choice of the configuration of paths does not matter as far as gauge-invariant quantities (such as the counterterm below) are concerned. Also, it is obvious from Eq. (2.21) that the gauge transformation of $\bar{z}_{\mu\nu}(n)$ takes the form $\bar{z}_{\mu\nu}(n)\to\bar{z}_{\mu\nu}(n)+\Delta_{\mu}z_{\nu}(n)-\Delta_{\nu}z_{\mu}(n)+q\bar{N}_{\mu\nu}(n)$, where $\bar{N}_{\mu\nu}(n)\in\mathbb{Z}$. Because of the flatness of $\bar{z}_{\mu\nu}(n)$, we thus infer that $\bar{N}_{\mu\nu}(n)$ is also flat, $\sum_{\nu,\rho,\sigma}\varepsilon_{\mu\nu\rho\sigma}\Delta_{\nu}\bar{N}_{\rho\sigma}(n)=0$.

Finally, when $\sum_{\nu,\rho,\sigma}\varepsilon_{\mu\nu\rho\sigma}\Delta_{\nu}\bar{z}_{\rho\sigma}(n)=\sum_{\nu,\rho,\sigma}\varepsilon_{\mu\nu\rho\sigma}\Delta_{\nu}\bar{N}_{\rho\sigma}(n)=0$, which can be expressed as $d\bar{z}^{(2)}=d\bar{N}=0$ in the notation of Appendix B, by employing the argument in Ref. Fujiwara:2000wn , one can see that the combination $\sum_{n\in\Lambda}\sum_{\mu,\nu,\rho,\sigma}\varepsilon_{\mu\nu\rho\sigma}\bar{z}_{\mu\nu}(n)\bar{z}_{\rho\sigma}(n+\hat{\mu}+\hat{\nu})$ shifts by $8q\mathbb{Z}$ (not $4q\mathbb{Z}$) under the gauge transformation.

Therefore, the counterterm with a finer coefficient is given by

$$e^{-S_{\text{counter}}}=\exp\left[\frac{2\pi ik}{8q}\sum_{n\in\Lambda}\sum_{\mu,\nu,\rho,\sigma}\varepsilon_{\mu\nu\rho\sigma}\bar{z}_{\mu\nu}(n)\bar{z}_{\rho\sigma}(n+\hat{\mu}+\hat{\nu})\right]$$

(4.5)

and this is gauge-invariant for $k\in\mathbb{Z}$ (note the difference in coefficients in Eqs. (4.4) and (4.5)). We expect that this is the finest gauge invariant coefficient from the corresponding result in the continuum theory Honda:2020txe .