Yet another lattice formulation of 2D $U(1)$ chiral gauge theory via bosonization

We consider $2\pi$ periodic real scalar fields on a 2D square lattice $\Gamma$. The lattice sites of $\Gamma$ are denoted as $n$, $m$, etc. For a moment, we assume that $\Gamma$ is a 2D torus and the scalar fields $e^{i\phi_{\alpha}(n)}$ obey periodic boundary conditions. We parametrize $e^{i\phi_{\alpha}(n)}$ by the logarithm in the principal branch:

$$\phi_{\alpha}(n):=\frac{1}{i}\ln e^{i\phi_{\alpha}(n)},\qquad-\pi<\phi_{\alpha}(n)\leq\pi.$$

(2.1)

This definition of $\phi_{\alpha}(n)$ is obviously invariant under the “$2\pi$ shift,” $e^{i\phi_{\alpha}(n)}\to e^{i\phi_{\alpha}(n)+2\pi i\mathbb{Z}}$.

Our lattice action for the scalar field Abe:2023uan is almost a literal transcription of the continuum action (1.1),⁴⁴4Lorentz indices $\mu$, $\nu$, etc. run over 1 and 2.

	$\displaystyle S_{\mathrm{B}}$	$\displaystyle=\sum_{\alpha}\sum_{n\in\Gamma}\Biggl{[}\frac{R^{2}}{4\pi}\sum_{\mu}D\phi_{\alpha}(n,\mu)D\phi_{\alpha}(n,\mu)+\frac{i}{2\pi}q_{V,\alpha}\sum_{\mu,\nu}\epsilon_{\mu\nu}A_{\mu}(\tilde{n})D\phi_{\alpha}(n+\hat{\mu},\nu)$
		$\displaystyle\qquad\qquad\qquad{}+\frac{i}{2}q_{V,\alpha}\sum_{\mu,\nu}\epsilon_{\mu\nu}N_{\mu\nu}(\tilde{n})\phi_{\alpha}(n+\hat{\mu}+\hat{\nu})\Biggr{]},$		(2.2)

except for the last “counter term,” which plays a crucial role to lead an ’t Hooft anomaly in the picture of Ref. Abe:2023uan and the simple form of the lattice gauge anomaly in the present context. For the continuum limit, the radius $R^{2}$ should be tuned to a specific value, not the classical value $1/2$ (see, e.g. Ref. Janke for this issue).

The explanation of various quantities in this expression is in order:

First, the scalar fields $\phi_{\alpha}(n)$ are coupled to a $U(1)$ gauge field; at this moment, the gauge field is regarded as background and later we will make it dynamical. For a technical reason caused by introducing the excision method at a later stage, we first introduce two $U(1)$ lattice gauge fields (link variables), one on $\Gamma$ and another on the dual lattice $\tilde{\Gamma}$, whose sites are defined by

$$\tilde{n}:=n+\frac{1}{2}\hat{1}+\frac{1}{2}\hat{2},$$

(2.3)

as

$$U(n,\mu)\in U(1),\qquad U(\tilde{n},\mu)\in U(1).$$

(2.4)

See Fig. 1. Then the latter gauge field is simply given as a copy of the former:

$$U(\tilde{n},\mu):=U(n,\mu).$$

(2.5)

Thus, introducing two gauge fields is meaningless at this stage but later when we introduce a magnetic object by the excised method Abe:2023uan , the correspondence (2.3) between the dual and original lattices and, correspondingly, the relation (2.5), must be modified. We parametrize the link variables by the logarithm in the principal branch:

	$\displaystyle A_{\mu}(n)$	$\displaystyle:=\frac{1}{i}\ln U(n,\mu),\qquad-\pi<A_{\mu}(n)\leq\pi,$
	$\displaystyle A_{\mu}(\tilde{n})$	$\displaystyle:=\frac{1}{i}\ln U(\tilde{n},\mu),\qquad-\pi<A_{\mu}(\tilde{n})\leq\pi.$		(2.6)

Of course, $A_{\mu}(\tilde{n})=A_{\mu}(n)$ at this stage.

The directional covariant difference, $D\phi_{\alpha}(\tilde{n},\mu)$ in Eq. (2.2), is defined by using the link variable on the original lattice, $U(n,\mu)$, as

	$\displaystyle D\phi_{\alpha}(n,\mu)$	$\displaystyle:=\frac{1}{i}\ln\left[e^{-i\phi_{\alpha}(n)}U(n,\mu)^{2q_{A,\alpha}}e^{i\phi_{\alpha}(n+\hat{\mu})}\right]$
		$\displaystyle=\Delta_{\mu}\phi_{\alpha}(n)+2q_{A,\alpha}A_{\mu}(n)+2\pi\ell_{\alpha,\mu}(n),$		(2.7)

where $\hat{\mu}$ denotes the unit vector in the positive $\mu$-direction and the branch of the logarithm is taken as $-\pi<D\phi_{\alpha}(n,\mu)\leq\pi$. Note that, when expressed in terms of the simple difference operator,

$$\Delta_{\mu}f(n):=f(n+\hat{\mu})-f(n),$$

(2.8)

we generally have the integer field $\ell_{\alpha,\mu}(n)\in\mathbb{Z}$ as in Eq. (2.7); $\ell_{\alpha,\mu}(n)$ is a local functional of $\phi_{\alpha}(n)$. By construction, the covariant difference (2.7) is invariant under the lattice gauge transformation:

	$\displaystyle\phi_{\alpha}(n)$	$\displaystyle\to\phi_{\alpha}(n)-2q_{A,\alpha}\Lambda(n),$
	$\displaystyle U(n,\mu)$	$\displaystyle\to e^{-i\Lambda(n)}U(n,\mu)e^{i\Lambda(n+\hat{\mu})},$
	$\displaystyle U(\tilde{n},\mu)$	$\displaystyle\to e^{-i\Lambda(\tilde{n})}U(\tilde{n},\mu)e^{i\Lambda(\tilde{n}+\hat{\mu})}.$		(2.9)

Because of Eq. (2.5), the gauge transformation functions are also identified as $\Lambda(\tilde{n})=\Lambda(n)$ at this stage. Under Eq. (2.9), we see that the gauge potentials in Eq. (2.6) are transformed as

	$\displaystyle A_{\mu}(n)$	$\displaystyle\to A_{\mu}(n)+\Delta_{\mu}\Lambda(n)+2\pi L_{\mu}(n),$
	$\displaystyle A_{\mu}(\tilde{n})$	$\displaystyle\to A_{\mu}(\tilde{n})+\Delta_{\mu}\Lambda(\tilde{n})+2\pi L_{\mu}(\tilde{n}),$		(2.10)

where we have integer fields $L_{\mu}(n)\in\mathbb{Z}$ and $L_{\mu}(\tilde{n})\in\mathbb{Z}$; $L_{\mu}(n)$ ($L_{\mu}(\tilde{n})$) is a local functional of $\Lambda(n)$ ($\Lambda(\tilde{n})$). Corresponding to the former, from the gauge invariance of Eq. (2.7), we have the transformation law of $\ell_{\alpha,\mu}(n)$:

$$\ell_{\alpha,\mu}(n)\to\ell_{\alpha,\mu}(n)-2q_{A,\alpha}L_{\mu}(n).$$

(2.11)

Finally, we introduce gauge invariant field strengths by

	$\displaystyle F_{\mu\nu}(n)$	$\displaystyle:=\frac{1}{i}\ln\left[U(n,\mu)U(n+\hat{\mu},\nu)U(n+\hat{\nu},\mu)^{-1}U(n,\nu)^{-1}\right],$
	$\displaystyle F_{\mu\nu}(\tilde{n})$	$\displaystyle:=\frac{1}{i}\ln\left[U(\tilde{n},\mu)U(\tilde{n}+\hat{\mu},\nu)U(\tilde{n}+\hat{\nu},\mu)^{-1}U(\tilde{n},\nu)^{-1}\right],$		(2.12)

where the branch of the logarithm is taken as $-\pi<F_{\mu\nu}(n)\leq\pi$ and $-\pi<F_{\mu\nu}(\tilde{n})\leq\pi$. Then, as per Eq. (2.7), we have

	$\displaystyle F_{\mu\nu}(n)$	$\displaystyle=\Delta_{\mu}A_{\nu}(n)-\Delta_{\nu}A_{\mu}(n)+2\pi N_{\mu\nu}(n),$
	$\displaystyle F_{\mu\nu}(\tilde{n})$	$\displaystyle=\Delta_{\mu}A_{\nu}(\tilde{n})-\Delta_{\nu}A_{\mu}(\tilde{n})+2\pi N_{\mu\nu}(\tilde{n}),$		(2.13)

where the integer fields, $N_{\mu\nu}(n)$ and $N_{\mu\nu}(\tilde{n})$, are local functionals of $U(n,\mu)$ and $U(\tilde{n},\mu)$, respectively. We note that under the gauge transformation (2.10), $N_{\mu\nu}(n)$ and $N_{\mu\nu}(\tilde{n})$ transform as

	$\displaystyle N_{\mu\nu}(n)$	$\displaystyle\to N_{\mu\nu}(n)-\Delta_{\mu}L_{\nu}(n)+\Delta_{\nu}L_{\mu}(n),$
	$\displaystyle N_{\mu\nu}(\tilde{n})$	$\displaystyle\to N_{\mu\nu}(\tilde{n})-\Delta_{\mu}L_{\nu}(\tilde{n})+\Delta_{\nu}L_{\mu}(\tilde{n}).$		(2.14)

This completes the explanation on quantities appearing in the lattice action (2.2).

As the correspondence between Eqs. (1.1) and (1.13) in continuum theory shows, in the bosonization, the axial-vector current is given by the “magnetic” current $\epsilon_{\mu\nu}\partial_{\nu}\phi_{\alpha}$. Its lattice counterpart,

$$\epsilon_{\mu\nu}D\phi_{\alpha}(n,\nu)$$

(2.15)

however, does not generally conserve, $\epsilon_{\mu\nu}\Delta_{\mu}D\phi_{\alpha}(n,\nu)\neq 0$, even when $A_{\mu}(n)=0$, because of the integer field $\ell_{\alpha,\mu}(n)$ in Eq. (2.7), i.e. generally $\epsilon_{\mu\nu}\Delta_{\mu}\ell_{\alpha,\nu}(n)\neq 0$. Then, we cannot define a conserved axial charge even if the target chiral gauge theory is anomaly-free. This failure can be evaded by imposing a certain smoothness condition on lattice fields as follows Abe:2023uan .

We require that possible configurations of the lattice fields satisfy the following gauge-invariant conditions (the latter two are the admissibility Luscher:1981zq ; Luscher:1998du ) for all flavor $\alpha$:

$$\sup_{n,\mu}\left|D\phi_{\alpha}(n,\mu)\right|<\epsilon,\qquad\sup_{n,\mu,\nu}\left|2q_{A,\alpha}F_{\mu\nu}(n)\right|<\delta,\qquad\sup_{\tilde{n},\mu,\nu}\left|q_{V,\alpha}F_{\mu\nu}(\tilde{n})\right|<\delta,$$

(2.16)

where

$$0<\epsilon<\frac{\pi}{2},\qquad 0<\delta<\min(\pi,2\pi-4\epsilon).$$

(2.17)

Then, from Eqs. (2.7) and (2.13), these conditions follow the bound Abe:2023uan ,

	$\displaystyle\left|\Delta_{\mu}\ell_{\alpha,\nu}(n)-\Delta_{\nu}\ell_{\alpha,\mu}(n)-2q_{A,\alpha}N_{\mu\nu}(n)\right|$
	$\displaystyle=\frac{1}{2\pi}\left|\Delta_{\mu}D\phi_{\alpha}(n,\nu)-\Delta_{\nu}D\phi_{\alpha}(n,\mu)-2q_{A,\alpha}F_{\mu\nu}(n)\right|$
	$\displaystyle<\frac{2}{\pi}\epsilon+\frac{1}{2\pi}\delta<1.$		(2.18)

Since the leftmost term is a sum of integers, this bound implies that

	$\displaystyle\Delta_{\mu}\ell_{\alpha,\nu}(n)-\Delta_{\nu}\ell_{\alpha,\mu}(n)$	$\displaystyle=2q_{A,\alpha}N_{\mu\nu}(n),$
	$\displaystyle\Delta_{\mu}D\phi_{\alpha}(n,\nu)-\Delta_{\nu}D\phi_{\alpha}(n,\mu)$	$\displaystyle=2q_{A,\alpha}F_{\mu\nu}(n).$		(2.19)

These are Bianchi identities in the presence of the gauge interaction. Note that these are invariant under the gauge transformation, Eqs. (2.11), (2.14), (2.9) and (2.10).

We now define a gauge-invariant magnetic charge inside a loop $C$ on $\Gamma$ by:⁵⁵5If the gauge field is turned off, this is the magnetic charge defined in Ref. Abe:2023uan .

	$\displaystyle m_{\alpha}$	$\displaystyle:=\frac{1}{2\pi}\sum_{(n,\mu)\in C}D\phi_{\alpha}(n,\mu)-\frac{2q_{A,\alpha}}{2\pi}F(C)$
		$\displaystyle=\sum_{(n,\mu)\in C}\ell_{\alpha,\mu}(n)-2q_{A,\alpha}N(C)\in\mathbb{Z},$		(2.20)

where the sum is taken along the loop. In this expression, we have introduced a logarithm of the Wilson loop along $C$,

$$F(C):=\frac{1}{i}\ln\left[\prod_{(n,\mu)\in C}U(n,\mu)\right]=\sum_{(n,\mu)\in C}A_{\mu}(n)+2\pi N(C),$$

(2.21)

where the branch is $-\pi<F(C)\leq\pi$ and $N(C)$ is an integer. In deriving the last expression of Eq. (2.20), we have recalled Eq. (2.7); from this, $m_{\alpha}$ is obviously an integer.

The magnetic charge $m_{\alpha}$ (2.20) is conserved in the sense that it is invariant under any deformation of the loop $C$ because of the Bianchi identity (2.19) (the change due to the integration of the field strengths is compensated by the change in the Wilson loop). This topological invariance however implies that $m_{\alpha}\equiv 0$ on a uniform lattice. We may nevertheless introduce a magnetically-charged object in the system by excising a certain region $\mathcal{D}$ from $\Gamma$, as in Fig. 2. This is the excision method of Ref. Abe:2023uan . The magnetic charge of the excised region $\mathcal{D}$ is given by setting $C=\partial\mathcal{D}$ in Eq. (2.20). For a later argument, we assume that for $C=\partial\mathcal{D}$, for all flavors $\alpha$,

$$\left|2q_{A,\alpha}F(\partial\mathcal{D})\right|\leq\delta^{\prime}$$

(2.22)

with a sufficiently small $\delta^{\prime}$ (we will later specify how $\delta^{\prime}$ should be small). Because of the first condition of Eq. (2.16) and this, the size of the region $\mathcal{D}$ is of order $1$ in lattice units and the magnetic object becomes point-like in the continuum limit; the precise shape of $\mathcal{D}$ is thus not important. Note that from the bosonization rule, the magnetic object carries the vector $U(1)$ charge but no axial $U(1)$ charge. This localized magnetic object can be identified with the vertex operator $e^{im_{\alpha}\tilde{\phi}_{\alpha}(x)}$ of the dual scalar field $\tilde{\phi}_{\alpha}(x)$ in the continuum theory. To realize this magnetic object in quantum theory, the functional integral over $\phi_{\alpha}(n)$ must be carried out under the boundary condition (2.20).

With the presence of an excised region $\mathcal{D}$ as depicted in Fig. 2, however, the correspondence between links of $\Gamma$ and those of $\tilde{\Gamma}$ is not obvious and the simple identification of gauge fields (2.5) does not work. For such a case with an excised region, we adopt the following rule: If the link $(\tilde{n},\mu)$ on $\tilde{\Gamma}$ possesses the corresponding link $(n,\mu)$ on $\Gamma$ with Eq. (2.3), we apply the simple identification (2.5). We find that precisely when the link $(\tilde{n},\mu)$ crosses the boundary of $\mathcal{D}$, $\partial\mathcal{D}$ (i.e., when $\tilde{n}=\tilde{n}_{*}$ or $\tilde{n}+\hat{\mu}=\tilde{n}_{*}$), such a correspondence breaks. For such a case, $U(\tilde{n},\mu)$ is not given by $U(n,\mu)$ and it should be treated as an independent variable:

$$\text{$U(\tilde{n},\mu)$ is an independent variable if the link $(\tilde{n},\mu)$ crosses the boundary of~{}$\mathcal{D}$}.$$

(2.23)

Also, the gauge transformation functions, $\Lambda(\tilde{n})$, can almost be given by $\Lambda(n)$ on $\Gamma$ by Eq. (2.3) but we find that the site $\tilde{n}_{*}$ is the exception, i.e.,

$$\text{$\Lambda(\tilde{n}_{*})$ is an independent gauge transformation variable}.$$

(2.24)

Now, let us study how our lattice action (2.2) changes under the lattice gauge transformation. In general, it should not be invariant in order to reproduce the gauge anomaly in the target chiral gauge theory. One may still hope that the lattice action becomes perfectly gauge invariant if the anomaly cancellation condition (1.17) matches. Remarkably, this actually occurs.

It is straightforward to see that under the gauge transformation the action $S_{\mathrm{B}}$ (2.2) changes into Abe:2023uan

	$\displaystyle S_{\mathrm{B}}+\frac{i}{2\pi}\sum_{\alpha}q_{V,\alpha}\sum_{n\in\Gamma}\sum_{\mu,\nu}\epsilon_{\mu\nu}\Delta_{\mu}\Lambda(\tilde{n})D\phi_{\alpha}(n+\hat{\mu},\hat{\nu})$
	$\displaystyle\qquad{}-i\sum_{\alpha}q_{V,\alpha}q_{A,\alpha}\sum_{n\in\Gamma}\sum_{\mu,\nu}\epsilon_{\mu\nu}\left[N_{\mu\nu}(\tilde{n})-\Delta_{\mu}L_{\nu}(\tilde{n})+\Delta_{\nu}L_{\mu}(\tilde{n})\right]\Lambda(n+\hat{\mu}+\hat{\nu})$
	$\displaystyle\qquad{}+2i\sum_{\alpha}q_{V,\alpha}q_{A,\alpha}\sum_{n\in\Gamma}\sum_{\mu,\nu}\epsilon_{\mu\nu}L_{\mu}(\tilde{n})A_{\nu}(n+\hat{\mu})$
	$\displaystyle\qquad{}+i\sum_{\alpha}q_{V,\alpha}\sum_{n\in\Gamma}\sum_{\mu,\nu}\epsilon_{\mu\nu}\Delta_{\nu}\left[L_{\mu}(\tilde{n})\phi_{\alpha}(n+\hat{\mu})\right]$
	$\displaystyle\qquad{}+2\pi i\sum_{\alpha}q_{V,\alpha}\sum_{n\in\Gamma}\sum_{\mu,\nu}\epsilon_{\mu\nu}L_{\mu}(\tilde{n})\ell_{\nu}(n+\hat{\mu}).$		(2.25)

The last term is $2\pi i\mathbb{Z}$ and is negligible in the partition function,

$$\int\prod_{\alpha}\prod_{n\in\Gamma}d\phi_{\alpha}(n)\,e^{-S_{\mathrm{B}}}.$$

(2.26)

Also, the total divergence term $\Delta_{\nu}[\dotsc]$ can be neglected because we are assuming the boundary conditions are periodic.

The second term of Eq. (2.25), $\sum_{n\in\Gamma}\sum_{\mu,\nu}\epsilon_{\mu\nu}\Delta_{\mu}\Lambda(\tilde{n})D\phi_{\alpha}(n+\hat{\mu},\hat{\nu})$, requires a close look Abe:2023uan . By considering combinations containing $\Lambda(\tilde{n})$ with a particular site $\tilde{n}$ (see Fig. 3), one finds that the second term can be written as

	$\displaystyle\frac{i}{2\pi}\sum_{\alpha}q_{V,\alpha}\sum_{n\in\Gamma}\sum_{\mu,\nu}\epsilon_{\mu\nu}\Delta_{\mu}\Lambda(\tilde{n})D\phi_{\alpha}(n+\hat{\mu},\hat{\nu})$
	$\displaystyle=-\frac{i}{4\pi}\sum_{\alpha}q_{V,\alpha}\sum_{n\in\Gamma}\sum_{\mu,\nu}\epsilon_{\mu\nu}\Lambda(\tilde{n})\left[\Delta_{\mu}D\phi_{\alpha}(n,\hat{\nu})-\Delta_{\nu}D\phi_{\alpha}(n,\hat{\mu})\right]$
	$\displaystyle=-\frac{i}{2\pi}\sum_{\alpha}q_{V,\alpha}q_{A,\alpha}\sum_{n\in\Gamma}\sum_{\mu,\nu}\epsilon_{\mu\nu}\Lambda(\tilde{n})F_{\mu\nu}(n),$		(2.27)

where, in the last step, we have noted Eq. (2.19). In total, the action changes into

	$\displaystyle S_{\mathrm{B}}+i\left(\sum_{\alpha}q_{V,\alpha}q_{A,\alpha}\right)\sum_{n\in\Gamma}\sum_{\mu,\nu}\epsilon_{\mu\nu}$
	$\displaystyle\qquad{}\times\biggl{\{}-\frac{1}{2\pi}\Lambda(\tilde{n})F_{\mu\nu}(n)-\left[N_{\mu\nu}(\tilde{n})-\Delta_{\mu}L_{\nu}(\tilde{n})+\Delta_{\nu}L_{\mu}(\tilde{n})\right]\Lambda(n+\hat{\mu}+\hat{\nu})$
	$\displaystyle\qquad\qquad{}+2L_{\mu}(\tilde{n})A_{\nu}(n+\hat{\mu})\biggr{\}},$		(2.28)

up to $2\pi i\mathbb{Z}$. Since the change is proportional to the anomaly coefficient (1.16), for anomaly-free chiral gauge theories, our lattice action and hence the regularized partition function (2.26) is exactly gauge invariant.

The above argument proceeds in an almost equal way, even with an excised region $\mathcal{D}$ which represents a magnetically charged object. For this case, we have the contribution from $\mathcal{D}$ in addition to Eq. (2.28) Abe:2023uan ,

	$\displaystyle-i\sum_{\alpha}q_{V,\alpha}\Lambda(\tilde{n}_{*})\sum_{(n,\mu)\in\partial\mathcal{D}}\left[\ell_{\alpha,\mu}(n)+\frac{2q_{A,\alpha}}{2\pi}A_{\mu}(n)\right]$
	$\displaystyle=-i\sum_{\alpha}q_{V,\alpha}\Lambda(\tilde{n}_{*})\left[\sum_{(n,\mu)\in\partial\mathcal{D}}\ell_{\alpha,\mu}(n)-2q_{A,\alpha}N(\partial\mathcal{D})\right]-\frac{i}{\pi}\left(\sum_{\alpha}q_{V,\alpha}q_{A,\alpha}\right)\Lambda(\tilde{n}_{*})F(\partial\mathcal{D})$
	$\displaystyle=-i\sum_{\alpha}q_{V,\alpha}m_{\alpha}\Lambda(\tilde{n}_{*}),$		(2.29)

where, in the first equality, we have used Eq. (2.21) with $C=\partial\mathcal{D}$. In the second equality, we have used the anomaly cancellation condition (1.17) and the definition of the magnetic charge (2.20) taking $C=\partial\mathcal{D}$. This breaking of the gauge symmetry owing to the magnetic object may be cured by connecting to the magnetic object an “open ’t Hooft line” in the dual lattice Abe:2023uan ,

$$\exp\left[-i\sum_{\alpha}q_{V,\alpha}m_{\alpha}\sum_{(\tilde{n},\mu)\in\tilde{P}}^{\tilde{n}_{*}}A_{\mu}(\tilde{n})\right],$$

(2.30)

where $\tilde{P}$ denotes a path on the dual lattice ending at $\tilde{n}_{*}$. In this way, under the gauge anomaly cancellation condition (1.17), the fermion sector of the target chiral gauge theory is formulated in a manifestly gauge invariant manner with the lattice regularization.

Under the gauge transformation (2.9), the vertex operator of the original scalar fields,

$$V_{\{n_{\alpha}\}}(n):=e^{i\sum_{\alpha}n_{\alpha}\phi_{\alpha}(n)},$$

(2.31)

changes as

$$V_{\{n_{\alpha}\}}(n)\to\exp\left[-i\sum_{\alpha}2q_{A,\alpha}n_{\alpha}\Lambda(n)\right]V_{\{n_{\alpha}\}}(n).$$

(2.32)

Thus, this vertex operator possesses the axial $U(1)$ charge $\sum_{\alpha}2q_{A,\alpha}n_{\alpha}$ while no vector $U(1)$ charge. We can make this vertex operator gauge invariant by connecting an open Wilson line

$$\exp\left[i\sum_{\alpha}2q_{A,\alpha}n_{\alpha}\sum_{(n,\mu)\in P}^{n}A_{\mu}(n)\right]$$

(2.33)

to it, where $P$ denotes a path on the lattice ending at the position of the vertex operator, $n$.

If we perform a constant shift of $\phi_{\alpha}(n)$, $\phi_{\alpha}(n)\to\phi_{\alpha}(n)+\xi_{\alpha}$, in the partition function (2.26), the lattice action $S_{\mathrm{B}}$ (2.2) produces a factor,

$$\exp\left[-i\sum_{\alpha}\xi_{\alpha}q_{V,\alpha}\sum_{\tilde{p}\in\tilde{\Gamma}}N_{12}(\tilde{p})\right]=\exp\left[-\frac{i}{2\pi}\sum_{\alpha}\xi_{\alpha}q_{V,\alpha}\sum_{\tilde{p}\in\tilde{\Gamma}}F_{12}(\tilde{p})\right],$$

(2.34)

where the lattice sum is taken over all plaquettes on the dual lattice $\tilde{\Gamma}$. This shows that, when the lattice first Chern number of the background gauge field,

$$\tilde{Q}:=\frac{1}{2\pi}\sum_{\tilde{p}\in\tilde{\Gamma}}F_{12}(\tilde{p})=\sum_{\tilde{p}\in\tilde{\Gamma}}N_{12}(\tilde{p})\in\mathbb{Z}$$

(2.35)

is non-zero, the partition function vanishes as expected from the index theorem in the corresponding gauge theory containing fermions. Here, we should note that the lattice first Chern number (2.35) can be non-trivial under the admissibility (2.16) Luscher:1998du . We can make the partition function (2.26) non-zero by inserting vertex operators (2.31) in an appropriate way. Since

$$V_{\{n_{\alpha}\}}(n)\to e^{i\sum_{\alpha}\xi_{\alpha}n_{\alpha}}V_{\{n_{\alpha}\}}(n),$$

(2.36)

under the constant shift, the condition for non-vanishingness is

$$\sum_{I}n_{I,\alpha}=\frac{q_{V,\alpha}}{2\pi}\sum_{\tilde{p}\in\tilde{\Gamma}}F_{12}(\tilde{p})=q_{V,\alpha}\tilde{Q},$$

(2.37)

where $I$ labels the inserted vertex operators.

On the other hand, for the magnetic object introduced by the excision method,

$$M_{\{m_{\alpha}\}}(\mathcal{D}),$$

(2.38)

the gauge transformation induces (see Eq. (2.29)),

$$M_{\{m_{\alpha}\}}(\mathcal{D})\to\exp\left[i\sum_{\alpha}q_{V,\alpha}m_{\alpha}\Lambda(\tilde{n}_{*})\right]M_{\{m_{\alpha}\}}(\mathcal{D}).$$

(2.39)

Therefore, this object possesses the vector $U(1)$ charge $-\sum_{\alpha}q_{V,\alpha}m_{\alpha}$ while possessing no axial $U(1)$ charge. For the magnetic objects, we have the consistency condition,

$$\sum_{\tilde{I}}m_{\tilde{I},\alpha}=-\frac{2q_{A,\alpha}}{2\pi}\sum_{p\in\Gamma-\sum_{\tilde{I}}\mathcal{D}_{\tilde{I}}}F_{12}(p)-\frac{2q_{A,\alpha}}{2\pi}\sum_{\tilde{I}}F(\partial\mathcal{D}_{\tilde{I}})=-2q_{A,\alpha}Q,$$

(2.40)

where $\tilde{I}$ labels the magnetic objects and the first lattice sum on the right-hand side is taken over the plaquette $p$ on $\Gamma-\sum_{\tilde{I}}\mathcal{D}_{\tilde{I}}$. This consistency is obtained by simply summing the Bianchi identity (2.19) over the region $\Gamma-\sum_{\tilde{I}}\mathcal{D}_{\tilde{I}}$ and noting the definition of the magnetic charge (2.20). In Eq. (2.40), we have introduced another lattice first Chern number:

	$\displaystyle Q$	$\displaystyle:=\frac{1}{2\pi}\sum_{p\in\Gamma-\sum_{\tilde{I}}\mathcal{D}_{\tilde{I}}}F_{12}(p)+\frac{1}{2\pi}\sum_{\tilde{I}}F(\partial\mathcal{D}_{\tilde{I}})$
		$\displaystyle=\sum_{p\in\Gamma-\sum_{\tilde{I}}\mathcal{D}_{\tilde{I}}}N_{12}(p)+\sum_{\tilde{I}}N(\partial\mathcal{D}_{\tilde{I}})\in\mathbb{Z},$		(2.41)

where, in the last line, we have used Eqs. (2.13) and (2.21). Although we have two lattice Chern numbers $\tilde{Q}$ (2.35) and $Q$ (2.41), one might expect that those two coincide in the continuum limit. In fact, since the difference between $\tilde{Q}$ and $Q$ arises only from the differences in the sum of $F_{12}(\tilde{n})$ and of $F_{12}(n)$ on a finite number of plaquettes around the magnetic objects and the sum of $F(\partial\mathcal{D}_{\tilde{I}})$, if we take $\delta$ in Eq. (2.16) and $\delta^{\prime}$ in Eq. (2.22) small enough, the equality of two integers, $\tilde{Q}=Q$, automatically holds. We assume this in what follows.

The elementary fermion fields in the target theory may be represented by “superposing” the vertex operator (2.31) and the magnetic object (2.38). For a flavor $\alpha$, we have

	$\displaystyle P_{R}\psi_{\alpha}:e^{+i\phi_{\alpha}(n)/2}M_{m_{\alpha}=-1}(\mathcal{D}),$
	$\displaystyle\bar{\psi}_{\alpha}P_{L}:e^{-i\phi_{\alpha}(n)/2}M_{m_{\alpha}=+1}(\mathcal{D}).$
	$\displaystyle P_{L}\psi_{\alpha}:e^{-i\phi_{\alpha}(n)/2}M_{m_{\alpha}=-1}(\mathcal{D}),$
	$\displaystyle\bar{\psi}_{\alpha}P_{R}:e^{+i\phi_{\alpha}(n)/2}M_{m_{\alpha}=+1}(\mathcal{D}).$		(2.42)

These representations can be identified by examining the transformation law under the gauge transformation. These representations may be used to compute correlation functions containing fermion fields. These elementary fields are not gauge invariant but we may dress them by open Wilson and ’t Hooft lines to render them gauge invariant. For instance,

$$\widehat{P_{L}\psi_{\alpha}}:e^{-i\phi_{\alpha}(n)/2}M_{m_{\alpha}=-1}(\mathcal{D})\exp\left[-iq_{A,\alpha}\sum_{(n,\mu)\in P}^{n}A_{\mu}(n)+iq_{V,\alpha}\sum_{(\tilde{n},\mu)\in\tilde{P}}^{\tilde{n}_{*}}A_{\mu}(\tilde{n})\right]$$

(2.43)

is gauge invariant, where the hat $\widehat{\phantom{x}}$ indicates the field is dressed. We may use such a gauge invariant field to saturate the selection rule. When $\tilde{Q}=Q$, the selection rules in Eqs. (2.37) and (2.40) imply that correlation functions vanish unless the (gauge-invariant) $|q_{L,\alpha}Q|$ product of

$$\begin{cases}P_{L}\psi_{\alpha}&\text{when $q_{L,\alpha}Q<0$},\\ \bar{\psi}_{\alpha}P_{R}&\text{when $q_{L,\alpha}Q>0$},\\ \end{cases}$$

(2.44)

and the (gauge-invariant) $|q_{R,\alpha}Q|$ product of

$$\begin{cases}P_{R}\psi_{\alpha}&\text{when $q_{R,\alpha}Q>0$},\\ \bar{\psi}_{\alpha}P_{L}&\text{when $q_{R,\alpha}Q<0$},\\ \end{cases}$$

(2.45)

are inserted in addition to pairs of $\psi_{\alpha}$ and $\bar{\psi}_{\alpha}$. In the target chiral gauge theory in continuum, one may define gauge-invariant fermion number currents of the left-handed and right-handed Weyl fermions, respectively, as (see Ref. Fujikawa:1994np and references cited therein)

$$J_{\mu}^{L,R}(x):=-\lim_{\Lambda\to\infty}\tr\gamma_{\mu}P_{L,R}\frac{1}{{\ooalign{\hfil/\hfil\crcr$D$}}_{L,R}}e^{{\ooalign{\hfil/\hfil\crcr$D$}}_{L,R}^{2}/\Lambda^{2}}\delta(x-y)|_{y\to x},$$

(2.46)

where ${\ooalign{\hfil/\hfil\crcr$D$}}_{L,R}:=\gamma_{\mu}(\partial_{\mu}+iq_{L,R}A_{\mu})$. Then the fermion number anomaly is given by⁶⁶6Our convention is $\gamma_{5}=i\gamma_{1}\gamma_{2}$ and $\epsilon_{12}=1$.

$\displaystyle\partial_{\mu}J_{\mu}^{L,R}(x)$

$\displaystyle=\mp\lim_{\Lambda\to\infty}\tr\gamma_{5}e^{{\ooalign{\hfil/\hfil\crcr$D$}}_{L,R}^{2}/\Lambda^{2}}\delta(x-y)|_{y\to x}=\mp\frac{q_{L,R}}{2\pi}F_{12}(x).$

(2.49)

The above selection rule in Eqs. (2.44) and (2.45) is precisely consistent with the 2D integral of these non-conservation laws.