Erratum to: Eur. Phys. J. C (2023) 83(6):481 https://doi.org/10.1140/epjc/s10052-023-11616-6

In this Erratum we do some corrections to equations in the continuum theory given in Sect. 2. Our discussions are falsely attributed to the \(2\pi \) periodicity of the compact scalar \(\chi \), and then the definition of the 4-form gauge field \(D^{(4)}\) is incorrect. These modifications do not modify any conclusion in the lattice theory.

A consistency check for the periodicity of the Lagrange multiplier \(\chi \) revealed that there was an error in the continuum Lagrangian. That is, Eq. (2.1) should be

$$\begin{aligned} {\mathcal {L}}&= \frac{1}{2g^2} \mathcal {F}\wedge \star \mathcal {F} + \frac{i\theta }{8\pi ^2} \mathcal {F}\wedge \mathcal {F} \nonumber \\&\quad + iq\chi \left( \frac{1}{8\pi ^2}\mathcal {F}\wedge \mathcal {F} - \frac{p}{2\pi }d C^{(3)}\right) + \frac{i\hat{\theta }}{2\pi } d C^{(3)}. \end{aligned}$$
(1)

Note that the third term in the right hand side is multiplied by q. Then, one can confirm the existence of the \(\mathbb {Z}_{pq}\) 3-form global symmetry.

Instead of Eq. (2.11), in order to maintain the periodicity of \(\chi \), we must impose the condition

$$\begin{aligned} \int \Bigg [\frac{q}{8\pi ^2}\left( \mathcal {F}'-B^{(2)}\right) \wedge \left( \mathcal {F}'-B^{(2)}\right) - \frac{pq}{2\pi } d C^{(3)}\Bigg ] \in \mathbb {Z}. \end{aligned}$$
(2)

The contribution from the quadratic term of \(B^{(2)}\) can in general take a value in \(\frac{1}{q}\mathbb {Z}\), but the other terms are integers. To admit of nontrivial configurations of \(B^{(2)}\), \(D^{(4)}\) (2.12) should be introduced as

$$\begin{aligned} pqD^{(4)} = d D^{(3)} - \frac{q}{4\pi } B^{(2)}\wedge B^{(2)} , \end{aligned}$$
(3)

where the 1-form transformation acts as

$$\begin{aligned} D^{(3)} \mapsto D^{(3)} - \frac{q}{2\pi } B^{(2)}\wedge \Lambda ^{(1)} + \frac{q}{4\pi }\Lambda ^{(1)}\wedge d\Lambda ^{(1)}. \end{aligned}$$
(4)

In particular, the \(2\pi \) periodicity of \(D^{(3)}\) is not violated by this gauge transformation. Therefore, Eq. (2.18) should be

$$\begin{aligned}&\displaystyle \int \Bigg [\frac{q}{8\pi ^2} \left( \mathcal {F}'-B^{(2)}\right) \wedge \left( \mathcal {F}'-B^{(2)}\right) \nonumber \\&\qquad - \frac{pq}{2\pi } d C^{(3)} + \frac{pq}{2\pi }D^{(4)}\Bigg ]\nonumber \\&\quad =\displaystyle \int \Bigg [\frac{q}{8\pi ^2} \mathcal {F}'\wedge \mathcal {F}'-\frac{q}{4\pi ^{2}}\mathcal {F}'\wedge B^{(2)} \nonumber \\&\qquad - \frac{pq}{2\pi } d C^{(3)} + \frac{1}{2\pi }dD^{(3)}\Bigg ]\in \mathbb {Z} . \end{aligned}$$
(5)

The shift \(\theta \rightarrow \theta +2\pi /p\) gives rise to a \(\mathbb {Z}_{p q^2}\) phase:

$$\begin{aligned} \int D^{(4)}&=\frac{2\pi }{pq^{2}} \left( q\int \frac{dD^{(3)}}{2\pi } +q\int \frac{\mathcal {F}'}{2\pi }\wedge \frac{qB^{(2)}}{2\pi } \nonumber \right. \\&\quad \left. -\frac{1}{2}\int \frac{qB^{(2)}}{2\pi }\wedge \frac{qB^{(2)}}{2\pi } \right) \in \frac{2\pi }{pq^{2}}\mathbb {Z}. \end{aligned}$$
(6)

Also, Eqs. (2.21) and (2.22) become

$$\begin{aligned}&pq\mathcal {D}^{(4)} = d D^{(3)} , \, \nonumber \\&pq\mathcal {D}^{(4)} \,\mapsto pq\mathcal {D}^{(4)} + \frac{q}{2\pi }B^{(2)}\wedge d\Lambda ^{(1)} + \frac{q}{4\pi }d\Lambda ^{(1)}\wedge d\Lambda ^{(1)} . \end{aligned}$$
(7)

These modifications do not modify any conclusion in the lattice theory. We would like to thank Ryo Yokokura for valuable discussions. The authors would like to apologize for this error.