Higher Gauging and Non-invertible Condensation Defects

Abstract

We discuss invertible and non-invertible topological condensation defects arising from gauging a discrete higher-form symmetry on a higher codimensional manifold in spacetime, which we define as higher gauging. A q-form symmetry is called p-gaugeable if it can be gauged on a codimension-p manifold in spacetime. We focus on 1-gaugeable 1-form symmetries in general 2+1d QFT, and gauge them on a surface in spacetime. The universal fusion rules of the resulting invertible and non-invertible condensation surfaces are determined. In the special case of 2+1d TQFT, every (invertible and non-invertible) 0-form global symmetry, including the ${\mathbb {Z}}_2$ electromagnetic symmetry of the ${\mathbb {Z}}_2$ gauge theory, is realized from higher gauging. We further compute the fusion rules between the surfaces, the bulk lines, and lines that only live on the surfaces, determining some of the most basic data for the underlying fusion 2-category. We emphasize that the fusion “coefficients” in these non-invertible fusion rules are generally not numbers, but rather 1+1d TQFTs. Finally, we discuss examples of non-invertible symmetries in non-topological 2+1d QFTs such as the free U(1) Maxwell theory and QED.

A More on the Fusion Rules

1.1 A.1 Fusion rules from the higher gauging of ${\mathbb {Z}}_N$ 1-form symmetries

In this appendix we will derive the fusion rule (5.22) between the higher quantum symmetry lines on the condensation surfaces that arise from a 1-gaugeable ${\mathbb {Z}}_N$ 1-form symmetry. The 0-anomaly of the 1-gaugeable ${\mathbb {Z}}_N$ 1-form symmetry is labeled by an integer k (see Sect. 5.1).

$$\begin{aligned} S_n \times S_{n'}\,{ fusion} \end{aligned}$$

As a warm up, we fist derive the fusion rule (5.13) between the condensation surfaces $S_n$, which come from the higher gauging of the ${\mathbb {Z}}_n$ 1-form symmetry subgroup with n|N. We will place the condensation surfaces on a genus-g surface $\Sigma _g$ with generating 1-cycles ${\textbf{A}}_{i=1,\dots ,g},{\textbf{B}}_{i=1,\dots ,g} \in H_1(\Sigma _g, {\mathbb {Z}}_N)$.

Let $n,n'$ be two divisors of N. The fusion of the corresponding condensation surfaces $S_n$ and $S_{n'}$ is

$$\begin{aligned} \begin{aligned}&S_{n}(\Sigma _g) \times S_{n'}(\Sigma _g) \\&\quad = \frac{1}{n^g {n'}^g} \sum _{\mu _i ,\nu _i \in {\mathbb {Z}}_n} \sum _{\mu '_i,\nu '_i \in {\mathbb {Z}}_{n'}} e^{2\pi i \frac{kN}{n n'} (\mu _i \nu '_i - \nu _i \mu '_i )} \, \eta \left( \sum _i N(\frac{\mu _i}{n}+\frac{\mu '_i}{n'}) {\textbf{A}}_i + N(\frac{\nu _i}{n}+\frac{\nu '_i}{n'}) {\textbf{B}}_i \right) \, . \end{aligned}\end{aligned}$$

(A.1)

Below we will do a change of variables on $\mu _i,\nu _i,\mu '_i,\nu '_i$ in order to simplify the above expression.

To proceed, we first note that for any pair of numbers $(\mu ,\mu ') \in {\mathbb {Z}}_n \times {\mathbb {Z}}_{n'}$ we can do the change of variable

$$\begin{aligned} \begin{aligned} \mu&= p \alpha + \frac{n}{\textrm{gcd}(n,n')}\lambda ~,\\ \mu '&= p' \alpha - \frac{n'}{\textrm{gcd}(n,n')}\lambda ~, \end{aligned}\end{aligned}$$

(A.2)

for $\alpha \in \left\{ 1, \dots , \textrm{lcm}(n,n') \right\}$ and $\lambda \in {\mathbb {Z}}_{\textrm{gcd}(n,n')}$, where

$$\begin{aligned} p \frac{n'}{\textrm{gcd}(n,n')} + p' \frac{n}{\textrm{gcd}(n,n')} = 1 \end{aligned}$$

(A.3)

for some fixed integers p and $p'$. Note that (A.2) defines a map from $\left\{ 1, \dots , \textrm{lcm}(n,n') \right\} \times {\mathbb {Z}}_{\textrm{gcd}(n,n')}$ to ${\mathbb {Z}}_n \times {\mathbb {Z}}_{n'}$ whose inverse is given by

$$\begin{aligned} \begin{aligned} \alpha&= \frac{n'}{\textrm{gcd}(n,n')}\mu + \frac{n}{\textrm{gcd}(n,n')} \mu ' \in {\mathbb {Z}}_{\textrm{lcm}(n,n')} ~,\\ \lambda&= p' \mu - p\mu ' \in {\mathbb {Z}}_{\textrm{gcd}(n,n')}~. \end{aligned}\end{aligned}$$

(A.4)

Therefore we conclude that the map is bijective and hence the change of variable is legitimate.

Now let us apply the change of variable (A.2) to $\mu _i,\mu _i',\nu _i,\nu _i'$ and write

$$\begin{aligned} \begin{aligned} \mu _i&= p \alpha _i + \frac{n}{\textrm{gcd}(n,n')}\lambda _i ~,\\ \mu '_i&= p' \alpha _i - \frac{n'}{\textrm{gcd}(n,n')}\lambda _i~,\\ \nu _i&= p \beta _i + \frac{n}{\textrm{gcd}(n,n')}\rho _i ~,\\ \nu '_i&= p' \beta _i - \frac{n'}{\textrm{gcd}(n,n')}\rho _i ~, \end{aligned}\end{aligned}$$

(A.5)

where $\alpha _i,\beta _i \in \left\{ 1, \dots , \textrm{lcm}(n,n') \right\}$, $\lambda _i,\rho _i \in {\mathbb {Z}}_{\textrm{gcd}(n,n')}$, and $pn'+p'n=\textrm{gcd}(n,n')$ for some fixed integers p and $p'$. Using these new set of variables we get

$$\begin{aligned} S_{n}(\Sigma _g)\times S_{n'}(\Sigma _g) = \frac{1}{n^g {n'}^g} \sum _{\begin{array}{c} 1 \le \alpha _i,\beta _i \le \textrm{lcm}(n,n') \\ \lambda _i,\rho _i \in {\mathbb {Z}}_{\textrm{gcd}(n,n')} \end{array}} e^{2\pi i \frac{k\ell }{\textrm{gcd}(n,n')} (\beta _i \lambda _i - \alpha _i \rho _i )} \, \eta \bigg ( \ell (\alpha _i \, {\textbf{A}}_i + \beta _i \, {\textbf{B}}_i ) \bigg ) \, , \end{aligned}$$

(A.6)

where $\ell = \frac{N}{\textrm{lcm}(n,n')}$. Summing over $\lambda _i,\rho _i \in {\mathbb {Z}}_{\textrm{gcd}(n,n')}$ gives the Kronecker deltas

$$\begin{aligned} \begin{aligned} \textrm{gcd}(n,n')^{2g} \, \delta _{\, \left( \frac{k\ell }{\textrm{gcd}(n,n')} \alpha _i \mod {\mathbb {Z}}\right) ,0} \, \delta _{\,\left( \frac{k\ell }{\textrm{gcd}(n,n')} \beta _i \mod {\mathbb {Z}}\right) ,0} \end{aligned}\end{aligned}$$

(A.7)

which forces $\alpha _i$ and $\beta _i$ to be divisible by $\frac{\textrm{gcd}(n,n')}{\textrm{gcd}(n,n',k\ell )}$. Therefore we do the replacement $\alpha _i,\beta _i \mapsto \frac{\textrm{gcd}(n,n')}{\textrm{gcd}(n,n',k\ell )} {\bar{\alpha }}_i, \frac{\textrm{gcd}(n,n')}{\textrm{gcd}(n,n',k\ell )}{\bar{\beta }}_i$ for $1 \le {\bar{\alpha }}_i,{\bar{\beta }}_i \le \frac{\textrm{gcd}(n,n',k\ell )nn'}{\textrm{gcd}(n,n')^2}$ and get

$$\begin{aligned} S_n(\Sigma _g)\times S_{n'}(\Sigma _g)&= \frac{\textrm{gcd}(n,n')^{2g}}{n^g{n'}^g} \sum _{ 1 \le {\bar{\alpha }}_i,{\bar{\beta }}_i \le \frac{\textrm{gcd}(n,n',k\ell )nn'}{\textrm{gcd}(n,n')^2} } \eta ^{\frac{\ell \, \textrm{gcd}(n,n')}{\textrm{gcd}(n,n',k\ell )}} \Bigg ({\bar{\alpha }}_i \, {\textbf{A}}_i+ {\bar{\beta }}_i \, {\textbf{B}}_i \Bigg ) \, , \nonumber \\&= \left( \textrm{gcd}(n,n',k\ell ) \right) ^g S_{\frac{\textrm{gcd}(n,n',k\ell )nn'}{\textrm{gcd}(n,n')^2}}(\Sigma _g) \, . \end{aligned}$$

(A.8)

We conclude that the fusion algebra is

$$\begin{aligned} S_n \times S_{n'} = \left( {\mathcal {Z}}_{\textrm{gcd}(n,n',k\ell )}\right) \, S_{\frac{\textrm{gcd}(n,n',k\ell )nn'}{\textrm{gcd}(n,n')^2}} \,, \end{aligned}$$

(A.9)

where $\left( {\mathcal {Z}}_{n}\right)$ stands for the 1+1d ${\mathbb {Z}}_n$ gauge theory.

$$\begin{aligned} {\hat{\eta }}_{n}^a {\tilde{\eta }}_{n}^b \times {\hat{\eta }}_{n'}^{a'} {\tilde{\eta }}_{n'}^{b'}\,{ fusion} \end{aligned}$$

Now we will derive the fusion between the higher quantum symmetry lines living on the surfaces (5.22). We put the composite defects ${\hat{\eta }}_{n}^a {\tilde{\eta }}_{n}^b$ of (5.21) on $({\bar{\gamma }}, \Sigma )$ and perform the computations covariantly without specifying a basis for $H_1(\Sigma , {\mathbb {Z}})$:

$$\begin{aligned} \begin{aligned}&{\hat{\eta }}_{n}^a {\tilde{\eta }}_{n}^b (\bar{\gamma }, \Sigma )\times {\hat{\eta }}_{n'}^{a'} {\tilde{\eta }}_{n'}^{b'} (\bar{\gamma }, \Sigma ) \\&\quad = \sum _{\begin{array}{c} \gamma \in H_1(\Sigma ,{\mathbb {Z}}_{n}) \\ \gamma ' \in H_1(\Sigma ,{\mathbb {Z}}_{n'}) \end{array}} \frac{e^{ 2\pi i \langle \bar{\gamma }, {a \over n}\gamma + {a' \over n'}\gamma ' \rangle + {2\pi ik \over N}\langle \frac{N}{n}\gamma + b\bar{\gamma }, \frac{N}{n'}\gamma ' + b'\bar{\gamma } \rangle }}{\sqrt{|H_1(\Sigma ,{\mathbb {Z}}_n)||H_1(\Sigma ,{\mathbb {Z}}_{n'})|}} \, \eta \left( \frac{N}{n}\gamma +\frac{N}{n'}\gamma ' + (b+b')\bar{\gamma } \right) \,. \end{aligned}\end{aligned}$$

(A.10)

Next, we apply the same change of variable as in (A.2) to $\gamma \in H_1(\Sigma ,{\mathbb {Z}}_{n})$ and $\gamma ' \in H_1(\Sigma ,{\mathbb {Z}}_{n'})$. We write

$$\begin{aligned} \begin{aligned} \gamma&= p \alpha + \frac{n}{\textrm{gcd}(n,n')}\lambda ~,\\ \gamma '&= p' \alpha - \frac{n'}{\textrm{gcd}(n,n')}\lambda ~, \end{aligned}\end{aligned}$$

(A.11)

for $\alpha \in \left\{ \sum _{i,j} \alpha _i {\textbf{A}}_i + \beta _j {\textbf{B}}_j \in H_1(\Sigma ,{\mathbb {Z}}) \mid \alpha _i,\beta _j \in \{ 1, \dots , \textrm{lcm}(n,n') \} \right\}$ and $\lambda \in H_1(\Sigma ,{\mathbb {Z}}_{\textrm{gcd}(n,n')})$, where the integers p and $p'$ satisfy $p n' + p'n = \textrm{gcd}(n,n')$. Here ${\textbf{A}}_{i},{\textbf{B}}_{j}$ are a fixed set of generators for $H_1(\Sigma ,{\mathbb {Z}})$.

In terms of these new variables we get:

$$\begin{aligned} \begin{aligned}&{\hat{\eta }}_{n}^a {\tilde{\eta }}_{n}^b(\bar{\gamma }, \Sigma ) \times {\hat{\eta }}_{n'}^{a'} {\tilde{\eta }}_{n'}^{b'} (\bar{\gamma }, \Sigma ) \\&\quad = \sum _{\begin{array}{c} \alpha \in H_1(\Sigma ,{\mathbb {Z}}_{\textrm{lcm}(n,n')}) \\ \lambda \in H_1(\Sigma ,{\mathbb {Z}}_{\textrm{gcd}(n,n')}) \end{array}} {e^{ {2\pi i \over \textrm{gcd}(n,n')} \langle (c-c')\bar{\gamma } - k\ell \alpha , \lambda \rangle + {2\pi i}\langle \bar{\gamma }, ( \frac{pc}{n} + \frac{p'c'}{n'} ) \alpha \rangle } \over \sqrt{|H_1(\Sigma ,{\mathbb {Z}}_n)||H_1(\Sigma ,{\mathbb {Z}}_{n'})|}} \; \eta \big ( \ell \alpha + (b+b')\bar{\gamma } \big ) \,, \end{aligned}\end{aligned}$$

(A.12)

where $c=a-kb'$, $c'=a'+kb$, and $\ell = {N \over \textrm{lcm}(n,n')}$. The sum over $\lambda$ gives a Kronecker delta, multiplied by $|H_1(\Sigma ,{\mathbb {Z}}_{\textrm{gcd}(n,n')})|$, that enforces $\alpha$ to satisfy

$$\begin{aligned} \begin{aligned} (c-c')\bar{\gamma } - k\ell \alpha = 0 \in H_1(\Sigma ,{\mathbb {Z}}_{\textrm{gcd}(n,n')}) \,. \end{aligned}\end{aligned}$$

(A.13)

Thus, we can rewrite the fusion as

$$\begin{aligned} \begin{aligned}&\frac{1}{|H_1(\Sigma ,{\mathbb {Z}}_{\textrm{gcd}(n,n')})|} \; {\hat{\eta }}_{n}^a {\tilde{\eta }}_{n}^b (\bar{\gamma }, \Sigma )\times {\hat{\eta }}_{n'}^{a'} {\tilde{\eta }}_{n'}^{b'} (\bar{\gamma }, \Sigma ) \\&\quad = \sum _{\begin{array}{c} \alpha \in H_1(\Sigma ,{\mathbb {Z}}_{\textrm{lcm}(n,n')}) \\ (c-c')\bar{\gamma } - k\ell \alpha = 0 \text { mod }\textrm{gcd}(n,n') \end{array}} {e^{ {2\pi i}\langle \bar{\gamma }, ( \frac{pc}{n} + \frac{p'c'}{n'} ) \alpha \rangle } \over \sqrt{|H_1(\Sigma ,{\mathbb {Z}}_n)||H_1(\Sigma ,{\mathbb {Z}}_{n'})|}} \; \eta \big ( \ell \alpha + (b+b')\bar{\gamma } \big ) \,. \end{aligned}\end{aligned}$$

(A.14)

If there is no solution to the constraint (A.13) we get zero on the righthand side. Otherwise, any solution can be written as

$$\begin{aligned} \alpha = {c-c' \over \textrm{gcd}(n,n',k\ell ) } \left( {k\ell \over \textrm{gcd}(n,n',k\ell )} \right) ^{-1}_{N \over x\ell } \bar{\gamma } + {\textrm{gcd}(n,n') \over \textrm{gcd}(n,n',k\ell ) } \bar{\alpha } \,, \qquad \bar{\alpha } \in H_1(\Sigma ,{\mathbb {Z}}_x)\,, \end{aligned}$$

(A.15)

where

$$\begin{aligned} \begin{aligned} x \equiv \frac{\textrm{gcd}(n,n',k\ell )nn'}{\textrm{gcd}(n,n')^2}\,, \end{aligned}\end{aligned}$$

(A.16)

and $\big ({k\ell \over \textrm{gcd}(n,n',k\ell )}\big )^{-1}_{N \over x\ell } \in ({\mathbb {Z}}_{N \over x\ell })^\times$ is the multiplicative inverse of ${k\ell \over \textrm{gcd}(n,n',k\ell )}$ modulo ${N \over x\ell } = {\textrm{gcd}(n,n') \over \textrm{gcd}(n,n',k\ell )}$. The multiplicative inverse always exists since ${k\ell \over \textrm{gcd}(n,n',k\ell )}$ is coprime with respect to ${\textrm{gcd}(n,n') \over \textrm{gcd}(n,n',k\ell )}$.

We substitute (A.15) into (A.14) to get

$$\begin{aligned} \begin{aligned}&\frac{1}{|H_1(\Sigma ,{\mathbb {Z}}_{\textrm{gcd}(n,n')})|} \; {\hat{\eta }}_{n}^a {\tilde{\eta }}_{n}^b(\bar{\gamma }, \Sigma ) \times {\hat{\eta }}_{n'}^{a'} {\tilde{\eta }}_{n'}^{b'} (\bar{\gamma }, \Sigma ) \\&\quad = \sum _{ \begin{array}{c} {\bar{\alpha }} \in H_1(\Sigma ,{\mathbb {Z}}_x) \\ {(c-c') \bar{\gamma } \over \textrm{gcd}(n,n',k\ell ) } \in H_1(\Sigma ,{\mathbb {Z}}) \end{array} } \frac{e^{ {2\pi i \over x} \frac{pcn'+p'c'n}{\textrm{gcd}(n,n')} \langle \bar{\gamma }, \bar{\alpha } \rangle }}{\sqrt{|H_1(\Sigma ,{\mathbb {Z}}_n)||H_1(\Sigma ,{\mathbb {Z}}_{n'})|}}\, \, \eta \Bigg ( \begin{array}{c} (b+b')\bar{\gamma } + {N \over x} {\bar{\alpha }} \\ +{\ell (c-c') \over \textrm{gcd}(n,n',k\ell ) } \big ({k\ell \over \textrm{gcd}(n,n',k\ell )} \big )^{-1}_{N \over x\ell } \bar{\gamma } \end{array} \Bigg ) \,. \end{aligned}\end{aligned}$$

(A.17)

Comparing this with (5.21) we find

$$\begin{aligned}\begin{aligned}&{\hat{\eta }}_{n}^a {\tilde{\eta }}_{n}^b(\bar{\gamma }, \Sigma ) \times {\hat{\eta }}_{n'}^{a'} {\tilde{\eta }}_{n'}^{b'} (\bar{\gamma }, \Sigma ) \\&\quad = \delta _{(c-c'){\bar{\gamma }} \,\text { mod gcd}(n,n',k\ell ) , 0} \, \sqrt{|H_1(\Sigma ,{\mathbb {Z}}_{\textrm{gcd}(n,n',k\ell )})|} \\&\qquad {\hat{\eta }}_{x}^{\frac{pcn'+p'c'n}{\textrm{gcd}(n,n')}} {\tilde{\eta }}_{x}^{{\ell (c-c') \over \textrm{gcd}(n,n',k\ell )}(\frac{k\ell }{\textrm{gcd}(n,n',k\ell )})^{-1}_{N \over x\ell }+b+b'}({\bar{\gamma }}, \Sigma ) \,. \end{aligned} \end{aligned}$$

The fusion coefficient $\delta _{(c-c'){\bar{\gamma }} \,\text { mod gcd}(n,n',k\ell ), 0} \, \sqrt{|H_1(\Sigma ,{\mathbb {Z}}_{\textrm{gcd}(n,n',k\ell )})|}$ is the partition function of the 1+1d ${\mathbb {Z}}_{\textrm{gcd}(n,n',k\ell )}$ gauge theory on $\Sigma$ with the Wilson line $W^{c-c'}$ on ${\bar{\gamma }}$ that we denote by $({\mathcal {Z}}_{\textrm{gcd}(n,n',k\ell )}, W^{c-c'})({\bar{\gamma }},\Sigma )$. This can be seen by

$$\begin{aligned} \begin{aligned} ({\mathcal {Z}}_N, W^{c}) ({\bar{\gamma }},\Sigma ) = {1\over \sqrt{|H_1(\Sigma , {\mathbb {Z}}_N)|}} \sum _{\gamma \in H_1(\Sigma ,{\mathbb {Z}}_N)} e^{\frac{2\pi i}{N} c \langle {\bar{\gamma }},\gamma \rangle } = \delta _{c{\bar{\gamma }} \,\text { mod }N , 0} \, \sqrt{|H_1(\Sigma ,{\mathbb {Z}}_N)|} \,. \end{aligned}\end{aligned}$$

(A.18)

We therefore conclude that

$$\begin{aligned} {\hat{\eta }}_{n}^a {\tilde{\eta }}_{n}^b \times {\hat{\eta }}_{n'}^{a'} {\tilde{\eta }}_{n'}^{b'} = \left( {\mathcal {Z}}_{\textrm{gcd}(n,n',k\ell )}, W^{c-c'} \right) \, {\hat{\eta }}_{x}^{\frac{pcn'+p'c'n}{\textrm{gcd}(n,n')}} {\tilde{\eta }}_{x}^{{\ell (c-c') \over \textrm{gcd}(n,n',k\ell )}\left( \frac{k\ell }{\textrm{gcd}(n,n',k\ell )}\right) ^{-1}_{N \over x\ell }+b+b'} \,. \end{aligned}$$

(A.19)

1.2 A.2 Fusion rules in the ${\mathbb {Z}}_p$ gauge theory

In this appendix we give the detail of the computation for the condensation surfaces in the ${\mathbb {Z}}_p$ gauge theory in Sect. 6.4.

$$\begin{aligned} {\textit{Fusion of condensation defects from higher gauging}}\,{\mathbb {Z}}_p \times {\mathbb {Z}}_p \end{aligned}$$

Let us choose $f,f' \in {\mathbb {Z}}_p$ and compute the fusion between $S_{{\mathbb {Z}}_p \times {\mathbb {Z}}_p, f}$ and $S_{{\mathbb {Z}}_p \times {\mathbb {Z}}_p, f'}$

$$\begin{aligned} \begin{aligned}&S_{{\mathbb {Z}}_p \times {\mathbb {Z}}_p, f}(\Sigma ) \times S_{{\mathbb {Z}}_p \times {\mathbb {Z}}_p, f'}(\Sigma ) \\&\quad = {1 \over {|H_1(\Sigma , {\mathbb {Z}}_p)|}^2 }\sum _{ \gamma _i, \gamma '_i} {e^{{2\pi i \over p} \left( {f \langle \gamma _1, \gamma _2\rangle + f' \langle \gamma '_1, \gamma '_2\rangle }\right) } } \, \eta _1(\gamma _1) \eta _2(\gamma _2) \, \eta _1(\gamma '_1)\eta _2(\gamma '_2) \\&\quad = {1 \over {|H_1(\Sigma , {\mathbb {Z}}_p)|}^2} \sum _{ \gamma _i, \gamma '_i} {e^{{2\pi i \over p} \left( {f \langle \gamma _1, \gamma _2\rangle + (f+f'+1) \langle \gamma '_1, \gamma '_2\rangle + (f+1) \langle \gamma _2, \gamma '_1 \rangle + f \langle \gamma '_2, \gamma _1 \rangle }\right) } } \, \eta _1(\gamma _1) \, \eta _2(\gamma _2) \,. \end{aligned}\end{aligned}$$

(A.20)

To simplifying the RHS, there are qualitatively two different cases depending on whether $f+f'+1 = 0 \pmod {p}$ or not:

Case 1 ($f+f'+1\ne 0$): The sum over $\gamma '_2$ results in the Kronecker delta function factor ${|H_1(\Sigma , {\mathbb {Z}}_p)|} \, \delta _{f\gamma _1 - [f+f'+1]\gamma '_1, 0}$ which sets $\gamma '_1 = \frac{f}{f+f'+1}\gamma _1$ and we get

$$\begin{aligned} \begin{aligned} S_{{\mathbb {Z}}_p \times {\mathbb {Z}}_p, f}(\Sigma ) \times S_{{\mathbb {Z}}_p \times {\mathbb {Z}}_p, f'}(\Sigma ) = \frac{1}{{|H_1(\Sigma , {\mathbb {Z}}_p)|}} \sum _{ \gamma _1, \gamma _2 } e^{\frac{2\pi i}{p} \frac{ff' }{f+f'+1} \langle \gamma _1, \gamma _2\rangle } \, \eta _1(\gamma _1) \, \eta _2(\gamma _2) \,. \end{aligned}\end{aligned}$$

(A.21)

This translates to the invertible fusion rule $S_{{\mathbb {Z}}_p \times {\mathbb {Z}}_p, f} \times S_{{\mathbb {Z}}_p \times {\mathbb {Z}}_p, f'} = S_{{\mathbb {Z}}_p \times {\mathbb {Z}}_p, \frac{ff'}{f+f'+1}}$. If we do the crucial change of variable $f=\frac{1}{m-1}$, the fusion rule simplifies to

$$\begin{aligned} S_{{\mathbb {Z}}_p \times {\mathbb {Z}}_p, \frac{1}{m-1}} \times S_{{\mathbb {Z}}_p \times {\mathbb {Z}}_p, \frac{1}{m'-1}} = S_{{\mathbb {Z}}_p \times {\mathbb {Z}}_p, \frac{1}{mm'-1}} ~. \end{aligned}$$

(A.22)

Case 2 ($f+f'+1=0$): In this case we can perform the sum over $\gamma '_1$ and $\gamma '_2$ to get

$$\begin{aligned} \begin{aligned} S_{{\mathbb {Z}}_p \times {\mathbb {Z}}_p, f}(\Sigma ) \times S_{{\mathbb {Z}}_p \times {\mathbb {Z}}_p, f'}(\Sigma )&= \sum _{ \gamma _1, \gamma _2 } \delta _{(f+1)\gamma _2,0} \, \delta _{f\gamma _1,0} \, e^{{2\pi i \over p} {f \langle \gamma _1, \gamma _2\rangle } } \, \eta _1(\gamma _1) \, \eta _2(\gamma _2) \\&= {\left\{ \begin{array}{ll} \sqrt{|H_1(\Sigma , {\mathbb {Z}}_p)|} \, S_{{\mathbb {Z}}_{p}^{(\infty )}}(\Sigma )\,, &{} f=0 \\ \sqrt{|H_1(\Sigma , {\mathbb {Z}}_p)|} \, S_{{\mathbb {Z}}_{p}^{(0)}}(\Sigma )\,, &{} f'=0 \\ 1\,, &{} f,f' \ne 0 \end{array}\right. } \end{aligned}\end{aligned}$$

(A.23)

For reasons that become clear later, we have introduced a notation where $S_{{\mathbb {Z}}_{p}^{(\infty )}}$ and $S_{{\mathbb {Z}}_{p}^{(0)}}$ correspond to the higher gauging of the first and second ${\mathbb {Z}}_p$ subgroups (generated by $\eta _1$ and $\eta _2$), respectively.

Putting everything together the fusion rules can be summarized as:

$$\begin{aligned} S_{{\mathbb {Z}}_p \times {\mathbb {Z}}_p, \frac{1}{m-1}} \times S_{{\mathbb {Z}}_p \times {\mathbb {Z}}_p, \frac{1}{m'-1}} = {\left\{ \begin{array}{ll} \left( {\mathcal {Z}}_p \right) S_{{\mathbb {Z}}_{p}^{(m)}} &{} m=\frac{1}{m'} =0 \text { or }\infty \\ S_{{\mathbb {Z}}_p \times {\mathbb {Z}}_p, \frac{1}{mm'-1}} &{} \text {otherwise} \end{array}\right. } ~, \end{aligned}$$

(A.24)

where we have introduced another notation of $S_{{\mathbb {Z}}_p \times {\mathbb {Z}}_p, \infty } \equiv 1$. Therefore, we see that the surfaces $S_{{\mathbb {Z}}_p \times {\mathbb {Z}}_p, f}$ with $f\ne 0,-1$ form a cyclic group ${\mathbb {Z}}_{p-1}$ under fusion. In contrast, the surface defects $S_{{\mathbb {Z}}_p \times {\mathbb {Z}}_p, 0}$ and $S_{{\mathbb {Z}}_p \times {\mathbb {Z}}_p, -1}$ are non-invertible.

$$\begin{aligned} {\textit{Fusion of condensation defects from higher gauging}}\,{\mathbb {Z}}_p \end{aligned}$$

The ${\mathbb {Z}}_p \times {\mathbb {Z}}_p$ 1-form symmetry has $p+1$ ${\mathbb {Z}}_p$ subgroups. Given any non-zero element $(m_1,m_2) \in {\mathbb {Z}}_p \times {\mathbb {Z}}_p$ associated with the topological line $\eta _1^{m_1} \eta _2^{m_2}$, we can take the ${\mathbb {Z}}_p$ subgroup generated by it and denote it by ${\mathbb {Z}}_p^{(m_1/m_2)}$. Note that ${\mathbb {Z}}_{p}^{(m)}$ only depend on $m=\frac{m_1}{m_2} \pmod {p}$, where $m=\infty$ and $m=0$ corresponds to the first and second ${\mathbb {Z}}_p$ subgroups which are 0-gaugeable. Therefore, from the results of Sect. 5.1, $S_{{\mathbb {Z}}_{p}^{(\infty )}}$ and $S_{{\mathbb {Z}}_{p}^{(0)}}$ are non-invertible and the rest are all invertible order 2 defects.

Denote the surface defect from the higher gauging of ${\mathbb {Z}}_{p}^{(m)}$ by

$$\begin{aligned} S_{{\mathbb {Z}}_{p}^{(m)}}(\Sigma ) = \frac{1}{\sqrt{|H_1(\Sigma , {\mathbb {Z}}_p)|}} \sum _{ \gamma \in H_1(\Sigma ,{\mathbb {Z}}_p) } \eta _1(m_1 \gamma ) \, \eta _2( m_2 \gamma ) \,. \end{aligned}$$

(A.25)

The fusion of this surface with itself has been computed in Sect. 5. Thus let us take $m \ne m'$ and compute the fusion of $S_{{\mathbb {Z}}_{p}^{(m)}}$ with $S_{{\mathbb {Z}}_{p}^{(m')}}$:

$$\begin{aligned} S_{{\mathbb {Z}}_{p}^{(m)}}(\Sigma )\times S_{{\mathbb {Z}}_{p}^{(m')}}(\Sigma )&= \sum _{ \gamma , \gamma ' } {e^{\frac{2\pi i}{p} m_2m'_1 \langle \gamma , \gamma '\rangle } \over {|H_1(\Sigma , {\mathbb {Z}}_p)|}} \, \eta _1(m_1 \gamma + m'_1\gamma ') \eta _2( m_2 \gamma + m'_2 \gamma ') \nonumber \\&= \sum _{ \gamma , \gamma ' } {e^{\frac{2\pi i}{p} \frac{m'}{m-m'} \langle m_1 \gamma + m'_1\gamma ', m_2 \gamma + m'_2 \gamma ' \rangle } \over {|H_1(\Sigma , {\mathbb {Z}}_p)|}} \nonumber \\&\quad \eta _1(m_1 \gamma + m'_1\gamma ') \eta _2( m_2 \gamma + m'_2 \gamma ') \nonumber \\&= S_{{\mathbb {Z}}_p \times {\mathbb {Z}}_p, \frac{1}{\frac{m}{m'}-1}} (\Sigma ) ~ . \end{aligned}$$

(A.26)

$$\begin{aligned} {\textit{Fusion between condensation defects from higher gauging}}\, {\mathbb {Z}}_p\,{\textit{ and }}\,{\mathbb {Z}}_p\times {\mathbb {Z}}_p \end{aligned}$$

Take $m=m_1/m_2$ and $f= 1/(m'-1)$, we have

$$\begin{aligned} \begin{aligned}&S_{{\mathbb {Z}}_{p}^{(m)}}(\Sigma )\times S_{{\mathbb {Z}}_{p} \times {\mathbb {Z}}_{p} , \frac{1}{m'-1}}(\Sigma ) \\ {}&= \frac{1}{\sqrt{|H_1(\Sigma , {\mathbb {Z}}_p)|}^3} \sum _{ \gamma , \gamma '_i } e^{\frac{2\pi i}{p} f \langle \gamma '_1, \gamma '_2 \rangle } \, \eta _1(m_1 \gamma ) \eta _2( m_2 \gamma ) \eta _1(\gamma _1') \eta _2( \gamma '_2) \\&= \frac{1}{\sqrt{|H_1(\Sigma , {\mathbb {Z}}_p)|}} \sum _{ \gamma '_1, \gamma '_2 } e^{\frac{2\pi i}{p} f \langle \gamma '_1, \gamma '_2 \rangle } \, \delta _{m_2(f+1) \gamma '_1, m_1f \gamma '_2} \; \eta _1(\gamma '_1) \eta _2(\gamma '_2) \\&= {\left\{ \begin{array}{ll} \left( {\mathcal {Z}}_p \right) S_{{\mathbb {Z}}_{p} \times {\mathbb {Z}}_{p} , \frac{1}{m'-1}}(\Sigma )\,, &{} m=m' \in \{0, \infty \} \\ S_{{\mathbb {Z}}_p^{(m/m')}}(\Sigma )\,, &{} \text {otherwise} \end{array}\right. } \,. \end{aligned}\end{aligned}$$

(A.27)

and

$$\begin{aligned} \begin{aligned}&S_{{\mathbb {Z}}_{p} \times {\mathbb {Z}}_{p} , \frac{1}{m'-1}}(\Sigma ) \times S_{{\mathbb {Z}}_{p}^{(m)}}(\Sigma ) \\ {}&= \frac{1}{\sqrt{|H_1(\Sigma , {\mathbb {Z}}_p)|}^3} \sum _{ \gamma , \gamma '_i } e^{\frac{2\pi i}{p} f \langle \gamma '_1, \gamma '_2 \rangle } \, \eta _1(\gamma _1') \eta _2( \gamma '_2) \eta _1(m_1 \gamma ) \eta _2( m_2 \gamma ) \\&= \frac{1}{\sqrt{|H_1(\Sigma , {\mathbb {Z}}_p)|}} \sum _{ \gamma '_1, \gamma '_2 } e^{\frac{2\pi i}{p} f \langle \gamma '_1, \gamma '_2 \rangle } \, \delta _{m_2f \gamma '_1, m_1(f+1) \gamma '_2} \; \eta _1(\gamma '_1) \eta _2(\gamma '_2) \\&= {\left\{ \begin{array}{ll} \left( {\mathcal {Z}}_p \right) S_{{\mathbb {Z}}_{p} \times {\mathbb {Z}}_{p} , \frac{1}{m'-1}}(\Sigma ) &{} m=1/m' \in \{0, \infty \} \\ S_{{\mathbb {Z}}_p^{(m'm)}}(\Sigma ) &{} \text {otherwise} \end{array}\right. } \,. \end{aligned} \end{aligned}$$

(A.28)

Putting everything together, we have derived the fusion rule (6.70) of the surfaces, which includes the invertible $D_{2(p-1)}$ 0-form symmetry, of the 2+1d ${\mathbb {Z}}_p$ gauge theory.