Non-invertible Condensation, Duality, and Triality Defects in 3+1 Dimensions

Abstract

We discuss a variety of codimension-one, non-invertible topological defects in general 3+1d QFTs with a discrete one-form global symmetry. These include condensation defects from higher gauging of the one-form symmetries on a codimension-one manifold, each labeled by a discrete torsion class, and duality and triality defects from gauging in half of spacetime. The universal fusion rules between these non-invertible topological defects and the one-form symmetry surface defects are determined. Interestingly, the fusion coefficients are generally not numbers, but 2+1d TQFTs, such as invertible SPT phases, ${\mathbb {Z}}_N$ gauge theories, and $U(1)_N$ Chern-Simons theories. The associativity of these algebras over TQFT coefficients relies on nontrivial facts about 2+1d TQFTs. We further prove that some of these non-invertible symmetries are intrinsically incompatible with a trivially gapped phase, leading to nontrivial constraints on renormalization group flows. Duality and triality defects are realized in many familiar gauge theories, including free Maxwell theory, non-abelian gauge theories with orthogonal gauge groups, ${{{\mathcal {N}}}}=1,$ and ${{{\mathcal {N}}}}=4$ super Yang-Mills theories.

Appendices

$(-1)^{Q}\times U(1)_N \,\hbox {=}\, U(1)_N$

Let M be an oriented three-manifold. In this appendix, we prove that, for even N, the $U(1)_{N}$ Chern-Simons theory is invariant under stacking the 2+1d ${\mathbb {Z}}_N^{(0)}$-SPT given by the Dijkgraaf-Witten term

$$\begin{aligned} \begin{aligned} (-1)^{Q(M,\Sigma )} = \exp \left( {i \pi \over N} \int _M A\cup \delta A\right) , \end{aligned} \end{aligned}$$

(A.1)

where $A=$PD$(\Sigma )$ is the Poincaré dual of the surface $\Sigma$ in M. This ${\mathbb {Z}}_N^{(0)}$-SPT corresponds to the order 2 element of $H^3(B{\mathbb {Z}}_N;U(1))\cong {\mathbb {Z}}_N$. More precisely, we mean the following

$$\begin{aligned} \begin{aligned} (-1)^{Q(M,\Sigma )} Z_{U(1)_N} [M ]=Z_{U(1)_N} [M ], \end{aligned} \end{aligned}$$

(A.2)

where $Z_{U(1)_N}[M]$ is the partition function of $U(1)_N$ on M. This fact was crucial for the triality fusion rule in (3.10) to be associative, where both the SPT $(-1)^Q$ and the TQFT $U(1)_N$ appear as the fusion “coefficients” of the defects.

We can schematically write the above equality as

$$\begin{aligned} \begin{aligned} \hbox {``} \, (-1)^Q \times U(1)_N = U(1)_N.\hbox {''} \end{aligned} \end{aligned}$$

(A.3)

Here, $(-1)^Q$, which is an SPT of PD$(\Sigma )$, and $U(1)_N$, which is a TQFT, are both regarded as special cases of an SET of a ${\mathbb {Z}}_N^{(0)}$ bundle on M, and the product above means the tensor product between two SETs. Indeed, most generally the fusion “coefficients” between two topological defects are SETs (see, for example, [19]).

Using the U(1) gauge field notations, the combined system of the Chern-Simons theory and this SPT phase can be expressed by the following Lagrangian:

$$\begin{aligned} \frac{iN}{4\pi } ada + \frac{iN}{4\pi } AdA + \frac{iN}{2\pi } A d{\tilde{a}} \end{aligned}$$

(A.4)

where a and ${\tilde{a}}$ are dynamical U(1) gauge fields and A is a classical background U(1) gauge field. The first term corresponds to the $U(1)_{N}$ Chern-Simons theory. In the last term, ${\tilde{a}}$ is a Lagrange multiplier enforcing A to be a ${\mathbb {Z}}_N$ gauge field. The second term is the Dijkgraaf-Witten term $(-1)^{Q(M,\Sigma )}$ for the ${\mathbb {Z}}_N$ gauge field A.

We can rewrite the Lagrangian as

$$\begin{aligned} \frac{iN}{4\pi } (a+A)d(a+A) + \frac{iN}{2\pi } A d({\tilde{a}}-a). \end{aligned}$$

(A.5)

By performing field redefinitions, $a' \equiv a+A$ and ${\tilde{a}}' \equiv {\tilde{a}} - a$, we see that this Lagrangian describes $U(1)_{N} \times (\text {trivial SPT}) = U(1)_{N}$. Thus, the $U(1)_{N}$ Chern-Simons theory is invariant under stacking the $(-1)^{Q(M,\Sigma )}$ SPT phase. This implies that the partition function of $U(1)_{N}$ on M must vanish if there exists a two-cycle $\Sigma \subset M$ such that $(-1)^{Q(M,\Sigma )} \ne 1$.

For example, take $N=2$, $M = {\mathbb {R}}P^3$, and $\Sigma = {\mathbb {R}}P^2 \subset {\mathbb {R}}P^3$. Then, we have $(-1)^{Q(M,\Sigma )} = -1$, and indeed one can verify that the partition function of the $U(1)_{2}$ Chern-Simons theory on ${\mathbb {R}}P^3$ vanishes.

More on the Triality Fusion Rules

In this appendix we give the detailed derivation of the triality fusion rules presented in Sects. 3.2 and 3.3. Below all the codimension-one defects are supported on a three-dimensional manifold M at $x=0$.

B.1. Even N Here we derive the fusion rule in Sect. 3.2 for the triality defect $\mathcal{D}_3$ and the condensation defects ${{{\mathcal {C}}}}_{N\ell \over 2}$ in any 3+1d QFT ${{\mathcal {Q}}}$ invariant under the ST gauging in the sense of (2.16) for the case of even N.

• $\overline{{\mathcal {D}}}_3 \times {\mathcal {D}}_3$: We have

  $$\begin{aligned} \begin{aligned} \overline{{\mathcal {D}}}_3 \times {\mathcal {D}}_3 :&{\mathcal {L}}[b_1] + \frac{2\pi i}{N}\left( -b_1 \cup b_2 - \frac{1}{2} q(b_2) + b_2 \cup B + \frac{1}{2} q(b_2) \right) \\ =&{\mathcal {L}}[b_1] + \frac{2\pi i}{N} \left( -b_1 \cup b_2 + b_2 \cup B \right) \,. \end{aligned} \end{aligned}$$

  (B.1)

  By flipping the sign of $b_2$, we see that $\overline{{\mathcal {D}}}_3 \times {\mathcal {D}}_3 = {\mathcal {C}}_0$.

• ${\mathcal {D}}_3 \times \overline{{\mathcal {D}}}_3$: We have

  $$\begin{aligned} {\mathcal {D}}_3 \times \overline{{\mathcal {D}}}_3 :{\mathcal {L}}[b_1] + \frac{2\pi i}{N} \left( b_1 \cup b_2 + \frac{1}{2} q(b_1) - b_2 \cup B - \frac{1}{2} q(B) \right) , \end{aligned}$$

  (B.2)

  which is our definition of ${\mathcal {C}}_{\frac{N}{2}}$. Thus, ${\mathcal {D}}_3 \times \overline{{\mathcal {D}}}_3 = {\mathcal {C}}_{\frac{N}{2}}$.

• ${\mathcal {D}}_3 \times {\mathcal {D}}_3$: We have

  $$\begin{aligned} {\mathcal {D}}_3 \times {\mathcal {D}}_3 :{\mathcal {L}}[b_1] + \frac{2\pi i}{N}\left( b_1 \cup b_2 + \frac{1}{2} q(b_1) +b_2 \cup B + \frac{1}{2} q(b_2) \right) . \end{aligned}$$

  (B.3)

  If we redefine the dynamical gauge fields as $b \equiv b_1$ and ${\tilde{b}} \equiv b_1 + b_2$, the expression becomes

  $$\begin{aligned} {\mathcal {D}}_3 \times {\mathcal {D}}_3 :\left( \frac{2\pi i}{2N} q({\tilde{b}}+B) \right) + \left( {\mathcal {L}}[b] + \frac{2\pi i}{N} \left( -b \cup B - \frac{1}{2} q(B) \right) \right) \,. \end{aligned}$$

  (B.4)

  In the first term we see that the ${\tilde{b}}$ gauge field is decoupled from other dynamical fields including the matter fields. The path integral over ${\tilde{b}}$ defines a 3+1d invertible field theory in the region $x>0$, with the Dirichlet boundary condition ${{\tilde{b}}}|=0$ imposed at $x=0$. This gives the $U(1)_{N}$ Chern-Simons theory living on M at $x=0$ as shown in [1, 79]. Its coupling to B only affects transverse junctions. The remaining terms correspond to the definition of $\overline{{\mathcal {D}}}_3$. Thus, we conclude ${\mathcal {D}}_3 \times {\mathcal {D}}_3 = U(1)_{N} \,\overline{{\mathcal {D}}}_3$.

• ${\mathcal {D}}_3 \times {\mathcal {C}}_{\frac{N\ell }{2}}$ and ${\mathcal {C}}_{\frac{N\ell }{2}} \times {\mathcal {D}}_3$: We have

  $$\begin{aligned}{} & {} {\mathcal {D}}_3 \times {\mathcal {C}}_{\frac{N\ell }{2}} :{\mathcal {L}}[b_1] + \frac{2\pi i}{N}\nonumber \\ {}{} & {} \left( b_1 \cup b_2 + \frac{1}{2} q(b_1) + b_2 \cup b_3 + \frac{1}{2} \ell q(b_2) - b_3 \cup B - \frac{1}{2} \ell q(B) \right) , \end{aligned}$$

  (B.5)

  and

  $$\begin{aligned}{} & {} {\mathcal {C}}_{\frac{N\ell }{2}} \times {\mathcal {D}}_3 :{\mathcal {L}}[b_1] + \frac{2\pi i}{N}\nonumber \\ {}{} & {} \left( b_1 \cup b_2 + \frac{1}{2} \ell q(b_1) - b_2 \cup b_3 - \frac{1}{2}\ell q(b_3) + b_3 \cup B + \frac{1}{2}q(b_3) \right) . \end{aligned}$$

  (B.6)

  By setting $\ell =0$ in Eq. (B.5) and redefining $b_3 \rightarrow -b_3$, we obtain ${\mathcal {D}}_3 \times {\mathcal {C}}_0 = {\mathcal {C}}_{\frac{N}{2}} \times {\mathcal {D}}_3$. Furthermore, if we let $b \equiv b_1$ and $b'_3 \equiv b_3 + b_1$, we get

  $$\begin{aligned} {\mathcal {D}}_3 \times {\mathcal {C}}_0 :\left( \frac{2\pi i}{N}(b_2 - B) \cup b'_3 \right) + \left( {\mathcal {L}}[b] + \frac{2\pi i}{N} \left( b \cup B + \frac{1}{2}q(b) \right) \right) . \end{aligned}$$

  (B.7)

  The first term gives the 2+1d $({\mathcal {Z}}_N)_0$ theory as we explained in Sect. 3.1, and the other terms give ${\mathcal {D}}_3$. Thus, ${\mathcal {D}}_3 \times {\mathcal {C}}_0 = {\mathcal {C}}_{\frac{N}{2}} \times {\mathcal {D}}_3 = ({\mathcal {Z}}_N)_0 \,{\mathcal {D}}_3$. Next, we have

  $$\begin{aligned} \begin{aligned}&{\mathcal {C}}_0 \times {\mathcal {D}}_3 :{\mathcal {L}}[b_1] + \frac{2\pi i}{N} \left( b_1 \cup b_2 - b_2 \cup b_3 + \frac{1}{2} q(b_3) + b_3 \cup B \right) \\&\quad = \frac{2\pi i}{N} \left( (-b'_2 + B) \cup b'_3 + \frac{1}{2} q(b'_3) \right) + \left( {\mathcal {L}}[b] + \frac{2\pi i}{N} \left( b \cup B + \frac{1}{2}q(b) \right) \right) \end{aligned} \end{aligned}$$

  (B.8)

  where $b \equiv b_1$, $b'_2 \equiv b_2 - b_1$, and $b'_3 \equiv b_3 - b_1$. The terms inside the first pair of parentheses define a 2+1d $({\mathcal {Z}}_N)_N$ gauge theory living on the defect [79]. To see this, we first set $B=0$ for simplicity, as the coupling to B only affects the transverse junctions. Next, following a similar discussion around Fig. 2, we integrate out $b'_2$ to force $b'_3$ to effectively become a gauge field $a \in H^1(M;{\mathbb {Z}}_N)$ living on M. The 4d twist term $\frac{2\pi i}{2N} q(b'_3)$ reduces to the level N 3d Dijkgraaf-Witten twist for the gauge field a as shown in [49]. We therefore obtain a 2+1d $({\mathcal {Z}}_N)_N$ gauge theory on M. Finally, remaining terms define the triality defect ${\mathcal {D}}_3$. We conclude ${\mathcal {C}}_0 \times {\mathcal {D}}_3 = ({\mathcal {Z}}_N)_N \, {\mathcal {D}}_3$. If instead $\ell =1$ in Eq. (B.5), we redefine the variables as $b_2 \rightarrow -b_3$ and $b_3 \rightarrow b_1 + b_2$ to obtain

  $$\begin{aligned} {\mathcal {D}}_3 \times {\mathcal {C}}_{\frac{N}{2}} :{\mathcal {L}}[b_1] + \frac{2\pi i}{N} \left( b_1 \cup b_2 - b_2 \cup b_3 + \frac{1}{2} q(b_3) + b_2 \cup B - \frac{1}{2}q(B) \right) .\nonumber \\ \end{aligned}$$

  (B.9)

  Let $b \equiv b_1$, $b'_2 \equiv b_2 - b_1$, and $b'_3 \equiv b_3 - b_1$, then we obtain

  $$\begin{aligned}{} & {} {\mathcal {D}}_3 \times {\mathcal {C}}_{\frac{N}{2}} :\frac{2\pi i}{N} \left( -(b'_2-B) \cup (b'_3 -B) + \frac{1}{2} q(b'_3 -B) \right) \nonumber \\{} & {} \quad + \left( {\mathcal {L}}[b] + \frac{2\pi i}{N} \left( b \cup B + \frac{1}{2}q(b) \right) \right) . \end{aligned}$$

  (B.10)

  We again see the decoupled 2+1d $({\mathcal {Z}}_N)_N$ gauge theory on the defect and also ${\mathcal {D}}_3$. The coupling of this $({\mathcal {Z}}_N)_N$ to the bulk symmetry background B is different from the previous case, but it only affects transverse junctions. We conclude that ${\mathcal {D}}_3 \times {\mathcal {C}}_{\frac{N}{2}} = ({\mathcal {Z}}_N)_N \,{\mathcal {D}}_3$.

• ${\mathcal {C}}_{\frac{N\ell _1}{2}} \times {\mathcal {C}}_{\frac{N\ell _2}{2}}$: We have already derived ${\mathcal {C}}_0 \times {\mathcal {C}}_0 = ({\mathcal {Z}}_N)_0 \,{\mathcal {C}}_0$ in Sect. 3.1. Now, consider the fusion

  $$\begin{aligned} {\mathcal {C}}_0 \times {\mathcal {C}}_{\frac{N}{2}} :{\mathcal {L}}[b_1] +\frac{2\pi i}{N}\left( b_1 \cup b_2 - b_2 \cup b_3 + b_3 \cup b_4 +\frac{1}{2} q(b_3) - b_4 \cup B - \frac{1}{2} q(B) \right) .\nonumber \\ \end{aligned}$$

  (B.11)

  Let $b'_4 \equiv b_4 - b_2$, then this becomes

  $$\begin{aligned} {\mathcal {C}}_0 \times {\mathcal {C}}_{\frac{N}{2}} :\frac{2\pi i}{N} \left( (b'_4 +B) \cup (b_3 -B) + \frac{1}{2} q(b_3 -B) \right) + \left( {\mathcal {L}}[b_1] + \frac{2\pi i}{N} \left( b_1 \cup b_2 - b_2 \cup B \right) \right) \nonumber \\ \end{aligned}$$

  (B.12)

  which gives ${\mathcal {C}}_0 \times {\mathcal {C}}_{\frac{N}{2}} = ({\mathcal {Z}}_N)_N \,{\mathcal {C}}_0$. Again, the coupling of the coefficient TQFT $({\mathcal {Z}}_N)_N$ to the bulk symmetry background B only affects transverse junctions. We can perform an alternative change of variables $b'_3 \equiv b_3 - b_1$ and $b'_2 \equiv -b_2 + b_1 + b_4$ to rewrite (B.11) as

  $$\begin{aligned}{} & {} {\mathcal {C}}_0 \times {\mathcal {C}}_{\frac{N}{2}} :\frac{2\pi i}{N} \left( b'_2 \cup b'_3 + \frac{1}{2} q(b'_3) \right) \nonumber \\{} & {} \quad + \left( {\mathcal {L}}[b_1] + \frac{2\pi i}{N} \left( b_1 \cup b_4 + \frac{1}{2} q(b_1) - b_4 \cup B - \frac{1}{2} q(B) \right) \right) \end{aligned}$$

  (B.13)

  which gives ${\mathcal {C}}_0 \times {\mathcal {C}}_{\frac{N}{2}} = ({\mathcal {Z}}_N)_N \,{\mathcal {C}}_{\frac{N}{2}}$. One can also verify that the two condensation defects commutes, and we obtain ${\mathcal {C}}_0 \times {\mathcal {C}}_{\frac{N}{2}} = {\mathcal {C}}_{\frac{N}{2}} \times {\mathcal {C}}_0 =({\mathcal {Z}}_N)_N \,{\mathcal {C}}_0 = ({\mathcal {Z}}_N)_N \,{\mathcal {C}}_{\frac{N}{2}}$. Finally, consider

  $$\begin{aligned} {\mathcal {C}}_{\frac{N}{2}} \times {\mathcal {C}}_{\frac{N}{2}} :{\mathcal {L}}[b_1] + \frac{2\pi i}{N} \left( b_1 \cup b_2 + \frac{1}{2} q(b_1) -b_2 \cup b_3 +b_3 \cup b_4 - b_4 \cup B - \frac{1}{2} q(B) \right) .\nonumber \\ \end{aligned}$$

  (B.14)

  Let $b'_4 \equiv b_4 - b_2$, then this reduces to

  $$\begin{aligned}{} & {} {\mathcal {C}}_{\frac{N}{2}} \times {\mathcal {C}}_{\frac{N}{2}} :\frac{2\pi i}{N} b'_4 \cup (b_3 - B)\nonumber \\ {}{} & {} + \left( {\mathcal {L}}[b_1] + \frac{2\pi i}{N} \left( b_1 \cup b_2 + \frac{1}{2} q(b_1) -b_2 \cup B -\frac{1}{2} q(B) \right) \right) , \end{aligned}$$

  (B.15)

  which gives ${\mathcal {C}}_{\frac{N}{2}} \times {\mathcal {C}}_{\frac{N}{2}} = ({\mathcal {Z}}_N)_0 \,{\mathcal {C}}_{\frac{N}{2}}$. To summarize, we have derived ${\mathcal {C}}_{\frac{N\ell _1}{2}} \times {\mathcal {C}}_{\frac{N\ell _2}{2}} = ({\mathcal {Z}}_N)_{N(\ell _1 + \ell _2)} {\mathcal {C}}_{\frac{N\ell _1}{2}} =({\mathcal {Z}}_N)_{N(\ell _1 + \ell _2)} {\mathcal {C}}_{\frac{N\ell _2}{2}}$ for general $\ell _1$ and $\ell _2$. The same fusion rules can also be derived from the definition of condensation defects given in Eq. (2.5) although we will not do so explicitly.

• ${\mathcal {D}}_3 \times {\mathcal {D}}_3 \times {\mathcal {D}}_3$: We have

  $$\begin{aligned} \begin{aligned} {\mathcal {D}}_3 \times {\mathcal {D}}_3 \times {\mathcal {D}}_3 :&{\mathcal {L}}[b_1] + \frac{2\pi i}{N} \left( b_1 \cup b_2 + \frac{1}{2} q(b_1) + b_2 \cup b_3 + \frac{1}{2} q(b_2) + b_3 \cup B + \frac{1}{2} q(b_3) \right) \\&= \frac{2\pi i}{2N} q({\tilde{b}}) + \left( {\mathcal {L}}[b_1] + \frac{2\pi i}{N} \left( b_1 \cup b'_3 - b'_3 \cup B \right) \right) , \end{aligned}\nonumber \\ \end{aligned}$$

  (B.16)

  where we performed field redefinitions ${\tilde{b}} \equiv b_1 + b_2 + b_3$ and $b'_3 \equiv - b_3$. The first term reduces to a 2+1d $U(1)_{N}$ Chern-Simons theory on the defect [1, 79], and the remaining terms define the condensation defect ${\mathcal {C}}_0$. Thus, ${\mathcal {D}}_3 \times {\mathcal {D}}_3 \times {\mathcal {D}}_3 = U(1)_{N} \,{\mathcal {C}}_0$. Alternatively, one can also define , in which case we obtain

  

  (B.17)

  from which we can deduce ${\mathcal {D}}_3 \times {\mathcal {D}}_3 \times {\mathcal {D}}_3 = U(1)_{N} \,{\mathcal {C}}_{\frac{N}{2}}$. Combining the above, we conclude ${\mathcal {D}}_3 \times {\mathcal {D}}_3 \times {\mathcal {D}}_3 = U(1)_{N} \,{\mathcal {C}}_0 = U(1)_{N} \,{\mathcal {C}}_{\frac{N}{2}}$.

B.2. Odd N Here we derive the fusion rule in Sect. 3.3 for the triality defect ${{{\mathcal {D}}}}_3$ and the condensation defects ${{{\mathcal {C}}}}_{0}$ in any 3+1d QFT ${{\mathcal {Q}}}$ invariant under the ST gauging in the sense of (2.16) for the odd N case.

• $\overline{{\mathcal {D}}}_3 \times {\mathcal {D}}_3$ and ${\mathcal {D}}_3 \times \overline{{\mathcal {D}}}_3$: First, we have

  $$\begin{aligned} \begin{aligned} \overline{{\mathcal {D}}}_3 \times {\mathcal {D}}_3 :&{\mathcal {L}}[b_1] + \frac{2\pi i}{N}\left( -b_1 \cup b_2 - \frac{N+1}{2} b_2 \cup b_2 + b_2 \cup B + \frac{N+1}{2} b_2 \cup b_2 \right) \\ =&{\mathcal {L}}[b_1] + \frac{2\pi i}{N} \left( -b_1 \cup b_2 + b_2 \cup B \right) \,. \end{aligned} \end{aligned}$$

  (B.18)

  By flipping the sign of $b_2$, we obtain $\overline{{\mathcal {D}}}_3 \times {\mathcal {D}}_3 = {\mathcal {C}}_0$. Next,

  $$\begin{aligned} \begin{aligned} {\mathcal {D}}_3 \times \overline{{\mathcal {D}}}_3 :&{\mathcal {L}}[b_1] + \frac{2\pi i}{N}\left( b_1 \cup b_2 + \frac{N+1}{2} b_1 \cup b_1 - b_2 \cup B - \frac{N+1}{2} B \cup B \right) \\ =&{\mathcal {L}}[b_1] + \frac{2\pi i}{N} \left( b_1 \cup (b'_2 + \frac{N+1}{2} B) - (b'_2 + \frac{N+1}{2} B) \cup B \right) \,, \end{aligned} \end{aligned}$$

  (B.19)

  where we have defined $b'_2 \equiv b_2 + \frac{N+1}{2} b_1$. The resulting defect obeys the same parallel fusion rule with $\eta$ as ${\mathcal {C}}_0$ (see Sect. 4.3). Thus, if we ignore the transverse junctions, we can further shift $b'_2$ by $\frac{N+1}{2} B$. This gives ${\mathcal {D}}_3 \times \overline{{\mathcal {D}}}_3 = {\mathcal {C}}_0$.

• ${\mathcal {D}}_3 \times {\mathcal {D}}_3$: We have

  $$\begin{aligned} \begin{aligned} {\mathcal {D}}_3 \times {\mathcal {D}}_3 :&{\mathcal {L}}[b_1] + \frac{2\pi i}{N}\left( b_1 \cup b_2 + \frac{N+1}{2} b_1 \cup b_1 + b_2 \cup B + \frac{N+1}{2} b_2 \cup b_2 \right) \\ =&\frac{2\pi i}{N} \frac{N+1}{2} ({\tilde{b}}+B)^2 + \left( {\mathcal {L}}[b] + \frac{2\pi i}{N} \left( -b \cup B - \frac{N+1}{2} B \cup B \right) \right) \,, \end{aligned} \end{aligned}$$

  (B.20)

  where we have defined $b \equiv b_1$ and ${\tilde{b}} \equiv b_1 + b_2$. The first term is a 3+1d invertible field theory living in $x>0$ with Dirichlet boundary condition ${{\tilde{b}}}|=0$ imposed at $x=0$. This gives the 2+1d $SU(N)_{-1}$ Chern-Simons theory living on the defect [79]. The remaining terms define $\overline{{\mathcal {D}}}_3$. Thus, we have ${\mathcal {D}}_3 \times {\mathcal {D}}_3 = SU(N)_{-1} \,\overline{{\mathcal {D}}}_3$. From this and the earlier fusion rule, we also find ${\mathcal {D}}_3 \times {\mathcal {D}}_3 \times {\mathcal {D}}_3 = SU(N)_{-1} \,{\mathcal {C}}_0$.

• ${\mathcal {D}}_3 \times {\mathcal {C}}_0$ and ${\mathcal {C}}_0 \times {\mathcal {D}}_3$: We have

  $$\begin{aligned} \begin{aligned} {\mathcal {D}}_3 \times {\mathcal {C}}_0 :&{\mathcal {L}}[b_1] + \frac{2\pi i}{N}\left( b_1 \cup b_2 + \frac{N+1}{2} b_1 \cup b_1 + b_2 \cup b_3 - b_3 \cup B \right) \\ =&\frac{2\pi i}{N} (b_2 - B) \cup b'_3 + \left( {\mathcal {L}}[b] + \frac{2\pi i}{N} \left( b \cup B + \frac{N+1}{2} b \cup b \right) \right) \,, \end{aligned} \end{aligned}$$

  (B.21)

  where $b \equiv b_1$ and $b'_3 \equiv b_3 + b_1$. The first term gives us the 2+1d $({\mathcal {Z}}_N)_0$ gauge theory living on the defect, and the remaining terms define ${\mathcal {D}}_3$. Thus, ${\mathcal {D}}_3 \times {\mathcal {C}}_0 = ({\mathcal {Z}}_N)_0 \, {\mathcal {D}}_3$. Similarly, we have

  $$\begin{aligned} \begin{aligned}&{\mathcal {C}}_0 \times {\mathcal {D}}_3 :{\mathcal {L}}[b_1] + \frac{2\pi i}{N}\left( b_1 \cup b_2 - b_2 \cup b_3 + b_3 \cup B + \frac{N+1}{2} b_3 \cup b_3 \right) \\&\quad = \frac{2\pi i}{N} (b'_2 + B) \cup b'_3 + \left( {\mathcal {L}}[b] + \frac{2\pi i}{N} \left( b \cup B + \frac{N+1}{2} b \cup b \right) \right) \,, \end{aligned} \end{aligned}$$

  (B.22)

  where $b \equiv b_1$, $b'_2 \equiv \frac{N+1}{2}b_1 -b_2 +\frac{N+1}{2}b_3$, and $b'_3 \equiv b_3 - b_1$. We see that ${\mathcal {C}}_0 \times {\mathcal {D}}_3 = ({\mathcal {Z}}_N)_0 \, {\mathcal {D}}_3$.

N-ality Defects From Mixed Anomalies

The Kramers-Wannier duality defect in 1+1d Ising model can be obtained from the invertible chiral symmetry defect in Majorana fermion by gauging the fermion parity symmetry [31, 101]. The chiral symmetry has a mixed anomaly with the fermion parity symmetry, and thus gauging the fermion parity symmetry “extends” the chiral symmetry to be a non-invertible symmetry generated by the Kramers-Wannier duality defect. Conversely, the Majorana fermion can be obtained from the Ising model by gauging a ${\mathbb {Z}}_2$ zero-form symmetry with a local counterterm [31, 102, 103].

We would like to understand whether the non-invertible N-ality defects discussed here can be obtained from invertible zero-form symmetry by gauging a one-form symmetry in a “parent theory”, generalizing the algorithm in [49]. In other words, we would like to understand whether the non-invertible symmetry defect becomes an invertible symmetry when we gauging a one-form symmetry with suitable local counterterm. As we will show in this appendix, this is in general not the case. Thus the non-invertible defect discussed here in general cannot be obtained from an invertible zero-form symmetry that has mixed anomaly with one-form symmetry by gauging the one-form symmetry. The condition that the non-invertible symmetry can be realized as an invertible symmetry upon gauging a discrete symmetry coincides with the condition that such non-invertible defect can be realized by invertible phases discussed in Sect. 5.

We define the codimension-one non-invertible defect in the same way as in Sect. 2 by gauging in half of the spacetime. On one side of the defect the theory has partition function Z[B] with the background two-form gauge field B. On the other side the partition function is $\sum _b Z[b]e^{iS[b,B]}$, where³¹

$$\begin{aligned} \begin{aligned} S[b,B]= {\left\{ \begin{array}{ll} {2\pi \over N}\int b\cup B+\frac{p}{2}q(b) &{} \text {even }N\\ \frac{2\pi }{N}\int b\cup B+ 2^{-1}p \, b\cup b &{} \text {odd }N \end{array}\right. } \end{aligned} \end{aligned}$$

(C.1)

where $2^{-1}=(N+1)/2$ is the inverse in ${\mathbb {Z}}_N$ for odd N.

Next, we gauge the one-form symmetry by promoting B to be dynamical, and add an additional local counterterm $S'[B,B']$ with a new background two-form gauge field $B'$,

$$\begin{aligned} S'[B,B']= \left\{ \begin{array}{ll} \frac{2\pi }{N}\int r B\cup B'+\frac{p'}{2}q(B)&{} \text {even }N\\ \frac{2\pi }{N}\int r B\cup B'+ p' B\cup B &{} \text {odd }N \end{array} \right. ~. \end{aligned}$$

(C.2)

The theories on the two sides of the defect are now

$$\begin{aligned}&\text {LHS}: \sum _B Z[B] e^{iS'[B,B']} \nonumber \\&\text {RHS}:\sum _{b,B}Z[b]e^{iS[b,B]+iS'[B,B']}~. \end{aligned}$$

(C.3)

If there is a choice of $S'$ such that we can integrate out B on the right hand side to obtain $\sum _b Z[b]e^{iS'[b,\ell B']}$ for some integer $\ell$ that is coprime with N and up to a local counterterm of the background field $B'$, then the defect becomes an invertible defect that generates zero-form symmetry in the theory with partition function $\sum _B Z[B]e^{iS'[B,B']}$. More specifically, this invertible symmetry acts on the dual ${\mathbb {Z}}_N$ one-form symmetry by $n\rightarrow n\ell$ for $n\in {\mathbb {Z}}_N$. If we gauge the dual symmetry generated by $\exp (i \oint B)$, such invertible zero-form symmetry becomes the non-invertible defect.

C.1. Duality Defect. Consider the Kramers-Wannier duality defect with $p=0$. We will take $\gcd (p',N)=1$.

C.1.1. Even N The partition function $\sum _{b,B}Z[b]e^{iS[b,B]+iS'[B,B']}$ is

$$\begin{aligned}&\sum _{b,B}Z[b]\exp \left( {2\pi i\over N}\int \left( b\cup B+r B\cup B'+\frac{p'}{2}q(B)\right) \right) \nonumber \\&\quad =\sum _{{{\tilde{b}}}=B+p'^{-1}(b+rB'), b} Z[b] \exp \left( \frac{2\pi i p'}{2N}\int q(\tilde{b})+\frac{2\pi i}{N}\int \right. \nonumber \\&\qquad \left. \left( -\frac{p'^{-1}}{2}q(b)-p'^{-1} r b\cup B'-\frac{p'^{-1}r^2}{2}q(B')\right) \right) ~. \end{aligned}$$

(C.4)

Thus we need to solve

$$\begin{aligned} p'^2=-1\text { mod }2N\quad (\text {bosonic});\quad p'^2=-1\text { mod }N\quad (\text {fermionic})~, \end{aligned}$$

(C.5)

for bosonic and fermionic theories, respectively.

The solutions are discussed in [22], in a different context of invertible phases that possess Kramers-Wannier duality defect. For instance, if the theory is bosonic, then the first equation does not have a solution and the Kramers-Wannier duality defect cannot be obtained from an invertible symmetry by gauging one-form symmetries.

On the other hand, if the theory is fermionic, and for instance take $N = 2$, then the equations have solution $p' = 1$. Thus the Kramers-Wannier duality non-invertible defect can be obtained from an invertible zero-form symmetry in the theory $\sum _B Z[B]e^{{\pi i\over 2} \int q(B)}$ by gauging the dual symmetry generated by $\exp (i\oint B)$. The dual symmetry and the invertible zero-form symmetry has a mixed anomaly: in the presence of the background gauge field $B'$ for the dual symmetry, the two sides of the defect differ by

$$\begin{aligned} -2\pi \frac{p'^{-1}r^2}{2N}\int q(B')~. \end{aligned}$$

(C.6)

The invertible symmetry acts on the one-form symmetry by multiplying the ${\mathbb {Z}}_N$ element with $\ell =-(p')^{-1}$.

C.1.2. Odd N From a similar discussion, we need to solve

$$\begin{aligned} 4p'^2=-1\text { mod }N~. \end{aligned}$$

(C.7)

The solutions are discussed in [22], in a different context of invertible phases that possess Kramers-Wannier duality defect. When the equation is solvable, the Kramers-Wannier duality defect can be obtained by gauging one-form symmetry in a theory with invertible symmetry. The invertible symmetry has mixed anomaly with the dual one-form symmetry: in the presence of background gauge field $B'$ for the dual one-form symmetry, the two sides of the defect differ by

$$\begin{aligned} -2\pi \frac{2^{-1}(2p')^{-1}r^2}{N}\int B'\cup B'~. \end{aligned}$$

(C.8)

The invertible symmetry acts on the one-form symmetry by multiplying the ${\mathbb {Z}}_N$ element with $\ell =-(2p')^{-1}$.

C.2. Triality Defect.

C.2.1. Even N We have $p=1$ and take $\gcd (p',N)=1$. The partition function $\sum _{b,B}Z[b]e^{iS[b,B]+iS'[B,B']}$ is

$$\begin{aligned}&\sum _{b,B}Z[b]\exp \left( {2\pi i\over N}\int \left( \frac{1}{2}q(b)+ b\cup B+r B\cup B'+\frac{p'}{2}q(B)\right) \right) \nonumber \\&\quad =\sum _{{{\tilde{b}}}=B+p'^{-1}(b+rB'), b} Z[b]\exp \left( \frac{2\pi i p'}{2N}\int q({{\tilde{b}}}) +\frac{2\pi i}{N}\right. \nonumber \\&\qquad \left. \int \left( \frac{1-p'^{-1}}{2}q(b)-p'^{-1} r b\cup B'-\frac{p'^{-1}r^2}{2}q(B')\right) \right) . \end{aligned}$$

(C.9)

Thus we need to solve

$$\begin{aligned} p'(p'-1)+1=0\text { mod }2N\quad (\text {bosonic});\quad p'(p'-1)+1=0\text { mod }N\quad (\text {fermionic}).\nonumber \\ \end{aligned}$$

(C.10)

Since $p'(p'-1)$ is even, and the above equation has definite even/odd parity, there is no solution. Thus the triality defect cannot be obtained from an invertible symmetry by gauging a one-form symmetry.

C.2.2. Odd N Here $p=N+1$ and we take $\gcd (p',N)=1$. The partition function $\sum _{b,B}Z[b]e^{iS[b,B]+iS'[B,B']}$ is

$$\begin{aligned}&\sum _{b,B}Z[b]\exp \,\left( {2\pi i\over N}\int \left( \frac{N+1}{2} (b\cup b)+ b\cup B+r B\cup B'+p'(B\cup B)\right) \right) \nonumber \\&\quad =\sum _{{{\tilde{b}}}=B+(2p')^{-1}(b+rB'), b} Z[b]\exp \left( \frac{2\pi ip'}{N}\int ({{\tilde{b}}}\cup {{\tilde{b}}}) \right) \nonumber \\&\qquad \times \exp \left( \frac{2\pi i}{N}\int \left( \left( \frac{N+1}{2}-2^{-1}(2p')^{-1}\right) \right. \right. \nonumber \\&\qquad \left. \left. (b\cup b)-(2p')^{-1} r b\cup B'-2^{-1}(2p')^{-1}r^2(B'\cup B')\right) \right) ~. \end{aligned}$$

(C.11)

Thus we need to solve

$$\begin{aligned} (2p')^2-2p'+1=0\text { mod }N~. \end{aligned}$$

(C.12)

There exits solution if and only if $N\in {{{\mathcal {X}}}}$ as defined in (5.12).

When $N\in {{{\mathcal {X}}}}$ and thus a solution exists, the coupling to $B'$ on the two sides of the invertible wall is r and $-(2p')^{-1}r$. The invertible symmetry has mixed anomaly with the dual symmetry: in the presence of background $B'$ for the dual symmetry generated by $\exp (i \oint B)$, the two sides of the defect differ by a local counterterm (take $r =1$)

$$\begin{aligned} -\frac{\pi (N+1)(2p')^{-1}}{N}\int (B'\cup B')~. \end{aligned}$$

(C.13)