Gauging (3+1)-dimensional topological phases: an approach from surface theories

TL;DR

This paper investigates (3+1)d bosonic topological phases with a $Z_2$ symmetry, analyzing their surface theories and gauged versions, revealing non-abelian excitations and connections to (2+1)d topological phases.

Contribution

It introduces a surface theory approach to classify (3+1)d topological phases and explores the effects of gauging $Z_2$ symmetry, including non-abelian excitations.

Findings

Surface partition functions yield modular matrices matching (2+1)d phases.

Gauging $Z_2$ symmetry produces non-abelian flux excitations.

Topological phases described by BF theories with abelian statistics.

Abstract

We discuss several bosonic topological phases in (3+1) dimensions enriched by a global $\mathbb{Z}_2$ symmetry, and gauging the $\mathbb{Z}_2$ symmetry. More specifically, following the spirit of the bulk-boundary correspondence, expected to hold in topological phases of matter in general, we consider boundary (surface) field theories and their orbifold. From the surface partition functions, we extract the modular $\mathcal{S}$ and $\mathcal{T}$ matrices and compare them with $(2+1)$d toplogical phase after dimensional reduction. As a specific example, we discuss topologically ordered phases in $(3+1)$ dimensions described by the BF topological quantum field theories, with abelian exchange statistics between point-like and loop-like quasiparticles. Once the $\mathbb{Z}_2$ charge conjugation symmetry is gauged, the $\mathbb{Z}_2$ flux becomes non-abelian excitation. The gauged topological phases we are considering here belong to the quantum double model with non-abelian group in $(3+1)$ dimensions.

Tables

Table 1. Table 1: The quantum dimensions d χ subscript 𝑑 𝜒 d_{\chi} , conformal dimensions h χ subscript ℎ 𝜒 h_{\chi} , and the number 𝒩 𝒩 \mathcal{N} of characters χ 𝜒 \chi from orbifolding the charge-conjugate ℤ 2 subscript ℤ 2 \mathbb{Z}_{2} symmetry of Eq. ( 2 ) when K K \mathrm{K} is odd. The conformal dimensions h χ subscript ℎ 𝜒 h_{\chi} are defined mod ℤ ℤ \mathbb{Z} . For χ a , b subscript 𝜒 𝑎 𝑏 \chi_{a,b} , if a ≠ b 𝑎 𝑏 a\neq b , we require 0 ≤ a < b ≤ K 0 𝑎 𝑏 K 0\leq a<b\leq\mathrm{K} here. If a = b 𝑎 𝑏 a=b , we require a ≤ ( K − 1 ) / 2 𝑎 K 1 2 a\leq(\mathrm{K}-1)/2 .

character $\chi$	$d_{\chi }$	$h_{\chi }$	$𝒩$
${\chi }_I=\frac{1}{2}{B}_{0,0}^K+\frac{1}{2}\left|\frac{\eta }{{\theta }_2}\right|$	$1$	$0$	$1$
${\chi }_j=\frac{1}{2}{B}_{0,0}^K-\frac{1}{2}\left|\frac{\eta }{{\theta }_2}\right|$	$1$	$0$	$1$
${\chi }_{a,b}=\frac{1}{2}{B}_{a,b}^K+\frac{1}{2}{B}_{K-a,K-b}^K$	$2$	$\frac{ab}{K}$	$\frac{K^2-1}{2}$
${\chi }_{\sigma }=\frac{1}{2}\left|\frac{\eta }{{\theta }_4}\right|+\frac{1}{2}\left|\frac{\eta }{{\theta }_3}\right|$	$K$	$0$	$1$
${\chi }_{\tau }=\frac{1}{2}\left|\frac{\eta }{{\theta }_4}\right|-\frac{1}{2}\left|\frac{\eta }{{\theta }_3}\right|$	$K$	$\frac{1}{2}$	$1$

Table 2. Table 2: The characters for the single-component BF theory after gauging the charge-conjugate ℤ 2 subscript ℤ 2 \mathbb{Z}_{2} symmetry when K K \mathrm{K} is odd: The quantum dimensions d χ subscript 𝑑 𝜒 d_{\chi} , spin statistics θ χ = e 2 ​ π ​ i ​ h χ subscript 𝜃 𝜒 superscript 𝑒 2 𝜋 𝑖 subscript ℎ 𝜒 \theta_{\chi}=e^{2\pi ih_{\chi}} , and the number 𝒩 𝒩 \mathcal{N} of characters χ 𝜒 \chi ( h χ subscript ℎ 𝜒 h_{\chi} are defined only modulo ℤ ℤ \mathbb{Z} .)

character $\chi$	$d_{\chi }$	$h_{\chi }$	$𝒩$
${\chi }_I=\frac{1}{2}{Z}_{0,0,0}+\frac{1}{2}{V}^{\frac{1}{2},0,0}$	$1$	$0$	$1$
${\chi }_j=\frac{1}{2}{Z}_{0,0,0}-\frac{1}{2}{V}^{\frac{1}{2},0,0}$	$1$	$0$	$1$
${\chi }_{a,b,c}=\frac{1}{2}{Z}_{a,b,c}+\frac{1}{2}{Z}_{K-a,K-b,K-c}$	$2$	$\frac{ab}{K}$	$\frac{K^3-1}{2}$
${\chi }_{{\sigma }_x}=\frac{1}{2}{V}^{0,\frac{1}{2},0}+\frac{1}{2}{V}^{\frac{1}{2},\frac{1}{2},0}$	$K$	$0$	$1$
${\chi }_{{\tau }_x}=\frac{1}{2}{V}^{0,\frac{1}{2},0}-\frac{1}{2}{V}^{\frac{1}{2},\frac{1}{2},0}$	$K$	$\frac{1}{2}$	$1$
${\chi }_{{\sigma }_y}=\frac{1}{2}{V}^{0,0,\frac{1}{2}}+\frac{1}{2}{V}^{\frac{1}{2},0,\frac{1}{2}}$	$K$	$0$	$1$
${\chi }_{{\tau }_y}=\frac{1}{2}{V}^{0,0,\frac{1}{2}}-\frac{1}{2}{V}^{\frac{1}{2},0,\frac{1}{2}}$	$K$	$0$	$1$
${\chi }_{{\sigma }_{xy}}=\frac{1}{2}{V}^{0,\frac{1}{2},\frac{1}{2}}+\frac{1}{2}{V}^{\frac{1}{2},\frac{1}{2},\frac{1}{2}}$	$K$	$0$	$1$
${\chi }_{{\tau }_{xy}}=\frac{1}{2}{V}^{0,\frac{1}{2},\frac{1}{2}}-\frac{1}{2}{V}^{\frac{1}{2},\frac{1}{2},\frac{1}{2}}$	$K$	$\frac{1}{2}$	$1$

Table 3. Table 3: The characters for the single-component BF theory after gauging the charge-conjugate ℤ 2 subscript ℤ 2 \mathbb{Z}_{2} symmetry when K K \mathrm{K} is even: The quantum dimensions d χ subscript 𝑑 𝜒 d_{\chi} , spin statistics θ χ = e 2 ​ π ​ i ​ h χ subscript 𝜃 𝜒 superscript 𝑒 2 𝜋 𝑖 subscript ℎ 𝜒 \theta_{\chi}=e^{2\pi ih_{\chi}} , and the number 𝒩 𝒩 \mathcal{N} of characters χ 𝜒 \chi ( h χ subscript ℎ 𝜒 h_{\chi} are defined only modulo ℤ ℤ \mathbb{Z} .)

character $\chi$	$d_{\chi }$	$h_{\chi }$	$𝒩$
${\chi }_I=\frac{1}{2}{Z}_{0,0,0}+V^{\frac{1}{2},0,0}$	$1$	$0$	$1$
${\chi }_j=\frac{1}{2}{Z}_{0,0,0}-V^{\frac{1}{2},0,0}$	$1$	$0$	$1$
${\chi }_{\frac{K}{2},0,0}^i=\frac{1}{2}{Z}_{\frac{K}{2},0,0}$	$1$	$0$	$2$
${\chi }_{0,\frac{K}{2},0}^i=\frac{1}{2}{Z}_{0,\frac{K}{2},0}$	$1$	$0$	$2$
${\chi }_{\frac{K}{2},\frac{K}{2},0}^i=\frac{1}{2}{Z}_{\frac{K}{2},\frac{K}{2},0}$	$1$	$\frac{K}{4}$	$2$
${\chi }_{0,\frac{K}{2},\frac{K}{2}}^i=\frac{1}{2}{Z}_{0,\frac{K}{2},\frac{K}{2}}$	$1$	$0$	$2$
${\chi }_{\frac{K}{2},0,\frac{K}{2}}^i=\frac{1}{2}{Z}_{\frac{K}{2},0,\frac{K}{2}}$	$1$	$0$	$2$
${\chi }_{0,0,\frac{K}{2}}^i=\frac{1}{2}{Z}_{0,0,\frac{K}{2}}$	$1$	$0$	$2$
${\chi }_{\frac{K}{2},\frac{K}{2},\frac{K}{2}}^i=\frac{1}{2}{Z}_{\frac{K}{2},\frac{K}{2},\frac{K}{2}}$	$1$	$\frac{K}{4}$	$2$
${\chi }_{a,b,c}=\frac{1}{2}{Z}_{a,b,c}+\frac{1}{2}{Z}_{K-a,K-b,K-c}$	$2$	$\frac{ab}{K}$	$\frac{K^3-8}{2}$
${\chi }_{{\sigma }_x}^m=\frac{1}{2}{V}^{0,\frac{1}{2},0}+\frac{1}{2}{V}^{\frac{1}{2},\frac{1}{2},0}$	$\frac{K}{2}$	$0$	$8$
${\chi }_{{\tau }_x}^m=\frac{1}{2}{V}^{0,\frac{1}{2},0}-\frac{1}{2}{V}^{\frac{1}{2},\frac{1}{2},0}$	$\frac{K}{2}$	$\frac{1}{2}$	$8$
${\chi }_{{\sigma }_y}^m=\frac{1}{2}{V}^{0,0,\frac{1}{2}}+\frac{1}{2}{V}^{\frac{1}{2},0,\frac{1}{2}}$	$\frac{K}{2}$	$0$	$8$
${\chi }_{{\tau }_y}^m=\frac{1}{2}{V}^{0,0,\frac{1}{2}}-\frac{1}{2}{V}^{\frac{1}{2},0,\frac{1}{2}}$	$\frac{K}{2}$	$0$	$8$
${\chi }_{{\sigma }_{xy}}^m=\frac{1}{2}{V}^{0,\frac{1}{2},\frac{1}{2}}+\frac{1}{2}{V}^{\frac{1}{2},\frac{1}{2},\frac{1}{2}}$	$\frac{K}{2}$	$0$	$8$
${\chi }_{{\tau }_{xy}}^m=\frac{1}{2}{V}^{0,\frac{1}{2},\frac{1}{2}}-\frac{1}{2}{V}^{\frac{1}{2},\frac{1}{2},\frac{1}{2}}$	$\frac{K}{2}$	$\frac{1}{2}$	$8$

Table 4. Table 4: The characters for the topological ℤ K × ℤ K subscript ℤ K subscript ℤ K \mathbb{Z}_{\mathrm{K}}\times\mathbb{Z}_{\mathrm{K}} theory in ( 3 + 1 ) 3 1 (3+1) d after gauging the ℤ 2 subscript ℤ 2 \mathbb{Z}_{2} symmetry: The quantum dimensions d χ subscript 𝑑 𝜒 d_{\chi} , spin statistics θ χ = e 2 ​ π ​ i ​ h χ subscript 𝜃 𝜒 superscript 𝑒 2 𝜋 𝑖 subscript ℎ 𝜒 \theta_{\chi}=e^{2\pi ih_{\chi}} , and the number 𝒩 𝒩 \mathcal{N} of characters χ 𝜒 \chi ( h χ subscript ℎ 𝜒 h_{\chi} are defined only modulo ℤ ℤ \mathbb{Z} .)

character $\chi$	$d_{\chi }$	$h_{\chi }$	$𝒩$
${\chi }_{n_0,n_1,n_2}^0=\frac{1}{2}{W}_{n_0,n_1,n_2}^{n_0,n_1,n_2}+\frac{1}{2}{Y}_{n_0,n_1,n_2}^{\frac{1}{2},0,0}$	$1$	$\frac{2n_0{n}_1}{K}$	$K^3$
${\chi }_{n_0,n_1,n_2}^1=\frac{1}{2}{W}_{n_0,n_1,n_2}^{n_0,n_1,n_2}-\frac{1}{2}{Y}_{n_0,n_1,n_2}^{\frac{1}{2},0,0}$	$1$	$\frac{2n_0{n}_1}{K}$	$K^3$
${\chi }_{n_0,n_1,n_2}^{l_0,l_1,l_2}=\frac{1}{2}{W}_{n_0,n_1,n_2}^{l_0,l_1,l_2}+\frac{1}{2}{W}_{l_0,l_1,l_2}^{n_0,n_1,n_2}$	$2$	$\frac{n_0{n}_1+l_0{l}_1}{K}$	$\frac{K^6-K^3}{2}$
${\chi }_{{\sigma }_x}^{m_0,m_1,m_2}=\frac{1}{4}{Y}_{m_i}^{0,\frac{1}{2},0}+\frac{1}{4}{Y}_{m_i}^{\frac{1}{2},\frac{1}{2},0}$	$K$	$\frac{m_0{m}_1}{2K}$	$K^3$
${\chi }_{{\tau }_x}^{m_0,m_1,m_2}=\frac{1}{4}{Y}_{m_i}^{0,\frac{1}{2},0}-\frac{1}{4}{Y}_{m_i}^{\frac{1}{2},\frac{1}{2},0}$	$K$	$\frac{K+m_0{m}_1}{2K}$	$K^3$
${\chi }_{{\sigma }_y}^{m_0,m_1,m_2}=\frac{1}{4}{Y}_{m_i}^{0,0,\frac{1}{2}}+\frac{1}{4}{Y}_{m_i}^{\frac{1}{2},0,\frac{1}{2}}$	$K$	$\frac{m_0{m}_1}{K}$	$K^3$
${\chi }_{{\tau }_y}^{m_0,m_1,m_2}=\frac{1}{4}{Y}_{m_i}^{0,0,\frac{1}{2}}-\frac{1}{4}{Y}_{m_i}^{\frac{1}{2},0,\frac{1}{2}}$	$K$	$\frac{m_0{m}_1}{K}$	$K^3$
${\chi }_{{\sigma }_{xy}}^{m_0,m_1,m_2}=\frac{1}{4}{Y}_{m_i}^{0,\frac{1}{2},\frac{1}{2}}+\frac{1}{4}{Y}_{m_i}^{\frac{1}{2},\frac{1}{2},\frac{1}{2}}$	$K$	$\frac{m_0{m}_1}{K}$	$K^3$
${\chi }_{{\tau }_{xy}}^{m_0,m_1,m_2}=\frac{1}{4}{Y}_{m_i}^{0,\frac{1}{2},\frac{1}{2}}-\frac{1}{4}{Y}_{m_i}^{\frac{1}{2},\frac{1}{2},\frac{1}{2}}$	$K$	$\frac{K+2m_0{m}_1}{2K}$	$K^3$

Table 5. Table 5: The quantum dimensions d χ subscript 𝑑 𝜒 d_{\chi} , spin statistics θ χ = e 2 ​ π ​ i ​ h χ subscript 𝜃 𝜒 superscript 𝑒 2 𝜋 𝑖 subscript ℎ 𝜒 \theta_{\chi}=e^{2\pi ih_{\chi}} and number 𝒩 𝒩 \mathcal{N} of characters χ 𝜒 \chi from orbifolding ℤ 2 subscript ℤ 2 \mathbb{Z}_{2} symmetry of boundary of the ℤ K × ℤ K subscript ℤ K subscript ℤ K \mathbb{Z}_{\mathrm{K}}\times\mathbb{Z}_{\mathrm{K}} gauge theory in ( 2 + 1 ) 2 1 (2+1) d. This table matches up with Table 4 after dimensional reduction.

character $\chi$	$d_{\chi }$	$h_{\chi }$	$𝒩$
${\chi }_{n_0,n_1}^0=\frac{1}{2}{W}_{n_0,n_1}^{n_0,n_1}+\frac{1}{2}{Y}_{n_0,n_1}^{\frac{1}{2},0}$	$1$	$\frac{2n_0{n}_1}{K}$	$K^2$
${\chi }_{n_0,n_1}^1=\frac{1}{2}{W}_{n_0,n_1}^{n_0,n_1}-\frac{1}{2}{Y}_{n_0,n_1}^{\frac{1}{2},0}$	$1$	$\frac{2n_0{n}_1}{K}$	$K^2$
${\chi }_{n_0,n_1}^{l_0,l_1,l_2}=\frac{1}{2}\left(W_{n_0,n_1}^{l_0,l_1}+W_{l_0,l_1}^{n_0,n_1}\right)$	$2$	$\frac{n_0{n}_1+l_0{l}_1}{K}$	$\frac{K^4-K^2}{2}$
${\chi }_{\sigma }^{m_0,m_1}=\frac{1}{4}{Y}_{m_i}^{0,\frac{1}{2}}+\frac{1}{4}{Y}_{m_i}^{\frac{1}{2},\frac{1}{2}}$	$K$	$\frac{m_0{m}_1}{2K}$	$K^2$
${\chi }_{\tau }^{m_0,m_1}=\frac{1}{4}{Y}_{m_i}^{0,\frac{1}{2}}-\frac{1}{4}{Y}_{m_i}^{\frac{1}{2},\frac{1}{2}}$	$K$	$\frac{1}{2}+\frac{m_0{m}_1}{2K}$	$K^2$

Table 6. Table 6: The quantum dimensions d χ subscript 𝑑 𝜒 d_{\chi} , spin statistics θ χ = e 2 ​ π ​ i ​ h χ subscript 𝜃 𝜒 superscript 𝑒 2 𝜋 𝑖 subscript ℎ 𝜒 \theta_{\chi}=e^{2\pi ih_{\chi}} and number 𝒩 𝒩 \mathcal{N} of characters χ 𝜒 \chi from orbifolding ℤ 2 subscript ℤ 2 \mathbb{Z}_{2} symmetry of surface of ℤ K × ℤ K subscript ℤ K subscript ℤ K \mathbb{Z}_{\mathrm{K}}\times\mathbb{Z}_{\mathrm{K}} twisted gauge theory in ( 3 + 1 ) 3 1 (3+1) d. n ~ 0 = n 0 + ( l 1 ​ n 2 − l 2 ​ n 1 ) / K subscript ~ 𝑛 0 subscript 𝑛 0 subscript 𝑙 1 subscript 𝑛 2 subscript 𝑙 2 subscript 𝑛 1 K \widetilde{n}_{0}=n_{0}+(l_{1}n_{2}-l_{2}n_{1})/\mathrm{K} and l ~ 0 = l 0 + ( n 1 ​ l 2 − n 2 ​ l 1 ) / K subscript ~ 𝑙 0 subscript 𝑙 0 subscript 𝑛 1 subscript 𝑙 2 subscript 𝑛 2 subscript 𝑙 1 K \widetilde{l}_{0}=l_{0}+(n_{1}l_{2}-n_{2}l_{1})/\mathrm{K} .

character $\chi$	$d_{\chi }$	$h_{\chi }$	$𝒩$
${\chi }_{n_0,n_1,n_2}^0=\frac{1}{2}{W}_{n_0,n_1,n_2}^{n_0,n_1,n_2}+\frac{1}{2}{Y}_{n_0,n_1,n_2}^{\frac{1}{2},0,0}$	$1$	$\frac{2n_0{n}_1}{K}$	$K^3$
${\chi }_{n_0,n_1,n_2}^1=\frac{1}{2}{W}_{n_0,n_1,n_2}^{n_0,n_1,n_2}-\frac{1}{2}{Y}_{n_0,n_1,n_2}^{\frac{1}{2},0,0}$	$1$	$\frac{2n_0{n}_1}{K}$	$K^3$
${\chi }_{n_0,n_1,n_2}^{l_0,l_1,l_2}=\frac{1}{2}\left(Z_{{\tilde{n}}_0,n_1,n_2}^{{\tilde{l}}_0,l_1,l_2}+Z_{{\tilde{l}}_0,l_1,l_2}^{{\tilde{n}}_0,n_1,n_2}\right)$	$2$	$\frac{{\tilde{n}}_0{n}_1+{\tilde{l}}_0{l}_1}{K}$	$\frac{K^6-K^3}{2}$
${\chi }_{{\sigma }_x}^{m_0,m_1,m_2}=\frac{1}{4}{Y}_{m_i}^{0,\frac{1}{2},0}+\frac{1}{4}{Y}_{m_i}^{\frac{1}{2},\frac{1}{2},0}$	$K$	$\frac{m_0{m}_1}{2K}$	$K^3$
${\chi }_{{\tau }_x}^{m_0,m_1,m_2}=\frac{1}{4}{Y}_{m_i}^{0,\frac{1}{2},0}-\frac{1}{4}{Y}_{m_i}^{\frac{1}{2},\frac{1}{2},0}$	$K$	$\frac{1}{2}+\frac{m_0{m}_1}{2K}$	$K^3$
${\chi }_{{\sigma }_y}^{m_0,m_1,m_2}=\frac{1}{4}{Y}^{0,0,\frac{1}{2}}+\frac{1}{4}{Y}^{\frac{1}{2},0,\frac{1}{2}}$	$K$	$\frac{m_0{m}_1}{K}$	$K^3$
${\chi }_{{\tau }_y}^{m_0,m_1,m_2}=\frac{1}{4}{Y}_{m_i}^{0,0,\frac{1}{2}}-\frac{1}{4}{Y}_{m_i}^{\frac{1}{2},0,\frac{1}{2}}$	$K$	$\frac{m_0{m}_1}{K}$	$K^3$
${\chi }_{{\sigma }_{xy}}^{m_0,m_1,m_2}=\frac{1}{4}{Y}_{m_i}^{0,\frac{1}{2},\frac{1}{2}}+\frac{1}{4}{Y}_{m_i}^{\frac{1}{2},\frac{1}{2},\frac{1}{2}}$	$K$	$\frac{m_0{m}_1}{K}$	$K^3$
${\chi }_{{\tau }_{xy}}^{m_0,m_1,m_2}=\frac{1}{4}{Y}_{m_i}^{0,\frac{1}{2},\frac{1}{2}}-\frac{1}{4}{Y}_{m_i}^{\frac{1}{2},\frac{1}{2},\frac{1}{2}}$	$K$	$\frac{1}{2}+\frac{m_0{m}_1}{K}$	$K^3$

Equations

$${\chi }_n^h=\text{mbox}{Tr}_h({\mathcal{P}}_n{e}^{-tH})$$

$$\mathcal{L}=\frac{1}{4\pi }({\partial }_t{\varphi }^I{\mathbf{K}}_{IJ}{\partial }_x{\varphi }^J+{\partial }_x{\varphi }^I{V}_{IJ}{\partial }_x{\varphi }^J),$$

$$B_{a,b}^{\mathrm{K}}(\tau )=\frac{1}{|\eta (\tau ){|}^2}\sum _{s,t\in \mathbb{Z}}{q}^{\frac{1}{4\mathrm{K}}(\mathrm{K}s+a+\mathrm{K}t+b{)}^2}{\bar{q}}^{\frac{1}{4\mathrm{K}}(\mathrm{K}t+a-\mathrm{K}s-b{)}^2}$$

$${\varphi }^{1,2}\to -{\varphi }^{1,2}.$$

$$Z^{\frac{1}{2},0}=\left|\frac{\eta }{{\theta }_2}\right|,Z^{0,\frac{1}{2}}=\left|\frac{\eta }{{\theta }_4}\right|,Z^{\frac{1}{2},\frac{1}{2}}=\left|\frac{\eta }{{\theta }_3}\right|,$$

$${\theta }_2=\sum _{n\in \mathbb{Z}}{q}^{\frac{1}{2}(n+\frac{1}{2}{)}^2},{\theta }_3=\sum _{n\in \mathbb{Z}}{q}^{\frac{n^2}{2}},{\theta }_4=\sum _{n\in \mathbb{Z}}(-1{)}^n{q}^{\frac{n^2}{2}}.$$

$$Z^{\frac{1}{2},0}\stackrel{\mathcal{S}}{⟷}{Z}^{0,\frac{1}{2}},Z^{\frac{1}{2},\frac{1}{2}}\stackrel{\mathcal{S}}{⟷}{Z}^{\frac{1}{2},\frac{1}{2}},$$

$$Z^{\frac{1}{2},0}\stackrel{\mathcal{T}}{⟷}{Z}^{\frac{1}{2},0},Z^{0,\frac{1}{2}}\stackrel{\mathcal{T}}{⟷}{Z}^{\frac{1}{2},\frac{1}{2}}.$$

$$\mathcal{S}=\frac{1}{\mathcal{D}}\left(\begin{array}{ccccc}1& 1& 2& \mathrm{K}& \mathrm{K}\\ 1& 1& 2& -\mathrm{K}& -\mathrm{K}\\ 2& 2& 4\cos [\frac{2\pi }{\mathrm{K}}(a{b}^{\prime }+b{a}^{\prime })]& 0& 0\\ \mathrm{K}& -\mathrm{K}& 0& \mathrm{K}& -\mathrm{K}\\ \mathrm{K}& -\mathrm{K}& 0& -\mathrm{K}& \mathrm{K}\end{array}\right)$$

$$S_{bulk}$$

$$b\to -b,a\to -a,$$

$$\mathcal{L}$$

$$-\frac{1}{2{\lambda }_1}({ϵ}_{ij}{\partial }_i{\zeta }_j{)}^2-\frac{1}{2{\lambda }_2}{G}^{ij}{\partial }_i\phi {\partial }_j\phi ,$$

$$\frac{(2\pi {)}^2}{{\mathrm{K}}^2{\lambda }_1{\lambda }_2}=1,{\lambda }_1=\frac{1}{\mathrm{K}},{\lambda }_2=\frac{(2\pi {)}^2}{\mathrm{K}}.$$

$$\exp [im\phi (t,x,y)],m\in \mathbb{Z}.$$

$$\oint d{x}^i{\partial }_i\phi =2\pi N_i,N_i\in \mathbb{Z},$$

$$\exp \left(im{\int }_{C}d{x}^i{\zeta }_i(t,x,y)\right),m\in \mathbb{Z},$$

$$\int dxdy{ϵ}_{ij}{\partial }_i{\zeta }_j=2\pi N_0$$

$$\phi (t,x+2\pi R_1,y)$$

$$\phi (t,x,y+2\pi R_2)$$

$$\int dxdy{ϵ}_{ij}{\partial }_i{\zeta }_j$$

$$Z^{n_0{n}_1{n}_2}=Z_{zero}^{n_0,n_1,n_2}{Z}_{osc}$$

$$Z_{zero}^{n_0{n}_1{n}_2}$$

$$\displaystyle\qquad+2\pi i\tau_{1}\mathrm{K}\tilde{N}_{0}\left[\tilde{N}_{1}+\beta\tilde{N}_{2}\right]+2\pi i\gamma\mathrm{K}\tilde{N}_{0}\tilde{N}_{2}\Big{\}},$$

$${\tilde{N}}_{\mu }:=N_{\mu }+n_{\mu }/\mathrm{K}.$$

$$Z_{osc}$$

$${\mathcal{S}}_{n_i,n_i^{\prime }}$$

$${\mathcal{T}}_{n_i,n_i^{\prime }}$$

$$\phi \to -\phi ,\zeta \to -\zeta .$$

$$V^{0,\frac{1}{2},0}$$

$$V^{0,0,\frac{1}{2}}$$

$$V^{\frac{1}{2},0,0}$$

$$V^{[\mu ],[\nu ],[\lambda ]}\stackrel{U_1^{\prime }}{⟶}{V}^{[\nu ],[\mu ],[\lambda ]},$$

$$V^{[\mu ],[\nu ],[\lambda ]}\stackrel{M}{⟶}{V}^{[\mu ],[\lambda ],[\nu ]},$$

$$V^{[\mu ],[\nu ],[\lambda ]}\stackrel{U_2}{⟶}{V}^{[\mu +\nu ],[\nu ],[\lambda ]},$$

Full text

1. I Introduction

2. II Orbifolding the $(1+1)$d boundary theory

3. II.1 Summary of orbifold CFT in $(1+1)$d

4. II.2 Orbifolding $\mathbb{Z}_{2}$ symmetry on the boundary of the $\mathbb{Z}_{\mathrm{K}}$ quantum double model

5. III Gauging $\mathbb{Z}_{2}$ charge-conjugation symmetry in (3+1)d $\mathbb{Z}_{\mathrm{K}}$ gauge theories

6. III.1 The BF surface theory

7. III.2 Gauging $\mathbb{Z}_{2}$ symmetry in the surface theory

8. IV Gauging $\mathbb{Z}_{2}$ symmetry in (3+1)d $\mathbb{Z}_{\mathrm{K}}\times\mathbb{Z}_{\mathrm{K}}$ gauge theories

9. IV.1 Orbifolding phases with non-trivial three-loop braiding statistics

10. V Physics in the bulk

11. VI Conclusion

12. A Theta functions

13. B Modular transformations on $T^{3}$

14. C Transformation properties of the characters in the gauged $\mathbb{Z}_{\mathrm{K}}\times\mathbb{Z}_{\mathrm{K}}$ gauge theory

I Introduction

Symmetry and topology intertwine in many phases of matter. Two prime examples are symmetry-protected topological (SPT) phases and symmetry-enriched topological (SET) phases. In SPT phases, symmetries play a crucial role, in that they are sharply distinct from trivial phases (i.e., product states) only in the presence of symmetries. Hasan and Kane (2010); Qi and Zhang (2011); Senthil (2014); Chen et al. (2013); Chiu et al. (2016) On the other hand, topologically ordered phases can be enriched by global symmetries. The seminal example is the charge fractionalization of Laughlin quasiparticles in the fractional quantum Hall effect. The relevant global symmetry in this case is $U(1)$ associated to particle number conservation. Varieties of SET phases have been discussed in the literature.Wilczek (1990); Fradkin (2013); Wen (2004)

A global symmetry in SPT or SET phases can be promoted to a local symmetry through gauging. Such “gauging” (in the bulk) or “orbifolding” (on the edge) is a useful tool to understand parent SPT and SET phases in $(2+1)$ dimensions.Levin and Gu (2012); Ryu and Zhang (2012); Sule et al. (2013); Hsieh et al. (2014) Gauging an SPT phase leads to a topological phase; thus the SPT phase is the parent of this topological phase. The topological class of the parent SPT phase can be inferred from, and, in fact, has one-to-one correspondence with the topological order (i.e., properties of anyons) which arises by the gauging procedure. We can also gauge a global (discrete) symmetry $\mathcal{G}$ in an SET phase, thereby giving rise to a new topological phase. In particular, if the global symmetry acts on emergent excitations (anyons) by permuting the anyon labelsKhan et al. (2014) in the parent SET, then gauging these symmetries will lead to more interesting ”twist liquids”. Barkeshli et al. (2014); Teo et al. (2015); Tarantino et al. (2016); Teo (2016)

The focus of this paper is to generalize the above idea to $(3+1)$ dimensions and discuss gauging/orbifolding global symmetries in $(3+1)$-dimensional bosonic topologically ordered phases. Starting from $(3+1)$-dimensional topologically ordered phases with Abelian topological order, which are described by (multi-component) BF theories, we gauge a global $\mathbb{Z}_{2}$ symmetry and show that the new topological phase is non-Abelian and related to a non-Abelian quantum double model.

Previous work has constructed line defects in $(3+1)$-dimensional topological phases.Mesaros et al. (2013); Bi et al. (2014); Ye et al. (2016) These semi-classical defects are analogous to twist defects in $(2+1)$d topological phases and can twist the anyon labels.Kitaev (2006); Bombin (2010); Kitaev and Kong (2012); You and Wen (2012); Barkeshli and Qi (2012); Barkeshli et al. (2013); Teo et al. (2014a, b) In this paper, we will fully gauge the discrete global symmetries so that these topological defects will become fully deconfined loop-like excitations. Our method for determining the resulting $(3+1)$d topological phase relies on the bulk-boundary correspondence. We work with the $(2+1)$-dimensional surface theories of the bulk $(3+1)$d BF theories, and consider the $\mathbb{Z}_{2}$ orbifold thereof. As in the context of $(1+1)$-dimensional conformal field theories (CFT), orbifolding a CFT amounts to considering the partition functions in the presence of twisted boundary conditions. Moore and Seiberg (1989); Dijkgraaf et al. (1989); Ginsparg (1988); Francesco et al. (1997) By putting the surface theory on the spacetime torus $T^{3}$ with proper boundary conditions in time and two spatial directions, we derive the transformation properties of the twisted partition function under the mapping class group (the large diffeomorphisms) of $T^{3}$.Hsieh et al. (2016); Chen et al. (2016) This procedure allows us to read off the modular $\mathcal{S}$ and $\mathcal{T}$ matrices, which encode the properties of anyons in the gauged surface theory. This, in turn, allows us to deduce the gauged bulk theory.

The rest of the paper is organized as follows: In Sec. II.1, we briefly review orbifold CFTs in $(1+1)$d. We further elucidate this with a simple example in the $\mathbb{Z}_{\mathrm{K}}$ quantum double model in Sec. II.2. In Sec. III, we consider three different topological phases in $(3+1)$d and study the corresponding orbifolded surface theories. In Sec. V, we discuss the bulk non-Abelian topological phase that results from gauging $\mathbb{Z}_{2}$ symmetry and make a connection with the surface orbifold theory. We summarize our results in Sec. VI.

II Orbifolding the $(1+1)$d boundary theory

II.1 Summary of orbifold CFT in $(1+1)$d

Topologically ordered phases in $(2+1)$d are often equipped (or enriched) with some global discrete symmetries. These symmetries may permute anyon labels, but leave the $\mathcal{S}$ and $\mathcal{T}$ matrices invariant. Gauging the discrete symmetry $\mathcal{G}$ will give rise to more interesting topological phases.

A way to understand this new topological phase is through studying the boundary theory. For a topological phase, the gapless boundary state can be described by a rational CFT $\mathcal{C}$. Gauging the global symmetry in the bulk corresponds to orbifolding the edge CFT by the symmetry. By orbifolding by symmetry $\mathcal{G}$, we project out the symmetry non-invariant states and, simultaneously, add some twist sectors to the Hilbert space. For the details of orbifold CFTs, see Ref. Moore and Seiberg, 1989; Dijkgraaf et al., 1989; Ginsparg, 1988; Francesco et al., 1997. The orbifold CFT $\mathcal{C}/\mathcal{G}$ can be understood by calculating the character for each primary field. The characters are the partition functions under the symmetry projection,

[TABLE]

where $\mathcal{P}_{n}$ is a projection operator, and $h\in\mathcal{G}$ defines the twist in the spatial direction. Here, the projection operator $\mathcal{P}_{n}$ depends on the set of phase factors $\{\omega\}$ known as the discrete torsion phases. E.g., if $\mathcal{G}$ is an abelian $\mathbb{Z}_{N}$ symmetry, the projection operator is simply written as $\mathcal{P}_{n}=\sum_{k=0}^{N-1}\omega^{-nk}g^{k}/N$ where $g$ is the generator of $\mathbb{Z}_{N}$ and $\omega$ is the $k$-th root of unity. The character $\chi^{h}_{n}$ can be understood as a linear combination of partition functions with fixed twist $h$ in the spatial direction and all allowed twists $g$ in the time direction. Here, if $\mathcal{G}$ is a non-Abelian group, we require $g$ belongs to centralizer subgroup of $h$ containing all $g$ that commutes with $h$, i.e., $gh=hg$.

The characters form a complete basis for a reprsentation of the group of modular transformations. The modular $\mathcal{S}$ and $\mathcal{T}$ matrices for the orbifold CFT, which are identical to the $\mathcal{S}$ and $\mathcal{T}$ matrices for the bulk topological order, thus encode topological information of the topological phase.

One interesting example is the toric code model. It has a global duality symmetry which exchanges the charge $e$ and flux $m$, while leaving $\psi=e\times m$ invariant. Gauging the $\mathbb{Z}_{2}$ duality symmetry leads to a non-Abelian topological phase. The calculation of the characters for the edge orbifold CFT shows that $\mathcal{S}$ and $\mathcal{T}$ matrices are equivalent to that for the $\mbox{Ising}\times\overline{\mbox{Ising}}$ CFT and suggests that the bulk topological phase has nine quasi-particles, including the Ising-like anyon excitation.Kitaev and Preskill (2006); Bais and Slingerland (2009)

II.2 Orbifolding $\mathbb{Z}_{2}$ symmetry on the boundary of the $\mathbb{Z}_{\mathrm{K}}$ quantum double model

Starting from an abelian topological phase, if we gauge a global symmetry $G$, the resulting non-abelian topological phase includes the following three types of excitations:Teo et al. (2015) First, for the anyon in a which is invariant under symmetry $G$, they remain as excitations in the twist liquid. They can also couple with gauge charge and form a composite particle. These excitations are abelian excitations with zero gauge flux and called Type $(1)$ excitations. Next, for anyons $\{{\bf a}_{i}\}$ which are not invariant under symmetry, they need to group together to form a superselection sector so that ${\bf a}_{1}+{\bf a}_{2}+\ldots$ is gauge invariant. Such Type (2) excitations are non-abelian quasi-particles with zero gauge flux. Finally, Type $(3)$ excitations are the most interesting ones: They carry non-trivial gauge flux and correspond to non-abelian twist defects before gauging. From the boundary field theory point of view, for these three types of excitations, we can construct the corresponding characters on the boundary.

We now work out an example completely explicitly in order to illuminate the general strategy: $\mathbb{Z}_{2}$ charge-conjugation symmetry in the $D(\mathbb{Z}_{\mathrm{K}})$ quantum double model. We here consider the case when $\mathrm{K}$ is an odd number. The detail for this method can be found in Ref. Chen et al., 2017.

Our use of a gapless edge CFT deserves further comment. For a single-component Chern-Simons theory, there is a chiral gapless mode on the boundary, which is stable and cannot be gapped out. This is because the single component Chern-Simons theory is anomalous and requires a gapless edge mode on the boundary to compensate the anomaly in the bulk. Meanwhile, for the non-chiral $D(\mathbb{Z}_{\mathrm{K}})$ quantum double model without anomaly, the boundary CFT can be gapped out if we do not impose any symmetry. Although the gapless CFT is not stable, it does encode topological data in the bulk. Thus, we can use the “fine-tuned” gapless CFT as an intermediate step to study the bulk topological phase via the bulk boundary correspondence. This is also true for the $(3+1)$d topological phase.

The $D(\mathbb{Z}_{\mathrm{K}})$ quantum double model has two fundamental quasi-particle excitations, $e$ and $m$. All the quasi-particle excitations can be written as $e^{a}m^{b}$, where $0\leq a,b<\mathrm{K}$. $e$ and $m$ are self-bosons, and have non-trivial mutual braiding statistics with braiding phase $e^{2\pi i/\mathrm{K}}$.

The $D(\mathbb{Z}_{\mathrm{K}})$ quantum double model has a global $\mathbb{Z}_{2}$ charge-conjugation symmetry which exchanges $e^{a}m^{b}$ and $e^{\mathrm{K}-a}m^{\mathrm{K}-b}$. If $\mathrm{K}$ is an odd number, there is no quasi-particle which is invariant under the charge-conjugation.

Once the $\mathbb{Z}_{2}$ symmetry is gauged, i.e., the global charge-conjugation symmetry is promoted to a local gauge symmetry, there is a $\mathbb{Z}_{2}$ bosonic charge $j$ which satisfies the fusion rule $j\times j=1$. On the other hand, the $\mathbb{Z}_{2}$ flux $\sigma$ is a non-Abelian quasi-particle. $\sigma$ can combine with $\mathbb{Z}_{2}$ charge to form the flux-charge composite quasi-particle, $\tau=\sigma\times j$. The original abelian anyons $e^{a}m^{b}$ will group together to form gauge invariant superselection sector $e^{a}m^{b}+e^{\mathrm{K}-a}m^{\mathrm{K}-b}$ with quantum dimension equal to 2.

Let us now take a look at the gauging procedure from the boundary field theory point of view. The relevant boundary theory is described by the following Lagrangian density

[TABLE]

where $(t,x)$ are the spacetime coordinates of the boundary; $\mathbf{K}=\mathrm{K}\sigma_{x}$, $\vec{\Phi}=(\phi^{1},\phi^{2})$ is a two-component boson, and $V$ is a symmetric and positive definite matrix that accounts for the interaction on the edge and is non-universal. This model has $\mathrm{K}^{2}$ characters and there is a choice of interaction $V_{IJ}$ for which they take the form

[TABLE]

where $a,b\in\mathbb{Z}\ \mbox{mod}\ \mathrm{K}$ are the anyon labels, $\tau$ is the modular parameter of the spacetime torus, $q=\exp(2\pi i\tau)$, and $\eta(\tau)$ is the Dedekind eta function. The details of the calculation can be found in Appendix.

The boundary theory (2) is invariant under the $\mathbb{Z}_{2}$ charge-conjugation symmetry

[TABLE]

Once orbifolded by the $\mathbb{Z}_{2}$ symmetry, $\phi^{1,2}$ can become $-\phi^{1,2}$ when the coordinates are taken around the time and spatial directions on the $(1+1)$d torus. Therefore orbifolding introduces anti-periodic boundary conditions in the $x$ and $t$ directions. The partition function with twisted boundary condition is labelled by $Z^{\mu\nu}$, where $\mu,\nu=0,\frac{1}{2}$ represents untwisted and twisted boundary condition in, respectively, the time and space directions. The twisted partition functions are given by

[TABLE]

where $\theta_{2,3,4}$ are Jacobi theta functions defined by

[TABLE]

One can readily check that under modular transformations,

[TABLE]

We use these twist blocks and the original characters $B_{a,b}^{\mathrm{K}}$ to construct the characters for the orbifold CFT and the result is shown in Table 1. From the table we can see that the quantum dimension for a $\mathbb{Z}_{2}$ flux $\sigma$ is $\mathrm{K}$, indicating that it is a non-Abelian quasi-particle. The $\mathcal{S}$ matrix is

[TABLE]

where the total quantum dimension is $\mathcal{D}=2\mathrm{K}$. The topologica spin $e^{2\pi ih}$ of anyonic excitations can be read off from the (eigenvalues of the) $\mathcal{T}$ matrix.

A trivial example is $\mathrm{K}=1$. Before gauging, the bulk has no topological order. After gauging the $\mathbb{Z}_{2}$ symmetry, the $\mathcal{S}$ and $\mathcal{T}$ matrix is the same as that for the toric code model, indicating the original phase is not a symmetry protected topological phase (SPT).Levin and Gu (2012) The twist fields $\sigma,~{}\tau$ have quantum dimension equal to one, and correspond to the Abelian $\mathbb{Z}_{2}$ flux in the bulk. For $\mathrm{K}\geq 3$, the gauged system has non-Abelian topological order in the bulk. This non-abelian topological order can be described by the $D(\mathcal{D}_{\mathrm{K}})$ quantum double model, where $\mathcal{D}_{\mathrm{K}}$ is the dihedral group of order $2\mathrm{K}$. Tarantino et al. (2016)

III Gauging $\mathbb{Z}_{2}$ charge-conjugation symmetry

in (3+1)d $\mathbb{Z}_{\mathrm{K}}$ gauge theories

In this section, we discuss gauging discrete symmetries in $(3+1)$d topologically ordered phases. A specific example we consider in this section is the topological $\mathbb{Z}_{\mathrm{K}}$ gauge theory. It is the long wavelength limit of the deconfined phase of the $\mathbb{Z}_{\mathrm{K}}$ gauge theory. A convenient description of the topological $\mathbb{Z}_{\mathrm{K}}$ gauge theory is given by the single-component $(3+1)$d BF theory, which is defined by the following action

[TABLE]

where $\mathcal{M}$ is a (3+1)d spacetime manifold; $a$ and $b$ are a one- and two-form, respectively; $\mathrm{K}$ is an integral parameter (“level”). This action describes the simplest topological phase in $(3+1)$d; its fundamental excitations are a particle excitation $e$ and a loop excitation $m$. They have non-trivial mutual braiding statistics with the braiding phase $e^{2\pi i/\mathrm{K}}$.

The BF theory (9) has $\mathbb{Z}_{2}$ charge-conjugation symmetry:

[TABLE]

and our goal in this section is to gauge this symmetry. We will show that the resulting gauge theory has a non-Abelian topological order.

Similar to the $(2+1)$d case, after gauging $\mathbb{Z}_{2}$ symmetry, there will be a $\mathbb{Z}_{2}$ charge $j$ which is a bosonic particle, and a $\mathbb{Z}_{2}$ flux which, in (3+1)d, is a vortex-line excitation. The original excitations $e^{a}m^{b}$ in the BF theory are not $\mathbb{Z}_{2}$ symmetry invariant and will be grouped together. We will denote them simply as $e^{a}m^{b}+e^{\mathrm{K}-a}m^{\mathrm{K}-b}$.

Our approach here is to generalize orbifolds of $(1+1)$d edge theories discussed in Sec. II, and discuss orbifolds of $(2+1)$ gapless boundary theories. In particular, we define a set of quantities which are analogous to the characters defined for $(1+1)$d CFTs. These characters are constructed by applying a projection operator on the partition function, which is equivalent to imposing a twisted boundary condition in the time direction. The characters form a complete basis under $SL(3,\mathbb{Z})$, the mapping class group of $T^{3}$, and the $\mathcal{T}$ matrix takes a diagonal form. By studying the characters on the boundary, we can extract information about the non-Abelian bulk topological order.

III.1 The BF surface theory

Our starting point is thus the boundary of the BF theory at level $\mathrm{K}$, which can be described by the following Lagrangian density

[TABLE]

where $i,j=x,y$, $\varphi$ is a scalar, and $\zeta_{i}$ is a one-form field (the temporal component of $\zeta$ is gauge-fixed to zero for convenience). We fix the coupling constant $\lambda_{1}$ and $\lambda_{2}$ according to

[TABLE]

for convenience. The boson field $\varphi$ is compact and satisfy $\varphi\equiv\varphi+2\pi.$ Hence, physical observables are exponentials

[TABLE]

The winding number of $\varphi$ is quantized in the absence of bulk quasiparticles, according to

[TABLE]

where $i=1,2$ and $i$ is not summed on the right hand side. On the other hand, the gauge field $\zeta_{i}$ is compact, meaning that physical observables are Wilson loops,

[TABLE]

where $C$ is a closed loop on $\partial\Sigma=T^{2}$. (Since the different components of $\zeta_{i}$ commute with each other, path-ordering is unecessary.) The flux associated to $\zeta_{i}$ is quantized, in the absence of bulk quasiparticles, according to

[TABLE]

where $N_{0}$ is an integer.

The surface theory (11) is put on a flat spacetime three-torus $T^{3}$, and $G^{ij}$ represents the spatial part of the metric. For the properties of the the flat $T^{3}$, and our parameterization of the metric, see Appendix B. Our flat three-torus is parameterized by six real parameters, $R_{0,1,2}$ and $\alpha,\beta,\gamma$. For example, $R_{0,1,2}$ are the periods in $t$, $x$, and $y$ directions, respectively. The mapping class group of $T^{3}$, ${SL}(3,\mathbb{Z})$, is generated by two transformations which we call $U_{1}$ and $U_{2}$. Any ${SL}(3,\mathbb{Z})$ transformation can be written as $U^{n_{1}}_{1}U^{n_{2}}_{2}U^{n_{3}}_{1}\cdots$. Coxeter and Moser (1980) We further decompose $U_{1}$ as $U_{1}=U^{\prime}_{1}M$, where $U_{1}^{\prime}$ corresponds to a $90^{\circ}$ rotation in the $\tau-x$ plane and $M$ is a $90^{\circ}$ rotation in the $x-y$ plane. In particular, $U^{\prime}_{1}$ sends $\tau\to-1/\tau$ where $\tau=\alpha+iR_{0}/R_{1}$. The two transformations $U^{\prime}_{1}$ and $U_{2}$ correspond respectively to modular $S$ and $T^{-1}$ transformations in the ${\tau}-x$ plane, generating the ${SL}(2,\mathbb{Z})$ subgroup of ${SL}(3,\mathbb{Z})$ group. Combined with $M$, they generate the whole ${SL}(3,\mathbb{Z})$ group. In the following, we denote $U^{\prime}_{1}M$ by $\mathcal{S}$ and $U_{2}$ by $\mathcal{T}^{-1}$.

The surface theory (11) can be studied in the presence of the following twisted boundary conditions:

[TABLE]

where $N_{0,1,2}\in\mathbb{Z}$, and $n_{0,1,2}=0,\ldots,\mathrm{K}-1$. We denote the corresponding partition functions as

[TABLE]

where $Z^{n_{0},n_{1},n_{2}}_{zero}$ and $Z_{osc}$ will be explained later. From the bulk point of view, these (twisted) boundary conditions correspond to insertion of Wilson loop and Wilson surfaces, i.e., bulk excitations. The bulk-boundary correspondence implies that, by studying the partition functions of the surface theories in the presence of these boundary conditions, in particular, their transformation law under $SL(3,\mathbb{Z})$, one can extract properties of bulk quasiparticles.Moradi and Wen (2015). The details for the calculation of surface partition functions can be found in Ref. Chen et al., 2016, where the bulk-(gapless) boundary correspondence is also discussed. Here we directly write down the partition functions for the surface. The zero mode part is

[TABLE]

where $2\mathrm{r}^{2}=\mathrm{K}$, and we have introduced the notation

[TABLE]

For the oscillator part,

[TABLE]

where $\Theta_{[a,b]}(\tau,m)$ is the massive theta function (see Appendix A). The total partition function for each sector is ${Z}^{n_{0}n_{1}n_{2}}=Z^{n_{0}n_{1}n_{2}}_{zero}Z_{osc}$. The modular $\mathcal{S}$ and $\mathcal{T}$ matrices are given by

[TABLE]

III.2 Gauging $\mathbb{Z}_{2}$ symmetry in

the surface theory

In terms of the surface theory (11), the $\mathbb{Z}_{2}$ charge conjugation symmetry is implemented as

[TABLE]

We now gauge the $\mathbb{Z}_{2}$ symmetry. We consider the cases of odd and even level $\mathrm{K}$ separately.

First, we include the twisted sectors obtained by twisting boundary conditions in $t$, $x$, and $y$ directions. The partition functions for these sectors are denoted by $V^{\mu,\nu,\lambda}$ where $\mu,\nu,\lambda=0$ $(1/2)$ represents the untwisted (twisted) boundary condition, for $t$, $x$ and $y$ directions, respectively. The partition functions with twisted boundary condition in $x,y,t$ directions can be readily computed and are given, respectively, by

[TABLE]

($s_{2}$ is an integer). All the other twisted sectors can be obtained by considering modular transformations:

[TABLE]

where $[a]=a\ \mbox{mod}\ \mathbb{Z}$. In total, there are 7 sectors with twisted boundary conditions. (To obtain this result, we note several properties of the massive theta functions $\Theta_{[a.b]}(\tau,m)$, listed in Appendix A.)

We use the twisted sectors $V^{\mu,\nu,\lambda}$ and $Z_{a,b,c}$ to build up the characters of the gauged $(2+1)$d surface theory. The result is listed in Table 2. Here, the subscript $h$ of $\chi_{h}$ denotes the type of bulk excitations. As mentioned previously, the characters are constructed by inserting projection operators in the partition function. This is similar to the construction of the minimal entangled state (MES) defined for $(3+1)$d topological phases.Moradi and Wen (2015); Jiang et al. (2014) There are $(\mathrm{K}^{3}+15)/2$ characters in total, the same as the bulk ground state degeneracy on $T^{3}$.

As in the $(1+1)$d edge theory, here we have the vacuum $I$, $\mathbb{Z}_{2}$ boson $j$ and superselection sector $a,b,c$. $\sigma_{x}$ corresponds to the bulk $\mathbb{Z}_{2}$ flux excitation which leaves a twist in the $x$ direction on the boundary. The character $\chi_{\sigma_{x}}$ includes partition functions with twisted boundary condition in the $x$ direction. $\tau_{x}$ can be understood as combining $\sigma_{x}$ with $\mathbb{Z}_{2}$ charge $j$ and therefore is a flux-charge composite excitation. $\chi_{\tau_{x}}$ is also linear combination of partition functions with twist boundary in $x$ direction. $\chi_{\sigma_{y}}$ and $\chi_{\tau_{y}}$ are the characters with twisted boundary condition in $y$ direction. $\chi_{\sigma_{xy}}$ is twisted in both the $x$ and $y$ directions. Under the transformation $\mathcal{T}$, $\chi_{h}$ will pick up a phase $\exp 2\pi ih_{\chi}$, where $h_{\chi}$ encodes information related to $(3+1)$d analogue of topological spins.

We now consider the $U_{1}^{\prime}$ matrix which, after dimensional reduction, is the $\mathcal{S}$ matrix for a $(1+1)$d CFT. The $U_{1}^{\prime}$ matrix encodes braiding information about bulk excitations,

[TABLE]

From the first column (row) in the above matrix, we can extract the “quantum dimension” of the bulk excitation $d_{\chi}$ shown in Table 2. When $\mathrm{K}=1$, the characters for the twist sectors have $d_{\chi}=1$. In this case, it is easy to verify that $U_{1}^{\prime}$ matrix and $h_{\chi}$ match up with that for the ordinary $(3+1)$d $\mathbb{Z}_{2}$ gauge theory (the $(3+1)$d toric code model). When $\mathrm{K}>1$, the characters for the twist sectors have $d_{\chi}>1$, suggesting that they are non-Abelian excitations. We will discuss these non-Abelian braiding statistics later in Sec. V.

We can similarly calculate the characters when $\mathrm{K}$ is even. Unlike in the case of $\mathrm{K}$ odd, there are several excitations in the original BF theory that are invariant under $\mathbb{Z}_{2}$ symmetry operation. They will remain in the gauged topological phase and can couple with the $\mathbb{Z}_{2}$ boson to form composite excitations. They will also provide species labels for $\sigma$ and $\tau$; the characters are shown in Table 3. There are $\mathrm{K}^{3}/2+60$ characters in total, which describes the ground state degeneracy in the bulk topological phase on $T^{3}$.

The new (3+1)d topological phase which we have obtained by gauging the $\mathbb{Z}_{2}$ symmetry is the $D(\mathcal{D}_{\mathrm{K}})$ quantum double model, where $\mathcal{D}_{\mathrm{K}}$ is the dihedral group of order $2\mathrm{K}$. The excitations for this quantum double model are labeled by $(\mathcal{C},\rho)$, where $\mathcal{C}$ denotes the conjugacy class and $\rho$ denotes the irreducible representation of the normalizer group for each conjugacy class. When $\mathrm{K}$ is odd, there are in total $\frac{n+3}{2}+\frac{n-1}{2}\times n+2=\frac{n^{2}+7}{2}$ excitations. When $\mathrm{K}$ is even, there are in total $(\frac{n}{2}+3)\times 2+(\frac{n}{2}-1)\times n+2\times 4=\frac{n^{2}+28}{2}$ excitations. These results are consistent with the calculation for the boundary orbifold theory. For instance, when $\mathrm{K}=3$, $\mathcal{D}_{3}$ is equivalent to the symmetric group $S_{3}$. There are eight excitations in the $D(S_{3})$ quantum double model, which agree with the above counting. Apart from the vacuum sector and two charge (particle) excitations, all the other five excitations are flux or flux-charge composite excitations with non-Abelian fusion rules. These fusion rules have been discussed in Ref. Moradi and Wen, 2015 and will not be analyzed further here.

More generically, for all (3+1)d Abelian topological phases considered in this paper, after gauging the $\mathbb{Z}_{2}$ symmetry, the new topological phases can always be described by the quantum double model with group $\widetilde{\mathcal{G}}=\mathcal{G}\rtimes\mathbb{Z}_{2}$, where $\mathcal{G}$ is the original abelian group. This is because the $\mathbb{Z}_{2}$ symmetry acts on both the charge and flux excitations independently and in the same way. If $\mathcal{G}=\mathbb{Z}_{\mathrm{K}}$, i.e., the $\mathbb{Z}_{\mathrm{K}}$ gauge theory, the gauged model is the $D(\mathcal{D}_{\mathrm{K}})$ quantum double model.

IV Gauging $\mathbb{Z}_{2}$ symmetry in

(3+1)d $\mathbb{Z}_{\mathrm{K}}\times\mathbb{Z}_{\mathrm{K}}$ gauge theories

In this section, we consider ordinary $\mathbb{Z}_{\mathrm{K}}\times\mathbb{Z}_{\mathrm{K}}$ topological gauge theory in $(3+1)$ dimensions with only particle-loop braiding statistics. The excitations in this model can be denoted by $e_{1}^{a_{1}}m_{1}^{b_{1}}e_{2}^{a_{2}}m_{2}^{b_{2}}$, where $e_{1},\ m_{1},\ e_{2},\ m_{2}$ are the fundamental excitations of this model. This model can be described by the two-component BF theory with both components at level $\mathrm{K}$.

The topological $\mathbb{Z}_{\mathrm{K}}\times\mathbb{Z}_{\mathrm{K}}$ gauge theory has a $\mathbb{Z}_{2}$ symmetry which exchanges

[TABLE]

Gauging this symmetry will lead to a non-Abelian topological phase.

The surface theory of the topological $\mathbb{Z}_{\mathrm{K}}\times\mathbb{Z}_{\mathrm{K}}$ gauge theory can be described by taking two copies of (11). The surface partition function can then be written down as

[TABLE]

where $n_{i},l_{i}\in\mathbb{Z}_{\mathrm{K}}$ and $i=0,1,2$.

After orbifolding the $\mathbb{Z}_{2}$ symmetry, the partition function with anti-periodic boundary condition in the $t$ direction is

[TABLE]

where $V^{[\mu],[\nu],[\lambda]}$ is defined in the previous section and $Z^{2\mathrm{K}}_{a,b,c}$ represents the surface partition function for $\mathbb{Z}_{2\mathrm{K}}$ gauge theory model. The term $Z^{2\mathrm{K}}_{2n_{i}}$ is obtained by identifying $n_{i}$ and $l_{i}$ in $Z_{n_{0},n_{1},n_{2}}Z_{l_{0},l_{1},l_{2}}$, which is imposed by the $\mathbb{Z}_{2}$ symmetry.

The other twisted partition functions $Y^{\mu,\nu,\lambda}_{n_{0},n_{1},n_{2}}$ can be obtained, starting from $Y^{\frac{1}{2},0,0}_{n_{0},n_{1},n_{2}}$, by applying modular transformations.

Schematically,

[TABLE]

The explicit form of other sectors are obtained by requiring them to be invariant (up to a phase) under $U_{1}^{\prime}$ transformation. They are given by

[TABLE]

where $p,q=0,1$ and $0\leq m_{i}<\mathrm{K}$. $W^{l_{0}l_{1}l_{2}}_{n_{0}n_{2}n_{2}}$ and $Y_{m_{0}m_{1}m_{2}}^{\nu\nu\lambda}$ can be properly combined to construct the characters. The result is summarized in Table 4. It is also instructive to consider the dimensional reduction. After dimensional reduction, the complete results for the characters on the $(1+1)$d edge are shown in Table 5.

Finally, by noting the transformation properties of the characters under $U_{1}^{\prime}$ listed in Appendix C, we read off the $U_{1}^{\prime}$ matrix:

[TABLE]

where $\mathcal{D}=2\mathrm{K}^{2}$ and $P=\left[1+(-1)^{m_{1}+m_{1}^{\prime}}+(-1)^{m_{0}+m_{0}^{\prime}}(1+(-1)^{m_{1}+m_{1}^{\prime}+\mathrm{K}})\right]$.

IV.1 Orbifolding phases with non-trivial

three-loop braiding statistics

For the $\mathbb{Z}_{\mathrm{K}}\times\mathbb{Z}_{\mathrm{K}}$ twisted gauge theory with non-trivial three loop braiding statistics Wang and Levin (2014); Jiang et al. (2014); Wang and Wen (2015), we can also construct the surface partition function and calculate $\mathcal{S}$ and $\mathcal{T}$ matrices. Chen et al. (2016) In this case, the quantum numbers $n_{0}$ and $l_{0}$ are shifted according to

[TABLE]

This model also has the $\mathbb{Z}_{2}$ exchange symmetry, which switches $n_{i}$ and $l_{i}$. Therefore we can gauge the $\mathbb{Z}_{2}$ symmetry. After orbifolding this symmetry on the surface, for the characters $\chi^{0}_{n_{0},n_{1},n_{2}}$ with $n_{i}=l_{i}$, they are still the same as for the untwisted $\mathbb{Z}_{\mathrm{K}}\times\mathbb{Z}_{\mathrm{K}}$ gauge theory. The character $\chi_{n_{i}}^{l_{i}}$ is modified slightly: the quantum numbers $n_{0},l_{0}$ replaced by $\widetilde{n}_{0}$ and $\widetilde{l}_{0}$. The characters for twisted sectors are still the same as before. The complete result is shown in Table 6. The $U_{1}^{\prime}$ matrix is very similar to the ordinary $\mathbb{Z}_{\mathrm{K}}\times\mathbb{Z}_{\mathrm{K}}$ gauge theory and we will not discuss it here.

V Physics in the bulk

In this section, we will study the bulk physics and discuss the non-Abelian braiding statistics of loop excitations.

Twist defects in (2+1)-dimensional topological phases

Before we discuss the $(3+1)$d case, it is instructive to review briefly the $(2+1)$d case.Kitaev (2006); Bombin (2010); Kitaev and Kong (2012); You and Wen (2012); Barkeshli and Qi (2012); Barkeshli et al. (2013); Teo et al. (2014a) For the $D(\mathbb{Z}_{\mathrm{K}})$ quantum double model, we can introduce a twofold twist defect which exchanges Abelian excitations $e^{a}m^{b}$ and $e^{\mathrm{K}-a}m^{\mathrm{K}-b}$. The twist defect is a point-like defect with a branch cut emanating from it. In Fig. 1, we depict several pairs of twist defects, with branch cuts extending between them. When an anyon $e^{a}m^{b}$ ($e^{\mathrm{K}-a}m^{\mathrm{K}-b}$) is dragged around the twist defect, it is transformed to $e^{\mathrm{K}-a}m^{\mathrm{K}-b}$ ($e^{a}m^{b}$) when it passes through the branch cut. Once the $\mathbb{Z}_{2}$ charge-conjugation symmetry is gauged, twist defects, where were treated above as non-dynamical objects, are deconfined $\mathbb{Z}_{2}$ flux excitations. Barkeshli et al. (2014); Teo et al. (2015); Tarantino et al. (2016) These $\mathbb{Z}_{2}$ fluxes can leave twists on the boundary, which correspond to the twist fields $\sigma$ or $\tau$ on the boundary discussed in the previous section. As we will see below, these $\mathbb{Z}_{2}$ fluxes are non-Abelian excitations.

The non-Abelian braiding statistics of these twist defects (before gauging) can be studied by calculating the fundamental unitary exchange operations, called B-symbols.Barkeshli and Qi (2012); Barkeshli et al. (2013); Teo et al. (2014a) Each B-move represents a counter-clockwise permutation of a pair of adjacent defects, which can result in a transformation of different ground states. The B-operations can be generated by a sequence of F and R-moves, and evaluated exactly once we specify the splitting space of the twist defects. Teo et al. (2014a) Here we show that twist defects are non-Abelian objects by directly deforming Wilson loop operators around a pair of twist defects shown in Fig. 1. Let us consider a system with $2N$ twist defects aligned on a line. The Hilbert space can be characterized by the non-contractible Wilson loop operators around the neighboring twist defects (Fig. 1 (a)). The Wilson loop operators are represented as the powers of fundamental $e$-loop and $m$-loop operators. It is important to note that the neighboring Wilson loop operators do not commute with each other due to the intersections (highlighted by brown dots). Therefore, the Hilbert space can be spanned by the eigenstates of Wilson loops $\{W_{1},W_{3},\ldots,W_{2N-1}\}$ where $W_{2j-1}$ can be either $e_{2j-1}$ or $m_{2j-1}$:

[TABLE]

On the other hand, Wilson loops $\{W_{2},W_{4},\ldots,W_{2N}\}$ act on the basis states as

[TABLE]

The non-commutative algebra between neighboring Wilson loop operators leads to non-Abelian braiding of twist defects. Barkeshli et al. (2013); You et al. (2013) In Fig. 1 (b), we consider four twist defects with $W_{1}=e^{a}m^{b},W_{2}=\mathbb{I}$. If we exchange the twist defects $\sigma_{2}$ and $\sigma_{3}$, $W_{1}$ will deform to $W_{1}W_{2}^{\prime}$ up to some phase with $W_{2}^{\prime}=e^{a}m^{b}$ and the original state $|0,0\rangle$ changes to $|a,b\rangle$.

Twist defects in (3+1)-dimensional topological phases

Similarly, we can understand the non-Abelian braiding of twist defects in three spatial dimensions. As shown in Fig. 2, the twist defect has a loop configuration in three spatial dimensions and has a branch sheet attached to it. Unlike in $(2+1)$d, this defect loop does not need to pair up with another twist defect since it is equivalent to a pair of extended defect lines that wrap around a non-contractible circle on $T^{3}$. For any excitation $e^{a}m^{b}$, as it goes through the branch sheet, it will become $e^{\mathrm{K}-a}m^{\mathrm{K}-b}$.

For a system with a finite number of twist defects, the Hilbert space is labeled by the non-contractible Wilson loop and surface operators as shown in Fig. 2. In Figs. 2 (a) and (b), we show the Wilson loop and surface operators defined in the presence of a single defect loop, while in (c) and (d), the Wilson operators are defined for a pair of loops. By counting these Wilson operators, we find that the quantum dimension of a defect loop is $\mathrm{K}^{2}$. Therefore the extended defect line has quantum dimension $\mathrm{K}$ and matches up with that for the twist field on the boundary theory in Table 2.

Here we shall use (b) and (c) to construct a subspace of the total Hilbert space and show that these loop excitations have non-Abelian braiding statistics. As shown in Fig. 3, the Wilson operators $W_{2i-1}$ and $W_{2j}$ are defined on a pair of defect loops. The adjacent Wilson operators do not commute with each other. As in the 2d case, the Hilbert space can be generated by acting with $\{W_{2j}\}$ operators on basis states in which $\{W_{2j-1}\}$ is diagonal. The braiding process of loop $2$ and $3$ is defined in Fig. 4. This exchange process can be better understood if we look at the dimensionally-reduced version in Fig. 5. $W_{1}$ under this process deforms into $W_{1}W_{2}$, suggesting that a defect loop is a non-Abelian object. Once $\mathbb{Z}_{2}$ symmetry is fully gauged, the defect loops will be the intrinsic non-Abelian $\mathbb{Z}_{2}$ flux excitations. These are loops in $D(\mathcal{D}_{\mathrm{K}})$ that carry flux equal to the conjugacy class of reflections in the dihedral group $\mathcal{D}_{\mathrm{K}}$. They fuse non-trivially because the composition of two reflections can be either the identity or a rotation; braiding transforms the system within the state space of these different fusion outcomes. For the $\mathbb{Z}_{\mathrm{K}}\times\mathbb{Z}_{\mathrm{K}}$ gauge theory (both with and without non-trivial three-loop braiding), using similar method, we can also show that the twist defect loop/$\mathbb{Z}_{2}$ flux excitations are non-Abelian objects.

VI Conclusion

In this paper, we gauge the $\mathbb{Z}_{2}$ symmetry in various Abelian topological phases in $(3+1)$ dimensions. By making use of the bulk-boundary correspondence, we discuss the orbifold theory on the $(2+1)$d surface state. We calculate the partition function on the $(2+1)$d torus with twisted boundary conditions and group them into characters. We further study how the characters transform under modular the $\mathcal{S}$ and $\mathcal{T}$ transformations which characterize the braiding statistics of particle and loops excitations in the bulk. Based on the topological data obtained on the boundary, we further analyze the defect loops/$\mathbb{Z}_{2}$ flux excitations in the bulk and use the Wilson loop algebra to show that these loop defects are non-Abelian.

Recently, it was shown that Abelian topological phases in $(3+1)$ dimensions, such as the $\mathbb{Z}_{\mathrm{K}}$ and $\mathbb{Z}_{\mathrm{K}}\times\mathbb{Z}_{\mathrm{K}}$ gauge theories discussed here, have flux line excitations carrying Cheshire charge, topological charge that cannot be localized to a point on the flux line or measured locally Else and Nayak (2017). Since many properties can be deduced from those of the parent Abelian theory, the gauging procedure discussed here may be an entry point for exploring the properties of Cheshire charge in non-Abelian topological phases in $(3+1)$ dimensions.

All the $(3+1)$d topological phases in this paper, obtained by gauging the $\mathbb{Z}_{2}$ symmetry, can be described by quantum double models with a non-Abelian group and their dimensional reductions are $(2+1)$d quantum double models. In the future, it would be interesting to explore non-Abelian topological phases in $(3+1)$d that go beyond quantum double models.

Acknowledgements.

We thank Michael Levin and Jeffrey Teo for useful discussions. We also thank KITP program “Symmetry, Topology, and Quantum Phases of Matter: From Tensor Networks to Physical Realizations”. This work is supported in part by the NSF under Grant No. DMR-1455296 and No. NSF PHY-1125915. XC was supported by a postdoctoral fellowship from the Gordon and Betty Moore Foundation, under the EPiQS initiative, Grant GBMF4304, at the Kavli Institute for Theoretical Physics.

Appendix A Theta functions

The Dedekind eta function $\eta(\tau)$ is defined by

[TABLE]

The massive theta function $\Theta_{[a,b]}(\tau,m)$ is defined by

[TABLE]

where

[TABLE]

The massive theta functions $\Theta_{[a.b]}(\tau,m)$ satisfy

[TABLE]

where $p,q\in\mathbb{Z}$.

Appendix B Modular transformations on $T^{3}$

In this appendix, we collect some necessary ingredients relating to a flat three-torus $T^{3}$ and its mapping class group.Hsieh et al. (2016) A flat three-torus is parameterized by six real parameters, $R_{0,1,2}$ and $\alpha,\beta,\gamma$. For a flat three-torus $T^{3}$, the dreibein can be factorized as

[TABLE]

and its inverse is given by

[TABLE]

such that ${e^{A}}_{\mu}{e^{\star}_{A}}^{\nu}={\delta_{\mu}}^{\nu}$ and ${e^{A}}_{\mu}{e^{\star}_{B}}^{\mu}={\delta^{A}}_{B}$. Here $R_{0}$, $R_{1}$, and $R_{2}$ are the radii for the directions $\tau$, $x$, and $y$, and $\alpha$, $\beta$, and $\gamma$ are related to the angles between directions $\tau$ and $x$, $x$ and $y$, and $\tau$ and $y$, respectively. The Euclidean metric is then given by

[TABLE]

and the line element is

[TABLE]

where $0\leq\theta^{\mu}\leq 2\pi$ are angular variables.

The group ${SL}(3,\mathbb{Z})$ is generated by two transformations:

[TABLE]

An ${SL}(3,\mathbb{Z})$ transformation acts as follows on the dreibein and metric:

[TABLE]

for any ${SL}(3,\mathbb{Z})$ element $L=U^{n_{1}}_{1}U^{n_{2}}_{2}U^{n_{3}}_{1}\cdots$. Under the $U_{2}$ transformation, the metric transforms according to

[TABLE]

which corresponds to the changes

[TABLE]

while $R_{0}$, $R_{1}$, $R_{2}$, and $\beta$ are unchanged.

On the other hand, the less trivial generator $U_{1}$ can be decomposed as

[TABLE]

where $U_{1}^{\prime}$ corresponds to the $90^{\circ}$ rotation in the $\tau-x$ plane and $M$ is the $90^{\circ}$ rotation in the $x-y$ plane. The generator $U^{\prime}_{1}$ acts on the metric as

[TABLE]

which corresponds to the changes

[TABLE]

where we have introduced

[TABLE]

Observe also that under $R_{0}\to R_{0}/|\tau|$ and $R_{1}\to R_{1}|\tau|$, $\tau_{2}\to\tau_{2}/|\tau|^{2}$. Hence, $U^{\prime}_{1}$ induces $\tau\to-1/\tau$.

Finally, the transformation $M$ acts on the metric as

[TABLE]

The two transformations $U^{\prime}_{1}$ and $U_{2}$ correspond respectively to modular $S$ and $T^{-1}$ transformations in the ${\tau}-x$ plane, generating the ${SL}(2,\mathbb{Z})$ subgroup of ${SL}(3,\mathbb{Z})$ group. Combined with $M$, they generate the whole ${SL}(3,\mathbb{Z})$ group. In the following, we denote $U^{\prime}_{1}M$ by $\mathcal{S}$ and $U_{2}$ by $\mathcal{T}^{-1}$.

Appendix C Transformation properties of the characters

in the gauged $\mathbb{Z}_{\mathrm{K}}\times\mathbb{Z}_{\mathrm{K}}$ gauge theory

In this appendix, we list the transformation properties of the characters of the topological $\mathbb{Z}_{\mathrm{K}}\times\mathbb{Z}_{\mathrm{K}}$ gauge theory after gauging the $\mathbb{Z}_{2}$ symmetry. From these transformation properties, one can construct the $\mathcal{S}$ matrix.

[TABLE]