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In the presence of symmetry-protection, topological invariants of topological phases of matter in free fermion systems, e.g., topological band insulators, can be directly computed via the properties of band structure. Nevertheless, it is usually difficult to extract topological invariants in strongly-correlated topological phases of matter in which band structure is not well-defined. One typical example is the fractional quantum Hall effect whose low-energy physics is governed by Chern-Simons topological gauge theory and Hall conductivity plateaus involve extremely fruitful physics of strong correlation. In this article, we focus on intrinsic topological order (iTO), symmetry-protected topological phases (SPT), and symmetry-enriched topological phases (SET) in boson and spin systems. Through gauge field-theoretical approach, we review some research progress on these topological phases of matter from the aspects of projective construction, low-energy effective theory and topological response theory.
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Keywords:
- strongly-correlated system/
- topological order/
- symmetry-protected topological state/
- topological quantum field theory
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拟设 完全被$f_1$填充的陈-能带 完全被$f_2$填充的陈-能带 自旋矢量$q^T_s$ 电荷矢量$q^T_c$ $A1$ $\uparrow+, \downarrow+, \uparrow-, \downarrow-$ $(2, 2)$ $\uparrow+, \downarrow+, \uparrow-, \downarrow-$ $(2, 2)$ $\left(\dfrac{1}{2}~~-\dfrac{1}{2}~~\dfrac{1}{2}~~-\dfrac{1}{2}~~\dfrac{1}{2}~~-\dfrac{1}{2}~~\dfrac{1}{2}~~-\dfrac{1}{2}\right)$ $(1~~1~~1~~1~~1~~1~~1~~1)$ $A2$ $\uparrow+, \downarrow-$ $(1, 1)$ $\uparrow+, \downarrow-$ $(1, 1)$ $\left( {1}/{2}~~- {1}/{2}~~ {1}/{2}~~ -{1}/{2}\right)$ $(1~~1~~1~~1)$ $A3$ $\uparrow+, \downarrow-$ $(1, 1)$ $\downarrow+, \uparrow-$ $(1, 1)$ $\left(1/{2}~~- {1}/{2}~~- {1}/{2}~~ {1}/{2}\right)$ $(1~~1~~1~~1)$ $A4$ $\uparrow+, \downarrow-$ $(1, 1)$ 无 $\left( {1}/{2}~~- {1}/{2}\right)$ $(1~~1)$ 
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U 任意一个格点上的物理希尔伯特空间基矢$ [f_1]n_{i, 1 \uparrow}, n_{i, 1 \downarrow}, n_{i, 2 \uparrow}, n_{i, 2 \downarrow}[f_2] $ 费米子填充总数要求 $U_1$ $(0, 0, 0, 0)$, $(0, 1, 0, 1)$, $(0, 1, 1, 0)$, $(1, 0, 0, 1)$, $(1, 0, 1, 0)$, $(1, 1, 1, 1)\, $ $N^{f1}=N^{f2} $ $U_2$ $(0, 0, 0, 0)$, $(0, 1, 1, 0)$, $(1, 0, 0, 1)$, $(1, 1, 1, 1)\, $ $N^{f1}_{\uparrow} = N^{f2}_{\downarrow}, $ $N^{f1}_{\downarrow}=N^{f2}_{\uparrow}$ $U_3$ $(0, 0, 0, 0)$, $(0, 1, 0, 1)$, $(1, 0, 1, 0)$, $(1, 1, 1, 1)\, $ $N^{f1}_{ \uparrow}=N^{f2}_{ \uparrow}, $ $N^{f1}_{ \downarrow}=N^{f2}_{\downarrow}$ $U_4$ $(0, 0, 1, 1)$, $(0, 1, 0, 1)$, $(1, 0, 1, 0)$, $(1, 1, 0, 0)$ $N^{f1}_{\uparrow}+N^{f2}_{\downarrow}=N_{\rm latt}$, $N^{f2}_{\uparrow}+ N^{f1}_{\downarrow}=N_{\rm att}$ $U_5$ $(1, 0, 0, 0)$, $(0, 1, 0, 0)$, $(0, 0, 1, 0)$, $(0, 0, 0, 1)$ $ N^{f1} + N^{f2}=N_{\rm latt}$ $U_6$ $(1, 0)$, $(0, 1)$ $ N^{f1} =N_{\rm latt}$ $U_7$ $(0, 0)$, $(1, 1)$ $N^{f1}_{\uparrow} = N^{f1}_{\downarrow }$ DownLoad:
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对称群G 拓扑规范场论与分类 $\mathbb{Z}_{N_1}\times\mathbb{Z}_{N_2}$ $\dfrac{1}{2{\text{π}}}\displaystyle\int \sum^2_Ib^I\wedge \, {\rm d}a^I+ p_1\displaystyle\int a^1\wedge a^2\wedge \, {\rm d}a^2~~(\mathbb{Z}_{N_{12}} )$ ; $\dfrac{1}{2{\text{π}}}\displaystyle\int \sum^2_Ib^I\wedge \, {\rm d}a^I+ p_2\displaystyle\int a^2\wedge a^1\wedge \, {\rm d}a^1 ~(\mathbb{Z}_{N_{12}}) $ $\mathbb{Z}_{N_1}\times\mathbb{Z}_{N_2}\times\mathbb{Z}_{N_3} $ $\dfrac{1}{2{\text{π}}}\displaystyle\int \sum^3_Ib^I\wedge \, {\rm d}a^I+ p_1 \displaystyle\int a^1\wedge a^2\wedge \, {\rm d}a^3~~(\mathbb{Z}_{N_{123}})$ ; $\dfrac{1}{2{\text{π}}}\displaystyle\int \sum^3_Ib^I\wedge \, {\rm d}a^I+ p_2 \displaystyle\int a^2\wedge a^3\wedge \, {\rm d}a^1~~(\mathbb{Z}_{N_{123}}) $ $\prod^4_I\mathbb{Z}_{N_I} $ $\dfrac{1}{2{\text{π}}}\displaystyle\int \sum^4_Ib^I\wedge \, {\rm d}a^I+p \displaystyle\int a^1\wedge a^2\wedge a^3\wedge a^4~~ ( \mathbb{Z}_{N_{1234}} )$ $\mathbb{Z}_{N_1}\times\mathbb{Z}_{N_2}\times {U}(1)$ $\dfrac{1}{2{\text{π}}}\displaystyle\int \sum^3_Ib^I\wedge \, {\rm d}a^I+p\displaystyle\int a^1\wedge a^2\wedge \, {\rm d}a^3~~ (\mathbb{Z}_{N_{12}})$ DownLoad:
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规范群$G_g$ twisted拓扑项 对称群$G_s$ SET分类 ${\mathbb{Z}}_2$ – ${\mathbb{Z}}_{2 n + 1}$ ${{\mathbb{Z}}_1}$ ${\mathbb{Z}}_2$ – ${\mathbb{Z}}_{2 n}$ $ {({\mathbb{Z}}_2)^2}\oplus {\mathbb{Z}}_1$ ${\mathbb{Z}}_3$ – ${\mathbb{Z}}_{3 n}$ $({\mathbb{Z}}_3)^2\oplus {\mathbb{Z}}_1 \oplus {\mathbb{Z}}_1$ ${\mathbb{Z}}_3$ – ${\mathbb{Z}}_{3 n + 1}$ ${\mathbb{Z}}_1$ ${\mathbb{Z}}_4$ – ${\mathbb{Z}}_{2 n + 1}$ ${\mathbb{Z}}_1$ ${\mathbb{Z}}_4$ – ${\mathbb{Z}}_{4 n+2}$ $ ({\mathbb{Z}}_2)^2\oplus {\mathbb{Z}}_1$ ${\mathbb{Z}}_4$ – ${\mathbb{Z}}_{4 n}$ $ ({\mathbb{Z}}_4)^2\oplus ({\mathbb{Z}}_2)^2\oplus {\mathbb{Z}}_1 \oplus {\mathbb{Z}}_1 $ ${\mathbb{Z}}_2\times {\mathbb{Z}}_2$ (0, 0) ${\mathbb{Z}}_{2 n}$ $({\mathbb{Z}}_2)^6\oplus ({\mathbb{Z}}_2)^2 \oplus ({\mathbb{Z}}_2)^2 \oplus ({\mathbb{Z}}_2)^2$ ${\mathbb{Z}}_2\times {\mathbb{Z}}_2$ (0, 0) ${\mathbb{Z}}_{2 n + 1}$ ${\mathbb{Z}}_1$ ${\mathbb{Z}}_2\times {\mathbb{Z}}_2$ (2, 0) ${\mathbb{Z}}_{2 n}$ $({\mathbb{Z}}_2)^6$ ${\mathbb{Z}}_2\times {\mathbb{Z}}_2$ (2, 0) ${\mathbb{Z}}_{2 n + 1}$ ${\mathbb{Z}}_1$ ${\mathbb{Z}}_2\times {\mathbb{Z}}_2$ (2, 2) ${\mathbb{Z}}_{2 n + 1}$ ${\mathbb{Z}}_1$ ${\mathbb{Z}}_2\times {\mathbb{Z}}_4$ (0, 0) ${\mathbb{Z}}_{2 n + 1}$ ${\mathbb{Z}}_1$ ${\mathbb{Z}}_2\times {\mathbb{Z}}_4$ (4, 0) ${\mathbb{Z}}_{2 n + 1}$ ${\mathbb{Z}}_1$ ${\mathbb{Z}}_2\times {\mathbb{Z}}_4$ (4, 4) ${\mathbb{Z}}_{2 n + 1}$ ${\mathbb{Z}}_1$ ${\mathbb{Z}}_2\times {\mathbb{Z}}_4$ (0, 0) ${\mathbb{Z}}_{4 n}$ $({\mathbb{Z}}_2)^4\times ({\mathbb{Z}}_4)^2\oplus 2({\mathbb{Z}}_4)^2\oplus4({\mathbb{Z}}_2)^2 \oplus ({\mathbb{Z}}_2)^6$ ${\mathbb{Z}}_2\times {\mathbb{Z}}_4$ (4, 0) ${\mathbb{Z}}_{4 n}$ $({\mathbb{Z}}_2)^4\times ({\mathbb{Z}}_4)^2$ ${\mathbb{Z}}_2\times {\mathbb{Z}}_4$ (4, 4) ${\mathbb{Z}}_{4 n}$ $({\mathbb{Z}}_2)^4\times ({\mathbb{Z}}_4)^2$ ${\mathbb{Z}}_2$ – ${\mathbb{Z}}_{2 m + 1} \times {\mathbb{Z}}_{2 n + 1}$ $({\mathbb{Z}}_{2\gcd(2 m + 1, 2 n + 1)})^2$ ${\mathbb{Z}}_2$ – ${\mathbb{Z}}_{2 m + 1} \times {\mathbb{Z}}_{2 n}$ $({\mathbb{Z}}_{\gcd(2 m + 1, 2 n)})^2 \oplus ({\mathbb{Z}}_{2\gcd(2 m + 1, 2 n)})^2 $ ${\mathbb{Z}}_2$ – ${\mathbb{Z}}_{2 m} \times {\mathbb{Z}}_{2 n}$ $({\mathbb{Z}}_2)^6\times({\mathbb{Z}}_{2\gcd(m, n)})^2\oplus ({\mathbb{Z}}_{2\gcd(2 m, n)})^2 \oplus ({\mathbb{Z}}_{2\gcd(m, 2 n)})^2 \oplus ({\mathbb{Z}}_{2\gcd(m, n)})^2$ ${\mathbb{Z}}_4$ – ${\mathbb{Z}}_{2 n + 1} \times {\mathbb{Z}}_{2 n + 1}$ $16({\mathbb{Z}}_{2 n + 1})^2$ ${\mathbb{Z}}_4$ – ${\mathbb{Z}}_{2(2 n + 1)} \times {\mathbb{Z}}_{2(2 n + 1)}$ $4({\mathbb{Z}}_2)^6 \times ({\mathbb{Z}}_{2(2 n + 1)})^2\oplus 12({\mathbb{Z}}_{2(2 n + 1)})^2$ ${\mathbb{Z}}_4$ – ${\mathbb{Z}}_{4 n} \times {\mathbb{Z}}_{4 n}$ $({\mathbb{Z}}_4)^6 \times ({\mathbb{Z}}_{4 n})^2\oplus 12({\mathbb{Z}}_{4 n})^2 \oplus 3[ ({\mathbb{Z}}_{4 n})^2\times ({\mathbb{Z}}_2)^6]$ ${\mathbb{Z}}_2\times {\mathbb{Z}}_2$ (0, 0) ${\mathbb{Z}}_{2}\times {\mathbb{Z}}_2$ $({\mathbb{Z}}_2)^{18}\oplus 6({\mathbb{Z}}_2)^8 \oplus 3({\mathbb{Z}}_2)^6 \oplus 6({\mathbb{Z}}_2)^4$ ${\mathbb{Z}}_2\times {\mathbb{Z}}_2$ (2, 0) ${\mathbb{Z}}_{2}\times {\mathbb{Z}}_2$ $({\mathbb{Z}}_2)^{18}$ ${\mathbb{Z}}_2\times {\mathbb{Z}}_2$ (2, 2) ${\mathbb{Z}}_{2}\times {\mathbb{Z}}_2$ $({\mathbb{Z}}_2)^{18}$ DownLoad:
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投影对称群(PSG) 规范群$G_g$ 对称群$G_s$ 三维体内($\varSigma^3$)
的规范理论表面($\partial\varSigma^3$)的反常
玻色理论二维平面($\varSigma^2$)的正常Chern-Simons理论的$K_G$-矩阵 ${\mathbb{Z}}_N \rtimes{\mathbb{Z}}^T_2$ ${\mathbb{Z}}_N$ ${\mathbb{Z}}^T_2$ $\dfrac{N}{4{\text{π}}} \epsilon^{\mu\nu\lambda\rho} B^c_{\mu\nu}\partial_\lambda A_\rho^c+$
$\dfrac{\theta_c}{8{\text{π}}^2} \epsilon^{\mu\nu\lambda\rho}\partial_\mu A^c_\nu \partial_\lambda A^c_\rho$${\mathbb{Z}}^T_2$破缺的 $\partial\varSigma^3$:
$\dfrac{N}{2{\text{π}}}\partial_0 \phi^c \epsilon^{ij}\partial_i \lambda^c_j$${\mathbb{Z}}^T_2$破缺的$\varSigma^2$:
$\left(\begin{array}{*{20}{c}} {2 p}&N\\ N&0 \end{array}\right)$${\mathbb{Z}}_N\!\times\!{\mathbb{Z}}^T_2$ ${\mathbb{Z}}_N$ ${\mathbb{Z}}^T_2$ $\dfrac{N}{4{\text{π}}} \epsilon^{\mu\nu\lambda\rho} B^s_{\mu\nu}\partial_\lambda A_\rho^s+$
$\dfrac{\theta_s}{8{\text{π}}^2} \epsilon^{\mu\nu\lambda\rho}\partial_\mu A^s_\nu \partial_\lambda A^s_\rho$${\mathbb{Z}}^T_2$破缺的$\partial\varSigma^3$:
$\dfrac{N}{2{\text{π}}}\partial_0 \phi^s \epsilon^{ij}\partial_i \lambda^s_j$${\mathbb{Z}}^T_2$破缺的$\varSigma^2$:
$ \left({\begin{array}{*{20}{c}} 2 p &N \\ N & 0 \end{array}} \right)$${\mathbb{Z}}_N \!\times\! [U(1)_{S^z}\rtimes{\mathbb{Z}}_2]$ ${\mathbb{Z}}_N\!\times\! U(1)_{S^z}$ ${\mathbb{Z}}_2$ $\dfrac{N}{4{\text{π}}} \epsilon^{\mu\nu\lambda\rho} B^c_{\mu\nu}\partial_\lambda A_\rho^c +$
$\dfrac{\theta_0}{4{\text{π}}^2} \epsilon^{\mu\nu\lambda\rho}\partial_\mu A^s_\nu \partial_\lambda A^c_\rho$${\mathbb{Z}}_2$破缺的$\partial\varSigma^3$:
$\dfrac{N}{2{\text{π}}}\partial_0 \phi^c \epsilon^{ij}\partial_i \lambda^c_j$${\mathbb{Z}}_2$破缺的$\varSigma^2$:
$ \left({\begin{array}{*{20}{c}} 2 p_1 &N & p_{12}& 0\\ N & 0 &0 & 0\\ p_{12} & 0 &2 p_2 & 0\\ 0 & 0 &0 & 0 \end{array}} \right)$$U(1)_C \!\times\! [{\mathbb{Z}}_N \rtimes{\mathbb{Z}}_2]$ $U(1)_C\!\times\!{\mathbb{Z}}_N$ ${\mathbb{Z}}_2$ $\dfrac{N}{4{\text{π}}} \epsilon^{\mu\nu\lambda\rho} B^s_{\mu\nu}\partial_\lambda A_\rho^s+$
$\dfrac{\theta_0}{4{\text{π}}^2} \epsilon^{\mu\nu\lambda\rho}\partial_\mu A^s_\nu \partial_\lambda A^c_\rho$${\mathbb{Z}}_2$破缺的$\partial\varSigma^3$:
$\dfrac{N}{2{\text{π}}}\partial_0 \phi^s \epsilon^{ij}\partial_i \lambda^s_j$${\mathbb{Z}}_2$破缺的$\varSigma^2$:
$ \left({\begin{array}{*{20}{c}} 2 p_1 &0 & p_{12}& 0\\ 0 & 0 &0 & 0\\ p_{12} & 0 &2 p_2 & N\\ 0 & 0 &N & 0 \end{array}} \right)$${\mathbb{Z}}_{N_1} \!\times\! [{\mathbb{Z}}_{N_2}\rtimes{\mathbb{Z}}_2]$ ${\mathbb{Z}}_{N_1}\!\times\! {\mathbb{Z}}_{N_2}$ ${\mathbb{Z}}_2$ $\dfrac{N_1}{4{\text{π}}} \epsilon^{\mu\nu\lambda\rho} B^c_{\mu\nu}\partial_\lambda A_\rho^c+$
$\dfrac{N_2}{4{\text{π}}} \epsilon^{\mu\nu\lambda\rho} B^s_{\mu\nu}\partial_\lambda A_\rho^s+$
$\dfrac{\theta_0}{4{\text{π}}^2} \epsilon^{\mu\nu\lambda\rho}\partial_\mu A^s_\nu \partial_\lambda A^c_\rho$${\mathbb{Z}}_2$破缺的$\partial\varSigma^3$:
$\dfrac{N_1}{2{\text{π}}}\partial_0 \phi^c \epsilon^{ij}\partial_i \lambda^c_j+$
$\dfrac{N_2}{2{\text{π}}}\partial_0 \phi^s \epsilon^{ij}\partial_i \lambda^s_j$${\mathbb{Z}}_2$破缺的$\varSigma^2$:
$\begin{aligned} & {}\\ & \left({\begin{array}{*{20}{c}} 2 p_1 &N_1 & p_{12}& 0\\ N_1 & 0 &0 & 0\\ p_{12} & 0 &2 p_2 & N_2\\ 0 & 0 &N_2 & 0 \end{array}} \right)\end{aligned}$DownLoad:
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轴子角 对称群 三维体内($\varSigma^3$)的响应 二维表面($\partial\varSigma^3$)的反常响应 二维平面($\varSigma^2$)的响应 $ \theta_c=2{\text{π}}+4{\text{π}} k$
(带电玻色系统)$U(1)_C\rtimes{\mathbb{Z}}^{\rm T}_2$ 电荷-威腾效应:
$N^c=n^c+N^c_m$量子电荷霍尔效应
(${\mathbb{Z}}^{\rm T}_2$破缺的$\partial\varSigma^3$):
$\widetilde{\sigma}^{c}=(1+2 k)\dfrac{1}{2{\text{π}}}$量子电荷霍尔效应
(${\mathbb{Z}}^{\rm T}_2$破缺的$\varSigma^2$)
$\sigma^c=2 k\dfrac{1}{2{\text{π}}}$$ \theta_s=2{\text{π}}+4{\text{π}} k$
(整数自旋的
玻色系统)$U(1)_{S^z} \times {\mathbb{Z}}^{\rm T}_2$ 自旋-威腾效应:
$N^s=\displaystyle \sum_i q_in_i^s+N^s_m\sum_{i}q_i^2$量子自旋霍尔效应
(${\mathbb{Z}}^{\rm T}_2$破缺的$\partial\varSigma^3$):
$\widetilde{\sigma}^{s}=(1+2 k)\dfrac{1}{2{\text{π} } }\displaystyle\sum_i{q_i^2}$量子自旋霍尔效应
(${\mathbb{Z}}^{\rm T}_2$破缺的$\varSigma^2$)
$\sigma^s=2 k\dfrac{1}{2{\text{π} } }\displaystyle\sum_i{q_i^2}$$ \theta_0={\text{π}}+2{\text{π}} k$
(带电和整数自旋
的玻色系统)$U(1)_C \!\times\! [U(1)_{S^z} \!\rtimes\! {\mathbb{Z}}_2]$ 交互-威腾效应: $N^c=n^c+\dfrac{1}{2}N^s_m$;
$N^s=n^s_{+}-n^s_{-}+\dfrac{1}{2}N^c_m$量子电荷-自旋/
自旋-电荷效应
(${\mathbb{Z}}_2$破缺的 $\partial\varSigma^3$):
$\widetilde{\sigma}^{cs}=\widetilde{\sigma}^{sc}=\left(\dfrac 1 2+k\right)\dfrac{1}{2{\text{π}}}$量子电荷-自旋/
效应 自旋-电荷
(${\mathbb{Z}}_2$破缺的$\varSigma^2$):
$\sigma^{cs}=\sigma^{sc}=k\dfrac{1}{2{\text{π}}}$DownLoad:
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时空维度 空间对称群$G_s$ 内部对称群$G_i$ 不可约的Wen-Zee拓扑项$S$ 角动量/自旋${\cal{J}}$ $(2 + 1)$维 $SO(2)$ $U(1)$ $\dfrac{k}{2{\text{π}}} \displaystyle\int \omega \wedge {\rm d}A$, $k \in \mathbb{Z}$ $\dfrac{k}{2{\text{π}}} \displaystyle\int_{M^2} {\rm d}A$ $(2 + 1)$维 $C_{N_0}$ $\mathbb{Z}_{N_1}$ $\dfrac{k}{2{\text{π}}} \displaystyle\int \omega \wedge {\rm d}A$, $k \in \mathbb{Z}_{N_{01}}$, $\dfrac{k}{2{\text{π}}} \displaystyle\int_{M^2} {\rm d}A$ $(2 + 1)$维 $C_{N_0}$ $\mathbb{Z}_{N_1} \times \mathbb{Z}_{N_2} $ $k \dfrac{ N_1 N_2}{(2{\text{π}})^2 N_{012}} \displaystyle\int \omega \wedge A^1 \wedge A^2$, $k \in \mathbb{Z}_{N_{012}}$ $k \dfrac{ N_1 N_2}{(2{\text{π}})^2 N_{012}} \displaystyle\int_{M^2} A^1 \wedge A^2$ $(3 + 1)$维 $C_{N_0}$ $\mathbb{Z}_{N_1}$ $k \dfrac{N_0 N_1}{ (2{\text{π}})^2 N_{01}} \displaystyle\int \omega \wedge A \wedge {\rm d}A$, $k \in \mathbb{Z}_{N_{01}}$ $k \dfrac{ N_1}{ (2{\text{π}})^2 N_{01}} \displaystyle\int _{M^3} A \wedge {\rm d}A$ $(3 + 1)$维 $C_{N_0}$ $\mathbb{Z}_{N_1}$ $k \dfrac{N_0 N_1}{(2{\text{π}})^2 N_{01}} \displaystyle\int A \wedge \omega \wedge {\rm d} \omega$, $k \in \mathbb{Z}_{N_{01}}$ $k \dfrac{ N_1}{2{\text{π}}^2 N_{01}} \displaystyle\int_{M^3} A \wedge {\rm d}\omega$ $(3 + 1)$维 $C_{N_0}$ $\mathbb{Z}_{N_1} \times U(1)$ $k \dfrac{N_0 N_1}{(2{\text{π}})^2 N_{01}} \displaystyle\int \omega \wedge A^1 \wedge {\rm d}A^2$, $k \in \mathbb{Z}_{N_{01}}$ $k \dfrac{N_1}{(2{\text{π}})^2 N_{01}} \displaystyle\int_{M^3} A^1 \wedge {\rm d}A^2$ $(3 + 1)$维 $SO(2)$ $\mathbb{Z}_{N_1} \times \mathbb{Z}_{N_2}$ $k \dfrac{N_1 N_2}{(2{\text{π}})^2 N_{12}} \displaystyle\int A^1 \wedge A^2 \wedge {\rm d} \omega$, $k \in \mathbb{Z}_{N_{12}}$ $k \dfrac{N_1 N_2}{(2{\text{π}})^2 N_{12}} \displaystyle\int_{M^3} {\rm d} (A^1 \wedge A^2)$ $(3 + 1)$维 $C_{N_0}$ $\mathbb{Z}_{N_1} \times \mathbb{Z}_{N_2}$ $k \dfrac{N_0 N_1}{(2{\text{π}})^2 N_{01}} \displaystyle\int \omega \wedge A^1 \wedge {\rm d}A^2$, $k \in \mathbb{Z}_{N_{012}}$ $k \dfrac{N_1}{(2{\text{π}})^2 N_{01}} \displaystyle\int_{M^3} A^1 \wedge {\rm d}A^2$ $(3 + 1)$维 $C_{N_0}$ $\mathbb{Z}_{N_1} \times \mathbb{Z}_{N_2}$ $k \dfrac{N_0 N_2}{(2{\text{π}})^2 N_{02}} \displaystyle\int \omega \wedge A^2 \wedge {\rm d}A^1$, $k \in \mathbb{Z}_{N_{012}}$ $k \dfrac{ N_2}{(2{\text{π}})^2 N_{02}} \displaystyle\int_{M^3} A^2 \wedge {\rm d}A^1$ $(3 + 1)$维 $C_{N_0}$ $\mathbb{Z}_{N_1} \times \mathbb{Z}_{N_2} \times \mathbb{Z}_{N_3}$ $k \dfrac{N_0 N_1 N_2 N_3}{(2{\text{π}})^3 N_{0123}} \displaystyle\int \omega \wedge A^1 \wedge A^2 \wedge A^3$,
$k \in \mathbb{Z}_{N_{0123}}$$k \dfrac{ N_1 N_2 N_3}{(2{\text{π}})^3 N_{0123}} \displaystyle\int_{M^3} A^1 \wedge A^2 \wedge A^3$ $(3 + 1)$维($*$) $SO(2)$ $U(1)$ $\dfrac{k}{2{\text{π}}} \displaystyle\int \omega \wedge {\rm d}B$, $k \in \mathbb{Z}$ $\dfrac{k}{2{\text{π}}} \displaystyle\int_{M^3} {\rm d}B$ $(3 + 1)$维($*$) $C_{N_0}$ ${\mathbb{Z}}_{N_1}$ $\dfrac{k}{2{\text{π}}} \displaystyle\int \omega \wedge {\rm d}B$, $k \in \mathbb{Z}_{N_{01}}$ $\dfrac{k}{2{\text{π}}} \displaystyle\int_{M^3} {\rm d}B$ $(3 + 1)$维($*$) $C_{N_0}$ $\mathbb{Z}_{N_1} \times \mathbb{Z}_{N_2}$ $k \dfrac{N_0 N_1 N_2}{(2{\text{π}})^2 N_{012}} \displaystyle\int \omega \wedge A \wedge B$, $k \in \mathbb{Z}_{N_{012}}$ $k \dfrac{N_1 N_2}{(2{\text{π}})^2 N_{012}} \displaystyle\int_{M^3} A \wedge B$ DownLoad:
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