Non-Abelian string and particle braiding in topological order: Modular $\mathrm{SL}(3,\mathbb{Z})$ representation and $(3+1)$-dimensional twisted gauge theory

String and particle braiding statistics are examined in a class of topological orders described by discrete gauge theories with a gauge group $G$ and a 4-cocycle twist ${\omega }_4$ of $G$'s cohomology group ${\mathcal{H}}^4(G,\mathbb{R}/\mathbb{Z})$ in three-dimensional space and one-dimensional time $(3+1\mathrm{D})$. We establish the topological spin and the spin-statistics relation for the closed strings and their multistring braiding statistics. The $3+1\mathrm{D}$ twisted gauge theory can be characterized by a representation of a modular transformation group, $\mathrm{SL}(3,\mathbb{Z})$. We express the $\mathrm{SL}(3,\mathbb{Z})$ generators ${\mathsf{S}}^{xyz}$ and ${\mathsf{T}}^{xy}$ in terms of the gauge group $G$ and the 4-cocycle ${\omega }_4$. As we compactify one of the spatial directions $z$ into a compact circle with a gauge flux $b$ inserted, we can use the generators ${\mathsf{S}}^{xy}$ and ${\mathsf{T}}^{xy}$ of an $\mathrm{SL}(2,\mathbb{Z})$ subgroup to study the dimensional reduction of the 3D topological order ${\mathcal{C}}^{3\mathrm{D}}$ to a direct sum of degenerate states of 2D topological orders ${\mathcal{C}}_b^{2\mathrm{D}}$ in different flux $b$ sectors: ${\mathcal{C}}^{3\mathrm{D}}={\oplus }_b{\mathcal{C}}_b^{2\mathrm{D}}$. The 2D topological orders ${\mathcal{C}}_b^{2\mathrm{D}}$ are described by 2D gauge theories of the group $G$ twisted by the 3-cocycle ${\omega }_{3\left(b\right)}$, dimensionally reduced from the 4-cocycle ${\omega }_4$. We show that the $\mathrm{SL}(2,\mathbb{Z})$ generators, ${\mathsf{S}}^{xy}$ and ${\mathsf{T}}^{xy}$, fully encode a particular type of three-string braiding statistics with a pattern that is the connected sum of two Hopf links. With certain 4-cocycle twists, we discover that, by threading a third string through two-string unlink into a three-string Hopf-link configuration, Abelian two-string braiding statistics is promoted to non-Abelian three-string braiding statistics.

Figure 1

The 3D topological order ${\mathcal{C}}^{3\mathrm{D}}$ can be regarded as the direct sum of 2D topological orders ${\mathcal{C}}_b^{2\mathrm{D}}$ in different sectors $b$, as ${\mathcal{C}}^{3\mathrm{D}}={\oplus }_b{\mathcal{C}}_b^{2\mathrm{D}}$, when we compactify a spatial direction $z$ into a circle. This idea is general and applicable to ${\mathcal{C}}^{3\mathrm{D}}$ without a gauge theory description. However, when ${\mathcal{C}}^{3\mathrm{D}}$ allows a gauge group $G$ description, $b$ stands for a group element (or the conjugacy class for the non-Abelian group) of $G$. Thus $b$ acts as a gauge flux along the dashed arrow in the compact direction $z$. Thus, ${\mathcal{C}}^{3\mathrm{D}}$ becomes the direct sum of different ${\mathcal{C}}_b^{2\mathrm{D}}$ values under distinct gauge fluxes $b$.

Figure 2

Mutual braiding statistics following the path $1\to 2\to 3\to 4$ along time evolution (see Sec. 3c2). (a) From a 2D viewpoint of dimensional reduced ${\mathcal{C}}_b^{2\mathrm{D}}$, the $2\pi$ braiding of two particles is shown. (b) The compact $z$ direction extends two particles to two closed (red, blue) strings. (c) An equivalent 3D view; the $b$ flux (along the dashed arrow) is regarded as the monodromy caused by the third (black) string. We identify the coordinates $x$ and $y$ and a compact $z$ to see that the full-braiding process is one (red) string going inside the loop of another (blue) string and then going back from the outside. For Abelian topological orders, the mutual braiding process between two excitations (A and B) in (a) yields a statistical Abelian phase $e^{\mathrm{i}{\theta }_{\left(\mathrm{A}\right)\left(\mathrm{B}\right)}}\propto {\mathsf{S}}_{\left(\mathrm{A}\right)\left(\mathrm{B}\right)}^{xy}$ proportional to the 2D ${\mathsf{S}}^{xy}$ matrix. The dimensional-extended equivalent picture (c) implies that the loop braiding yields a phase $e^{\mathrm{i}{\theta }_{\left(\mathrm{A}\right)\left(\mathrm{B}\right),b}}\propto {\mathsf{S}}_{b\left(\mathrm{A}\right)\left(\mathrm{B}\right)}^{xy}$ of Eq. (34) (up to a choice of canonical basis), where $b$ is the flux of the black string. We clarify that in both (b) and (c) our strings may carry both flux and charge. If a string carries only a pure charge, then it is effectively a point particle in three dimensions. If a string carries a pure flux, then it is effectively a loop of a pure string in three dimensions. If a string carries both charge and flux (as a dyon in two dimensions), then it is a loop with string fluxes attached to some charged particles in three dimensions. Therefore the string-string braiding in (c) actually represents several braiding processes—particle-particle, particle-loop, and loop-loop braidings; all processes are threaded with a background (black) string.

Figure 3

Illustration for ${\mathsf{O}}_{\left(\mathrm{A}\right)\left(\mathrm{B}\right)}=\langle {\Psi }_{\mathrm{A}}|\widehat{\mathsf{O}}|{\Psi }_{\mathrm{B}}\rangle$. Evolution from an initial-state configuration $|{\Psi }_{\mathrm{in}}\rangle$ on the spatial manifold (from the top) along the time direction (dashed line) to the final state $|{\Psi }_{\mathrm{out}}\rangle$ (at the bottom). For the spatial ${\mathbb{T}}^d$ torus, the mapping class group $\mathrm{MCG}\left({\mathbb{T}}^d\right)$ is the modular $\mathrm{SL}(d,\mathbb{Z})$ transformation. We show schematically the time evolution on the spatial ${\mathbb{T}}^2$ and ${\mathbb{T}}^3$. The ${\mathbb{T}}^3$ is shown as a ${\mathbb{T}}^2$ attached a $S^1$ circle on each point.

Figure 4

Combining the reasoning in Eq. (18) and Fig. 1, we obtain the physical meaning of dimensional reduction from a 3 + 1D twisted gauge theory as a 3D topological order to several sectors of 2D topological orders: ${\mathcal{C}}_{G,{\omega }_4}^{3\mathrm{D}}={\oplus }_b{\mathcal{C}}_{G,{\omega }_{3\left(b\right)}}^{2\mathrm{D}}$. Here $b$ stands for the gauge flux (Wilson line operator) of gauge group $G$. Here ${\omega }_3$ are dimensionally reduced 3-cocycles from 4-cocycles ${\omega }_4$. Note that there is a zero-flux $b=0$ sector with ${\mathcal{C}}_{G,\left(\mathrm{untwist}\right)}^{2\mathrm{D}}={\mathcal{C}}_G^{2\mathrm{D}}$.

Figure 5

Both process (a) and process (b) start from the creation of a pair of particle $q$ and antiparticle $\overline{q}$, but the worldlines evolve along time to the bottom differently. Process (a) produces a phase $e^{\mathrm{i}2\pi s}$ due to $2\pi$ rotation of $q$, with spin $s$. Process (b) produces a phase $e^{\mathrm{i}\Theta }$ due to the exchange statistics. The homotopic equivalence by deformation implies $e^{\mathrm{i}2\pi s}=e^{\mathrm{i}\Theta }$.

Figure 6

Topological spin of (a) a particle by $2\pi$-self rotation in two dimensions; (b) a framed closed string by $2\pi$-self rotation in three dimensions with a compact $z$; (c) a closed string [(blue) loop] by $2\pi$-self flipping, threaded by a background (black) string creating monodromy $b$ flux (along the dashed arrow), under a single Hopf link $2_1^2$ configuration. All above equivalent pictures describe the physics of topological spin in terms of ${\mathsf{T}}_b^{xy}$. For Abelian topological orders, the spin of an excitation (say A) in (a) yields an Abelian phase $e^{\mathrm{i}{\Theta }_{\left(\mathrm{A}\right)}}={\mathsf{T}}_{\left(\mathrm{A}\right)\left(\mathrm{A}\right)}^{xy}$ proportional to the diagonal of the 2D ${\mathsf{T}}^{xy}$ matrix. The dimensional-extended equivalent picture (c) implies that the loop-flipping yields a phase $e^{\mathrm{i}{\Theta }_{\left(\mathrm{A}\right),b}}={\mathsf{T}}_{b\left(\mathrm{A}\right)\left(\mathrm{A}\right)}^{xy}$ of Eq. (31) (up to a choice of canonical basis), where $b$ is the flux of the black string.

Figure 7

Exchange statistics of (a) two identical particles at positions 1 and 2 by a $\pi$ winding (half-winding), (b) two identical strings by a $\pi$ winding in three dimensions with a compact $z$, and (c) two identical closed strings [(blue) loops] with a $\pi$-winding around, both threaded by a background (black) string creating monodromy $b$ flux, under the Hopf links $2_1^2\#2_1^2$ configuration. (a)–(c) describe the equivalent physics in three dimensions with a compact $z$ direction. The physics of exchange statistics of a closed string turns out to be related to the topological spin in Fig. 6, discussed in Sec. 3c3.

Figure 8

Number of particle types = GSD on $S^{d-1}\times S^1$.

Figure 9

For the 3 + 1D type IV ${\omega }_{4,\mathrm{IV}}$ twisted gauge theory ${\mathcal{C}}_{G,{\omega }_{4,\mathrm{IV}}}^{3\mathrm{D}}$: (a) the two-string statistics in the unlinked $0_1^2$ configuration is Abelian (the $b=0$ sector as ${\mathcal{C}}_G^{2\mathrm{D}}$); and (b) the three-string statistics in the two Hopf links $2_1^2\#2_1^2$ configuration is non-Abelian (the $b\ne 0$ sector in ${\mathcal{C}}_b^{2\mathrm{D}}={\mathcal{C}}_{G,{\omega }_{3,\mathrm{III}}}^{2\mathrm{D}}$). The $b\ne 0$ flux sector creates a monodromy effectively acting as the third (black) string threading the two (red and blue) strings.

Figure 10

Under Alexander-Briggs notation, an unknot is $0_1$, and two unknots can form an unlink $0_1^2$. A Hopf link is $2_1^2$, and the connected sum of two Hopf links is $2_1^2\#2_1^2$.

Figure 11

The trefoil knot is $3_1$. Some other simplest three-string links (beyond Hopf links $2_1^2\#2_1^2$) are shown: $6_1^3$, $6_2^3$ (Borromean rings), $6_3^3$. From the spin-statistics relation of a closed string discussed in Sec. 3c, where the topological spin of certain knot/link configurations ($0_1$ for the monodromy flux $b=0$ and $2_1^2$ for $b\ne 0$) is equivalent to the exchange statistics of certain knot/link configurations ($0_1^2$ for $b=0$ and $2_1^2\#2_1^2$ for $b\ne 0$) under Eq. (35). Therefore, we may further conjecture that the topological spin of a trefoil knot $3_1$ may relate to the braiding statistics of $6_1^3$, $6_2^3$, $6_3^3$.

Figure 12

spacetime complex ${\mathbb{T}}^3\times I$, where $I=[0,1]$ is the time direction. ${\mathbb{T}}^3\times \left\{0\right\}$ and ${\mathbb{T}}^3\times \left\{1\right\}$ are shown. Gray (blue) lines illustrate how the two ${\mathbb{T}}^3$'s are connected for $t\in (0,1)$. Note that the two ${\mathbb{T}}^3$'s differ by a rotation ${\mathsf{S}}^{xyz}$. In other words, when time forms a loop, the two ${\mathbb{T}}^3$'s are glued together as $1\to 1^{\prime }$, $2\to 2^{\prime }$, $3\to 3^{\prime }$, $4\to 4^{\prime }$, $5\to 5^{\prime }$, $6\to 6^{\prime }$, $7\to 7^{\prime }$, and $8\to 8^{\prime }$.

Figure 13

Complex $M_1$.

Figure 14

Complex $M_2$.

Figure 15

Complex $M_3$.

Figure 16

Complex $M_4$.

Figure 17

Complex $M_1^{\prime }$.

Figure 18

Complex $M_1^{″}$, which is formed by one 4-simplex. Note that all the vertices in (a) are in the same time slice, but (curved) edge $\left(2^{\prime }{7}^{\prime }\right)$ is in an earlier time slice and (curved) edge $\left(3^{\prime }{6}^{\prime }\right)$ is in a later time slice. To realize this using straight edges, we put the vertex $6^{\prime }$ in a later time slice, and this gives us the 4-simplex in (b).

Figure 19

Complex $M_1^{\prime }$ is formed by eight 4-simplexes.

Figure 20

Complex $M_2^{\prime }$, which is formed by eight 4-simplexes.

Figure 21

Complex $M_2^{″}$, which is formed by one 4-simplex. Note that all the vertices in (a) are in the same time slice, but (curved) edge $\left(2^{\prime }{7}^{\prime }\right)$ is in an earlier time slice and (curved) edge $\left(3^{\prime }{6}^{\prime }\right)$ is in a later time slice. To realize this using straight edges, we put vertex $6^{\prime }$ in a later time slice, and this gives us the 4-simplex in (b).