G-Crossed Modularity of Symmetry Enriched Topological Phases

Abstract

The universal properties of $(2+1)$D topological phases of matter enriched by a symmetry group G are described by G-crossed extensions of unitary modular tensor categories (UMTCs). While the fusion and braiding properties of quasiparticles associated with the topological order are described by a UMTC, the G-crossed extensions further capture the properties of the symmetry action, fractionalization, and defects arising from the interplay of the symmetry with the topological order. We describe the relation between the G-crossed UMTC and the topological state spaces on general surfaces that may include symmetry defect branch lines and boundaries that carry topological charge. We define operators in terms of the G-crossed UMTC data that represent the mapping class transformations for such states on a torus with one boundary, and show that these operators provide projective representations of the mapping class groups. This allows us to represent the mapping class group on general surfaces and ensures a consistent description of the corresponding symmetry enriched topological phases on general surfaces. Our analysis also enables us to prove that a faithful G-crossed extension of a UMTC is necessarily G-crossed modular.

Appendices

Appendix A: Review of G-Crossed UBTC

In this appendix, we provide a brief review of G-crossed UBTCs (working with skeletonizations) following Ref. [9], to which we direct the reader for more details and derivations.

The fusion structure of a G-crossed UBTC $\mathcal {B}_{G}^{\times }$ (i.e. ignore the braiding), is described by a G-graded unitary fusion tensor category (UFTC)

$$\begin{aligned} \mathcal {B}_{G} = \bigoplus _{\textbf{g} \in G} \mathcal {B}_\textbf{g} . \end{aligned}$$

(A1)

This is a UFTC for which the fusion rules are additionally required to be G-graded. In more detail, each topological charge (simple object) is assigned a particular element of the symmetry group G and the fusion rules must respect the group multiplication of G. We denote the identity element of G as $\textbf{0}$ and inverses as $\bar{\textbf{g}} =\textbf{g}^{-1}$ Topological charges assigned the group element $\textbf{g}$ correspond to the distinct types of $\textbf{g}$-defects, and we recognize $\textbf{0}$-defects as the quasiparticles of the topological phase. In this way, we write topological charges corresponding to $\textbf{g}$-defects as $a_\textbf{g} \in \mathcal {B}_\textbf{g}$³ and the associative fusion algebra takes the form

$$\begin{aligned} a_\mathbf{{g}} \otimes b_\mathbf{{h}} = \sum _{c \in \mathcal {B}_\mathbf{{gh}}} N_{a_\textbf{g} b_\textbf{h}}^{c_\textbf{gh}} c_\mathbf{{gh}}. \end{aligned}$$

(A2)

The fusion multiplicities $N_{a b}^c$ are non-negative integers indicating how topological charges can combine or split. We require a unique vacuum charge, which we denote as 0 for which $N_{a 0}^{c} = N_{0 a}^{c} \delta _{a c}$. For each charge $a_\mathbf{{g}}$, we require a unique conjugate charge $\overline{a_\mathbf{{g}}} \in \mathcal {B}_{\bar{\textbf{g}}}$ for which $N_{a b}^{0} = \delta _{b \bar{a}}$.

Each fusion product has an associated fusion and splitting vector space, for which we write the basis states as

(A3)

(A4)

where $\text {dim} V_{ab}^{c} = \text {dim} V^{ab}_{c} = N_{ab}^{c}$, and $\mu = 1,\ldots , N_{ab}^{c}$. The normalization factors in translating between diagrammatic and bra/ket notation are given in terms of the quantum dimensions $d_{a}$ of the respective charges, which are chosen to make bending lines unitary transformations. More general states and operators can be formed diagrammatically by stacking together trivalent vertices such that lines glued together have the same topological charge.

Diagrams can be reduced using the inner product

(A5)

which relates fusion and splitting spaces as duals. We require the inner product to be Hermitian, which requires that we choose the quantum dimensions to be equal to the Frobenius-Perron dimensions, i.e. $d_{a}$ equals the largest (positive) eigenvalue of the matrix $\textbf{N}_{a}$ defined by $[\textbf{N}_{a}]_{bc} = N_{a b}^c$. This Hermitian inner product provides the fusion category with a pivotal structure, i.e. it enables us to bend the the lines, with appropriate unitary transformations. It also makes it spherical, i.e. $d_a = d_{\bar{a}}$. The quantum dimensions obey the relation

$$\begin{aligned} d_{a_\mathbf{{g}}} d_{b_\mathbf{{h}}} = \sum _{c_\textbf{gh}} N_{a b}^c d_{c_\mathbf{{gh}}}. \end{aligned}$$

(A6)

We also define the total quantum dimension of a $\textbf{g}$-sector $\mathcal {B}_\textbf{g}$ to be

$$\begin{aligned} \mathcal {D}_\mathbf{{g}} = \sqrt{\sum _{a_\mathbf{{g}}} d_{a_\mathbf{{g}}}^2}. \end{aligned}$$

(A7)

For any $\mathbf{{g}} \in G$ with nonempty $\mathcal {B}_\mathbf{{g}} \ne \emptyset$, we have $\mathcal {D}_\mathbf{{g}} = \mathcal {D}_\mathbf{{0}}$.

The inner product allows us to write the partition of identity for a pair of charges $a_\textbf{g}$ and $b_\textbf{h}$ as

(A8)

The notion of associativity of fusion on the state space is encoded in the F-moves

(A9)

These may be viewed as changes of bases, and are required to satisfy consistency conditions known as the pentagon equations. We make a canonical gauge choice such that $F^{abc}_{d} = \mathbb {1}$ whenever any of a, b, c is the vacuum charge 0. For a unitary FTC, the F-moves are required to be unitary transformations, i.e. the F-symbols satisfy

$$\begin{aligned} \left[ \left( F_{d}^{abc}\right) ^{-1}\right] _{\left( f,\mu ,\nu \right) \left( e,\alpha ,\beta \right) }= & {} \left[ \left( F_{d}^{abc}\right) ^{\dagger }\right] _{\left( f,\mu ,\nu \right) \left( e,\alpha ,\beta \right) } = \left[ F_{d}^{abc}\right] _{\left( e,\alpha ,\beta \right) \left( f,\mu ,\nu \right) }^{*}. \end{aligned}$$

(A10)

Line bending can be related to F-moves through the relation

(A11)

from which we see that

$$\begin{aligned}{}[F^{a \bar{a} a}_a]_{00} = \frac{\varkappa _{a}}{d_a}. \end{aligned}$$

(A12)

Here, $\varkappa _{a} = \varkappa _{\bar{a}}^{-1}$ is a phase, not necessarily equal to one. When $a=\bar{a}$, $\varkappa _{a} = \pm 1$ is a gauge invariant quantity known as the Frobenius-Shur indicator.

G-crossed braiding is a generalization of regular braiding that incorporates symmetry action and fractionalization of the group G. Unlike regular braiding, the topological charges of the braided objects need not remain fixed, and compatibility of braiding with fusion does not simply require that sliding lines over or under fusion vertices be trivial. In order to incorporate G-crossed braiding, we select a group action $\rho : G \mapsto \text {Aut}(\mathcal {B}_{G}^{\times })$, where the details of what it means to be an element of $\text {Aut}(\mathcal {B}_{G}^{\times })$ can be imposed as additional consistency conditions, which we will state below. We write a shorthand for the symmetry action on topological charges as

$$\begin{aligned} \,^\textbf{k}a_\textbf{g} = \rho _\textbf{k}(a_\textbf{g}) \in \mathcal {B}_{\textbf{kg}\bar{\textbf{k}}}. \end{aligned}$$

(A13)

At the level of fusion rules, these permutations of topological charge must satisfy

$$\begin{aligned} N_{ \,^\textbf{k}a_\textbf{g} \,^\textbf{k}b_\textbf{h}}^{\,^\textbf{k}c_\textbf{gh}} = N_{a_\textbf{g} b_\textbf{h}}^{c_\textbf{gh}}, \end{aligned}$$

(A14)

which also implies

$$\begin{aligned} d_{a_\textbf{g}} = d_{\,^\textbf{k}a_\textbf{g}}. \end{aligned}$$

(A15)

The basic data satisfies consistency conditions that allows us to interpret the U-symbols as corresponding to the symmetry action and the $\eta$-symbols as corresponding to symmetry fractionalization. Compatibility of the F-moves with sliding lines over and under fusion vertices yields

$$\begin{aligned}{} & {} \sum _{\alpha ',\beta ',\mu '\nu '} \left[ U_\textbf{k}(\,^\textbf{k}a,\,^\textbf{k}b;\,^\textbf{k}e )\right] _{\alpha \alpha '} \left[ U_\textbf{k}(\,^\textbf{k}e, \,^\textbf{k}c;\,^\textbf{k}d)\right] _{\beta \beta '} \left[ F_{\,^\textbf{k}d}^{ \,^\textbf{k}a \,^\textbf{k}b \,^\textbf{k}c}\right] _{(^\textbf{k}e,\alpha ',\beta ')(^\textbf{k}f,\mu ',\nu ')} \nonumber \\{} & {} \qquad \times \left[ U_\textbf{k}(\,^\textbf{k}b,\,^\textbf{k}c;\,^\textbf{k}f)^{-1}\right] _{\mu ' \mu } \left[ U_\textbf{k}(\,^\textbf{k}a,\,^\textbf{k}f;\,^\textbf{k}d)^{-1}\right] _{\nu ' \nu } = \left[ F_{d}^{abc}\right] _{(e,\alpha ,\beta )(f,\mu ,\nu )},\nonumber \\{} & {} \end{aligned}$$

(A16)

and

$$\begin{aligned} \eta _{^{\bar{\textbf{g}}}x}\left( \textbf{h}, \textbf{k} \right) \eta _{x}\left( \textbf{g}, \textbf{hk} \right) = \eta _{x}\left( \textbf{g}, \textbf{h} \right) \eta _{x}\left( \textbf{gh}, \textbf{k} \right) . \end{aligned}$$

(A17)

Compatibility of the R-moves with sliding lines over and under fusion vertices yields the G-crossed Yang-Baxter equation

(A18)

and

$$\begin{aligned}&\sum _{\mu ',\nu '} \left[ U_\textbf{k}(\,^\textbf{k}b_\textbf{h} , \,^{\textbf{k} \bar{\textbf{h}}}a_\textbf{g} ; \,^\textbf{k} c_\textbf{gh} ) \right] _{\mu \mu '} \left[ R^{ \,^\textbf{k}a_\textbf{g} \,^\textbf{k}b_\textbf{h}}_{\,^\textbf{k}c_\textbf{gh}} \right] _{\mu ' \nu '} \left[ U_\textbf{k} (\,^\textbf{k}a_\textbf{g} , \,^\textbf{k}b_\textbf{h} ; \,^\textbf{k}c_\textbf{gh} )^{-1} \right] _{\nu ' \nu } \nonumber \\&\qquad \qquad \qquad \qquad \qquad = \frac{\eta _{\,^\textbf{k}a_\textbf{g}}(\textbf{k},\textbf{h} )}{\eta _{\,^\textbf{k}a_\textbf{g}}({\textbf{kh}\bar{\textbf{k}}},\textbf{k})} \left[ R^{a_\textbf{g} b_\textbf{h}}_{c_\textbf{gh}} \right] _{\mu \nu }. \end{aligned}$$

(A19)

Finally, compatibility of sliding two vertices over and under each other yields

$$\begin{aligned}&\sum _{\alpha , \beta } \left[ U_\textbf{k}\left( a ,b ;c \right) ^{-1} \right] _{\mu \alpha } \left[ U_\textbf{l}\left( \,^{\bar{\textbf{k}}}a ,\,^{\bar{\textbf{k}}}b ;\,^{\bar{\textbf{k}}}c \right) ^{-1} \right] _{\alpha \beta } \left[ U_\textbf{kl}\left( a ,b ;c \right) \right] _{\beta \nu } \nonumber \\&\qquad \qquad \qquad \qquad = \frac{ \eta _{a}\left( \textbf{k}, \textbf{l} \right) \eta _{b}\left( \textbf{k}, \textbf{l} \right) }{\eta _{c}\left( \textbf{k}, \textbf{l} \right) } \delta _{\mu \nu }. \end{aligned}$$

(A20)

BTCs have important gauge invariant quantities known as the topological S-matrix and the topological twists. For G-crossed BTCs, the similarly defined quantities are no longer gauge invariant (in the case of the defects), but they remain important. The topological twists are phases defined by

(A21)

The topological S-matrix is defined as

(A22)

One should note that the quanitity above is only well-defined if $\mathbf{gh = hg }$, otherwise the loops will not be able to close back upon themselves, as the topological charge values would change upon braiding. In the third equality, we have used the G-crossed ribbon identity

$$\begin{aligned} \sum _\lambda \left[ R^{b_\textbf{h} \,^{\bar{\textbf{h}}}a_\textbf{g}}_{c_\textbf{gh}}\right] _{\mu \lambda } \left[ R^{a_\textbf{g} b_\textbf{h}}_{c_\textbf{gh}} \right] _{\lambda \nu } = \frac{\theta _{c_\textbf{gh}}}{\theta _{a_\textbf{g}}\theta _{b_\textbf{h}}} \frac{\left[ U_\textbf{gh}(a_\textbf{g},b_\textbf{h};c_\textbf{gh})\right] _{\mu \nu } }{\eta _{a_\textbf{g}}(\textbf{g,h}) \eta _{b_\textbf{h}}(\textbf{h},\,^{\bar{\textbf{h}}}{} \textbf{g})} . \end{aligned}$$

(A23)

Additionally, we define the “punctured torus” $S^{(z)}$-matrix (corresponding to a torus with boundary carrying topological charge z) as

(A24)

In this case, $\textbf{g}$ and $\textbf{h}$ are not required to commute. It is straightforward to see that $S^{(0)} = S$.

The definition of the S-matrix gives the property

(A25)

Applying this twice gives the relation

$$\begin{aligned} \frac{S_{x_\textbf{k} a_\textbf{g}}}{S_{x_\textbf{k} 0}} \frac{S_{x_\textbf{k} b_\textbf{h}}}{S_{x_\textbf{k} 0}} = \sum _{c \in \mathcal {B}_\textbf{gh}^\textbf{k} , \mu } \frac{[U_{\bar{\textbf{k}}}(a_\textbf{g}, b_\textbf{h} ; c_\textbf{gh})]_{\mu \mu }}{\eta _{\overline{x_\textbf{k}} }(\textbf{g}, \textbf{h})} \frac{S_{x_\textbf{k} c_\textbf{gh}}}{S_{x_\textbf{k} 0}} , \end{aligned}$$

(A26)

when $^\textbf{k}a_\textbf{g} = a_\textbf{g}$, $^\textbf{k}b_\textbf{h} = b_\textbf{h}$, and $^\textbf{g}x_\textbf{k} = ^\textbf{h}x_\textbf{k} = x_\textbf{k}$.

We will end this review section with a collection of additional derived properties that will be useful for our calculations.

Equation (A16) with $e=f=0$ yields the relation

$$\begin{aligned} \frac{\varkappa _{\,^\textbf{k}a} }{ \varkappa _{a} } =\frac{ \left[ F_{\,^\textbf{k}a}^{ \,^\textbf{k}a \,^\textbf{k}\bar{a} \,^\textbf{k}a}\right] _{00} }{\left[ F_a^{a\bar{a}a}\right] _{00}} =\frac{U_\textbf{k}(\,^\textbf{k}\bar{a},\,^\textbf{k}a ;0)}{ U_\textbf{k}(\,^\textbf{k}a, \,^\textbf{k}\bar{a};0) }. \end{aligned}$$

(A27)

When $a = \bar{a}$, it follows from Eq. (A27) that $\varkappa _{a}=\varkappa _{\,^\textbf{k}a}$. When $^\textbf{k}a = a$ is $\textbf{k}$-invariant, it follows from Eq. (A27) that

$$\begin{aligned} U_\textbf{k}(a,\bar{a};0) = U_\textbf{k}(\bar{a},a;0). \end{aligned}$$

(A28)

Using Eq. (A18) with the definition of the twist, we find the general relation between $\theta _{a}$ and $\theta _{\,^\textbf{k}a}$ is

$$\begin{aligned} \theta _{a_\textbf{g}} = \frac{\eta _{\,^\textbf{k}a_\textbf{g}}({\textbf{kg} \bar{\textbf{k}} , \textbf{k}}) }{\eta _{\,^\textbf{k}a_\textbf{g}}(\textbf{k, g})} \theta _{\,^\textbf{k}a_\textbf{g}} = \frac{\eta _{a_\textbf{g}}({\bar{\textbf{k}} , \textbf{kg}\bar{\textbf{k}} }) }{\eta _{a_\textbf{g}}({\textbf{g}, \bar{\textbf{k}}})} \theta _{\,^\textbf{k}a_\textbf{g}}. \end{aligned}$$

(A29)

When $^\textbf{k}a = a$, it follows that

$$\begin{aligned} \eta _{a_\textbf{g}}(\textbf{g, k}) = \eta _{a_\textbf{g}}(\textbf{k, g}). \end{aligned}$$

(A30)

We also note that Eq. (A17) gives $\eta _{^\textbf{k}x} (\textbf{k},{\bar{\textbf{k}} }) = \eta _{x} ({\bar{\textbf{k}} }, \textbf{k})$ for any x and $\textbf{k}$, so we also have

$$\begin{aligned} \eta _{a_\textbf{g}}({\textbf{k},\bar{\textbf{k}}}) = \eta _{a_\textbf{g}}({\bar{\textbf{k}},\textbf{k}}) \end{aligned}$$

(A31)

when $^\textbf{k}a = a$.

The topological S-matrix for defects can also transform nontrivially under symmetry action, as it satisfies the relation

$$\begin{aligned} S_{\,^\textbf{k}a_\textbf{g} \,^\textbf{k} b_\textbf{h} }= \frac{\eta _{\,^\textbf{k}\bar{a}}(\textbf{k,h}) \eta _{\,^\textbf{k}b}({\textbf{k} , \bar{\textbf{g}} })}{\eta _{\,^\textbf{k}\bar{a}}({\textbf{kh}\bar{\textbf{k}},\textbf{k}}) \eta _{\,^\textbf{k}b}({ \textbf{k}\bar{\textbf{g}}\bar{\textbf{k}} , \textbf{k} }) }S_{a_\textbf{g} b_\textbf{h} }. \end{aligned}$$

(A32)

Unlike a BTC, it is not necessarily the case that $\theta _{a_\textbf{g}}$ and $\theta _{\overline{a_\textbf{g}}}$ are equal in a G-crossed BTC. In particular, we have

$$\begin{aligned} \theta _{a_\textbf{g}}= U_\textbf{g}(\overline{a_\textbf{g}},a_\textbf{g};0) \eta _{\overline{a_\textbf{g}}}({\bar{\textbf{g}}},{\textbf{g}}) \theta _{\overline{a_\textbf{g}}}. \end{aligned}$$

(A33)

and

$$\begin{aligned} \theta _{a_\textbf{g}}= & {} U_\textbf{g}(a_\textbf{g},\overline{a_\textbf{g}};0) \varkappa _{a_\textbf{g}} \left( R^{ \overline{a_\textbf{g}} a_\textbf{g} }_{0}\right) ^{-1} = \eta _{a_\textbf{g}}(\textbf{g}, \bar{\textbf{g}})^{-1} \varkappa _{a_\textbf{g}}^{-1} \left( R^{a_\textbf{g} \overline{a_\textbf{g}} }_{0}\right) ^{-1} \end{aligned}$$

(A34)

Equation (A20) when $\textbf{l}={\bar{\textbf{k}}}$ gives

$$\begin{aligned}{}[U_{\bar{\textbf{k}}}(^{\bar{\textbf{k}}}a, ^{\bar{\textbf{k}}}b; ^{\bar{\textbf{k}}}c)]_{\mu \nu } = \frac{\eta _c({\textbf{k}, \bar{\textbf{k}}})}{\eta _a({\textbf{k}, \bar{\textbf{k}}}) \eta _b({\textbf{k}, \bar{\textbf{k}}})} [U_\textbf{k}(a, b; c)^{-1}]_{\mu \nu } \end{aligned}$$

(A35)

and when $c=0$ gives

$$\begin{aligned} \eta _{a}(\textbf{k, l})\eta _{\bar{a}}(\textbf{k,l}) = \frac{U_\textbf{kl}(a, \bar{a};0)}{U_\textbf{k}(a, \bar{a};0) U_\textbf{l}(^{\bar{\textbf{k}}}a, ^{\bar{\textbf{k}}}\bar{a};0)}. \end{aligned}$$

(A36)

Equations (A22), (A6), and (A33) yield the property

(A37)

Appendix B: Sequences of Relations for Sect. 6

See Figs. 10, 11, 12 and 13.

Fig. 10

Diagrammatic steps used in deriving Eq. (46)

Fig. 11

Diagrammatic steps used in deriving Eq. (48)

Fig. 12

Diagrammatic steps used in deriving Eq. (49)

Fig. 13

Diagrammatic steps used in deriving Eq. (51)