Noninvertible anomalies and mapping-class-group transformation of anomalous partition functions

Recently, it was realized that anomalies can be completely classified by topological orders, symmetry protected topological orders, and symmetry enriched topological orders in one higher dimension. The anomalies that people used to study are invertible anomalies that correspond to invertible topological orders and/or symmetry protected topological orders in one higher dimension. In this paper, we introduce a notion of noninvertible anomaly, which describes the boundary of generic topological order. It is characterized by two features. First, a theory with noninvertible anomaly has a multicomponent partition function. Second, under the mapping class group transformation of space-time, the vector of partition functions transform covariantly. In fact, the anomalous partition functions transform in the same way as the degenerate ground states of the corresponding topological order in one higher dimension. This general theory of noninvertible anomaly may have wide applications. As an example, we show that the irreducible gapless boundary of 2+1D double-semion topological order must have central charge $c=\overline{c}>\frac{25}{28}$.

Figure 1

(a) A particular time $t$ evolution produces a particular ground state in the degenerate ground-state subspace on the space $S^1\times S^1$. (b) A particular extension of a space-time $S^1\times S^1$ as the boundary of a bulk $D^2\times S^1$ produces a particular anomalous partition function in the vector space of partition functions on space-time $S^1\times S^1$ (i.e., the boundary).

Figure 2

(a) Space-time $D^2\times S^1$ (solid cylinder). (b) $I\times S^1\times S^1$ (cylinder) and $D^2\times S^1$ (solid cylinder). (c) Gluing the cylinder with solid cylinder, along the $S^1\times S^1=T^2$ boundary, reproduces the space-time $D^2\times S^1$. The tensor networks on the solid cylinder and the cylinder define the path integral. The tensors on the inner solid cylinder are the bulk tensors that describe a topological path integral. The tensors on the outer cylinder can be anything, which may describe a gapless CFT at long distance. Different choices of boundary tensor network on the outer cylinder give rise to different types of boundaries.

Figure 3

The space-time $D^2\times S^1$ with a world line of type-$i$ topological excitation, wrapping in the $S^1$ direction. The path integral on the inner solid cylinder is a topological path integral with world line, as described in Appendix pp3-s5.

Figure 4

The lattice $\mathrm{\Gamma }$ formed by points $(a,b)$. Each point corresponds to a U(1) vertex operator with scaling dimension $(h,\overline{h})=(\frac{1}{2}{a}^2,\frac{1}{2}{b}^2)$. The “$\times$” points give rise to $|{\chi }_0^{u{1}_4}{|}^2$. The “$\circ$” points give rise to $|{\chi }_{\mathbf{1}}^{u{1}_4}{|}^2$. The “+” points give rise to $|{\chi }_2^{u{1}_4}{|}^2$. The “$\diamond$” points give rise to $|{\chi }_3^{u{1}_4}{|}^2$. We also mark the directions of the $l$ and $m$ labels. The shaded points carry the $Z_2$ charge $l+m=1$ mod 2, and the unshaded points carry the $Z_2$ charge $l+m=0$ mod 2.

Figure 5

The modular transformations on the partition functions $Z_{M_n},n=\mathbf{1},2,3$, for a gapless boundary of a 2+1D $Z_2$ topological order. For example, the two red lines to the right represent the following $T$ transformations: $M_2\to M_3:Z_{M_2}(\tau +1,\overline{\tau }+1)=Z_{M_3}(\tau ,\overline{\tau })$ and $M_3\to M_2:Z_{M_3}(\tau +1,\overline{\tau }+1)=Z_{M_2}(\tau ,\overline{\tau })$. The blue lines represent the $S$ transformations. The pattern of the transformations characterizes an 1+1D noninvertible gravitational anomaly described by 2+1D $Z_2$ topological order.

Figure 6

The modular transformations on the partition functions $Z_n,n=1,2,\cdots ,6$, for a gapless boundary of 2+1D DS topological order. For example, the red lines in the middle represent the following $T$ transformations: $Z_4\to Z_2:Z_4(\tau +1,\overline{\tau }+1)=Z_2(\tau ,\overline{\tau })$ and $Z_5\to Z_4:Z_5(\tau +1,\overline{\tau }+1)=Z_4(\tau ,\overline{\tau })$. The blue lines represent the $S$ transformations. The pattern of the transformations characterizes an 1+1D noninvertible gravitational anomaly described by 2+1D DS topological order.

Figure 7

A two-dimensional complex. The vertices (0-simplices) are labeled by $i$. The edges (1-simplices) are labeled by $\langle ij\rangle$. The faces (2-simplices) are labeled by $\langle ijk\rangle$. The degrees of freedoms may live on the vertices (labeled by $v_i)$, on the edges (labeled by $e_{ij})$ and on the faces (labeled by ${\varphi }_{ijk})$.

Figure 8

Two branched simplices with opposite orientations. (a) A branched simplex with positive orientation and (b) a branched simplex with negative orientation.

Figure 9

The tensor $C_{v_0{v}_1{v}_2{v}_3;{\varphi }_{013}{\varphi }_{123}}^{e_{01}{e}_{02}{e}_{03}{e}_{12}{e}_{13}{e}_{23};{\varphi }_{012}{\varphi }_{023}}$ is associated with a tetrahedron, which has a branching structure. If the vertex-0 is above the triangle-123, then the tetrahedron will have an orientation $s_{0123}=*$. If the vertex-0 is below the triangle-123, the tetrahedron will have an orientation $s_{0123}=1$. The branching structure gives the vertices a local order: the $i\mathrm{th}$ vertex has $i$ incoming edges.

Figure 10

A retriangulation of a 3D complex.

Figure 11

A retriangulation of another 3D complex.

Figure 12

The RG flow of $\delta S={\sum }_{i=1,2,3}\int d^{2}x{g}_i{J}_i{\overline{J}}_i$. There are only fixed lines, shown as the blue lines. There are no stable sheets or regions, as indicated by the orange flow arrows. The corners are labeled by $(s_1,s_2,s_3)$.