Non-invertible anomalies and mapping-class-group transformation of anomalous partition functions

TL;DR

The paper develops a framework for non-invertible gravitational anomalies realized as boundary phenomena of non-invertible 2+1D topological orders. It treats boundary partition functions as a vector space that transforms covariantly under the bulk’s modular data, unifying gapped and gapless boundaries through a common ZST constraint. By detailed analysis of 2+1D $\mathbb{Z}_2$, double-semion, single-semion, and Fibonacci orders, it provides explicit boundary constructions, modular-covariant partition functions, and degeneracy structures, and it derives a lower bound $c\ge\frac{25}{28}$ for irreducible gapless boundaries of the double-semion order. The work also clarifies how non-invertible anomalies relate to, yet extend beyond, 't Hooft anomalies by showing that such anomalies can be realized on lattices with nonlocal Hilbert spaces, and offers systematic tools to detect and classify boundary anomalies from 1+1D partition functions. Overall, it provides a unifying perspective on anomaly, topological order, and boundary physics with concrete criteria for possible boundary theories and their modular properties.

Abstract

Recently, it was realized that anomalies can be completely classified by topological orders, symmetry protected topological (SPT) 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 non-invertible anomaly, which describes the boundary of generic topological order. A key feature of non-invertible anomaly is that it has several partition functions. Under the mapping class group transformation of space-time, those partition functions transform in a certain way characterized by the data of the corresponding topological order in one higher dimension. 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 non-invertible anomaly may have wide applications. As an example, we show that the irreducible gapless boundary of 2+1D double-semion (DS) topological order must have central charge $c=\bar c \geq \frac{25}{28}$.

Figures (12)

• 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 $S^1\times S^1$ ( 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 \ref{['toppathW']}.
• Figure 4: The lattice $\Ga$ 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 $|\chi_0^{u1_4}|^2$. The "$\circ$" points give rise to $|\chi_\one^{u1_4}|^2$. The "$+$" points give rise to $|\chi_2^{u1_4}|^2$. The "$\diamond$" points give rise to $|\chi_3^{u1_4}|^2$. We also mark the directions of the $l$-label and $m$-label. 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 $Z_{M_n}$, $n=\one,2,3$, 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 non-invertible gravitational anomaly described by 2+1D $Z_2$ topological order.