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

TL;DR

This paper introduces the concept of non-invertible anomalies as boundaries of topological orders, characterizing their partition functions and transformation properties under mapping class group actions, with implications for gapless boundary states.

Contribution

It defines non-invertible anomalies, linking their partition functions to topological order data and transformation behaviors, expanding the understanding of anomalies beyond invertible cases.

Findings

Non-invertible anomalies have multiple partition functions.

Partition functions transform under mapping class group similarly to ground states.

Gapless boundary of 2+1D double-semion order has central charge ≥ 25/28.

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

Equations

$$S=\int \mathrm{d}x\mathrm{d}t{\psi }^{†}({\partial }_t+\mathrm{i}{A}_t-{\partial }_x-\mathrm{i}{A}_x)\psi ,$$

$$Z(g\cdot g_{\mu \nu },i)$$

$$Z(g\cdot g_{\mu \nu })=Z(g_{\mu \nu }).$$

$$Z_{\text{TN}}(M^D,\mathcal{T})={\mathrm{e}}^{-{\int }_{M^D}ϵ{\mathrm{d}}^{D}x}{Z}_{\text{TN}}^{\text{top}}(M^D,\mathcal{T}).$$

$$\text{phase change}={\mathrm{e}}^{\mathrm{i}2\pi {\oint }_I\alpha }={\mathrm{e}}^{\mathrm{i}2\pi {\oint }_{M^D\times I}\Omega }$$

$$Z_{\text{field}}^{\text{top}}(M^3,g_{\mu \nu })={\mathrm{e}}^{\mathrm{i}\frac{2\pi \Delta c}{24}{\oint }_{M^D}{\omega }_3}$$

$$K_{E_8}$$

$$Z(\tau )={\chi }^{E_8}(\tau )=\frac{{\Theta }_{K_{E_8}}(q)}{{\eta }^8(q)},q\equiv {\mathrm{e}}^{2\mathrm{i}\pi \tau }$$

$${\Theta }_K(q)$$

$${\chi }^{E_8}=q^{-1/3}(1+248q+4124q^2+O(q^3)),$$

$${\chi }^{E_8}(-1/\tau )={\chi }^{E_8}(\tau ),{\chi }^{E_8}(\tau +1)={\mathrm{e}}^{-\mathrm{i}\frac{2\pi }{3}}{\chi }^{E_8}(\tau )$$

$$\underset{q\to 0}{\lim }Z(q)=\text{integer}\times q^{-\frac{c}{24}}{\overline{q}}^{-\frac{\overline{c}}{24}},\Delta c=c-\overline{c}.$$

$$Z(B^d;M^D,\mathcal{T})=Z_{\text{TN}}^{\text{top}}(M^D,\mathcal{T}),B^d=\partial M^D.$$

$$Z(B^d,{\mathcal{T}}_B;M^D,\mathcal{T})=Z_{\text{TN}}(B^d,{\mathcal{T}}_B)Z_{\text{TN}}^{\text{top}}(M^D,\mathcal{T})$$

$$Z(T^2;D^2\times S^1)=Z_{\text{TN}}^{\text{top}}(D^2\times S^1),\partial D^2=S^1.$$

$$Z(T^2;D_i^2\times S^1)=Z_{\text{TN}}^{\text{top}}(D_i^2\times S^1).$$

$$Z(T^2,|{\psi }_i⟩)=Z_{\text{TN}}^{\text{top}}(D_i^2\times S^1).$$

$$Z(\tau ,\overline{\tau },|{\psi }_i⟩)=Z_{\text{TN}}^{\text{top}}(D_i^2\times S^1).$$

$$Z(\tau ,\overline{\tau })=Z(\tau +1,\overline{\tau }+1).$$

$$Z(\tau ,\overline{\tau },T_{ij}^{\text{top}}|{\psi }_j⟩)=Z(\tau +1,\overline{\tau }+1,|{\psi }_i⟩),$$

$$Z(\tau ,\overline{\tau },S_{ij}^{\text{top}}|{\psi }_j⟩)=Z(-1/\tau ,-1/\overline{\tau },|{\psi }_i⟩),$$

$$Z(\tau ,\overline{\tau },M_{ij}|{\psi }_j⟩)=M_{ij}Z(\tau ,\overline{\tau },|{\psi }_j⟩)$$

$$⟨\varphi |{\psi }_i⟩=Z(\tau ,\overline{\tau },|{\psi }_i⟩).$$

$$T_{ij}^{\text{top}}Z(\tau ,\overline{\tau };j)=$$

$$S_{ij}^{\text{top}}Z(\tau ,\overline{\tau };j)=$$

$$Z(i)=T_{ij}^{\text{top}}Z(j),Z(i)=S_{ij}^{\text{top}}Z(j).$$

$$Z(\tau ,\overline{\tau },1)=Z(\tau ,\overline{\tau },m)=1,Z(\tau ,\overline{\tau },e)=Z(\tau ,\overline{\tau },f)=0.$$

$$T_{Z_2}^{\text{top}}$$

$$Z(\tau ,\overline{\tau },1)=Z(\tau ,\overline{\tau },e)=2,Z(\tau ,\overline{\tau },m)=Z(\tau ,\overline{\tau },f)=0.$$

$$Z(\tau ,\overline{\tau },1)=Z(\tau ,\overline{\tau },e)=1,Z(\tau ,\overline{\tau },m)=Z(\tau ,\overline{\tau },f)=0,$$

$$H=\int \mathrm{d}x{\lambda }_R\mathrm{i}{\partial }_x{\lambda }_R-{\lambda }_L\mathrm{i}{\partial }_x{\lambda }_L.$$

Full text

1. I Introduction
2. II Topological invariant and properties of boundary partition function
3. III A non-invertible bosonic global gravitational anomaly from 2+1D $Z_{2}$ topological order
4. IV A non-invertible bosonic global gravitational anomaly from 2+1D DS topological order
5. V Non-invertible gravitational anomaly and “non-locality” of Hilbert space
6. VI Systematical search of gapped and gapless boundaries of a 2+1D topological order
7. VII Detect anomalies from 1+1D partition functions
8. VIII Summary
9. A Characters of chiral CFTs
10. B Non-on-site $Z_{2}$ symmetry transformations
11. C Topological path integral on a space-time with world lines domain walls
12. D Renormalization group flow of marginal perturbations of $SU(2)_{1}$ CFT

I Introduction

A classical field theory described by an action may have a gauge symmetry if the action is gauge invariant. The corresponding theory is called a classical gauge theory. A gauge anomaly is an obstruction to quantize the classical gauge theory, since the path integral measure may not be gauge invariant.Adler (1969); Bell and Jackiw (1969) Similarly, a classical action may have a diffeomorphism invariance. Then a gravitational anomaly is an obstruction to have a diffeomorphic invariant path integral.Witten (1985) So the standard point of view of anomaly corresponds to the obstruction to go from classical theory to quantum theory. This kind of gauge anomaly and gravitational anomaly are always invertible, *i.e. *can be canceled by another anomalous theory. The examples include 1+1D $U(1)$-gauged chiral fermion theory

[TABLE]

which has both perturbative $U(1)$ gauge anomaly and perturbative gravitational anomaly. We like to remark that such defined anomaly is not a property of physical systems, but a property of a formalism trying to convert a classical theory to a quantum theory.

There is another invertible anomaly – ’t Hooft anomaly, that can be defined within a quantum system with a global symmetry, and is a property of physical systems. It is not an obstruction to go from classical theory to quantum theory, but rather an obstruction to gauge a global symmetry within a quantum system.’t Hooft (1980) It is quite amazing that the obstruction to quantize a classical gauge theory (gauge anomaly) is closely related to the obstruction to gauge a global symmetry within a quantum system.

Motivated by some early results,Callan and Harvey (1985); Ryu et al. (2012) in recent years, we started to have a new understanding of anomaly as a physical property of quantum systems:Wen (2013); Kong and Wen (2014); Kong et al. (2015)

The anomaly in a theory directly corresponds to the topological orderWen (1989, 1990) and/or symmetry protected topological (SPT) orderGu and Wen (2009); Chen et al. (2011, 2013) (with on-site symmetry) in one higher dimension. Such an anomalous theory is realized by a boundary of the corresponding topological order and/or SPT order (see Fig. 1b).

So an anomaly is nothing but a topological order and/or a SPT order in one higher dimension, and the anomalies can be classified via the classification of topological orders and SPT orders in one higher dimension.Wen (2013); Kapustin and Thorngren (2014) The boundary of topological order realizes all possible gravitational anomaly, and the boundary of SPT order realizes all possible ’t Hooft anomaly and mixed gravity/’t Hooft anomaly. This point of viewed of anomaly plus Atiyah formulation of topological quantum field theoryAtiyah (1988) allow us to develop a general theory of anomaly.Wen (2013); Kong and Wen (2014); Kong et al. (2015)

But the anomaly from this new point of view is not the same as the previously defined anomaly before 2013, and is more general. This is because topological orders are usually not invertible.Kong and Wen (2014); Freed (2014) Hence, the anomaly realized by the boundary of topological orders may be non-invertible as well (*i.e. *cannot be canceled by any other anomaly). In contrast, the standard anomalies are always invertible. Thus the standard anomalies are classified by invertible topological orders and/or SPT orders in one higher dimension, and are realized by the boundary of the invertible topological orders and/or the SPT orders. But there are more general topological orders, which are not invertible. Those non-invertible topological orders will give rise to a new kind of gravitational anomalies on the boundary, which will be called non-invertible anomaly. For example, the chiral conformal field theory (CFT) on the boundary of a generic Chern-Simons theory is an example of non-invertible anomaly.Witten (1989)

We like to mention that, in addition to the above general point of view proposed in LABEL:W1313,KW1458,KZ150201690 that include non-invertible anomalies, LABEL:FV14095723,M14107442 also proposed a similar general point of view based mathematical category theory and cobordism theory. In particular, those general points of view suggest that the partition function becomes a vector in a vector space for a theory with non-invertible anomaly, as suggested in LABEL:A8875. In fact, the vector space that contains partition functions of an anomalous theory can be identified with the degenerate ground state subspaceWen (1989, 1990) (see Fig. 1a) of the topological order in one higher dimension that characterizes the anomaly. This is because if we regard a space direction as time direction, the space manifold can be viewed as a boundary of space-time, and the ground degenerate subspace becomes the vector space of partition functions (see Fig. 1).Kong and Wen (2014); Wen and Wang (2018)

In this paper, we will study some simplest non-invertible anomalies – bosonic global gravitational anomalies in 1+1D which correspond to a 2+1D bosonic topological order. We will first give a general discussion, in particular the physical meaning of “partition function” as a vector in a vector space. Then we will discuss some examples of 1+1D bosonic theories with non-invertible gravitational anomalies that correspond to:

2+1D bosonic $Z_{2}$ topological orderRead and Sachdev (1991); Wen (1991) (*i.e. *the topological order described by the $Z_{2}$-gauge theory). 2. 2.

2+1D bosonic double-semion (DS) topological order.Freedman et al. (2004); Levin and Wen (2005) 3. 3.

2+1D bosonic single-semion (SS) topological order (i.e. $\nu=1/2$ quantum Hall state).Laughlin (1983) 4. 4.

2+1D bosonic Fibonacci topological order.Freedman et al. (2004); Levin and Wen (2005)

We will also discuss an application of invertible and non-invertible anomalies. There is a general belief that a gapless CFT has a partition function that is invariant under mapping class group (MCG) transformations of the space-time (the modular transformations for 2-dimensional space-time), provided that the CFT can be put on a lattice. Being able to put a CFT on a lattice is nothing but the anomaly-free condition. This suggests that the MCG invariance of the partition function corresponds to the anomaly-free condition. So an anomalous CFT will have a partition function which is not MCG invariant, but MCG covariant.111The partition functions of a fermionic CFT on a $d+1$D lattice are MCG covariant in a special way, so that the sum of the partition functions are still invariant under the fermionic MCG. For example, in 1+1D, the fermionic MCG is generated by $T^{2}$ and $S$. Since the anomaly corresponds to a topological order in one higher dimension that is described by a higher category, the change of anomalous partition function can be described by the data of this higher category. In this paper, we will derive one such result.

Consider a CFT in $d$-dimensional closed space-time $M^{d}$, whose gravitational anomaly is described a $(d+1)$D topological order. The $(d+1)$D topological order has $N$-fold degenerate ground states on $M^{d}$. Let $G_{M^{d}}$ be the MCG for $M^{d}$. Under a MCG transformation $g\in G_{M^{d}}$, the degenerate ground states transform according to a representation $R^{\text{top}}(g)$ of $G_{M^{d}}$.Wen (1990); Keski-Vakkuri and Wen (1993); Rowell et al. (2009); Kong and Wen (2014) Such a representation $R^{\text{top}}(g)$ is the data that characterize the $(d+1)$D topological order (and hence the anomaly). It was conjecturedWen (1990) that such data fully characterize the topological order. From the correspondence described in Fig. 1, we find that

the anomalous CFT in $d$-dimensional space-time has several partition functions $Z(g_{\mu\nu},i)$, $i=1,2,\cdots,\dim(R^{\text{top}})$, which transform as:

$\displaystyle Z(g\cdot g_{\mu\nu},i)$ $\displaystyle=R^{\text{top}}_{ij}(g)Z(g_{\mu\nu},j),$

(2)

where $g_{\mu\nu}$ is the metrics on the $d$-dimensional space-time $M^{d}$, which describes the shape of $M^{d}$, and $g\cdot g_{\mu\nu}$ is the MCG action on $g_{\mu\nu}$.

When $d=2$, eqn. (2) becomes eqn. (II.5), which we will explain in some detail.

For an anomaly-free CFT, the corresponding $(d+1)$D topological order is trivial and $R^{\text{top}}=1$ are 1-by-1 matrices. In the case, the above becomes the usual MCG invariant condition on the partition function:

[TABLE]

It is likely that the MCG invariant partition functions on $M^{d}$ completely classify anomaly-free CFTs. Thus, it is also likely that that the modular covariant partition function (2) completely classify anomalous CFTs (*i.e. *the boundaries of $(d+1)$D topological order described by $R^{\text{top}}(g)$).

We like to point out that eqn. (2) also covers the cases of gapped boundaries of $(d+1)$D topological order. In this case $Z(g_{\mu\nu},i)=Z(i)$ becomes $g_{\mu\nu}$ independent. The $d=2$ case is studied in detail in LABEL:LWW1414,dwall, where $Z(i)$ is denoted as $W^{\mathbf{1}i}$ and is called fusion matrix or wavefunction overlap. Thus eqn. (2) is a unified description for both gapped and gapless boundary.

As another application, we point out that anomaly-free fermionic theories exactly correspond a subset of the bosonic theories with the non-invertible gravitational anomaly described by the bosonic $Z_{2}$ topological order with emergent fermion (*i.e. *the twisted $Z_{2}$ gauge theory) in one higher dimension. Thus we can construct anomaly-free fermionic theories, such as their partition functions, by constructing bosonic theories and their partition functions with this particular non-invertible gravitational anomaly.

We want to mention that LABEL:KZ170501087 has given a very general and complete theory of purely chiral CFT on the boundary of a 2+1D topological order, based on tensor category theory. The general theory developed here works for both purely chiral and non-chiral CFT on the boundary. When the boundary is purely chiral, our theory is just a subset of the full theory developed in LABEL:KZ170501087. Both theories provide a unified approach for gapped and gapless boundaries. Recently, some non-chiral CFT’s on the boundary of 2+1D topological order were studied in LABEL:CZ190312334. In particular, multi-component partition functions on a 1+1D gapless boundary of 2+1D double-Ising topological order were calculated. A connection between the modular transformation of boundary partition functions and the $S,T$ matrices that characterize the modular tensor category for the 2+1D bulk topological order was noticed. The appearance of many sectors in anomalous CFT was also pointed out in LABEL:LS190404833. Our paper generalizes those results and provides a more systematic discussion.

We also like to remark that, in the presence of symmetry, there are also several partition functions from the different symmetry twisting boundary conditions in $d$-dimensional space-time.Ryu and Zhang (2012); Tiwari et al. (2018) If the anomaly is not invertible, there will be several partition functions for each twisted boundary condition. Those partition functions also transform covariantly under MCG transformations. This generalization is discussed in LABEL:STanomaly, for $d=2$ case.

II Topological invariant and properties of boundary partition function

First, let us describe the topological path integral that can realize various topological orders. The boundaries of those topological orders realize invertible and non-invertible anomalous theories. This way, we can relate anomalies with topological invariants in one higher dimensions.

II.1 Topological partition function as topological invariant

A very general way to characterize a topologically ordered phase is via its partition function $Z(M^{D})$ on closed spactime $M^{D}$ with all possible topologies. A detailed discussion on how to define the partition function via tensor network is given in LABEL:KW1458 and in Appendix C. From this careful definition, we see that the partition function also depends on the branched triangulation of the space-time (see Appendix C), as well as the tensor associated with each simplex. We collectively denote the triangulation, the branching structure, and the tensors as ${\cal T}$. Thus the partition function should be more precisely denoted as $Z_{\text{TN}}(M^{D},{\cal T})$. In a very fine triangulation limit (*i.e. *the thermodynamic limit), we believe that the partition function depends on ${\cal T}$ via an effective metric tensor $g_{\mu\nu}$ of the spacetime manifold, if the tensor network describes a “liquid” state, as opposed to a foliated state (a non-liquid state).Chamon (2005); Haah (2011); Zeng and Wen (2015); Shirley et al. (2018a, b) Thus, the partition function can be denoted as $Z_{\text{field}}(M^{D},g_{\mu\nu})$ in the thermodynamic limit. $Z_{\text{field}}(M^{D},g_{\mu\nu})$ correspond to the partition function of a field theory where the different lattice regularizations ${\cal T}$ are not important as long as they produce the same equivalent metric $g_{\mu\nu}$. Here, $g_{\mu\nu}$ and $g^{\prime}_{\mu\nu}$ are regarded as equivalent if they differ by a diffeomorphim since $Z_{\text{field}}(M^{D},g_{\mu\nu})=Z_{\text{field}}(M^{D},g^{\prime}_{\mu\nu})$. Let ${\cal M}_{M^{D}}$ be the space formed by all metrics $g_{\mu\nu}$ of $M^{D}$ (up to diffeomorphic equivalence), which is called the moduli space of $M^{D}$. Thus the partition function $Z_{\text{field}}(M^{D},g_{\mu\nu})$ is a complex function on the moduli space ${\cal M}_{M^{D}}\xrightarrow{Z_{\text{field}}(M^{D},-)}\mathbb{C}$.

However, $Z_{\text{TN}}(M^{D},{\cal T})$ (or $Z_{\text{TN}}(M^{D},g_{\mu\nu})$) is not a topological invariant since it contains a so-called volume term $\hskip 1.0pt\mathrm{e}^{-\int_{M^{D}}\epsilon\,\hskip 1.0pt\mathrm{d}^{D}x}$ where $\epsilon$ is the energy density. But this problem can be fixed, by factoring out the volume term. This way, we can obtain a topological partition function $Z_{\text{TN}}^{\text{top}}(M^{D})$ which is believed to be a topological invariant:Kong and Wen (2014); Wen and Wang (2018)

[TABLE]

Appendix C.4 describes the way to fine-tune the tensors to make the volume term vanishes (i.e. $\epsilon=0$). In this case, the path integral directly produces the topological partition function. Such a topological invariant may completely characterize the topological order.

Let us describe topological invariant, the topological partition function of the field theory, $Z_{\text{field}}^{\text{top}}(M^{D},g_{\mu\nu})$ (i.e. $Z_{\text{TN}}^{\text{top}}(M^{D},{\cal T})$) in more details. The “topological property” of $Z_{\text{field}}^{\text{top}}(M^{D},g_{\mu\nu})$ may appear in two ways:Kong and Wen (2014)

$Z_{\text{field}}^{\text{top}}(M^{D},g_{\mu\nu})$ is a local constant function on ${\cal M}_{M^{D}}$. In this case, the topological partition function only depends on $\pi_{0}({\cal M}_{M^{D}})$: $\pi_{0}({\cal M}_{M^{D}})\xrightarrow{Z_{\text{field}}(M^{D},-)}\mathbb{C}$. Such a complex function on $\pi_{0}({\cal M}_{M^{D}})$ is a topological invariant, since $Z_{\text{field}}^{\text{top}}(M^{D},g_{\mu\nu})$ does not depend on any smooth change of $g_{\mu\nu}$. In this case, the boundary has a global gravitational anomaly. 2. 2.

The reduction from lattice partition function $Z_{\text{TN}}^{\text{top}}(M^{D},{\cal T})$ to the field theory partition function $Z_{\text{field}}^{\text{top}}(M^{D},g_{\mu\nu})$ may have a phase ambiguity. However, we can define the change of phase for $Z_{\text{field}}^{\text{top}}(M^{D},g_{\mu\nu})$ as we go along a segment $I$ in ${\cal M}_{M^{D}}$ without ambiguity:

[TABLE]

where $\alpha$ is a 1-form on ${\cal M}_{M^{D}}$ and $\Omega$ is closed $D+1$ form constructed from the curvature tensor on $M^{D}\times I$. In this case, the boundary has a perturbative gravitational anomaly. For example, when $D=3$, $\Omega=\frac{\Delta c}{24}p_{1}$, where $p_{1}$ is the first Pontryagin class on 4-manifold. $Z_{\text{field}}^{\text{top}}(M^{3},g_{\mu\nu})$ is given by

[TABLE]

where the 3-form $\omega_{3}$ satisfies $\hskip 1.0pt\mathrm{d}\omega_{3}=p_{1}$ and corresponds to the gravitational Chern-Simons term. The coefficient $\Delta c$ is the chiral central charge of the boundary state. In this case, $Z_{\text{field}}^{\text{top}}(M^{3},g_{\mu\nu})$ depends on the smooth change of $g_{\mu\nu}$ and is not a topological invariant in the usual sense.

II.2 Invertible and non-invertible topological order

Most topological orders are not invertible under the stacking operation. (Here, by definition, an invertible orderKong and Wen (2014); Freed (2014) can be canceled by another order, *i.e. *the stacking of the two orders gives rise to the trivial order.) The invertible topological orders form a subset of topological orders. The topological invariant for invertible topological orders, $Z_{\text{TN}}^{\text{top}}(M^{D},{\cal T})$, is a pure phase factor: $|Z_{\text{TN}}^{\text{top}}(M^{D},{\cal T})|=1$.Wen (2014); Hung and Wen (2014); Kapustin (2014); Kong and Wen (2014)

The boundaries of invertible topological orders have the standard gravitational anomalies. The gravitational anomalies in literature all belong to this case. The boundaries of non-invertible topological orders represent the structures that are different from the usual anomalies. We will call those structures as non-invertible gravitational anomalies, and call the standard gravitational anomalies as invertible gravitational anomalies. In this paper, we will concentrate on the non-invertible anomalies.

To give an example of invertible anomalies, let us consider a $E_{8}$ bosonic quantum hall state described by the following $K$-matrixWen and Zee (1992); Wen (1995)

[TABLE]

which has an invertible topological order, since det$(K_{E_{8}})=1$. Its boundary is described by the $(E_{8})_{1}$ CFT that has a perturbative gravitational anomaly, due to its non-zero chiral central charge $c=8$. It is a chiral CFT whose partition function has a single character,

[TABLE]

where $\eta(q)=q^{\frac{1}{24}}\prod_{n=1}^{\infty}(1-q^{n})$ is the Dedekind eta function, and $\Theta_{K}$ is the theta function for a lattice characterized by an integer symmetric matrix $K$:

[TABLE]

and $K_{E_{8}}$ is the $E_{8}$ root lattice, given by eqn. (LABEL:KE8). The first a few terms in the expansion is

[TABLE]

where the $248$ generators of $E_{8}$ are counted in the second term in this single sector. $\chi^{E_{8}}$ transforms according to the one-dimensional representation of the modular group

[TABLE]

The 1+1D perturbative gravitational anomaly characterized by the chiral central charge $\Delta c$ constrains the boundary partition function in 1+1D:

[TABLE]

Thus, knowing the 1+1D boundary partition function, we can also determine its perturbative gravitational anomaly $\Delta c$. In this paper, we will try to go one step further. We like to determine the global anomaly from the partition function.

II.3 Properties of boundary partition function

To concentrate on global anomaly, we will assume that there is no perturbative anomaly. In this case, the global anomaly is characterized by the bulk topological invariant $Z_{\text{field}}^{\text{top}}(M^{D},{\cal T})$, which can be realized by the topological path integral described in Appendix C.4.Kong and Wen (2014) In this paper, we assume the bulk theory is always described by the topological path integral, whose partition function directly corresponds to the topological invariant $Z_{\text{field}}^{\text{top}}(M^{D},{\cal T})$.

To link such a topological invariant (*i.e. *topological path integral), $Z_{\text{TN}}^{\text{top}}(M^{D},{\cal T})$ to the partition function on the boundary $B^{d}$, $d=D-1$, we note that the boundary partition function is given by (Fig. 2a)Kong and Wen (2014); Wen and Wang (2018)

[TABLE]

The boundary is the so-called natural boundary described in Appendix C.3, but here we sum over the boundary degrees of freedom. We note that the bulk is gapped. Thus, the low energy properties of the boundary (below the bulk gap) are described by the above $Z(B^{d},{\cal T}_{B})$.

We may obtain a more general boundary by stacking a $d$-dimensional system described by a $d$-dimensional tensor network, $Z_{\text{TN}}(B^{d},{\cal T}_{B})$, to the boundary (see Fig. 2b). The resulting boundary partition function has a form

[TABLE]

We may also allow the boundary and bulk degrees of freedom to interact with each other by gluing the boundary to the bulk as in Fig. 2c. We see that the boundary partition function $Z(B^{d},{\cal T}_{B};M^{D},{\cal T})$ is not purely given by a tensor network on the boundary $B^{d}$, which gives rise to a partition function $Z_{\text{TN}}(B^{d},{\cal T}_{B})$. The boundary partition function also contain a bulk topological term $Z_{\text{TN}}^{\text{top}}(M^{D},{\cal T})$. This makes the boundary quantum system defined by $Z(B^{d},{\cal T}_{B};M^{D},{\cal T})$ to be potentially anomalous. If the boundary partition function is given purely by a tensor network $Z_{\text{TN}}(B^{d},{\cal T}_{B})$ on the boundary (*i.e. *when $Z_{\text{TN}}^{\text{top}}(M^{D},{\cal T})=1$), such a quantum system will be anomaly-free.

II.4 1+1D anomalous theory on space-time torus $T^{2}$

In this section, we will concentrate on an 1+1D anomalous theory. To define its partition function on a space-time torus $T^{2}$, we consider a 2+1D tensor network path integral (see Appendix C) on $D^{2}\times S^{1}$ (see Fig. 2c)

[TABLE]

The tensors on the inner solid cylinder define a topological path integral described in Appendix C which realize a topological order that corresponds to the anomaly under consideration. The tensors on the outer cylinder (see Fig. 2b) can be anything, which determine the different types of boundaries.

We can define a more general partition function for 1+1D anomalous theory by inserting a world-line (see Fig. 3)

[TABLE]

We note that the surface of the inner solid cylinder in Fig. 3 (after integrating out only the bulk degrees of freedom as in Appendix C.3) corresponds to a wavefunction, $|\psi_{i}\rangle$, that describes one of the degenerate ground states of the bulk topological order on the torus. If the path integral on the inner solid cylinder is a topological path integral, $|\psi_{i}\rangle$ automatically normalizes to 1: $\langle\psi_{i}|\psi_{i}\rangle=1$ (as discussed in LABEL:dwall). Thus, more precisely, the 1+1D partition function for an anomalous theory is given by

[TABLE]

Here the degenerate ground state wave functions $|\psi_{i}\rangle$ are labeled by the type-$i$ of the topological excitations. For the trivial excitations labeled by $\mathbf{1}$, $Z(T^{2},|\psi_{\mathbf{1}}\rangle)$ correspond to the partition function for the space-time in Fig. 2c without the insertion of the world-line.

The dependence on the ground state wave function $|\psi_{i}\rangle$ of the topological order on the torus is the key character of anomalous partition function:

If $|\psi_{i}\rangle$ is a product state, then $Z(T^{2},|\psi_{i}\rangle)$ is a partition function of an anomaly-free theory. 2. 2.

If $|\psi_{i}\rangle$ is unique (*i.e. *the topological order has a non-degenerate ground state on the torus), then $Z(T^{2},|\psi_{i}\rangle)$ is a partition function of a theory with invertible anomaly. 3. 3.

If $|\psi_{i}\rangle$ is not unique (*i.e. *the topological order has degenerate ground states on torus), then $Z(T^{2},|\psi_{i}\rangle)$ is a partition function of a theory with non-invertible anomaly.

II.5 Modular transformations of the partition function for an anomalous

theory

Let us fine tune the action of the 1+1D anomalous theory, so that it has a vanishing ground state energy density. In this case, its partition function on $T^{2}$ will not depend on the size of the space-time, but only depend on the shape of the space-time. The shape of a torus $T^{2}$ can described by a complex number $\tau$. Thus we may write the 1+1D partition function as

[TABLE]

However, $\tau$ and $\tau^{\prime}=\tau+1$ describe the same shape after a coordinate transformation. For an anomaly-free 1+1D theory, we have

[TABLE]

However, for an anomalous 1+1D theory, we have

[TABLE]

since the coordinate transformation acts non-trivially on the ground state wavefunction $|\psi_{i}\rangle$ on torus. Here the unitary matrix $T^{\text{top}}_{ij}$ describes such a non-trivial action, which is a modular transformation of the torus ground states of the 2+1D bulk topological order.Wen (1990); Keski-Vakkuri and Wen (1993) Similarly, $\tau$ and $\tau^{\prime}=-1/\tau$ also describe the same shape after a coordinate transformation. Thus

[TABLE]

where the unitary matrix $S^{\text{top}}_{ij}$ describes another modular transformation of the torus ground states of the bulk topological order.

We note that the partition function $Z(\tau,\overline{\tau},|\psi_{i}\rangle)$ depends on $|\psi_{i}\rangle$ in a linear fashion

[TABLE]

This is because the path integral that sums over the degrees of freedom in the bulk and the outer surface of outer cylinder (see Fig. 2b) gives rise to a wave function $\langle\phi|$ that lives on the inner surface of the outer cylinder. The partition function $Z(\tau,\overline{\tau},|\psi_{i}\rangle)$ is simply

[TABLE]

Thus $Z(\tau,\overline{\tau},|\psi_{i}\rangle)$ is a linear function of $|\psi_{i}\rangle$. As a result, eqn. (20) and eqn. (21) can be rewritten as

[TABLE]

where $Z(\tau,\overline{\tau},i)\equiv Z(\tau,\overline{\tau},|\psi_{i}\rangle)$. Eqn. (II.5) is another key result of this paper. It describes the modular transformation properties of the partition functions for anomalous theory.

We stress that, for an anomalous theory, its partition functions are vectors in a vector space ${\cal Z}$. The anomalous theory itself only determines such a vector space ${\cal Z}$. When ${\cal Z}$ is one dimensional, the anomaly is invertible. When the dimension of ${\cal Z}$ is more than one, the anomaly is non-invertible.

For gapped anomalous theory, the partition functions do not depend on $\tau$. Eqn. (II.5) becomes

[TABLE]

We recover a condition for gapped boundary of a topological order obtained in LABEL:LWW1414,dwall, where $Z(i)$ was denoted as $W^{\mathbf{1}i}$. Note that for the gapped case, the partition functions $Z(i)$ are ground state degeneracy of the systems and are non-negative integers.

The above is a general discussion of 1+1D anomalous theory, which can have a non-invertible anomaly. In particular, the boundary CFT may have separate right-moving part and left-moving part, and each part transforms according to certain $S_{R,L}$ and $T_{R,L}$ matrices. Those boundary $S_{R,L}$ and $T_{R,L}$ matrices may be different from the $S^{\text{top}}$ and $T^{\text{top}}$ matrices for the bulk topological order. However, after we combine the right movers and left movers to construct multi component partition functions $Z(\tau,\overline{\tau};i)$, we find that $Z(\tau,\overline{\tau};i)$ transform according to the bulk $S^{\text{top}}$ and $T^{\text{top}}$ matrices. In the following, we will discuss some simple examples of 1+1D non-invertible anomaly.

III A non-invertible bosonic global gravitational

anomaly from 2+1D $Z_{2}$ topological order

A 2+1D $Z_{2}$ topological order has four type of excitations, $\mathbf{1},e,m,f$, where $e,m$ are bosons and $f$ is a fermion. $e,m,f$ are topological excitations with $\pi$ mutual statistics respect to each other. (Remember that a topological excitation is defined as the excitation that cannot be created by any local operators). Such a topological order can have many different boundaries, which all carry the same non-invertible gravitational anomaly. In this section, we will discuss some of those boundary theories.Kong and Wen (2014)

III.1 Two gapped boundaries of the 2+1D $Z_{2}$ topological order

A gapped boundary of the 2+1D $Z_{2}$ topological order is induced by $m$ particle condensation. This boundary has only one type of topological excitations $e$. The topological excitation $e$ has a $Z_{2}$ fusion $e\otimes e=1$, and is described by a symmetric fusion category ${{\cal R}\text{ep}}(Z_{2})$ (which is the fusion category formed by the representations of $Z_{2}$ group). Such a boundary described by ${{\cal R}\text{ep}}(Z_{2})$ has a non-degenerate ground state. Its partition function is given by $Z(\tau,\overline{\tau},\mathbf{1})=1$ (where $\mathbf{1}$ means that there is no insertion of world-line, i.e. $i=\mathbf{1}$ in Fig. 3).

The insertion of a world-line of $m$-type topological excitations (see Fig. 3) produces another boundary, where $e$ on the boundary $S^{1}$ acquires a $\pi$-phase as it goes around the boundary. The partition function for such a boundary is still given by $Z(\tau,\overline{\tau},m)=1$.

If we insert a world-line of $e$-type or a $f$-type, the resulting boundary will carry an un-paired $e$ excitations. Such an un-paired $e$ costs a finite energy $\epsilon_{e}$. These boundaries will have partition functions Z(\tau,\overline{\tau},e)=Z(\tau,\overline{\tau},f)=\#\hskip 1.0pt\mathrm{e}^{-\epsilon_{e}\beta}\big{|}_{\beta\to\infty}=0, when the size of space-time $\beta$ approaches to infinity.

So the first gapped boundary of $Z_{2}$ topological order is described by four partition functions in the excitations basis $(\mathbf{1},e,m,f)$

[TABLE]

They can be viewed as the partition function for an anomalous $c=0$ CFT (*i.e. *a gapped theory). One can check that these four partition functions in the excitations basis satisfy eqn. (25),Lan et al. (2015, ) since for $Z_{2}$ topological order, $S^{\text{top}},T^{\text{top}}$ are given by

[TABLE]

Let us obtain another gapped boundary of the 2+1D $Z_{2}$ topological order, by lowering the energy of $e$ to a negative value. This will drive a “$Z_{2}$ symmetry” breaking transition and obtain an $e$-condensed state, which have a 2-fold ground state degeneracy on a ring. (If we condensed $e$ particle on an open segment on the boundary, we will also get a 2-fold ground state degeneracy.) This new boundary is described by the following four partition functions

[TABLE]

They again satisfy eqn. (II.5).

Here $Z(\tau,\overline{\tau},\mathbf{1})=2$ means the $Z_{2}$ topological order on $D^{2}$ (*i.e. *the boundary state on $S^{1}$, see Fig. 2c) has a 2-fold degeneracy. This 2-fold degeneracy come form the emergent mod-2 conservation of $e$-particles on the boundary, and subsequently the spontaneous breaking of this emergent $Z_{2}$ symmetry. However, since the $Z_{2}$ symmetry is emergent, when the boundary $S^{1}$ has a finite density $n_{e}$ of the $e$-particles, the emergent mod-2 conservation may be explicitly broken by an amount $\hskip 1.0pt\mathrm{e}^{-1/n_{e}\xi}$ where $\xi$ is a length scale. In this case, the 2-fold degeneracy is lifted by an amount $\hskip 1.0pt\mathrm{e}^{-1/n_{e}\xi}$. So the boundary described by eqn. (28) is unstable. After the lifting of the degeneracy, the boundary is actually described by

[TABLE]

which correspond to the boundary of the 2+1D $Z_{2}$ topological order induced by $e$ condensation (while the boundary induced by $m$ condensation is described by eqn. (26)).

III.2 A gapless boundary of the 2+1D $Z_{2}$ topological order

A gapless boundary of the 2+1D $Z_{2}$ topological order is given by a 1+1D gapless system described by a Majorana fermion field

[TABLE]

We like to stress that such a 1+1D gapless system is actually a bosonic system where the states in the many-body Hilbert are all bosonic (i.e. contain an even number of Majorana fermions). We refer such a 1+1D gapless system as the boson-restricted Majorana fermion theory. It is different from the usual Majorana fermion theory.

We can give the Majorana fermion a mass gap to obtain a gapped boundary:

[TABLE]

Such a gapped boundary correspond to the gapped boundary described above. If we lower $m$ to a negative value, we should drive the “$Z_{2}$ symmetry” breaking transition described above and obtain a 2-fold ground state degeneracy on a ring. This is different from the standard Majorana fermion theory where the negative $m$ also gives rise to non-degenerate ground state. So for our boson-restricted Majorana fermion theory, a positive $m$ gives rise to non-degenerate ground state while a negative $m$ gives rise to a 2-fold ground state degeneracy on a ring. If we only change the sign of $m$ on an open segment, then both the standard Majorana fermion theory and our bosonic Majorana fermion theory will give rise to a 2-fold ground state degeneracy.

So when $m=0$ the gapless bosonic Majorana fermion theory describes the critical point of the $Z_{2}$ symmetry breaking phase transition mentioned above. The gapless boson-restricted Majorana fermion theory describes a conformal field theory (CFT) with a non-invertible gravitational anomaly. In this paper, we like to understand this anomalous CFT in detail. In particular, we would like to compute its partition function and their properties under modular transformation.

To understand the critical CFT for the “$Z_{2}$ symmetry” breaking transition, let us introduce a 1d lattice Hamiltonian on a ring to describe the gapped boundary in Section III.1

[TABLE]

where $\sigma^{l},\ l=x,y,z$ are Pauli matrices. Here an up-spin $\sigma^{z}_{i}=1$ correspond to an empty site and an down-spin $\sigma^{z}_{i}=-1$ correspond to a site occupied with an $e$ particle. Since number of the $e$ particles is always even, thus the Hilbert space ${\cal V}$ of our model is formed by states with even numbers of down spins $\sigma^{z}_{i}=-1$. Note that our Hilbert space is non-local, *i.e. *it does not have a tensor product decomposition:

[TABLE]

where ${\cal V}_{i}$ is the two dimensional Hilbert space for site-$i$. It is this property that make our model to have a non-invertible gravitational anomaly.

We like to mention that, we can view the 2+1D $Z_{2}$ topological order as a gauged $Z_{2}$ symmetric state with a trivial SPT order. The boundary of the 2+1D $Z_{2}$ symmetric state can be described by a transverse Ising model eqn. (32) with the standard Hilbert space (*i.e. *without the $\prod_{i}\sigma^{z}_{i}=1$ constraint). The boundary can be in a symmetric phase (described by eqn. (32) with $U\gg J$) or a $Z_{2}$ symmetry breaking phase (described by eqn. (32) with $U\ll J$). We see that after gauging the $Z_{2}$ symmetry to obtain the $Z_{2}$ topological order in the bulk, the only change in the boundary theory is the addition of the constraint $\prod_{i}\sigma^{z}_{i}=1$,Yang et al. (2014) that changes the many-body Hilbert space to make it non-local (*i.e. *make the boundary theory to have a non-invertible gravitational anomaly).

In our model (32) for the boundary, the $J$ term is allowed since the $e$ particles have only a mod-2 conservation. In the $U\gg J$ limit, the above lattice model describes the gapped phase in Section III.1. As we change $U$ to $U\ll J$, we will drive a “$Z_{2}$ symmetry” breaking phase transition. The critical point at $U=J$ is described by a CFT with non-invertible gravitational anomaly. Such a CFT is described by the boson-restricted Majorana fermion theory mentioned in Section III.2. The Majorana fermion theory is obtained from eqn. (32) via the Jordan-Wigner transformation.

To obtain the partition function of the anomalous CFT, let us first consider the partition function of the transverse Ising model (32) at critical point $U=J$. There are four partition functions $Z_{a_{x},a_{t}}$ for the transverse Ising model, with different $\mathbb{Z}_{2}$ boundary conditions $a_{x}=\pm 1$ and $a_{t}=\pm 1$. The partition functions are given by the characters $\chi_{\mathbf{1}}(\tau),\chi_{\psi}(\tau),\chi_{\sigma}(\tau),$ of two Ising CFTs (see Appendix A.1), one for right movers and the other for the left movers. We find

[TABLE]

This means that the partition functions for the even and odd $Z_{2}$ sectors are given by

[TABLE]

For the anomalous CFT on the boundary of 2+1D $Z_{2}$ topological order, its partition function is given by the partition function of the Ising model for the even $Z_{2}$ sector

[TABLE]

If we insert the $e$ world-line in the bulk (see Fig. 2), the corresponding partition function $Z(\tau,\overline{\tau},e)$ is given by $Z_{\text{odd}}(\tau,\overline{\tau})$:

[TABLE]

Similarly, we find

[TABLE]

and

[TABLE]

We find that the above partition functions $Z(\tau,\overline{\tau},i)$, $i=\mathbf{1},e,m,f$, indeed satisfy eqn. (II.5). Those partition functions describe a 1+1D gapless theoy with a non-invertible gravitational anomaly, which can appear as a boundary of the 2+1D $Z_{2}$ topological order.

IV A non-invertible bosonic global gravitational

anomaly from 2+1D DS topological order

Now let us consider the boundary of the 2+1D DS topological order. Since the DS topological order can be viewed as a gauged 2+1D $Z_{2}$ symmetric state with the non-trivial $Z_{2}$ SPT order, we will first consider the boundary theory of the 2+1D $Z_{2}$ SPT state on a 1d ring with even number of sites:Chen and Wen (2012)

[TABLE]

The above Hamiltonian has a non-on-site $Z_{2}$ symmetry generated by

[TABLE]

where $CZ_{ij}$ acts on two spins as

[TABLE]

From Appendix B, we see that the above Hamiltonian in eqn. (IV) is $Z_{2}$ symmetric. But the $Z_{2}$ symmetry has a ’t Hooft anomaly.

To have a theory that is defined on rings with both even and odd sites, we should consider different (but equivalent) non-on-site $Z_{2}$ symmetry:

[TABLE]

where $s_{ij}$ acts on two spins as

[TABLE]

The $Z_{2}$ transformation has a simple picture: it flips all the spins and include a $(-)^{N_{\uparrow\to\downarrow}}$ phase, where $N_{\uparrow\to\downarrow}$ is the number of $\uparrow\to\downarrow$ domain wall . From Appendix B, we see that the Hamiltonian eqn. (IV) is also invariant under the new $Z_{2}$ transformation.

The boundary of 2+1D $Z_{2}$ SPT state described by eqn. (IV) has a symmetry breaking phase when $U\gg J$. The boundary can also be gapless described by a $c=\overline{c}=1$ CFT when $U=0$. Eqn. (IV) has no symmetric gapped phase, since the $Z_{2}$ symmetry is not on-site (*i.e. *has a ’t Hooft anomaly).Chen et al. (2011)

When $U=0$, the model (IV) can be mapped to the XY-model on 1d lattice:Chen and Wen (2012)

[TABLE]

In this case the anomalous 1+1D theory is gapless and is described by a $u1_{4}\times\overline{u1}_{4}$ CFT (see Appendix A.2). It has a partition function

[TABLE]

with primary fields of dimension $(h_{i},\overline{h}_{i})=(\frac{i^{2}}{8},\frac{i^{2}}{8})$. The above partition can be rewritten as

[TABLE]

where $(a,b)$ form a lattice $\Gamma$ (see Fig. 4):

[TABLE]

So all the $U(1)$ vertex operators in our XY-model can be labeled by $(l,m)$ which have scaling dimension

[TABLE]

This is the labeling scheme used in LABEL:CW1235. It was found that the $U(1)$ vertex operator labeled by $(l,m)$ carry the $Z_{2}$-charge $l+m$ mod 2.

We see that each character $|\chi_{i}^{u1_{4}}|^{2}$ contains $U(1)$ vertex operators with different $Z_{2}$-charges. Thus it is more convenient to rewrite the partition function in terms of the $u1_{16}$ characters

[TABLE]

The $U(1)$ vertex operators in $|\chi_{2i}^{u1_{16}}|^{2}$ carry the $Z_{2}$-charge $i$ mod 2. The $U(1)$ vertex operators in $\overline{\chi}_{2i+4}^{u1_{16}}\chi_{2i}^{u1_{16}}$ carry the $Z_{2}$-charge $i+1$ mod 2.

In the presence of the $Z_{2}$-symmetry, we can define 4-partition functions for different $Z_{2}$-symmetry twists in the space and time directions $(a_{x},a_{t})=(\pm 1,\pm 1)$. Z_{\text{Z_{2}-SPT}} is the partition function with no symmetry twist $(a_{x},a_{t})=(1,1)$:

[TABLE]

with a $Z_{2}$-symmetry twist in time directions the terms with $Z_{2}$-charge 1 acquire a $-$ sign:

[TABLE]

After an $S$-transformation of $u1_{16}$ (see Apendix A.2), we get

[TABLE]

From $Z_{-1,1}$ we find

[TABLE]

by adding a $-$ sign to the terms with $Z_{2}$-charge 1.

Now we gauge the $Z_{2}$ on-site symmetry in the 2+1D SPT state to obtain the 2+1D DS topological order. The 2+1D DS topological order has a gapped boundary which contains topological excitation $s$ that satisfies a $Z_{2}$ fusion role $s\otimes s=1$. The 1d particles with $Z_{2}$ fusion role are described by one of the two fusion categories. The first one is ${{\cal R}\text{ep}}(Z_{2})$ mentioned in the last section. The second one is a different fusion category, which we refer as the semion fusion category.Freedman et al. (2004); Levin and Wen (2005) Such a gapped boundary can be described by eqn. (IV) in $U\gg J$ limit (*i.e. *in the $Z_{2}$ symmetry breaking phase), where the $Z_{2}$ domain walls correspond to the boundary particle $s$. The fusion of those domain walls is described by the semion fusion category, provided that the fusion processes preserve the non-on-site $Z_{2}$ symmetry (43),

However, there is one problem with the above picture: in the $Z_{2}$ symmetry breaking phase, all the domain wall configurations have 2-fold degeneracy induced by the $Z_{2}$ transformation eqn. (43). To fix this problem, we need to modify the many-body Hilbert space on a ring by imposing the constraint

[TABLE]

*i.e. *we include only even $Z_{2}$-charge states in our many-body Hilbert space. The model eqn. (IV), together with the $Z_{2}$-even Hilbert space, describes the boundary of the 2+1D DS topological order. Such a 1+1D theory has a non-invertible gravitational anomaly described by 2+1D DS topological order.

Now we see that using the partition functions $Z_{a_{x},a_{t}}$ of the model eqn. (IV) with different $Z_{2}$-symmetry twists, we can construct the four partition functions for the gapless boundary of 2+1D DS topological order. For example, the partition function of the model eqn. (IV) in the even $Z_{2}$ charge sector, $\frac{Z_{1,1}+Z_{1,-1}}{2}$, corresponds to the partition function for the boundary of the DS topological order without an insertion, $Z(\tau,\overline{\tau},\mathbf{1})$. Note that the DS topological order has four types of excitations: trivial excitation $\mathbf{1}$, semion $s$, conjugate semion $s^{*}$, and topological boson $b$. Thus the boundary has four partition functions $Z(\tau,\overline{\tau},\mathbf{1})$, $Z(\tau,\overline{\tau},s)$, $Z(\tau,\overline{\tau},s^{*})$, and $Z(\tau,\overline{\tau},b)$, which are given by

[TABLE]

The 2+1D DS topological order is characterized by (in the basis of $\mathbf{1},s,s^{*},b$)

[TABLE]

Using the above $S^{\text{top}}_{\text{DS}}$ and $T^{\text{top}}_{\text{DS}}$ and the modular transformations of $u1_{16}$ in Appendix A.2, we can explicitly check that the four boundary partition functions eqn. (IV) indeed satisfy the modular covariance eqn. (II.5).

V Non-invertible gravitational anomaly and “non-locality” of

Hilbert space

In LABEL:W1313, it was stressed that the ’t Hooft anomaly of a global symmetry in a theory is not an obstruction for the theory to have an ultra-violate (UV) completion (*i.e. *to have a lattice realization). Such an anomalous theory can still be realized on a lattice, however, the global symmetry has to be realized as an non-on-site symmetry in the lattice model.

In this section, we would like to propose that a non-invertible gravitational anomaly in a theory is not an obstruction for the theory to have a UV completion. The anomalous theory can still be realized on a lattice, if there is no perturbative gravitational anomaly. However, the Hilbert space of UV theory ${\cal V}$ is not given by the lattice Hilbert space ${\cal V}_{\text{latt}}$: ${\cal V}\neq{\cal V}_{\text{latt}}$.

The lattice Hilbert space ${\cal V}_{\text{latt}}$ has a tensor product decomposition

[TABLE]

where ${\cal V}_{i}$ is the Hilbert space for each lattice site. We call the Hilbert space with such a tensor product decomposition as local Hilbert space. A system with such a local Hilbert space is free of gravitational anomaly, by definition.

In contrast to an anomaly-free theory, the UV completion of a theory with non-invertible gravitational anomaly does not have a local Hilbert space (*i.e. *with the above tensor product decomposition). In other words, a non-invertible gravitational anomaly is not an obstruction to have a UV completion, but for the UV completion to have a local Hilbert space. This understanding of non-invertible gravitational anomaly is supported by the example discussed in the last section.

In last section, we pointed out the boundary of 2+1D $Z_{2}$ topological order (which has a non-invertible gravitational anomaly) has a UV completion described by a lattice model (32), with a constraint on the Hilbert space $\prod_{i}\sigma^{z}_{i}=1$. It is the constraint $\prod_{i}\sigma^{z}_{i}=1$ that makes the Hilbert space non-local.

Let us describe the above result using a more physical reasoning. One boundary of 2+1D $Z_{2}$ topological order has a single type of topological excitations $e$, which have a mod 2 conservation. The Hilbert space always has an even number of $e$ particles. On the other hand, when there is no $e$-particle excitations, the boundary ground state is not degenerate. Here we like to point out that

the even-particle constraint (i.e. $Z_{2}$ fusion) plus the non-degeneracy of the ground state is a sign of 1+1D non-invertible gravitational anomaly.

The example in the last section supports such a claim. The even-particle constraint is imposed by $\prod_{i}\sigma^{z}_{i}=1$. The non-degenerate ground state is given by $\otimes_{i}|\sigma^{z}=1\rangle$. Such a theory describes a boundary of 2+1D $Z_{2}$ topological order and has a non-invertible anomaly. The Ising model may also in the symmetry breaking phase. Due to the constraint $\otimes_{i}|\sigma^{z}=1\rangle$, the symmetry breaking phase also has a unique ground state $\otimes_{i}|\sigma^{x}=1\rangle+\otimes_{i}|\sigma^{x}=-1\rangle$. In such a symmetry breaking phase, there are always an even number of domain walls, that correspond to an even number of topological excisions.

On the other hand, even-particle constraint plus two-fold degenerate ground states will lead to an anomaly-free theory. We can consider an Ising model in symmetry breaking phase and without the constraint on Hilbert space. Such a phase has two-fold degenerate ground states and the number of domain walls (which correspond to the $e$-particles) is always even. Thus even-particle constraint plus two-fold degenerate ground states can be realized by a lattice model with local Hilbert space and is thus anomaly-free.

There is also a mathematical way to understand the above claim. The $e$-particles with mod 2 conservation in 1+1D can be described by a fusion category with a $Z_{2}$ fusion ring. There are only two different fusion categories with a $Z_{2}$ fusion ring, both have 1+1D non-invertible anomaly. One fusion category describes the boundary of 2+1D $Z_{2}$ topological order, and the other describes the boundary of DS topological order. Both non-invertible anomalies can be described by the following Ising model

[TABLE]

but with different constraints on Hilbert space. The anomaly corresponds to the $Z_{2}$ topological order has a constraint $\prod_{i}\sigma^{x}_{i}=1$, and the anomaly corresponds to the DS topological order has a constraint $\prod_{i}\sigma^{x}_{i}\prod_{i}s_{i,i+1}=1$ (see eqn. (43)).

VI Systematical search of gapped and gapless boundaries of a 2+1D

topological order

VI.1 Boundaries of 2+1D topological order

In this section, we want to systematically find gapped and gapless boundaries of a 2+1D topological order by solving eqn. (II.5) from the data $S^{\text{top}},T^{\text{top}}$ of the bulk topological order. This is a generalization of finding possible 1+1D critical theories via finding modular invariant partition functions. Note that, regardless of whether the boundary is gapped or gapless, it always has the same anomaly characterized by the bulk topological order.

To solve eqn. (II.5), we may start with a CFT with partition functions $Z_{\text{bdy}}(\tau,\overline{\tau},I)$, which transform as

[TABLE]

We then construct $Z(\tau,\overline{\tau},i)$ via

[TABLE]

Now eqn. (II.5) becomes

[TABLE]

We see that $M_{iI}$ must satisfy

[TABLE]

We also note that, for a fixed $i$, $Z(\tau,\overline{\tau},i)$ can be zero, indicating the always presence of gapped excitations on the boundary. $Z(\tau,\overline{\tau},i)$ can also be a $\tau$-independent positive integer. It means that the ground states are gapped and have a degeneracy given by $Z(\tau,\overline{\tau},i)$. Otherwise, $Z(\tau,\overline{\tau},i)$ has an expansion

[TABLE]

where $(h_{i},\overline{h}_{i})$ are the scaling dimensions for the type-$i$ topological excitation. Such an expansion describes the many-body spectrum of the gapless boundary of the disk $D^{2}_{i}$, with a type-$i$ topological excitation at the center of the disk. Here the subscript $i$ in $D^{2}_{i}$ indicates the type-$i$ excitation on the disk. Let us assume the boundary $S^{1}=\partial D^{2}_{i}$ has a length $L$. Then $D_{n,\overline{n}}(i)$ is number of many-body states on $D^{2}_{i}$ with energy $(n+h_{i}+\overline{n}+\overline{h}_{i})\frac{2\pi}{L}$, and momentum $(n+h_{i}-\overline{n}-\overline{h}_{i})\frac{2\pi}{L}$. Here we have assumed that velocity of the gapless excitations is $v=1$. Thus $D_{n,\overline{n}}(i)$ are non-negative integers.

Also $D_{0,0}(i)$ is the ground state degeneracy on the boundary of the disk $D^{2}_{i}$. Since the boundary can be gapless, the ground state degeneracy needs to be defined carefully. Here, we view two energy levels with an energy difference of order $2\pi/L$ as non-degenerate. We view two energy levels with an energy difference smaller than $(2\pi/L)^{\alpha}$, $\alpha>1$, as degenerate. It is in this sense we define the ground state degeneracy $D_{0,0}(i)$ for a gapless system in $L\to\infty$ limit. We believe that the ground state degeneracy on disk $D^{2}$ is always 1. Therefore, we like to impose a nondegeneracy condition on the boundary $D_{0,0}(\mathbf{1})=1$. $Z_{\text{bdy}}(\tau,\overline{\tau},I)$ satisfies a similar quantization condition.

From eqn. (61), we see that $M_{iI}$ is the multiplicity of the number of energy levels in the many-body spectrum of the boundary theory. Therefore, for a fixed $i$, if $M_{iI}\neq 0$, then

[TABLE]

In practice, to find $M_{iI}$, we may compute the eigenvectors of $T^{\text{top}}\otimes T^{*}+S^{\text{top}}\otimes S^{*}$ with eigenvalue 2, that satisfy the above quantization condition.

VI.2 $Z_{2}$ topological order

To find a CFT that describes a boundary of 2+1D $Z_{2}$ topological order, we need to solve eqn. (II.5) with $S^{\text{top}},\ T^{\text{top}}$ given by eqn. (27) that characterize the 2+1D $Z_{2}$ topological order. Let us first try to find gapped boundaries by choosing $Z_{\text{bdy}}(\tau,\overline{\tau})=1$, the partition function of a trivial gapped 1+1D state. Now eqn. (63) reduces to

[TABLE]

So we need to find common eigenvectors of $S^{\text{top}}_{Z_{2}}$ and $T^{\text{top}}_{Z_{2}}$, both with eigenvalue 1. We also require the solutions to satisfy the quantization condition eqn. (VI.1), *i.e. *the components of the solutions are all non-negative integers. The condition $D_{0,0}(\mathbf{1})=1$ becomes $Z(\mathbf{1})=1$. This agrees with the fact that the ground state of 2+1D $Z_{2}$ topological order on a disk $D^{2}$ is non-degenerate if there is no accidental degeneracy. This can be achieved by finding eigenvectors of $S^{\text{top}}_{Z_{2}}+T^{\text{top}}_{Z_{2}}$ that satisfy eqn. (VI.1).

We find that $S^{\text{top}}_{Z_{2}}+T^{\text{top}}_{Z_{2}}$ has two eigenvectors with eigenvalue 2, given by

[TABLE]

where $i=(\mathbf{1},e,m,f)$. They are the only two non-negative integral eigenvectors with $Z(\mathbf{1})=1$. Thus the 2+1D $Z_{2}$ topological order has only two types of gapped boundaries, an $e$ condensed boundary described by $Z_{e}(i)$ and an $m$ condensed boundary described by $Z_{m}(i)$.Lan et al. (2015)

If we choose $Z_{\text{bdy}}(\tau,\overline{\tau},I)$ to be the partition functions (the characters) of $\text{Is}\otimes\overline{\text{Is}}$ CFT (see Appendix A.1), then $S,T$ will be $9\times 9$ matrices:

[TABLE]

where $S_{\text{Is}},T_{\text{Is}}$ are given in eqn. (A.1). We find eigenvalue 2 for $T^{\text{top}}_{Z_{2}}\otimes T_{\text{Is}\otimes\overline{\text{Is}}}^{*}+S^{\text{top}}_{Z_{2}}\otimes S_{\text{Is}\otimes\overline{\text{Is}}}^{*}$ to be 3-fold degenerate. We obtain the following three solutions of eqn. (II.5)

[TABLE]

that satisfy the quantization condition eqn. (VI.1).

The first two solutions correspond to the two gapped boundaries of the 2+1D $Z_{2}$ topological order induced by $e$ and $m$ condensation respectively, and then stacking with a transverse Ising model at critical point. So the first two solutions are regarded as gapped boundaries. Here we would like introduce the notion of reducible boundary. If the partition functions $Z(\tau,\overline{\tau},i)$ of a boundary has a form

[TABLE]

then we say the boundary is reducible. We will call the boundary described by $Z^{\prime}(\tau,\overline{\tau},i)$ as the reduced boundary. Here $Z(\tau,\overline{\tau},i)$ and $Z^{\prime}(\tau,\overline{\tau},i)$ are partition functions satisfying (II.5) and eqn. (VI.1), and $Z_{\text{inv}}(\tau,\overline{\tau})$ is a modular invariant partition function satisfying eqn. (VI.1). Noticing that $|\chi_{\mathbf{1}}(\tau)|^{2}+|\chi_{\psi}(\tau)|^{2}+|\chi_{\sigma}(\tau)|^{2}$ is modular invariant, so the first two boundaries are reducible and their reduced boundary are gapped boundaries described by eqn. (67).

The third solution (71) corresponds to an irreducible gapless boundary. Now we like to consider the stability of such $c=\overline{c}=\frac{1}{2}$ gapless boundary. But before that, we want review the stability of the critical point of transverse Ising model described by

[TABLE]

From the above partition function, we see that there are two relavant operators: $\overline{\psi}\psi$ with scaling dimension $(h,\overline{h})=(\frac{1}{2},\frac{1}{2})$, and $\overline{\sigma}\sigma$ with scaling dimension $(h,\overline{h})=(\frac{1}{16},\frac{1}{16})$. Among the two, $\overline{\sigma}\sigma$ is odd under the $Z_{2}$ symmetry of the transverse Ising model.

Similarly, to examine the stability of the gapless boundary (71), we examine the partition function $Z(\tau,\overline{\tau},\mathbf{1})$. We do not consider other partition functions, since the partition function $Z(\tau,\overline{\tau},\mathbf{1})$ describes the physical boundary of Fig. 2 without the insertion of the world-line. From $Z(\tau,\overline{\tau},\mathbf{1})$, we see that gapless boundary (71) has only one relavant operator $\overline{\psi}\psi$ with scaling dimension $(h,\overline{h})=(\frac{1}{2},\frac{1}{2})$. So the gapless boundary (71) can be the phase transition point between two gapped boundaries. In fact, according to the discussion in Section III.2, the third gapless boundary is the critical transition point between the gapped $e$ condensed boundary and the $m$ condensed boundary.

We can also use the characters $\chi^{m4}_{h}$ of the $(4,5)$ minimal model (or tricritical Ising model Francesco et al. (2012)) with central charge $c=\overline{c}=\frac{7}{10}$, to construct the boundary partition functions $Z_{\text{bdy}}(\tau,\overline{\tau},I)$ that have the $Z_{2}$ non-invertible anomaly (*i.e. *satisfy eqn. (II.5)). We obtain

[TABLE]

If we choose $Z_{\text{bdy}}(\tau,\overline{\tau},I)$ to be built from the characters of $u(1)_{M}\otimes\overline{u(1)}_{M}$ CFT, we obtain the following simple solution of gapless boundary

[TABLE]

Note that $Z(\tau,\overline{\tau},e)$ and $Z(\tau,\overline{\tau},m)$ are no longer identical, but differ by a charge conjugation, whose action induce on the characters is $C:\chi_{i}^{u1_{M}}\overline{\chi}_{j}^{u1_{M}}\rightarrow\chi_{i}^{u1_{M}}\overline{\chi}_{M-j}^{u1_{M}}$.

VI.3 Double-semion topological order

To find gapped boundaries of 2+1D DS topological order, we need to solve

[TABLE]

where $T^{\text{top}}_{\text{DS}},S^{\text{top}}_{\text{DS}}$ are given by eqn. (57). We find that $S^{\text{top}}_{\text{DS}}+T^{\text{top}}_{\text{DS}}$ has only one eigenvector with eigenvalue 2, given by

[TABLE]

where $i=(\mathbf{1},s,s^{*},b)$. Thus the 2+1D DS topological order has only one type of gapped boundary, a $b$ condensed boundary.Lan et al. (2015)

Next, we consider possible gapless boundaries of DS topological order described by $\text{Is}\otimes\overline{\text{Is}}$ CFT, by solving eqn. (63) for solutions satisfying eqn. (VI.1). We find only one eigenvector for $T^{\text{top}}_{\text{DS}}\otimes T_{\text{Is}\otimes\overline{\text{Is}}}^{*}+S^{\text{top}}_{\text{DS}}\otimes S_{\text{Is}\otimes\overline{\text{Is}}}^{*}$ with eigenvalue $2$. We obtain the following unique solution of eqn. (II.5)

[TABLE]

Such a solution corresponds to the gapped boundary of 2+1D DS topological order, and then stacking with a transverse Ising model at critical point. So this solution is regarded as a gapped boundary. There is no irreducible gapless boundary described by $\text{Is}\otimes\overline{\text{Is}}$.

Actually, we can obtain an even stronger result

2+1D DS topological order has no irreducible gapless boundary with central charge $c=\overline{c}<\frac{25}{28}$.

This result is obtained by realizing that the DS anomalous partition functions, for irreducible gapless boundary, has a non-zero component $Z(\tau,\overline{\tau},s)$. Otherwise, the gapless boundary can be viewed as a gapped boundary stacked with an anomaly-free 1+1D CFT. The condition (II.5) for the $T^{\text{top}}$-transformation requires that the excitations in the partition function has topological spin $h-\overline{h}=\frac{1}{4}$ mod 1. This constraint the central charge of the anomalous CFT. If the CFT has a central charge $c=\overline{c}<1$, then the boundary CFT must be given by a chiral-anti-chiral minimal model $C^{\text{ft}}_{p,p+1}\times\overline{C}^{\text{ft}}_{p,p+1}$. The topological spin for the operators in such CFT is given by $s_{r,s,r^{\prime},s^{\prime}}=h_{r,s}-h_{r^{\prime},s^{\prime}}$ (see eqn. (107)). We find that, for $p<7$, $s_{r,s,r^{\prime},s^{\prime}}$ cannot be $\frac{1}{4}$ mod 1. Thus the condition eqn. (II.5) cannot by satisfied for $T^{\text{top}}$ transformation.

Last, we consider possible gapless boundary theories of DS topological order described by $u1_{M}\otimes\overline{u1}_{M}$ CFT, by solving eqn. (63) for solutions satisfying eqn. (VI.1). This includes many cases, one for each choice of $M$. So we need to consider each case separately.

For $M=16$, we have found an irreducible gapless boundary described by $u1_{16}\otimes\overline{u1}_{16}$ CFT:

[TABLE]

However, it is not clear if there are other irreducible gapless boundaries described by $u1_{16}\otimes\overline{u1}_{16}$ CFT.

From the partition function $Z(\tau,\overline{\tau},\mathbf{1})$, we see that there is an relevant operator with scaling dimension $(h,\overline{h})=(\frac{4^{2}}{2\times 16},\frac{4^{2}}{2\times 16})=(\frac{1}{2},\frac{1}{2})$. So the gapless boundary (79) is unstable. It describes the transition point between two gapped phases in eqn. (IV). One gapped phase for $U>0$ and other gapped phase for $U<0$. The gapless critical point is described by $U=0$.

For $M=4$, we find that there is no irreducible gapless boundary described by $u1_{4}\otimes\overline{u1}_{4}$ CFT.

For $M=2$, we find that there is only one irreducible gapless boundary described by $u1_{2}\otimes\overline{u1}_{2}$ CFT:

[TABLE]

There is no other irreducible gapless boundary described by $u1_{2}\otimes\overline{u1}_{2}$ CFT. But there is a reducible gapless boundary described by

[TABLE]

which is a stacking of a gapped boundary described by eqn. (77) and the CFT for spin-1/2 Heisenberg chain.

From the partition function $Z(\tau,\overline{\tau},\mathbf{1})$, we find that the irreducible boundary (80) has no relevant operator. It has only several marginal operators, such as $\overline{J}J$, $\hskip 1.0pt\mathrm{e}^{\pm\hskip 1.0pt\mathrm{i}\hskip 1.0pt\sqrt{2}\phi}\hskip 1.0pt\mathrm{e}^{\pm\hskip 1.0pt\mathrm{i}\hskip 1.0pt\sqrt{2}\overline{\phi}}$ with scaling dimension $(h,\overline{h})=(1,1)$. Here $J$ is the $U(1)$ current operator and $\hskip 1.0pt\mathrm{e}^{\pm\hskip 1.0pt\mathrm{i}\hskip 1.0pt\sqrt{2}\phi}$ are $U(1)$-charged operators. Those operators can be marginally relavent. If there is only one marginally relavent operator $g\hat{O}$ in the Hamiltonian, the renomalization group (RG) flow of the coupling constant $g$ is given by

[TABLE]

We see that regardless the sign of $\alpha$, there is a finite region of $g$ where $g$ flows to zero. In this case, the CFT can be stable. When there are many marginally relevant operators $g_{i}\hat{O}_{i}$, RG flow of the coupling constants $g_{i}$ is given byCardy (1996)

[TABLE]

In Appendix D, we discuss the above RG equation in more details and show that generic coupling constants $g_{i}$ always flow to infinite. Thus, the CFT is unstable, and the 2+1D DS topological order always has a gapped boundary without fine tuning.

We like remark that from this gapless boundary of DS topological order, and apply the relations (IV), we can find another galpess boundary theory of $\mathbb{Z}_{2}$ SPT, whose partition function is given by

[TABLE]

which is different from eqn. (46). The partition function (84) can also be rewritten as

[TABLE]

where $(a,b)$ form a lattice $\Gamma^{u1_{2}}$,

[TABLE]

The $\mathbb{Z}_{2}$ charges of the vectex operators in $|\chi_{i}^{u1_{2}}|^{2}$ is $i\mod 2$, or $(l+m)\mod 2$ on the lattice.

From the $Z_{2}$-even partition function $|\chi_{0}^{u1_{2}}(q)|^{2}$, we find that the gapless boundary (84) has no $Z_{2}$-even relevant operator. However, as mentioned above, there may be many marginally relavent operators, and it is not clear if the gapless boundary (84) of 2+1D $Z_{2}$-SPT order is perturbatively stable or not. In some previous studies, a gapless boundary (46) for the same 2+1D $Z_{2}$-SPT order is found to be perturbatively unstable against $Z_{2}$ symmetric perturbations,Chen and Wen (2012); Levin and Gu (2012) via relavent perturbations (with total scaling dimension less than 2). In this paper, we found a gapless boundary of $Z_{2}$-SPT state (84), which is more stable against $Z_{2}$ symmetric perturbations, in the sense that the instability only come from potentially marginally relevant operators (with total scaling dimension equal to 2).

VI.4 Single-semion topological order

There is a close relative of 2+1D DS topological order – 2+1D single-semion (SS) topological order, which has only two types of excitations: trivial excitation $\mathbf{1}$ and semion $s$. The 2+1D SS topological order can be realized by $\nu=1/2$ bosonic Laughlin state.

Let us describe the data that characterizes the 2+1D SS topological order. The topological spins and the quantum dimensions of $\mathbf{1}$ and $s$ are $(s_{\mathbf{1}},s_{s})=(0,\frac{1}{4})$ and $(d_{\mathbf{1}},d_{s})=(1,1)$. The topological $S^{\text{top}}_{\text{SS}},T^{\text{top}}_{\text{SS}}$ matrices are

[TABLE]

To obtain the possible boundaries of 2+1D SS topological order, we just need to solve eqn. (II.5). We find a simple boundary described by the following partition function (in terms of $u1_{2}$ characters (109))

[TABLE]

The 1+1D theory described by the above partition functions has both perturbative and global gravitational anomaly.

VI.5 Fibonacci topological order

Another simple 2+1D topological order is the Fibonacci topological order. It is characterized by the following topological data. The central charge is $\frac{14}{5}\mod 8$. There are two types of excitations $\mathbf{1}$ and $\gamma$. Their topological spins and the quantum dimensions are $(s_{\mathbf{1}},s_{\gamma})=(0,\frac{2}{5})$ and $(d_{\mathbf{1}},d_{s})=(1,\phi)$, where $\phi=\frac{\sqrt{5}+1}{2}$, the golden ratio. The topological $S^{\text{top}}_{\text{Fib}},T^{\text{top}}_{\text{Fib}}$ matrices are

[TABLE]

Solving eqn. (II.5), we can find several gapless boundary of the Fibonacci topological order:

• •

$(G_{2})_{1}$ CFT with central charge $(c,\overline{c})=\left(\frac{14}{5},0\right)$, with the partition functions

[TABLE]

where $\chi_{i}^{G2_{1}}(\tau)$ are the characters of level-1 $G_{2}$ current algebra, see Appendix A.4. The first multiplicity equaling $7$ appearing in $Z(\tau,\gamma)$ implies that when there is a Fibonacci anyon in the bulk, the boundary has $7$-fold degeneracy. The degeneracy cannot be split unless the anyon is moved to the boundary.

• •

$su(2)_{3}\times u(1)_{M}$ CFT has a central charge $c=\frac{9}{5}+1=\frac{14}{5}$ and $\overline{c}=0$. The $su2_{3}$ CFT has four chiral characters $\chi_{j}^{su2_{3}}$, labeled by the spin $j=0,\frac{1}{2},1,\frac{3}{2}$ (see Appendix A.3) with $S,T$-matrices

[TABLE]

When $M=2$, we find a solution of eqn. (II.5):

[TABLE]

In fact, we find the expansion of the $Z(\tau,i)$ in eqn. (92) in terms of modular parameter $q=\hskip 1.0pt\mathrm{e}^{\hskip 1.0pt\mathrm{i}\hskip 1.0pt2\pi\tau}$ to be the same as that of eqn. (• ‣ VI.5).

• •

The same result also arises in $su(2)_{28}$ with $c=\frac{14}{5}$,

[TABLE]

and see Appendix A.3 for explicit forms of characters.

• •

$(E_{8})_{1}\times\overline{(F_{4})_{1}}$ CFT, with central charge $(c,\overline{c})=\left(8,\frac{26}{5}\right)$, and $c-\overline{c}=\frac{14}{5}$. $(F_{4})_{1}$ CFT has the $S,T$ matrices

[TABLE]

Therefore, $(E_{8})_{1}\times\overline{(F_{4})_{1}}$ is also a gapless boundary of the Fibonacci topological order.

[TABLE]

VII Detect anomalies from 1+1D partition functions

So far, we have discussed how to use anomaly to constraint the structure of 1+1D partition function. In this section, we are going to consider a different problem: given a partition function, how to determine its anomaly? We have mentioned that the 1+1D perturbative gravitational anomaly can be partially detected via $q\to 0$ limit of partition function (see eqn. (12)). So here we will concentrate on global gravitational anomalies.

Let us consider partition functions constructed using the characters of Ising CFT:

[TABLE]

Under modular transformation $Z_{M}$ transforms as

[TABLE]

where $S_{\text{Is}},T_{\text{Is}}$ are given by eqn. (A.1).

Let us consider a particular partition function

[TABLE]

which is not modular invariant. Starting from $M_{\mathbf{1}}$, the modular transformations (VII) generate two other partition functions described by

[TABLE]

The actions of modular transformations on $Z_{M_{\mathbf{1}}}$, $Z_{M_{2}}$, and $Z_{M_{3}}$ are described by Fig. 5. Such orbits of modular transformations can be used to characterize the anomaly in the partition function. However, it is not clear if such a characterization is complete or not, *i.e. *it is not clear if different anomalies always have different orbits. However, the orbits in Fig. 5 are consistent with the 1+1D anomaly described by 2+1D $Z_{2}$ topological order. This is because the $S^{\text{top}}_{Z_{2}},T^{\text{top}}_{Z_{2}}$ transformations of the 2+1D $Z_{2}$ topological order (27), when acting on

[TABLE]

will generate

[TABLE]

The actions of $S^{\text{top}}_{Z_{2}},T^{\text{top}}_{Z_{2}}$ on $|\mathbf{1}\rangle$, $|2\rangle$, and $|3\rangle$ will generate the same orbits as in Fig. 5.

We may also consider a partition function constructed using $u1_{16}$ characters:

[TABLE]

Starting from $Z(\tau,\overline{\tau},\mathbf{1})=Z_{1}(\tau,\overline{\tau})$, using modular transformations $S,T$ in eqn. (110), we can generate five additional partition functions $Z_{n}(\tau,\overline{\tau})$, $n=2,3,4,5,6$. Under the modular transformations $S,T$, the partition functions $Z_{n}(\tau,\overline{\tau}),\ n=1,\cdots,6$ change into each other. The orbits are described by Fig. 6. Such orbits are consistent with the 1+1D anomaly described by 2+1D DS topological order

VIII Summary

In this paper, we study non-invertible gravitational anomalies that correspond to non-invertible topological orders in one higher dimension. A theory with a non-invertible anomaly can have many partition functions, which are linear combinations of $N$ partition functions. For 1+1D non-invertible anomaly, $N$ is the number types of the topological excitations in the corresponding 2+1D topological order. The anomalous 1+1D partition functions $Z(\tau,\overline{\tau},i)$, $i=1,\cdots,N$, are not invariant under the modular transformation, but transform in a non-trivial way described by the modular matrices $S^{\text{top}}_{ij}$ and $T^{\text{top}}_{ij}$ that characterize the corresponding 2+1D topological order. Similarly, anomalous theory on an arbitrary close space-time manifold $M^{d}$ also has many partition functions $Z(M^{d},i)$, which transforms according to a representation $R_{M_{d}}$ of the mapping class group $G_{M^{d}}$ of $M^{d}$. The $G_{M^{d}}$ representation $R_{M_{d}}$ describes how the ground states of the corresponding $(d+1)$D topological order transform on a spatial manifold $M^{d}$. As an application of our theory of non-invertible anomaly, we show that for 2+1D DS topological order, its irreducible gapless boundary must have central charge $c=\overline{c}\geq\frac{25}{28}$.

At the beginning of the paper, we mentioned that ’t Hooft anomaly is an obstruction to gauge a global symmetry. However, if we include theories with non-invertible anomaly, then even global symmetry with ’t Hooft anomaly can be gauged, which will result in a theory with a non-invertible anomaly. This is because a theory with ’t Hooft anomaly can be realized as a boundary of SPT state in one dimension higher, where the global symmetry is realized as an on-site-symmetry. We can gauge the global on-site-symmetry in bulk and turn the SPT state into a topologically ordered state. The boundary of the resulting topologically ordered state is the theory obtained by gauging the anomalous global symmetry. This connection between ’t Hooft anomaly and non-invertible gravitational anomaly allows us to use the theory on non-invertible gravitational anomaly developed in this paper to systematically understand ’t Hooft anomaly and its effect on low energy properties. Those issues will be studied in LABEL:STanomaly.

We like to thank Tian Lan and Samuel Monnier for many helpful discussions. This research is partially supported by NSF grant DMS-1664412.

Appendix A Characters of chiral CFTs

A.1 The minimal model CFT

The chiral CFTs with central charge $c<1$ are called the minimal models. They are labeled by two integers $p,p^{\prime}$ with $p,p^{\prime}>2$ and an equivalence $(p,p^{\prime})\sim(p^{\prime},p)$. We demote those CFTs as $C^{\text{ft}}_{p,p^{\prime}}$. The central charge and the dimensions of primary fields are given by

[TABLE]

which satisfy

[TABLE]

The CFTs are unitary if and only if $|p^{\prime}-p|=1$. In this case, the character for the primary field $(r,s)$ is given by

[TABLE]

where $\chi_{r,s}(q)=\chi_{p-r,p^{\prime}-s}(q)$. The $S$-matrix is

[TABLE]

For unitary minimal models $(p,p^{\prime})=(p,p+1)$, we have

[TABLE]

For $c=1/2$ Ising CFT, $p=3,\ p^{\prime}=4$. $(r,s)=(1,1)$ and $(2,3)$ correspond to the identity primary field $1$. $(r,s)=(1,2)$ and $(2,2)$ correspond to primary field $\sigma$ with $h_{\sigma}=\frac{1}{16}$. $(r,s)=(1,3)$ corresponds to primary field $\psi$ with $h_{\psi}=\frac{1}{2}$. In the basis of $\{\chi_{\mathbf{1}},\chi_{\psi},\chi_{\sigma}\}$, the modular transformation is given by

[TABLE]

A.2 $u1_{M}$ CFT

$u1_{M}$ current algebra is generated by the current $\partial_{z}\varphi(z)$ and $\hskip 1.0pt\mathrm{e}^{\hskip 1.0pt\mathrm{i}\hskip 1.0pt\sqrt{M}\varphi}$. The primary fields of the current algebra are $\hskip 1.0pt\mathrm{e}^{\hskip 1.0pt\mathrm{i}\hskip 1.0pt\frac{m}{\sqrt{M}}\varphi},0\leq m\leq M-1$. The character $\chi^{u1_{M}}_{m}$ of $u1_{M}$ CFT is given by

[TABLE]

where $0\leq m<M$ and $R^{2}=M$. Under modular transformation $S$ and $T$, the characters transform as follows,

[TABLE]

In the case of semion model, the left-moving part has two sectors, the vacuum and semion sector. They are primary fields of $u1_{2}$ current algebra.

[TABLE]

A.3 $su2_{k}$ CFT

The CFT of the level-$k$ $SU(2)$ current algebra, $su2_{k}$, has characters $\chi_{j}^{su2_{k}}(\tau)$:

[TABLE]

where $j\in\left\{0,\frac{1}{2},\cdots,\frac{k}{2}\right\}$. Their modular transformations are

[TABLE]

A.4 Exceptional current algebra CFT

Both $(G_{2})_{1}$ and $(F_{4})_{1}$ characters have the form as follows Mathur et al. (1989),

[TABLE]

where $\lambda(\tau)=\left(\frac{\theta_{2}(\tau)}{\theta_{3}(\tau)}\right)^{4}$, in terms of theta functions,

[TABLE]

Under modular transformation, $T:\lambda\rightarrow\lambda(\lambda-1)$, and $S:\lambda\rightarrow 1-\lambda$.

[TABLE]

is the hypergeometric function defined for $|z|<1$, and $x=1+\frac{c}{2}$. The parameters for some examples are

[TABLE]

Appendix B Non-on-site $Z_{2}$ symmetry transformations

The first non-on-site $Z_{2}$ symmetry transformation (41) transforms $\sigma^{x}_{i}$ in the following way (see eqn. (IV))

[TABLE]

The second non-on-site $Z_{2}$ symmetry transformation (43) transforms $\sigma^{x}_{i}$ in the same way (see eqn. (IV)):

[TABLE]

Appendix C Topological path integral on a space-time with world

lines domain walls

C.1 Space-time lattice and branching structure

To find the conditions on the domain-wall data, we need to use extensively the space-time path integral. So we will first describe how to define a space-time path integral. We first triangulate the $3$-dimensional space-time to obtain a simplicial complex ${\cal M}^{3}$ (see Fig. 7). Here we assume that all simplicial complexes are of bounded geometry in the sense that the number of edges that connect to one vertex is bounded by a fixed value. Also, the number of triangles that connect to one edge is bounded by a fixed value, *etc *.

In order to define a generic lattice theory on the space-time complex ${\cal M}^{3}$, it is important to give the vertices of each simplex a local order. A nice local scheme to order the vertices is given by a branching structure.Costantino (2005); Chen et al. (2013) A branching structure is a choice of the orientation of each edge in the $n$-dimensional complex so that there is no oriented loop on any triangle (see Fig. 8).

The branching structure induces a local order of the vertices on each simplex. The first vertex of a simplex is the vertex with no incoming edges, and the second vertex is the vertex with only one incoming edge, *etc *. So the simplex in Fig. 8a has the following vertex ordering: $0<1<2<3$.

The branching structure also gives the simplex (and its sub simplexes) an orientation denoted by $s_{ij\cdots k}=1,*$. Fig. 8 illustrates two $3$-simplices with opposite orientations $s_{0123}=1$ and $s_{0123}=*$. The red arrows indicate the orientations of the $2$-simplices which are the subsimplices of the $3$-simplices. The black arrows on the edges indicate the orientations of the $1$-simplices.

The degrees of freedom of our lattice model live on the vertices (denoted by $v_{i}$ where $i$ labels the vertices), on the edges (denoted by $e_{ij}$ where $\langle ij\rangle$ labels the edges), and on other high dimensional simplicies of the space-time complex (see Fig. 7).

C.2 Discrete path integral

In this paper, we will only consider a type of 2+1D path integral that can be constructed from a tensor set $T$ of two real and one complex tensors: $T=(w_{v_{0}},{d}_{e_{01}}^{v_{0}v_{1}},{C}_{v_{0}v_{1}v_{2}v_{3};\phi_{013}\phi_{123}}^{e_{01}e_{02}e_{03}e_{12}e_{13}e_{23};\phi_{012}\phi_{023}})$. The complex tensor ${C}_{v_{0}v_{1}v_{2}v_{3};\phi_{013}\phi_{123}}^{e_{01}e_{02}e_{03}e_{12}e_{13}e_{23};\phi_{012}\phi_{023}}$ can be associated with a tetrahedron, which has a branching structure (see Fig. 9). A branching structure is a choice of an orientation of each edge in the complex so that there is no oriented loop on any triangle (see Fig. 9). Here the $v_{0}$ index is associated with the vertex-0, the $e_{01}$ index is associated with the edge-$01$, and the $\phi_{012}$ index is associated with the triangle-$012$. They represent the degrees of freedom on the vertices, edges, and triangles.

Using the tensors, we can define the path integral on any 3-complex that has no boundary:

[TABLE]

where $\sum_{v_{0},\cdots;e_{01},\cdots;\phi_{012},\cdots}$ sums over all the vertex indices, the edge indices, and the face indices, $s_{0123}=1$ or $*$ depending on the orientation of tetrahedron (see Fig. 9). We believe such type of path integral can realize any 2+1D topological order.

C.3 Path integral on space-time with natural boundary

On the complex ${\cal M}^{3}$ with boundary: ${\cal B}^{2}=\partial{\cal M}^{3}$, the partition function is defined differently:

[TABLE]

where $\sum_{v_{i};e_{ij};\phi_{ijk}}$ only sums over the vertex indices, the edge indices, and the face indices that are not on the boundary. The resulting $Z({\cal M}^{3})$ is actually a complex function of $v_{i}$’s, $e_{ij}$’s, and $\phi_{ijk}$’s on the boundary ${\cal B}^{2}$: $Z({\cal M}^{3};\{v_{i};e_{ij};\phi_{ijk}\})$. Such a function is a vector in the vector space ${\cal V}_{{\cal B}^{2}}$. (The vector space ${\cal V}_{{\cal B}^{2}}$ is the space of all complex function of the boundary indices on the boundary complex ${\cal B}^{2}$: $\Psi(\{v_{i};e_{ij};\phi_{ijk}\})$.) We will denote such a vector as $|\Psi({\cal M}^{3})\rangle$. boundary) are attached with the tensors $w_{v_{i}}$ and ${d}_{e_{01}}^{v_{0}v_{1}}$. The boundary (122) defined above is called a natural boundary of the path integral.

We also note that only the vertices and the edges in the bulk (*i.e. *not on the boundaries.) When we glue two boundaries together, those tensors $w_{v_{i}}$ and ${d}_{e_{ij}}^{v_{i}v_{j}}$ are added back. For example, let ${\cal M}^{3}$ and ${\cal N}^{3}$ to have the same boundary (with opposite orientations)

[TABLE]

which give rise to wavefunctions on the boundary $|\Psi({\cal M}^{3})\rangle$ and $\langle\Psi({\cal N}^{3})|$ after the path integral in the bulk. Gluing two boundaries together corresponds to the inner product $\langle\Psi({\cal N}^{3})|\Psi({\cal M}^{3})\rangle$. So the tensors $w_{v_{i}}$ and ${d}_{e_{ij}}^{v_{i}v_{j}}$ defines the inner product in the boundary Hilbert space ${\cal V}_{{\cal B}^{2}}$. Therefore, we require $w_{v_{i}}$ and ${d}_{e_{ij}}^{v_{i}v_{j}}$ to satisfy the following unitary condition

[TABLE]

C.4 Topological path integral

We notice that the above path integral is defined for any space-time lattice. The partition function $Z({\cal M}^{3})$ depends on the choices of the space-time lattice. For example, $Z({\cal M}^{3})$ depends on the number of the cells in space-time, which give rise to the leading volume dependent term, in the large space-time limit (*i.e. *the thermodynamic limit)

[TABLE]

where $V$ is the space-time volume, $\epsilon$ is the energy density of the ground state, and $Z^{\text{top}}({\cal M}^{3})$ is the volume independent partition function. It was conjectured that the volume independent partition function $Z^{\text{top}}({\cal M}^{3})$ in the thermodynamic limit, as a function of closed space-time ${\cal M}^{3}$, is a topological invariant that can fully characterize topological order.Kong and Wen (2014); Wen and Wang (2018) So it is very desirable to fine tune the path integral to make the energy density $\epsilon=0$. This can be achieved by fine-tuning the tensors $w_{v_{i}}$ and ${d}_{e_{ij}}^{v_{i}v_{j}}$. But we can do better. We can choose the tensor $(w_{v_{0}}$, ${d}_{e_{01}}^{v_{0}v_{1}}$, ${C}_{v_{0}v_{1}v_{2}v_{3};\phi_{013}\phi_{123}}^{e_{01}e_{02}e_{03}e_{12}e_{13}e_{23};\phi_{012}\phi_{023}})$ to be the fixed-point tensor-set under the renormalization group flow of the tensor network.Levin and Nave (2007); Gu and Wen (2009) In this case, not only the volume factor $\hskip 1.0pt\mathrm{e}^{-\epsilon V}$ disappears, the volume independent partition function $Z^{\text{top}}({\cal M}^{3})$ is also re-triangulation invariant, for any size of space-time lattice. In this case, we refer the path integral as a topological path integral, and denote the resulting partition function as $Z^{\text{top}}({\cal M}^{3})$. $Z^{\text{top}}$ is also referred as the volume independent the partition function, which is a very important concept, since only the volume independent partition functions correspond to topological invariants. In particular, it was conjectured that such kind of topological path integrals describes all the topological order with gappable boundary. For details, see LABEL:KW1458,WW180109938.

The invariance of partition function $Z$ under the re-triangulation in Fig. 10 and 11 requires that

[TABLE]

[TABLE]

We would like to mention that there are other similar conditions for different choices of the branching structures. The branching structure of a tetrahedron affects the labeling of the vertices. For more details, see LABEL:ZW180809394.

C.5 Topological path integral with world-lines

In this paper, we also need to use the space-time path integral with world-lines of topological excitations. We denote the resulting partition function as

[TABLE]

where $i,j,k,\cdots\in\{1,2,\cdots,N\}$ label the type of topological excitations, and $\alpha,\beta,\gamma$ label the different fusion channels (*i.e. *different choices of actions at the junction of three world-lines). The world lines are defined via a different choice of tensors for simplexes that touch the world-lines. In this paper, we will choose the tensors very carefully, so that the path integral with world-lines is also re-triangulation invariant (even for the re-triangulations that involve the world-lines). The different choices of re-triangulation-invariant world-lines are labeled by the different types of topological excitations. In this paper, we will only consider those topological path integrals with re-triangulation invariance.

Appendix D Renormalization group flow of marginal perturbations of $SU(2)_{1}$ CFT

There are in total $9$ terms of marginal perturbations in $SU(2)_{1}$ CFT, composed of left and right currents. Let us first consider the following three couplings:

[TABLE]

The renormalization group (RG) equations have the form

[TABLE]

where $\alpha_{ijk}$ is proportional to the operator product expansion,

[TABLE]

It follows that

[TABLE]

The solution of the beta function has $4$ fixed lines. To solve them, take the form $g_{i}(t)=\lambda_{i}f(t)$, and one finds $\frac{\lambda_{1}\lambda_{2}}{\lambda_{3}}=\frac{\lambda_{2}\lambda_{3}}{\lambda_{1}}=\frac{\lambda_{3}\lambda_{1}}{\lambda_{2}}$. Therefore, $\lambda_{i}=s_{i}\alpha$, where $\alpha\geq 0$, $s_{i}=\pm 1$ to be determined. The RG equations become

[TABLE]

where $s=s_{1}s_{2}s_{3}$. The solution is

[TABLE]

And this leads to the RG solution of fixed lines

[TABLE]

We find that

• •

when $s>0$, the following four fixed lines flow towards infinity

[TABLE]

• •

when $s<0$, the following four fixed lines flow towards $g_{1}=g_{2}=g_{3}=0$

[TABLE]

This allows us to show that there are no stable regions or sheets in the $(g_{1},g_{2},g_{3})$ parameter space, as illustrated in Fig. 12.

Through the above example, we see a general pattern. If there is only one marginally relevant coupling, *i.e. *if we are on a fixed line, then there is a finite region, such that all the couplings in that region flow to zero. This finite region represents the region of stable gapless phase. If there are two marginally relevant couplings, *i.e. *if we are on a plane spanned by two fixed lines, then there is no finite region where the couplings flow to zero. When there are more marginally relevant couplings, the system is getting even more unstable. So we believe that, for our case with 9 marginally relevant couplings, the corresponding CFT is unstable.