arXiv:2601.04318

Framing Anomaly in Lattice Chern-Simons-Maxwell Theory

Ze-An Xu, Jingyuan Chen

Published Jan 7, 2026• 1 citation

Abstract

Framing anomaly is a key property of $(2+1)d$ chiral topological orders, for it reveals that the chirality is an intrinsic bulk property of the system, rather than a property of the boundary between two systems. Understanding framing anomaly in lattice models is particularly interesting, as concrete, solvable lattice models of chiral topological orders are rare. In a recent work, we defined and solved the $U(1)$ Chern-Simons-Maxwell theory on spacetime lattice, showing its chiral edge mode and the associated gravitational anomaly on boundary. In this work, we show its framing anomaly in the absence of boundary, by computing the expectation of a lattice version of the modular $T$ operator in the ground subspace on a spatial torus, from which we extract that $\langle T \rangle$ has a universal phase of $-2π/12$ as expected: $-2π/8$ from the Gauss-Milgram sum of the topological spins of the ground states, and $2π/24$ from the framing anomaly; we can also extract the $2π/24$ framing anomaly phase alone from the full spectrum of $T$ in the ground subspace by computing $\langle T^m \rangle$. This pins down the last and most crucial property required for a valid lattice definition of $U(1)$ Chern-Simons theory.

DOI: 10.48550/arxiv.2601.04318

Related papers

Chiral Phases on the Lattice

Michael Austin DeMarco

arXiv:2203.01427Mar 2, 20220 citations

A Lattice Chiral Boson Theory in $1+1$d

DeMarco, Michael, Ethan Lake, Xiao-Gang Wen

arXiv:2305.03024May 4, 20230 citations

Electric polarization and discrete shift from boundary and corner charge in crystalline Chern insulators

Yuxuan Zhang, Maissam Barkeshli

arXiv:2410.03821Physical review. B./Physical review. BFeb 27, 20254 citations

Geometric and computational aspects of chiral topological quantum matter

Omri Golan

arXiv:2106.10897Jun 21, 20212 citations

Topological phases and quantum criticality from $SU(2)$ Chern-Simons-matter theories

Yunchao Hao, Yingcheng Li, Kangle Li +1 more

arXiv:2608.14180Aug 14, 20260 citations

Chern insulator boundary criticality

Benjamin Moy, Eduardo Fradkin

arXiv:2608.18199Aug 18, 20260 citations

arXiv:2601.04318

Framing Anomaly in Lattice Chern-Simons-Maxwell Theory

Ze-An Xu, Jingyuan Chen

Published Jan 7, 2026• 1 citation

arXiv HTML PDF

Abstract

Framing anomaly is a key property of

(2+1)d

chiral topological orders, for it reveals that the chirality is an intrinsic bulk property of the system, rather than a property of the boundary between two systems. Understanding framing anomaly in lattice models is particularly interesting, as concrete, solvable lattice models of chiral topological orders are rare. In a recent work, we defined and solved the

U(1)

Chern-Simons-Maxwell theory on spacetime lattice, showing its chiral edge mode and the associated gravitational anomaly on boundary. In this work, we show its framing anomaly in the absence of boundary, by computing the expectation of a lattice version of the modular

T

operator in the ground subspace on a spatial torus, from which we extract that

\langle T \rangle

has a universal phase of

-2π/12

as expected:

-2π/8

from the Gauss-Milgram sum of the topological spins of the ground states, and

2π/24

from the framing anomaly; we can also extract the

2π/24

framing anomaly phase alone from the full spectrum of

T

in the ground subspace by computing

\langle T^m \rangle

. This pins down the last and most crucial property required for a valid lattice definition of

U(1)

Chern-Simons theory.

DOI: 10.48550/arxiv.2601.04318

Related papers

Chiral Phases on the Lattice

Michael Austin DeMarco

While chiral quantum field theories (QFTs) describe a wide range of physical systems, from the standard model to topological quantum matter, the realization of chiral QFTs on a lattice has proved to be difficult due to the Nielsen-Ninomiya theorem and the possible presence of quantum anomalies. In this thesis, we use the connection between chiral phases of matter and chiral quantum field theories (QFTs) to define chiral QFTs on a lattice and allow a huge class of exotic field theories to be simulated numerically. Our work builds on the 'mirror fermion' approach to the problem of defining chiral theories on a lattice, which defines chiral field theories as the edge modes of chiral phases. We begin by reviewing the deep connections between chiral phases of matter, chiral field theories, and anomalies. We then develop numerical treatments of an $SU(2)$ chiral field theory, and provide a semiclassically solvable definition of Abelian $2+1$ chiral topological orders. This leads to an exactly solvable definition of chiral $U(1)$ SPT phases with zero correlation length, which we use to extract the edge chiral field theories exactly. These zero-correlation length models are vastly more simple than previous approaches to defining chiral field theories on the lattice.While chiral quantum field theories (QFTs) describe a wide range of physical systems, from the standard model to topological quantum matter, the realization of chiral QFTs on a lattice has proved to be difficult due to the Nielsen-Ninomiya theorem and the possible presence of quantum anomalies. In this thesis, we use the connection between chiral phases of matter and chiral quantum field theories (QFTs) to define chiral QFTs on a lattice and allow a huge class of exotic field theories to be simulated numerically. Our work builds on the 'mirror fermion' approach to the problem of defining chiral theories on a lattice, which defines chiral field theories as the edge modes of chiral phases. We begin by reviewing the deep connections between chiral phases of matter, chiral field theories, and anomalies. We then develop numerical treatments of an $SU(2)$ chiral field theory, and provide a semiclassically solvable definition of Abelian $2+1$ chiral topological orders. This leads to an exactly solvable definition of chiral $U(1)$ SPT phases with zero correlation length, which we use to extract the edge chiral field theories exactly. These zero-correlation length models are vastly more simple than previous approaches to defining chiral field theories on the lattice.

arXiv:2203.01427Mar 2, 20220 citations

A Lattice Chiral Boson Theory in $1+1$ dA Lattice Chiral Boson Theory in $1+1$d

DeMarco, Michael, Ethan Lake, Xiao-Gang Wen

Chiral field theories describe large classes of matter, from the edges of Quantum Hall systems to the electroweak sector of the Standard Model, but defining them on the lattice has been an ongoing challenge due to a no-go theorem precluding free local models, the potential of symmetry anomalies, and sign problems. Some approaches define a $1+1$ d chiral field theory as the edge of a $2+1$ d system and argue that the edge decouples from the bulk, but this can be difficult to verify due to finite size effects and strong interactions. On the other hand, recent work has shown how to define the $2+1$ d bulk theory as an exactly solvable model with zero correlation length, in which case the edge theory may be extracted exactly. We use these techniques to derive a lattice field theory on a $1+1$ d spacetime lattice which carries an anomalous chiral $U(1)$ symmetry with zero chiral central charge. The lattice theory with anomalous chiral $U(1)$ symmetry is always gapless, regardless of lattice interactions. We demonstrate the chiral anomaly by coupling to a background gauge field, develop a field theory which demonstrates the chiral behavior, and show how to assemble a chiral, anomaly-free theory where the gauge field may be taken to be dynamical.Chiral field theories describe large classes of matter, from the edges of Quantum Hall systems to the electroweak sector of the Standard Model, but defining them on the lattice has been an ongoing challenge due to a no-go theorem precluding free local models, the potential of symmetry anomalies, and sign problems. Some approaches define a $1+1$d chiral field theory as the edge of a $2+1$d system and argue that the edge decouples from the bulk, but this can be difficult to verify due to finite size effects and strong interactions. On the other hand, recent work has shown how to define the $2+1$d bulk theory as an exactly solvable model with zero correlation length, in which case the edge theory may be extracted exactly. We use these techniques to derive a lattice field theory on a $1+1$d spacetime lattice which carries an anomalous chiral $U(1)$ symmetry with zero chiral central charge. The lattice theory with anomalous chiral $U(1)$ symmetry is always gapless, regardless of lattice interactions. We demonstrate the chiral anomaly by coupling to a background gauge field, develop a field theory which demonstrates the chiral behavior, and show how to assemble a chiral, anomaly-free theory where the gauge field may be taken to be dynamical.

arXiv:2305.03024May 4, 20230 citations

Electric polarization and discrete shift from boundary and corner charge in crystalline Chern insulators

Yuxuan Zhang, Maissam Barkeshli

Recently, it has been shown how topological phases of matter with crystalline symmetry and $U(1)$ charge conservation can be partially characterized by a set of many-body invariants, the discrete shift ${\mathcal{S}}_{\mathrm{o}}$ and electric polarization ${\stackrel{P\vec}{\mathcal{P}}}_{\mathrm{o}}$, where $\mathrm{o}$ labels a high-symmetry point. Crucially, these can be defined even with nonzero Chern number and/or magnetic field. One manifestation of these invariants is through quantized fractional contributions to the charge in the vicinity of a lattice disclination or dislocation. In this paper, we show that these invariants can also be extracted from the length and corner dependence of the total charge (mod 1) on the boundary of the system. We provide a general formula in terms of ${\mathcal{S}}_{\mathrm{o}}$ and ${\stackrel{P\vec}{\mathcal{P}}}_{\mathrm{o}}$ for the total charge of any subregion of the system which can include full boundaries or bulk lattice defects, unifying boundary, corner, disclination, and dislocation charge responses into a single general theory. These results hold for Chern insulators, despite their gapless chiral edge modes, and for which an unambiguous definition of an intrinsically two-dimensional electric polarization has been unclear until recently. We also discuss how our theory can fully characterize the topological response of quadrupole insulators.Recently, it has been shown how topological phases of matter with crystalline symmetry and $U(1)$ charge conservation can be partially characterized by a set of many-body invariants, the discrete shift ${\mathcal{S}}_{\mathrm{o}}$ and electric polarization ${\stackrel{P\vec}{\mathcal{P}}}_{\mathrm{o}}$, where $\mathrm{o}$ labels a high-symmetry point. Crucially, these can be defined even with nonzero Chern number and/or magnetic field. One manifestation of these invariants is through quantized fractional contributions to the charge in the vicinity of a lattice disclination or dislocation. In this paper, we show that these invariants can also be extracted from the length and corner dependence of the total charge (mod 1) on the boundary of the system. We provide a general formula in terms of ${\mathcal{S}}_{\mathrm{o}}$ and ${\stackrel{P\vec}{\mathcal{P}}}_{\mathrm{o}}$ for the total charge of any subregion of the system which can include full boundaries or bulk lattice defects, unifying boundary, corner, disclination, and dislocation charge responses into a single general theory. These results hold for Chern insulators, despite their gapless chiral edge modes, and for which an unambiguous definition of an intrinsically two-dimensional electric polarization has been unclear until recently. We also discuss how our theory can fully characterize the topological response of quadrupole insulators.

arXiv:2410.03821Physical review. B./Physical review. BFeb 27, 20254 citations

Geometric and computational aspects of chiral topological quantum matter

Omri Golan

In this thesis, we study chiral topological phases of 2+1 dimensional quantum matter. Such phases are abstractly characterized by their non-vanishing chiral central charge $c$ , a topological invariant which appears as the coefficient of a gravitational Chern-Simons (gCS) action in bulk, and of corresponding gravitational anomalies at boundaries. The chiral central charge is of particular importance in chiral superfluids and superconductors (CSF/Cs), where $U(1)$ particle-number symmetry is broken, and $c$ is, in some cases, the only topological invariant characterizing the system. However, as opposed to invariants which can be probed by gauge fields in place of gravity, the concrete physical implications of $c$ in the context of condensed matter physics is quite subtle, and has been the subject of ongoing research and controversy. The first two parts of this thesis are devoted to the physical interpretation of the gCS action and gravitational anomalies in the context of CSF/Cs, where they are of particular importance, but have nevertheless remained poorly understood. We then turn to a seemingly unrelated aspect of chiral topological phases - their computational complexity. The infamous $sign\ problem$ leads to an exponential complexity in Monte Carlo simulations of generic many-body quantum systems. Nevertheless, many phases of matter are known to admit a sign-problem-free representative, allowing an efficient classical simulation. The possibility of $intrinsic$ sign problems, where a phase of matter admits no sign-problem-free representative, was recently raised but remains largely unexplored. Here, we establish the existence of an intrinsic sign problem in a broad class of chiral topological phases, both bosonic and fermionic, defined by the requirement that $e^{2πi c/24}$ is $not$ the topological spin of an anyon.In this thesis, we study chiral topological phases of 2+1 dimensional quantum matter. Such phases are abstractly characterized by their non-vanishing chiral central charge $c$, a topological invariant which appears as the coefficient of a gravitational Chern-Simons (gCS) action in bulk, and of corresponding gravitational anomalies at boundaries. The chiral central charge is of particular importance in chiral superfluids and superconductors (CSF/Cs), where $U(1)$ particle-number symmetry is broken, and $c$ is, in some cases, the only topological invariant characterizing the system. However, as opposed to invariants which can be probed by gauge fields in place of gravity, the concrete physical implications of $c$ in the context of condensed matter physics is quite subtle, and has been the subject of ongoing research and controversy. The first two parts of this thesis are devoted to the physical interpretation of the gCS action and gravitational anomalies in the context of CSF/Cs, where they are of particular importance, but have nevertheless remained poorly understood. We then turn to a seemingly unrelated aspect of chiral topological phases - their computational complexity. The infamous $sign\ problem$ leads to an exponential complexity in Monte Carlo simulations of generic many-body quantum systems. Nevertheless, many phases of matter are known to admit a sign-problem-free representative, allowing an efficient classical simulation. The possibility of $intrinsic$ sign problems, where a phase of matter admits no sign-problem-free representative, was recently raised but remains largely unexplored. Here, we establish the existence of an intrinsic sign problem in a broad class of chiral topological phases, both bosonic and fermionic, defined by the requirement that $e^{2πi c/24}$ is $not$ the topological spin of an anyon.

arXiv:2106.10897Jun 21, 20212 citations

Topological phases and quantum criticality from $SU(2)$ Chern-Simons-matter theoriesTopological phases and quantum criticality from $SU(2)$ Chern-Simons-matter theories

Yunchao Hao, Yingcheng Li, Kangle Li +1 more

Motivated by recent numerical studies where various $SU(2)$ Chern-Simons-matter theories emerge, we analytically study topological phases and quantum criticality in two-dimensional systems described by such theories. First, we classify $SU(2)_k$ topological orders in all lattice spin systems with a $p4\times SO(3)$ symmetry, where $k$ is an arbitrary nonzero integer. We find that for each odd $k$ , the topological order can emerge in systems with an arbitrary Lieb-Schultz-Mattis (LSM) anomaly, and the symmetry cannot permute anyons. If the system has a nontrivial (respectively, trivial) LSM anomaly, then there is exactly one (respectively, nine) symmetry-enriched topological (SET) phases. On the other hand, $SU(2)_k$ topological order with any even $k$ can only emerge in systems with a trivial LSM anomaly. If $k\notin\{6, 10, 14, \cdots\}$ , the symmetry cannot permute anyons, and there are 16 SET phases. If $k\in\{6, 10, 14, \cdots\}$ , there are 4 different ways how the symmetry can permute anyons, and there are 64 SET phases. Next, we analyze the $SU(2)_k$ Chern-Simons theories coupled to $N_f$ flavors of gapless matter fields that can be either bosonic or fermionic. For both types of theories, we consider a joint large- $N_f$ and large- $k$ limit with $N_f/k$ fixed, and compute the scaling dimensions of the bilinear operators of the bosons or fermions to the order of $1/N_f$ . These results sharpen our understanding of these emergent exotic topological phases and quantum criticality, and provide useful guidance to explore them further.Motivated by recent numerical studies where various $SU(2)$ Chern-Simons-matter theories emerge, we analytically study topological phases and quantum criticality in two-dimensional systems described by such theories. First, we classify $SU(2)_k$ topological orders in all lattice spin systems with a $p4\times SO(3)$ symmetry, where $k$ is an arbitrary nonzero integer. We find that for each odd $k$, the topological order can emerge in systems with an arbitrary Lieb-Schultz-Mattis (LSM) anomaly, and the symmetry cannot permute anyons. If the system has a nontrivial (respectively, trivial) LSM anomaly, then there is exactly one (respectively, nine) symmetry-enriched topological (SET) phases. On the other hand, $SU(2)_k$ topological order with any even $k$ can only emerge in systems with a trivial LSM anomaly. If $k\notin\{6, 10, 14, \cdots\}$, the symmetry cannot permute anyons, and there are 16 SET phases. If $k\in\{6, 10, 14, \cdots\}$, there are 4 different ways how the symmetry can permute anyons, and there are 64 SET phases. Next, we analyze the $SU(2)_k$ Chern-Simons theories coupled to $N_f$ flavors of gapless matter fields that can be either bosonic or fermionic. For both types of theories, we consider a joint large-$N_f$ and large-$k$ limit with $N_f/k$ fixed, and compute the scaling dimensions of the bilinear operators of the bosons or fermions to the order of $1/N_f$. These results sharpen our understanding of these emergent exotic topological phases and quantum criticality, and provide useful guidance to explore them further.

arXiv:2608.14180Aug 14, 20260 citations

Chern insulator boundary criticality

Benjamin Moy, Eduardo Fradkin

We investigate signatures of chirality at Chern insulator transitions in the presence of a boundary. The transition between a trivial insulator and a Chern insulator with Chern number $C=1$ is described by a massless Dirac fermion whose parity anomaly gives a critical Hall conductivity $σ_{xy}=\frac{1}{2}\frac{e^2}{h}$ . Using a Dirac mass domain wall construction, we show that the chiral edge mode delocalizes into the bulk at criticality, but the boundary fermion correlation function retains a chiral structure and acquires the scaling dimension of the bulk fermion. We compute current correlation functions and demonstrate that the anomaly of the bulk Hall response is matched by delocalized chiral modes near the boundary. Using only the residual conformal symmetry of a (2+1)d conformal field theory (CFT) in a half-space, we identify parity-odd terms in current and energy-momentum tensor correlation functions that encode these modes and determine their electromagnetic and gravitational anomaly coefficients. Our analysis therefore applies to general time-reversal breaking (2+1)d CFTs, beyond the free Dirac transition. In particular, the analysis of the gravitational anomaly is also applicable to the free Majorana CFT governing the transition between a trivial superconductor and a topological superconductor. We also extend our results to more general Chern number changing transitions and to the transition between a (3+1)d topological insulator and a trivial insulator.We investigate signatures of chirality at Chern insulator transitions in the presence of a boundary. The transition between a trivial insulator and a Chern insulator with Chern number $C=1$ is described by a massless Dirac fermion whose parity anomaly gives a critical Hall conductivity $σ_{xy}=\frac{1}{2}\frac{e^2}{h}$. Using a Dirac mass domain wall construction, we show that the chiral edge mode delocalizes into the bulk at criticality, but the boundary fermion correlation function retains a chiral structure and acquires the scaling dimension of the bulk fermion. We compute current correlation functions and demonstrate that the anomaly of the bulk Hall response is matched by delocalized chiral modes near the boundary. Using only the residual conformal symmetry of a (2+1)d conformal field theory (CFT) in a half-space, we identify parity-odd terms in current and energy-momentum tensor correlation functions that encode these modes and determine their electromagnetic and gravitational anomaly coefficients. Our analysis therefore applies to general time-reversal breaking (2+1)d CFTs, beyond the free Dirac transition. In particular, the analysis of the gravitational anomaly is also applicable to the free Majorana CFT governing the transition between a trivial superconductor and a topological superconductor. We also extend our results to more general Chern number changing transitions and to the transition between a (3+1)d topological insulator and a trivial insulator.

arXiv:2608.18199Aug 18, 20260 citations

Framing anomaly is a key property of

(2+1)d

U(1)

T

operator in the ground subspace on a spatial torus, from which we extract that

\langle T \rangle

has a universal phase of

-2π/12

as expected:

-2π/8

from the Gauss-Milgram sum of the topological spins of the ground states, and

2π/24

from the framing anomaly; we can also extract the

2π/24

framing anomaly phase alone from the full spectrum of

T

in the ground subspace by computing

\langle T^m \rangle

. This pins down the last and most crucial property required for a valid lattice definition of

U(1)

Chern-Simons theory.

SU(2)

chiral field theory, and provide a semiclassically solvable definition of Abelian

2+1

chiral topological orders. This leads to an exactly solvable definition of chiral

U(1)

SPT phases with zero correlation length, which we use to extract the edge chiral field theories exactly. These zero-correlation length models are vastly more simple than previous approaches to defining chiral field theories on the lattice.

A Lattice Chiral Boson Theory in

1+1

1+1

d chiral field theory as the edge of a

2+1

d system and argue that the edge decouples from the bulk, but this can be difficult to verify due to finite size effects and strong interactions. On the other hand, recent work has shown how to define the

2+1

d bulk theory as an exactly solvable model with zero correlation length, in which case the edge theory may be extracted exactly. We use these techniques to derive a lattice field theory on a

1+1

d spacetime lattice which carries an anomalous chiral

U(1)

symmetry with zero chiral central charge. The lattice theory with anomalous chiral

U(1)

symmetry is always gapless, regardless of lattice interactions. We demonstrate the chiral anomaly by coupling to a background gauge field, develop a field theory which demonstrates the chiral behavior, and show how to assemble a chiral, anomaly-free theory where the gauge field may be taken to be dynamical.

Recently, it has been shown how topological phases of matter with crystalline symmetry and

U(1)

charge conservation can be partially characterized by a set of many-body invariants, the discrete shift

{\mathcal{S}}_{\mathrm{o}}

and electric polarization ${\stackrel{P\vec}{\mathcal{P}}}_{\mathrm{o}}$, where

\mathrm{o}

labels a high-symmetry point. Crucially, these can be defined even with nonzero Chern number and/or magnetic field. One manifestation of these invariants is through quantized fractional contributions to the charge in the vicinity of a lattice disclination or dislocation. In this paper, we show that these invariants can also be extracted from the length and corner dependence of the total charge (mod 1) on the boundary of the system. We provide a general formula in terms of

{\mathcal{S}}_{\mathrm{o}}

and ${\stackrel{P\vec}{\mathcal{P}}}_{\mathrm{o}}$ for the total charge of any subregion of the system which can include full boundaries or bulk lattice defects, unifying boundary, corner, disclination, and dislocation charge responses into a single general theory. These results hold for Chern insulators, despite their gapless chiral edge modes, and for which an unambiguous definition of an intrinsically two-dimensional electric polarization has been unclear until recently. We also discuss how our theory can fully characterize the topological response of quadrupole insulators.

In this thesis, we study chiral topological phases of 2+1 dimensional quantum matter. Such phases are abstractly characterized by their non-vanishing chiral central charge

c

, a topological invariant which appears as the coefficient of a gravitational Chern-Simons (gCS) action in bulk, and of corresponding gravitational anomalies at boundaries. The chiral central charge is of particular importance in chiral superfluids and superconductors (CSF/Cs), where

U(1)

particle-number symmetry is broken, and

c

is, in some cases, the only topological invariant characterizing the system. However, as opposed to invariants which can be probed by gauge fields in place of gravity, the concrete physical implications of

c

in the context of condensed matter physics is quite subtle, and has been the subject of ongoing research and controversy. The first two parts of this thesis are devoted to the physical interpretation of the gCS action and gravitational anomalies in the context of CSF/Cs, where they are of particular importance, but have nevertheless remained poorly understood. We then turn to a seemingly unrelated aspect of chiral topological phases - their computational complexity. The infamous

sign\ problem

leads to an exponential complexity in Monte Carlo simulations of generic many-body quantum systems. Nevertheless, many phases of matter are known to admit a sign-problem-free representative, allowing an efficient classical simulation. The possibility of

intrinsic

sign problems, where a phase of matter admits no sign-problem-free representative, was recently raised but remains largely unexplored. Here, we establish the existence of an intrinsic sign problem in a broad class of chiral topological phases, both bosonic and fermionic, defined by the requirement that

e^{2πi c/24}

not

the topological spin of an anyon.

Topological phases and quantum criticality from

SU(2)

Chern-Simons-matter theories

Motivated by recent numerical studies where various

SU(2)

Chern-Simons-matter theories emerge, we analytically study topological phases and quantum criticality in two-dimensional systems described by such theories. First, we classify

SU(2)_k

topological orders in all lattice spin systems with a

p4\times SO(3)

symmetry, where

k

is an arbitrary nonzero integer. We find that for each odd

k

, the topological order can emerge in systems with an arbitrary Lieb-Schultz-Mattis (LSM) anomaly, and the symmetry cannot permute anyons. If the system has a nontrivial (respectively, trivial) LSM anomaly, then there is exactly one (respectively, nine) symmetry-enriched topological (SET) phases. On the other hand,

SU(2)_k

topological order with any even

k

can only emerge in systems with a trivial LSM anomaly. If

k\notin\{6, 10, 14, \cdots\}

, the symmetry cannot permute anyons, and there are 16 SET phases. If

k\in\{6, 10, 14, \cdots\}

, there are 4 different ways how the symmetry can permute anyons, and there are 64 SET phases. Next, we analyze the

SU(2)_k

Chern-Simons theories coupled to

N_f

flavors of gapless matter fields that can be either bosonic or fermionic. For both types of theories, we consider a joint large-

N_f

and large-

k

limit with

N_f/k

fixed, and compute the scaling dimensions of the bilinear operators of the bosons or fermions to the order of

1/N_f

. These results sharpen our understanding of these emergent exotic topological phases and quantum criticality, and provide useful guidance to explore them further.

We investigate signatures of chirality at Chern insulator transitions in the presence of a boundary. The transition between a trivial insulator and a Chern insulator with Chern number

C=1

is described by a massless Dirac fermion whose parity anomaly gives a critical Hall conductivity

σ_{xy}=\frac{1}{2}\frac{e^2}{h}

. Using a Dirac mass domain wall construction, we show that the chiral edge mode delocalizes into the bulk at criticality, but the boundary fermion correlation function retains a chiral structure and acquires the scaling dimension of the bulk fermion. We compute current correlation functions and demonstrate that the anomaly of the bulk Hall response is matched by delocalized chiral modes near the boundary. Using only the residual conformal symmetry of a (2+1)d conformal field theory (CFT) in a half-space, we identify parity-odd terms in current and energy-momentum tensor correlation functions that encode these modes and determine their electromagnetic and gravitational anomaly coefficients. Our analysis therefore applies to general time-reversal breaking (2+1)d CFTs, beyond the free Dirac transition. In particular, the analysis of the gravitational anomaly is also applicable to the free Majorana CFT governing the transition between a trivial superconductor and a topological superconductor. We also extend our results to more general Chern number changing transitions and to the transition between a (3+1)d topological insulator and a trivial insulator.