$2d$ 't Hooft Anomaly, Orbifolding, and Boundary States

can be seen from the noncommutativity of two symmetry lines inserted along the nontrivial cycles of two-torus and we propose a criterion to detect the anomaly, which agrees with the truncated modular

S

-matrix approach. The obstruction for orbifolding has been recently interpreted as a mixed anomaly between

G

and large diffeomorphisms. We clarify the relations among anomaly-free conditions, orbifoldable conditions, and invariant boundary state condition, focusing on Wess-Zumino-Witten models.

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Two-dimensional Anomaly, Orbifolding, and Boundary States

Ken Kikuchi, Yang Zhou

We study anomalies of discrete internal global symmetry $G$ in two-dimensional rational conformal field theories based on twisted torus partition functions. The anomaly of $G$ can be seen from the noncommutativity of two symmetry lines inserted along the nontrivial cycles of two-torus and we propose a criterion to detect the anomaly, which agrees with the truncated modular $S$ -matrix approach. The obstruction for orbifolding has been recently interpreted as a mixed anomaly between $G$ and large diffeomorphisms. We clarify the relations among anomaly-free conditions, orbifoldable conditions, and invariant boundary state condition, focusing on Wess-Zumino-Witten models.We study anomalies of discrete internal global symmetry $G$ in two-dimensional rational conformal field theories based on twisted torus partition functions. The anomaly of $G$ can be seen from the noncommutativity of two symmetry lines inserted along the nontrivial cycles of two-torus and we propose a criterion to detect the anomaly, which agrees with the truncated modular $S$-matrix approach. The obstruction for orbifolding has been recently interpreted as a mixed anomaly between $G$ and large diffeomorphisms. We clarify the relations among anomaly-free conditions, orbifoldable conditions, and invariant boundary state condition, focusing on Wess-Zumino-Witten models.

arXiv:1908.02918Aug 8, 20197 citations

From orbifolding conformal field theories to gauging topological phases

Xiao Chen, Abhishek Roy, Jeffrey C. Y. Teo +1 more

Topological phases of matter in (2+1) dimensions are commonly equipped with global symmetries, such as electric-magnetic duality in gauge theories and bilayer symmetry in fractional quantum Hall states. Gauging these symmetries into local dynamical ones is one way of obtaining exotic phases from conventional systems. We study this using the bulk-boundary correspondence and applying the orbifold construction to the (1+1)-dimensional edge described by a conformal field theory (CFT). Our procedure puts twisted boundary conditions into the partition function and predicts the fusion, spin, and braiding behavior of anyonic excitations after gauging. We demonstrate this for the electric-magnetic self-dual ${\mathbb{Z}}_{N}$ gauge theory, the twofold symmetric $\text{SU}{(3)}_{1}$ , and the ${S}_{3}$ -symmetric $\text{SO}{(8)}_{1}$ Wess-Zumino-Witten theories.Topological phases of matter in (2+1) dimensions are commonly equipped with global symmetries, such as electric-magnetic duality in gauge theories and bilayer symmetry in fractional quantum Hall states. Gauging these symmetries into local dynamical ones is one way of obtaining exotic phases from conventional systems. We study this using the bulk-boundary correspondence and applying the orbifold construction to the (1+1)-dimensional edge described by a conformal field theory (CFT). Our procedure puts twisted boundary conditions into the partition function and predicts the fusion, spin, and braiding behavior of anyonic excitations after gauging. We demonstrate this for the electric-magnetic self-dual ${\mathbb{Z}}_{N}$ gauge theory, the twofold symmetric $\text{SU}{(3)}_{1}$, and the ${S}_{3}$-symmetric $\text{SO}{(8)}_{1}$ Wess-Zumino-Witten theories.

arXiv:1706.00557Physical review. B./Physical review. BSep 27, 201714 citations

Boundary conditions and anomalies of conformal field theories in 1 + 1 dimensions

Linhao Li, Chang-Tse Hsieh, Yuan Yao +1 more

We study a relationship between conformally invariant boundary conditions and anomalies of conformal field theories (CFTs) in $1+1$ dimensions. For a given CFT with a global symmetry, we consider symmetric ``gapping potentials'', which are relevant perturbations to the CFT. If a gapping potential is introduced only in a subregion of the system, it provides a certain boundary condition to the CFT. From this equivalence, if there exists a Cardy boundary state, which is invariant under a symmetry, then the CFT can be gapped with a unique ground state by adding the corresponding gapping potential. This means that the symmetry of the CFT is anomaly free. Using this approach, we systematically deduce the anomaly-free conditions for various types of CFTs with several different symmetries. They include the free compact boson theory, Wess-Zumino-Witten models, and unitary minimal models. When the symmetry of the CFT is anomalous, it implies a Lieb-Schultz-Mattis type ingappability of the system. Our results are consistent with, where available, known results in the literature. Moreover, we extend the discussion to other symmetries including spin groups and generalized time-reversal symmetries.We study a relationship between conformally invariant boundary conditions and anomalies of conformal field theories (CFTs) in $1+1$ dimensions. For a given CFT with a global symmetry, we consider symmetric ``gapping potentials'', which are relevant perturbations to the CFT. If a gapping potential is introduced only in a subregion of the system, it provides a certain boundary condition to the CFT. From this equivalence, if there exists a Cardy boundary state, which is invariant under a symmetry, then the CFT can be gapped with a unique ground state by adding the corresponding gapping potential. This means that the symmetry of the CFT is anomaly free. Using this approach, we systematically deduce the anomaly-free conditions for various types of CFTs with several different symmetries. They include the free compact boson theory, Wess-Zumino-Witten models, and unitary minimal models. When the symmetry of the CFT is anomalous, it implies a Lieb-Schultz-Mattis type ingappability of the system. Our results are consistent with, where available, known results in the literature. Moreover, we extend the discussion to other symmetries including spin groups and generalized time-reversal symmetries.

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Global Anomaly Detection in Two-Dimensional Symmetry-Protected Topological Phases

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Edge theories of symmetry-protected topological phases are well known to possess global symmetry anomalies. In this Letter we focus on two-dimensional bosonic phases protected by an on-site symmetry and analyze the corresponding edge anomalies in more detail. Physical interpretations of the anomaly in terms of an obstruction to orbifolding and constructing symmetry-preserving boundaries are connected to the cohomology classification of symmetry-protected phases in two dimensions. Using the tensor network and matrix product state formalism we numerically illustrate our arguments and discuss computational detection schemes to identify symmetry-protected order in a ground state wave function.Edge theories of symmetry-protected topological phases are well known to possess global symmetry anomalies. In this Letter we focus on two-dimensional bosonic phases protected by an on-site symmetry and analyze the corresponding edge anomalies in more detail. Physical interpretations of the anomaly in terms of an obstruction to orbifolding and constructing symmetry-preserving boundaries are connected to the cohomology classification of symmetry-protected phases in two dimensions. Using the tensor network and matrix product state formalism we numerically illustrate our arguments and discuss computational detection schemes to identify symmetry-protected order in a ground state wave function.

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Anomalies and bounds on charged operators

Ying-Hsuan Lin, Shu-Heng Shao

We study the implications of the 't Hooft anomaly (i.e., obstruction to gauging) on conformal field theory, focusing on the case when the global symmetry is ${\mathbb{Z}}_{2}$ . Using the modular bootstrap, universal bounds on ( $1+1$ )-dimensional bosonic conformal field theories with an internal ${\mathbb{Z}}_{2}$ global symmetry are derived. The bootstrap bounds depend dramatically on the 't Hooft anomaly. In particular, there is a universal upper bound on the lightest ${\mathbb{Z}}_{2}$ odd operator if the symmetry is anomalous, but there is no bound if the symmetry is nonanomalous. In the nonanomalous case, we find that the lightest ${\mathbb{Z}}_{2}$ odd state and the defect ground state cannot both be arbitrarily heavy. We also consider theories with a $U(1)$ global symmetry, and comment that there is no bound on the lightest $U(1)$ charged operator if the symmetry is nonanomalous.We study the implications of the 't Hooft anomaly (i.e., obstruction to gauging) on conformal field theory, focusing on the case when the global symmetry is ${\mathbb{Z}}_{2}$. Using the modular bootstrap, universal bounds on ($1+1$)-dimensional bosonic conformal field theories with an internal ${\mathbb{Z}}_{2}$ global symmetry are derived. The bootstrap bounds depend dramatically on the 't Hooft anomaly. In particular, there is a universal upper bound on the lightest ${\mathbb{Z}}_{2}$ odd operator if the symmetry is anomalous, but there is no bound if the symmetry is nonanomalous. In the nonanomalous case, we find that the lightest ${\mathbb{Z}}_{2}$ odd state and the defect ground state cannot both be arbitrarily heavy. We also consider theories with a $U(1)$ global symmetry, and comment that there is no bound on the lightest $U(1)$ charged operator if the symmetry is nonanomalous.

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Fusion Category Symmetry I: Anomaly In-Flow and Gapped Phases

Ryan Thorngren, Yifan Wang

We study generalized discrete symmetries of quantum field theories in 1+1D generated by topological defect lines with no inverse. In particular, we describe 't Hooft anomalies and classify gapped phases stabilized by these symmetries, including new 1+1D topological phases. The algebra of these operators is not a group but rather is described by their fusion ring and crossing relations, captured algebraically as a fusion category. Such data defines a Turaev-Viro/Levin-Wen model in 2+1D, while a 1+1D system with this fusion category acting as a global symmetry defines a boundary condition. This is akin to gauging a discrete global symmetry at the boundary of Dijkgraaf-Witten theory. We describe how to "ungauge" the fusion category symmetry in these boundary conditions and separate the symmetry-preserving phases from the symmetry-breaking ones. For Tambara-Yamagami categories and their generalizations, which are associated with Kramers-Wannier-like self-dualities under orbifolding, we develop gauge theoretic techniques which simplify the analysis. We include some examples of CFTs with fusion category symmetry derived from Kramers-Wannier-like dualities as an appetizer for the Part II companion paper.We study generalized discrete symmetries of quantum field theories in 1+1D generated by topological defect lines with no inverse. In particular, we describe 't Hooft anomalies and classify gapped phases stabilized by these symmetries, including new 1+1D topological phases. The algebra of these operators is not a group but rather is described by their fusion ring and crossing relations, captured algebraically as a fusion category. Such data defines a Turaev-Viro/Levin-Wen model in 2+1D, while a 1+1D system with this fusion category acting as a global symmetry defines a boundary condition. This is akin to gauging a discrete global symmetry at the boundary of Dijkgraaf-Witten theory. We describe how to "ungauge" the fusion category symmetry in these boundary conditions and separate the symmetry-preserving phases from the symmetry-breaking ones. For Tambara-Yamagami categories and their generalizations, which are associated with Kramers-Wannier-like self-dualities under orbifolding, we develop gauge theoretic techniques which simplify the analysis. We include some examples of CFTs with fusion category symmetry derived from Kramers-Wannier-like dualities as an appetizer for the Part II companion paper.

arXiv:1912.02817Dec 5, 2019107 citations

2d

't Hooft Anomaly, Orbifolding, and Boundary States

We study anomalies of discrete internal global symmetry

G

in two-dimensional rational conformal field theories based on twisted torus partition functions. The anomaly of

G

can be seen from the noncommutativity of two symmetry lines inserted along the nontrivial cycles of two-torus and we propose a criterion to detect the anomaly, which agrees with the truncated modular

S

-matrix approach. The obstruction for orbifolding has been recently interpreted as a mixed anomaly between

G

and large diffeomorphisms. We clarify the relations among anomaly-free conditions, orbifoldable conditions, and invariant boundary state condition, focusing on Wess-Zumino-Witten models.

We study anomalies of discrete internal global symmetry

G

in two-dimensional rational conformal field theories based on twisted torus partition functions. The anomaly of

G

can be seen from the noncommutativity of two symmetry lines inserted along the nontrivial cycles of two-torus and we propose a criterion to detect the anomaly, which agrees with the truncated modular

S

-matrix approach. The obstruction for orbifolding has been recently interpreted as a mixed anomaly between

G

and large diffeomorphisms. We clarify the relations among anomaly-free conditions, orbifoldable conditions, and invariant boundary state condition, focusing on Wess-Zumino-Witten models.

{\mathbb{Z}}_{N}

gauge theory, the twofold symmetric

\text{SU}{(3)}_{1}

, and the

{S}_{3}

-symmetric

\text{SO}{(8)}_{1}

Wess-Zumino-Witten theories.

We study a relationship between conformally invariant boundary conditions and anomalies of conformal field theories (CFTs) in

1+1

dimensions. For a given CFT with a global symmetry, we consider symmetric ``gapping potentials'', which are relevant perturbations to the CFT. If a gapping potential is introduced only in a subregion of the system, it provides a certain boundary condition to the CFT. From this equivalence, if there exists a Cardy boundary state, which is invariant under a symmetry, then the CFT can be gapped with a unique ground state by adding the corresponding gapping potential. This means that the symmetry of the CFT is anomaly free. Using this approach, we systematically deduce the anomaly-free conditions for various types of CFTs with several different symmetries. They include the free compact boson theory, Wess-Zumino-Witten models, and unitary minimal models. When the symmetry of the CFT is anomalous, it implies a Lieb-Schultz-Mattis type ingappability of the system. Our results are consistent with, where available, known results in the literature. Moreover, we extend the discussion to other symmetries including spin groups and generalized time-reversal symmetries.

We study the implications of the 't Hooft anomaly (i.e., obstruction to gauging) on conformal field theory, focusing on the case when the global symmetry is

{\mathbb{Z}}_{2}

. Using the modular bootstrap, universal bounds on (

1+1

)-dimensional bosonic conformal field theories with an internal

{\mathbb{Z}}_{2}

global symmetry are derived. The bootstrap bounds depend dramatically on the 't Hooft anomaly. In particular, there is a universal upper bound on the lightest

{\mathbb{Z}}_{2}

odd operator if the symmetry is anomalous, but there is no bound if the symmetry is nonanomalous. In the nonanomalous case, we find that the lightest

{\mathbb{Z}}_{2}

odd state and the defect ground state cannot both be arbitrarily heavy. We also consider theories with a

U(1)

global symmetry, and comment that there is no bound on the lightest

U(1)

charged operator if the symmetry is nonanomalous.