We study the boundary states for the rational points in the moduli spaces of c=1 conformal and c=3/2 superconformal field theories, including the isolated Ginsparg points. We use the orbifold and simple-current techniques to relate the boundary states of different theories and to obtain symmetry-breaking, non-Cardy boundary states. We show some interesting examples of fractional and twisted branes on orbifold spaces.

Jan 1, 20027 citations

arXiv:1908.02918

Two-dimensional Anomaly, Orbifolding, and Boundary States

Ken Kikuchi, Yang Zhou

Published Aug 8, 2019• 7 citations

arXiv HTML PDF

Abstract

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.

DOI: 10.48550/arxiv.1908.02918

Related papers

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

Aug 8, 20191 citation

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.

arXiv:2205.11190Physical review. B./Physical review. BJul 11, 20249 citations

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

Nick Bultinck, Robijn Vanhove, Jutho Haegeman +1 more

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.

arXiv:1710.02314Physical Review LettersApr 10, 201818 citations

Modular Bootstrap and Discrete Symmetries in 2D CFTs

DARIA UNGOLO

Two-dimensional Conformal Field Theories (CFTs) allow us to describe quantum field theories at critical points and in general systems without a characteristic scale. It is possible to implement such theories on topologically non-trivial surfaces. If we formulate a two-dimensional CFT on a toroidal surface, a new symmetry emerges known as modular invariance. This symmetry corresponds to the set of transformations that act without changing the toroidal surface, consequently, the partition function of the theory must remain invariant under these transformations. This invariance is used as a constraint by the modular bootstrap method to obtain information about the spectrum without using the Lagrangian and operator content of the theory. By imposing that the theory is consistent with modular symmetry, it is possible to determine the limits of the spectrum beyond which modular invariance can no longer be satisfied, thus constraining the space of admissible two-dimensional conformal field theories. To study CFTs that present discrete global symmetries, it is essential to introduce Topological Defect Lines (TDLs). These are one-dimensional, topologically invariant operators that implement the action of the symmetry group on the Hilbert space. The presence of TDLs modifies the topology of the system and introduces new partition functions associated with different symmetry sectors. Under modular transformations, the TDLs mix among themselves, and consequently, the corresponding partition functions also mix, thus promoting the modular constraint to a vector-valued condition. This formalism makes it possible to implement discrete symmetries also considering the possible presence of anomalies. The objective of this thesis is therefore to understand how discrete symmetries and anomalies affect the spectrum of a CFT. Based on the modular bootstrap framework developed by Lin and Shao for the Z2 symmetry, the analysis was first analytically extended to the Abelian case Z3. This was useful in extending the discussion to a more complex case, namely the non-Abelian symmetry S3. This extension constitutes the original theoretical contribution of the thesis, providing the explicit formulation of modular bootstrap equations for CFTs with non-Abelian discrete symmetry. The analytical modular constraints were reformulated numerically, bridging the modular bootstrap framework with semidefinite programming in SDPB. This analysis paves the way for a numerical study based on convex optimization techniques to explore the constraints arising from modular bootstrap equations.Two-dimensional Conformal Field Theories (CFTs) allow us to describe quantum field theories at critical points and in general systems without a characteristic scale. It is possible to implement such theories on topologically non-trivial surfaces. If we formulate a two-dimensional CFT on a toroidal surface, a new symmetry emerges known as modular invariance. This symmetry corresponds to the set of transformations that act without changing the toroidal surface, consequently, the partition function of the theory must remain invariant under these transformations. This invariance is used as a constraint by the modular bootstrap method to obtain information about the spectrum without using the Lagrangian and operator content of the theory. By imposing that the theory is consistent with modular symmetry, it is possible to determine the limits of the spectrum beyond which modular invariance can no longer be satisfied, thus constraining the space of admissible two-dimensional conformal field theories. To study CFTs that present discrete global symmetries, it is essential to introduce Topological Defect Lines (TDLs). These are one-dimensional, topologically invariant operators that implement the action of the symmetry group on the Hilbert space. The presence of TDLs modifies the topology of the system and introduces new partition functions associated with different symmetry sectors. Under modular transformations, the TDLs mix among themselves, and consequently, the corresponding partition functions also mix, thus promoting the modular constraint to a vector-valued condition. This formalism makes it possible to implement discrete symmetries also considering the possible presence of anomalies. The objective of this thesis is therefore to understand how discrete symmetries and anomalies affect the spectrum of a CFT. Based on the modular bootstrap framework developed by Lin and Shao for the Z2 symmetry, the analysis was first analytically extended to the Abelian case Z3. This was useful in extending the discussion to a more complex case, namely the non-Abelian symmetry S3. This extension constitutes the original theoretical contribution of the thesis, providing the explicit formulation of modular bootstrap equations for CFTs with non-Abelian discrete symmetry. The analytical modular constraints were reformulated numerically, bridging the modular bootstrap framework with semidefinite programming in SDPB. This analysis paves the way for a numerical study based on convex optimization techniques to explore the constraints arising from modular bootstrap equations.

Dec 4, 20250 citations

Boundary States of c=1 and 3/2 Rational Conformal Field Theories

Andrea Cappelli A, Giuseppe D’Appollonio

Jan 1, 20027 citations

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.

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.

{\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.