Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:1904.04833 · Physical review. D/Physical review. D.

Anomalies and bounds on charged operators

Ying-Hsuan Lin, Shu-Heng Shao

Published Jul 25, 2019• 100 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physrevd.100.025013

Related papers

ZN symmetries, anomalies, and the modular bootstrap

Ying-Hsuan Lin, Shu-Heng Shao

We explore constraints on $(1+1)d$ unitary conformal field theory with an internal ${\mathbb{Z}}_{N}$ global symmetry, by bounding the lightest symmetry-preserving scalar primary operator using the modular bootstrap. Among the other constraints we have found, we prove the existence of a ${\mathbb{Z}}_{N}$-symmetric relevant/marginal operator if $N\ensuremath{-}1\ensuremath{\le}c\ensuremath{\le}9\ensuremath{-}N$ for $N\ensuremath{\le}4$, with the end points saturated by various Wess-Zumino-Witten models that can be embedded into $({\mathfrak{e}}_{8}{)}_{1}$. Its existence implies that robust gapless fixed points are not possible in this range of $c$ if only a ${\mathbb{Z}}_{N}$ symmetry is imposed microscopically. We also obtain stronger, more refined bounds that depend on the `t Hooft anomaly of the ${\mathbb{Z}}_{N}$ symmetry.

arXiv:2101.08343•Physical review. D/Physical review. D.•Jun 1, 2021•48 citations

Bootstrapping Lieb-Schultz-Mattis anomalies

Ryan A. Lanzetta, Lukasz Fidkowski

We incorporate the microscopic assumptions that lead to a certain generalization of the Lieb-Schultz-Mattis (LSM) theorem for one-dimensional spin chains into the conformal bootstrap. Our approach accounts for the ``LSM anomaly'' possessed by these spin chains through a combination of modular bootstrap and correlator bootstrap of symmetry defect operators. Specifically, we obtain universal bounds on the local operator content of $(1+1)d$ conformal field theories (CFTs) that could describe translationally invariant lattice Hamiltonians with a ${\mathbb{Z}}_{N}\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{N}$ symmetry realized projectively at each site. We assume, for such models, that in the CFT the lattice translation symmetry is realized as an emanant internal ${\mathbb{Z}}_{N}$ symmetry. We present bounds on local operators both with and without refinement by their global symmetry representations. Interestingly, we can obtain nontrivial bounds on charged operators when $N$ is odd, which turns out to be impossible with modular bootstrap alone. Our bounds exhibit distinctive kinks, some of which are approximately saturated by known theories and others that are unexplained. We discuss additional scenarios with the properties necessary for our bounds to apply, including certain multicritical points between $(1+1)d$ symmetry protected topological phases, where we argue that the anomaly studied in our bootstrap calculations should emerge.

arXiv:2207.05092•Physical review. B./Physical review. B•May 19, 2023•13 citations

Bootstrapping noninvertible symmetries

Ying-Hsuan Lin, Shu-Heng Shao

Using the numerical modular bootstrap, we constrain the space of $1+1\mathrm{d}$ conformal field theories (CFTs) with a finite noninvertible global symmetry described by a fusion category $\mathcal{C}$. We derive universal and rigorous upper bounds on the lightest $\mathcal{C}$-preserving scalar local operator for fusion categories such as the Ising and Fibonacci categories. These numerical bounds constrain the possible robust gapless phases protected by a noninvertible global symmetry, which commonly arise from microscopic lattice models such as the anyonic chains. We also derive bounds on the lightest $\mathcal{C}$-violating local operator. Our bootstrap equations naturally arise from a ``slab construction,'' where the CFT is coupled to the $2+1\mathrm{d}$ Turaev-Viro topological quantum field theory , also known as the symmetry TFT.

arXiv:2302.13900•Physical review. D/Physical review. D.•Jun 28, 2023•54 citations

$2d$ '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.

•Aug 8, 2019•1 citation

Universal Bounds in Even-Spin CFTs

Joshua D. Qualls

We prove using invariance under the modular $S$- and $ST$-transformations that every unitary two-dimensional conformal field theory (CFT) of only even-spin operators (with no extended chiral algebra and with central charges $c,\tilde{c}>1$) contains a primary operator with dimension $Δ_1$ satisfying $0 < Δ_1 < (c+\tilde{c})/24 + 0.09280...$ After deriving both analytical and numerical bounds, we discuss how to extend our methods to bound higher conformal dimensions before deriving lower and upper bounds on the number of primary operators in a given energy range. Using the AdS$_3$/CFT$_2$ dictionary, the bound on $Δ_1$ proves the lightest massive excitation in appropriate theories of 3D matter and gravity with cosmological constant $Λ< 0$ can be no heavier than $1/(8G_N)+O(\sqrt{-Λ})$; the bounds on the number operators are related via AdS/CFT to the entropy of states in the dual gravitational theory. In the flat-space approximation, the limiting mass is exactly that of the lightest BTZ black hole.

arXiv:1412.0383•Dec 1, 2014•3 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.

•Dec 4, 2025•0 citations
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 Z2{\mathbb{Z}}_{2}Z2​. Using the modular bootstrap, universal bounds on (1+11+11+1)-dimensional bosonic conformal field theories with an internal Z2{\mathbb{Z}}_{2}Z2​ 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 Z2{\mathbb{Z}}_{2}Z2​ 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 Z2{\mathbb{Z}}_{2}Z2​ odd state and the defect ground state cannot both be arbitrarily heavy. We also consider theories with a U(1)U(1)U(1) global symmetry, and comment that there is no bound on the lightest U(1)U(1)U(1) charged operator if the symmetry is nonanomalous.
We explore constraints on (1+1)d(1+1)d(1+1)d unitary conformal field theory with an internal ZN{\mathbb{Z}}_{N}ZN​ global symmetry, by bounding the lightest symmetry-preserving scalar primary operator using the modular bootstrap. Among the other constraints we have found, we prove the existence of a ZN{\mathbb{Z}}_{N}ZN​-symmetric relevant/marginal operator if $N\ensuremath{-}1\ensuremath{\le}c\ensuremath{\le}9\ensuremath{-}N$ for $N\ensuremath{\le}4$, with the end points saturated by various Wess-Zumino-Witten models that can be embedded into (e8)1({\mathfrak{e}}_{8}{)}_{1}(e8​)1​. Its existence implies that robust gapless fixed points are not possible in this range of ccc if only a ZN{\mathbb{Z}}_{N}ZN​ symmetry is imposed microscopically. We also obtain stronger, more refined bounds that depend on the `t Hooft anomaly of the ZN{\mathbb{Z}}_{N}ZN​ symmetry.
We incorporate the microscopic assumptions that lead to a certain generalization of the Lieb-Schultz-Mattis (LSM) theorem for one-dimensional spin chains into the conformal bootstrap. Our approach accounts for the ``LSM anomaly'' possessed by these spin chains through a combination of modular bootstrap and correlator bootstrap of symmetry defect operators. Specifically, we obtain universal bounds on the local operator content of (1+1)d(1+1)d(1+1)d conformal field theories (CFTs) that could describe translationally invariant lattice Hamiltonians with a ${\mathbb{Z}}_{N}\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{N}$ symmetry realized projectively at each site. We assume, for such models, that in the CFT the lattice translation symmetry is realized as an emanant internal ZN{\mathbb{Z}}_{N}ZN​ symmetry. We present bounds on local operators both with and without refinement by their global symmetry representations. Interestingly, we can obtain nontrivial bounds on charged operators when NNN is odd, which turns out to be impossible with modular bootstrap alone. Our bounds exhibit distinctive kinks, some of which are approximately saturated by known theories and others that are unexplained. We discuss additional scenarios with the properties necessary for our bounds to apply, including certain multicritical points between (1+1)d(1+1)d(1+1)d symmetry protected topological phases, where we argue that the anomaly studied in our bootstrap calculations should emerge.
Using the numerical modular bootstrap, we constrain the space of 1+1d1+1\mathrm{d}1+1d conformal field theories (CFTs) with a finite noninvertible global symmetry described by a fusion category C\mathcal{C}C. We derive universal and rigorous upper bounds on the lightest C\mathcal{C}C-preserving scalar local operator for fusion categories such as the Ising and Fibonacci categories. These numerical bounds constrain the possible robust gapless phases protected by a noninvertible global symmetry, which commonly arise from microscopic lattice models such as the anyonic chains. We also derive bounds on the lightest C\mathcal{C}C-violating local operator. Our bootstrap equations naturally arise from a ``slab construction,'' where the CFT is coupled to the 2+1d2+1\mathrm{d}2+1d Turaev-Viro topological quantum field theory , also known as the symmetry TFT.
2d2d2d 't Hooft Anomaly, Orbifolding, and Boundary States
We study anomalies of discrete internal global symmetry GGG in two-dimensional rational conformal field theories based on twisted torus partition functions. The anomaly of GGG 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 SSS-matrix approach. The obstruction for orbifolding has been recently interpreted as a mixed anomaly between GGG 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 prove using invariance under the modular SSS- and STSTST-transformations that every unitary two-dimensional conformal field theory (CFT) of only even-spin operators (with no extended chiral algebra and with central charges c,c~>1c,\tilde{c}>1c,c~>1) contains a primary operator with dimension Δ1Δ_1Δ1​ satisfying 0<Δ1<(c+c~)/24+0.09280...0 < Δ_1 < (c+\tilde{c})/24 + 0.09280...0<Δ1​<(c+c~)/24+0.09280... After deriving both analytical and numerical bounds, we discuss how to extend our methods to bound higher conformal dimensions before deriving lower and upper bounds on the number of primary operators in a given energy range. Using the AdS3_33​/CFT2_22​ dictionary, the bound on Δ1Δ_1Δ1​ proves the lightest massive excitation in appropriate theories of 3D matter and gravity with cosmological constant Λ<0Λ< 0Λ<0 can be no heavier than 1/(8GN)+O(−Λ)1/(8G_N)+O(\sqrt{-Λ})1/(8GN​)+O(−Λ​); the bounds on the number operators are related via AdS/CFT to the entropy of states in the dual gravitational theory. In the flat-space approximation, the limiting mass is exactly that of the lightest BTZ black hole.
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.