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:2609.20813

A comparison between statistics and 't Hooft anomaly

Hanyu Xue

Published Sep 17, 2026• 0 citations
arXivHTMLPDF

Abstract

Anomalies of generalized symmetries are closely related to the statistics of topological excitations. Truncating a symmetry transformation to a finite region produces an operator that creates symmetry defects on its boundary; we call this a symmetry patch operator. Such operators resemble hopping operators that create topological excitations at the boundary of their support, but this geometric similarity does not justify identifying the two. For finite Abelian groups in one spatial dimension, we prove that if the same family of local operators defines both valid hopping operators and compatible symmetry patches, the type-III component of the anomaly is $-2$ times that of the statistics. We establish a one-to-one correspondence between statistics and anomaly by treating hopping operators as symmetric operators that commute with the global symmetry, following the description of symmetry through the algebra of local symmetric operators. We explain this correspondence through gauging by coupling boundary matter to a Dijkgraaf-Witten gauge field in one higher spatial dimension. Hopping operators act on the gauge-invariant degrees of freedom. After fixing the background gauge field, the residual gauge transformations become the global symmetry, and the gauge invariance of hopping operators becomes the requirement that they commute with this symmetry.

DOI
10.48550/arxiv.2609.20813

Related papers

Symmetry as a shadow of topological order and a derivation of topological holographic principle

Arkya Chatterjee, Xiao-Gang Wen

Symmetry is usually defined via transformations described by a (higher) group. But a symmetry really corresponds to an algebra of local symmetric operators, which directly constrains the properties of the system. In this paper, we point out that the algebra of local symmetric operators contains a special class of extended operators---transparent patch operators, which reveal the selection sectors and hence the corresponding symmetry. The algebra of those transparent patch operators in $n$-dimensional space gives rise to a nondegenerate braided fusion $n$-category, which happens to describe a topological order in one higher dimension (for finite symmetry). Such a holographic theory not only describes (higher) symmetries, it also describes anomalous (higher) symmetries, noninvertible (higher) symmetries (also known as algebraic higher symmetries), and noninvertible gravitational anomalies. Thus, topological order in one higher dimension, replacing group, provides a unified and systematic description of the above generalized symmetries. This is referred to as symmetry/topological-order (Symm/TO) correspondence. Our approach also leads to a derivation of topological holographic principle: boundary uniquely determines the bulk, or more precisely, the algebra of local boundary operators uniquely determines the bulk topological order. As an application of the Symm/TO correspondence, we show the equivalence between ${\mathbb{Z}}_{2}\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{2}$ symmetry with mixed anomaly and ${\mathbb{Z}}_{4}$ symmetry, as well as between many other symmetries, in 1-dimensional space.

arXiv:2203.03596•Physical review. B./Physical review. B•Apr 24, 2023•88 citations

2+1D symmetry-topological-order from local symmetric operators in 1+1D

Kansei Inamura, Xiao-Gang Wen

A generalized symmetry (defined by the algebra of local symmetric operators) can go beyond group or higher group description. A theory of generalized symmetry (up to holo-equivalence) was developed in terms of symmetry-TO -- a bosonic topological order (TO) with gappable boundary in one higher dimension. We propose a general method to compute the 2+1D symmetry-TO from the local symmetric operators in 1+1D systems. Our theory is based on the commutant patch operators, which are extended operators constructed as products and sums of local symmetric operators. A commutant patch operator commutes with all local symmetric operators away from its boundary. We argue that topological invariants associated with anyon diagrams in 2+1D can be computed as contracted products of commutant patch operators in 1+1D. In particular, we give concrete formulae for several topological invariants in terms of commutant patch operators. Topological invariants computed from patch operators include those beyond modular data, such as the link invariants associated with the Borromean rings and the Whitehead link. These results suggest that the algebra of commutant patch operators is described by 2+1D symmetry-TO. Based on our analysis, we also argue briefly that the commutant patch operators would serve as order parameters for gapped phases with finite symmetries.

arXiv:2310.05790•Oct 9, 2023•4 citations

Invariants of Sequential Circuits and Generalized Non-Abelian Statistics

Shintaro Sato, Yoshimasa Hidaka, Ryohei Kobayashi

Non-invertible symmetries in quantum many-body systems generally give rise to sequential unitary circuits that move symmetry defects. In this paper, we investigate invariants defined by sequences of such circuits, which move non-invertible defects and generate a Berry phase evaluated on quantum states with defects. We show that this Berry phase generally defines an invariant under local deformations, provided that the sequential circuits preserve the locality of those deformations. This invariant also rules out a short-range-entangled state that preserves the non-invertible symmetry, thereby signaling the 't Hooft anomaly of a non-invertible symmetry purely in terms of unitary operators acting on a state. We then apply this framework to loop excitations in three spatial dimensions and identify a new loop excitation in the (3+1)D $\mathbb{D}_4$ topological order, which we dub a non-Abelian fermionic loop. Using the invariant of sequential circuits, we characterize the statistics of non-Abelian fermionic loops. In addition, we find a new (3+1)D mixed topological order with a single non-Abelian fermionic loop, whose long-range entanglement is protected by an invariant of sequential circuits.

cond-mat.str-elCross-listedarXiv:2606.11527•Jun 11, 2026•0 citations

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.

arXiv:1912.02817•Dec 5, 2019•107 citations

Gauging Modulated Symmetries: Bond Algebras, Higher-Form Symmetries, and Symmetry-Enriched Topological Order

Gustavo M. Yoshitome, Pedro R. S. Gomes, Guilherme Delfino

We develop a constructive framework for characterizing modulated symmetries and their gauging in $d$-dimensional lattice systems. The modulation profiles are organized as a finite-dimensional representation of the space group acting on a finite Abelian internal symmetry group. Since the corresponding automorphism group is finite, the nontrivial spatial action factors through a finite quotient of the space group, leading to periodic modulation patterns and providing a classification framework for discrete Abelian modulated symmetries. The symmetry-allowed local interactions are systematically obtained from a kernel construction, which also yields an algebra of open operators. Upon gauging, its elements are transmuted into extended Wilson and boundary operators while retaining their algebraic relations. In the deconfined regime, the resulting algebra realizes a mixed 't Hooft anomaly, providing a microscopic route to Abelian crystalline symmetry-enriched topological phases (SETs). These SETs do not mix electric and magnetic sectors, which transform in mutually contragredient representations, thereby preserving their braiding pairing. We illustrate the framework through explicit examples and show that several models present in the literature fit naturally within our construction.

arXiv:2610.10764•Oct 7, 2026•0 citations

Higher-group symmetry in finite gauge theory and stabilizer codes

Maissam Barkeshli, Yu-An Chen, Po-Shen Hsin +1 more

A large class of gapped phases of matter can be described by topological finite group gauge theories. In this paper, we show how such gauge theories possess a higher-group global symmetry, which we study in detail. We derive the d d -group global symmetry and its ’t Hooft anomaly for topological finite group gauge theories in (d+1) ( d + 1 ) space-time dimensions, including non-Abelian gauge groups and Dijkgraaf-Witten twists. We focus on the 1-form symmetry generated by invertible (Abelian) magnetic defects and the higher-form symmetries generated by invertible topological defects decorated with lower dimensional gauged symmetry-protected topological (SPT) phases. We show that due to a generalization of the Witten effect and charge-flux attachment, the 1-form symmetry generated by the magnetic defects mixes with other symmetries into a higher group. We describe such higher-group symmetry in various lattice model examples. We discuss several applications, including the classification of fermionic SPT phases in (3+1)D for general fermionic symmetry groups, where we also derive a simpler formula for the [O_5] ∈ H^5(BG, U(1)) [ O 5 ] ∈ H 5 ( B G , U ( 1 ) ) obstruction that has appeared in prior work. We also show how the d d -group symmetry is related to fault-tolerant non-Pauli logical gates and a refined Clifford hierarchy in stabilizer codes. We discover new logical gates in stabilizer codes using the d d -group symmetry, such as a controlled Z gate in the (3+1) D \mathbb{Z}_2 ℤ 2 toric code.

arXiv:2211.11764•SciPost Physics•Apr 3, 2024•48 citations
Anomalies of generalized symmetries are closely related to the statistics of topological excitations. Truncating a symmetry transformation to a finite region produces an operator that creates symmetry defects on its boundary; we call this a symmetry patch operator. Such operators resemble hopping operators that create topological excitations at the boundary of their support, but this geometric similarity does not justify identifying the two. For finite Abelian groups in one spatial dimension, we prove that if the same family of local operators defines both valid hopping operators and compatible symmetry patches, the type-III component of the anomaly is −2-2−2 times that of the statistics. We establish a one-to-one correspondence between statistics and anomaly by treating hopping operators as symmetric operators that commute with the global symmetry, following the description of symmetry through the algebra of local symmetric operators. We explain this correspondence through gauging by coupling boundary matter to a Dijkgraaf-Witten gauge field in one higher spatial dimension. Hopping operators act on the gauge-invariant degrees of freedom. After fixing the background gauge field, the residual gauge transformations become the global symmetry, and the gauge invariance of hopping operators becomes the requirement that they commute with this symmetry.
Symmetry is usually defined via transformations described by a (higher) group. But a symmetry really corresponds to an algebra of local symmetric operators, which directly constrains the properties of the system. In this paper, we point out that the algebra of local symmetric operators contains a special class of extended operators---transparent patch operators, which reveal the selection sectors and hence the corresponding symmetry. The algebra of those transparent patch operators in nnn-dimensional space gives rise to a nondegenerate braided fusion nnn-category, which happens to describe a topological order in one higher dimension (for finite symmetry). Such a holographic theory not only describes (higher) symmetries, it also describes anomalous (higher) symmetries, noninvertible (higher) symmetries (also known as algebraic higher symmetries), and noninvertible gravitational anomalies. Thus, topological order in one higher dimension, replacing group, provides a unified and systematic description of the above generalized symmetries. This is referred to as symmetry/topological-order (Symm/TO) correspondence. Our approach also leads to a derivation of topological holographic principle: boundary uniquely determines the bulk, or more precisely, the algebra of local boundary operators uniquely determines the bulk topological order. As an application of the Symm/TO correspondence, we show the equivalence between ${\mathbb{Z}}_{2}\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{2}$ symmetry with mixed anomaly and Z4{\mathbb{Z}}_{4}Z4​ symmetry, as well as between many other symmetries, in 1-dimensional space.
A generalized symmetry (defined by the algebra of local symmetric operators) can go beyond group or higher group description. A theory of generalized symmetry (up to holo-equivalence) was developed in terms of symmetry-TO -- a bosonic topological order (TO) with gappable boundary in one higher dimension. We propose a general method to compute the 2+1D symmetry-TO from the local symmetric operators in 1+1D systems. Our theory is based on the commutant patch operators, which are extended operators constructed as products and sums of local symmetric operators. A commutant patch operator commutes with all local symmetric operators away from its boundary. We argue that topological invariants associated with anyon diagrams in 2+1D can be computed as contracted products of commutant patch operators in 1+1D. In particular, we give concrete formulae for several topological invariants in terms of commutant patch operators. Topological invariants computed from patch operators include those beyond modular data, such as the link invariants associated with the Borromean rings and the Whitehead link. These results suggest that the algebra of commutant patch operators is described by 2+1D symmetry-TO. Based on our analysis, we also argue briefly that the commutant patch operators would serve as order parameters for gapped phases with finite symmetries.
Non-invertible symmetries in quantum many-body systems generally give rise to sequential unitary circuits that move symmetry defects. In this paper, we investigate invariants defined by sequences of such circuits, which move non-invertible defects and generate a Berry phase evaluated on quantum states with defects. We show that this Berry phase generally defines an invariant under local deformations, provided that the sequential circuits preserve the locality of those deformations. This invariant also rules out a short-range-entangled state that preserves the non-invertible symmetry, thereby signaling the 't Hooft anomaly of a non-invertible symmetry purely in terms of unitary operators acting on a state. We then apply this framework to loop excitations in three spatial dimensions and identify a new loop excitation in the (3+1)D D4\mathbb{D}_4D4​ topological order, which we dub a non-Abelian fermionic loop. Using the invariant of sequential circuits, we characterize the statistics of non-Abelian fermionic loops. In addition, we find a new (3+1)D mixed topological order with a single non-Abelian fermionic loop, whose long-range entanglement is protected by an invariant of sequential circuits.
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 develop a constructive framework for characterizing modulated symmetries and their gauging in ddd-dimensional lattice systems. The modulation profiles are organized as a finite-dimensional representation of the space group acting on a finite Abelian internal symmetry group. Since the corresponding automorphism group is finite, the nontrivial spatial action factors through a finite quotient of the space group, leading to periodic modulation patterns and providing a classification framework for discrete Abelian modulated symmetries. The symmetry-allowed local interactions are systematically obtained from a kernel construction, which also yields an algebra of open operators. Upon gauging, its elements are transmuted into extended Wilson and boundary operators while retaining their algebraic relations. In the deconfined regime, the resulting algebra realizes a mixed 't Hooft anomaly, providing a microscopic route to Abelian crystalline symmetry-enriched topological phases (SETs). These SETs do not mix electric and magnetic sectors, which transform in mutually contragredient representations, thereby preserving their braiding pairing. We illustrate the framework through explicit examples and show that several models present in the literature fit naturally within our construction.
A large class of gapped phases of matter can be described by topological finite group gauge theories. In this paper, we show how such gauge theories possess a higher-group global symmetry, which we study in detail. We derive the d d -group global symmetry and its ’t Hooft anomaly for topological finite group gauge theories in (d+1) ( d + 1 ) space-time dimensions, including non-Abelian gauge groups and Dijkgraaf-Witten twists. We focus on the 1-form symmetry generated by invertible (Abelian) magnetic defects and the higher-form symmetries generated by invertible topological defects decorated with lower dimensional gauged symmetry-protected topological (SPT) phases. We show that due to a generalization of the Witten effect and charge-flux attachment, the 1-form symmetry generated by the magnetic defects mixes with other symmetries into a higher group. We describe such higher-group symmetry in various lattice model examples. We discuss several applications, including the classification of fermionic SPT phases in (3+1)D for general fermionic symmetry groups, where we also derive a simpler formula for the [O_5] ∈ H^5(BG, U(1)) [ O 5 ] ∈ H 5 ( B G , U ( 1 ) ) obstruction that has appeared in prior work. We also show how the d d -group symmetry is related to fault-tolerant non-Pauli logical gates and a refined Clifford hierarchy in stabilizer codes. We discover new logical gates in stabilizer codes using the d d -group symmetry, such as a controlled Z gate in the (3+1) D \mathbb{Z}_2 ℤ 2 toric code.