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:1404.4385 · Journal of High Energy Physics

Framed Wilson operators, fermionic strings, and gravitational anomaly in 4d

Ryan Thorngren

Published Feb 1, 2015• 66 citations
arXivHTMLPDF

Abstract

We study gapped systems with anomalous time-reversal symmetry and global gravitational anomaly in three and four spacetime dimensions. These systems describe topological order on the boundary of bosonic Symmetry Protected Topological (SPT) Phases. Our description of these phases is via the recent cobordism proposal for their classification. In particular, the behavior of these systems is determined by the geometry of Stiefel-Whitney classes. We discuss electric and magnetic operators defined by these classes, and new types of Wilson lines and surfaces that sit on their boundary. The lines describe fermionic particles, while the surfaces describe a sort of fermionic string. We show that QED with a fermionic monopole exhibits the 4d global gravitational anomaly and has a fermionic π-flux.

DOI
10.1007/jhep02(2015)152

Related papers

Framed Wilson Operators on the Boundaries of Novel SPT Phases

Ryan Thorngren

We study topological order on the boundary of Symmetry Pro-tected Topological Phases protected by time-reversal symmetry via the recent cobordism description of these theories. We find that the quasiparticle and quasistring excitations in these systems enjoy spe-cial braiding statistics and other properties which are determined by the behavior of certain Stiefel-Whitney classes. The statistics of qua-sistrings realize a recent description of loop braiding statistics in 3+1d. 1

•Apr 16, 2014•2 citations

Symmetry-protected topological phases and quantum anomalies

Chang-Tse Hsieh

We present the correspondence between symmetry-protected topological (SPT) phases and their anomalous boundary states, based on examples in various spacetime dimensions. Through the study of the effect of interactions on these SPT phases, we discovered a new formalism of quantum anomalies, associated with discrete spacetime (such as time-reversal and spatial reflection) symmetries in particular, to classify distinct interacting topological phases. An example is the Z2 classification of the (2+1)d topological insulator protected by charge U(1) and time-reversal (or CP) symmetries, which can be deduced by the form of the global U(1) gauge anomalies on its edge theories defined on closed unorientable manifolds. In this case, the nontrivial phase (in free systems) is robust against electron interactions. Another example is the (3+1)d topological superconductor protected by only time-reversal or reflection symmetry. For this system, we identified the bulk phase by studying the global gravitational anomalies of the surface theories formulated on unorientable spacetime manifolds, and also discussed its connection to the collapse of the non-interacting classification by an integer Z to Z16, in the presence of interactions. We also revisit the problem of gauging a discrete internal symmetry in theories of chiral (Weyl) fermions in 3+1 dimensions – which have not been fully understood so far – from the perspective of fermionic SPT phases in 4+1 dimensions. Comparing with the previous results, we give a complete answer for the anomalies constraints on the discrete symmetry, as our approach is based on purely geometrical considerations, namely, our assumption is more fundamental and general. Furthermore, our result also provides an understanding of gapped states of fermions with anomalous discrete symmetries, and we present a model, based on weak coupling, for constructing these anomalous gapped states.

•Jul 11, 2017•0 citations

(3+1)d boundaries with gravitational anomaly of (4+1)d invertible topological order for branch-independent bosonic systems

Zheyan Wan, Juven Wang, Xiao-Gang Wen

We study bosonic systems on a space-time lattice (with imaginary time) defined by path integrals of commuting fields. We introduce a concept of branch-independent bosonic (BIB) systems, whose path integral is independent of the branch structure of the space-time simplicial complex, even for a space-time with boundaries. In contrast, a generic lattice bosonic (GLB) system's path integral may depend on the branch structure. We find the invertible topological order characterized by the Stiefel-Whitney cocycle [such as $(4+1)\mathrm{d}$ ${\mathrm{w}}_{2}{\mathrm{w}}_{3}$] to be nontrivial for branch-independent bosonic systems, but this topological order and a trivial gapped tensor product state belong to the same phase (via a smooth deformation without any phase transition) for generic lattice bosonic systems. This implies that the invertible topological orders in generic lattice bosonic systems on a space-time lattice are not classified by the oriented cobordism. The branch dependence on a lattice may be related to the orthonormal frame of smooth manifolds and the framing anomaly of continuum field theories. In general, the branch structure on a discretized lattice may be related to a frame structure on a smooth manifold that trivializes any Stiefel-Whitney classes. We construct branch-independent bosonic systems to realize the ${\mathrm{w}}_{2}{\mathrm{w}}_{3}$ topological order, and its $(3+1)\mathrm{d}$ gapped or gapless boundaries. One of the gapped boundaries is a $(3+1)\mathrm{d}$ ${\mathbb{Z}}_{2}$ gauge theory with (1) fermionic ${\mathbb{Z}}_{2}$ gauge charge particle which trivializes ${\mathrm{w}}_{2}$ and (2) ``fermionic'' ${\mathbb{Z}}_{2}$ gauge flux line trivializes ${\mathrm{w}}_{3}$. In particular, if the flux loop's world sheet is unorientable, then the orientation-reversal one-dimensional world line must correspond to a fermion world line that does not carry the ${\mathbb{Z}}_{2}$ gauge charge. We also explain why Spin and Spin${}^{c}$ structures trivialize the ${\mathrm{w}}_{2}{\mathrm{w}}_{3}$ nonperturbative global pure gravitational anomaly to zero [which helps to construct the anomalous $(3+1)\mathrm{d}$ gapped ${\mathbb{Z}}_{2}$ and gapless all-fermion U(1) gauge theories], but the Spin${}^{h}$ and Spin${\ifmmode\times\else\texttimes\fi{}}_{{\mathbb{Z}}_{2}}$Spin$(n\ensuremath{\ge}3)$ structures modify the ${\mathrm{w}}_{2}{\mathrm{w}}_{3}$ into a nonperturbative global mixed gauge-gravitational anomaly, which helps to constrain grand unifications (e.g., $n=10,18$) or construct new models.

arXiv:2112.12148•Physical review. B./Physical review. B•Jul 20, 2022•12 citations

Bosonic Topological Insulators and Paramagnets: a view from cobordisms

Anton Kapustin

We classify Bosonic Topological Insulators and Paramagnets in D<=4 spatial dimensions using the cobordism approach. For D<4 we confirm that the only such phase which does not fit into the group cohomology classification is the 3D Bosonic Topological Insulator protected by time-reversal symmetry whose surface admits an all-fermion topologically ordered state. For D=4 there is a unique "beyond group cohomology" phase. It is protected by gravitational anomalies of the boundary theory and is stable without any additional symmetry.

arXiv:1404.6659•Apr 26, 2014•52 citations

Definition and classification of Fermi surface anomalies

Da-Chuan Lu, Juven Wang, Yi‐Zhuang You

We propose that the Fermi surface anomaly of symmetry group $G$ in any dimension is universally classified by $G$-symmetric interacting fermionic symmetry-protected topological (SPT) phases in $(0+1)$-dimensional spacetime. The argument is based on the perspective that the gapless fermions on the Fermi surface can be viewed as the topological boundary modes of Chern insulators in the phase space (position-momentum space). Given the noncommutative nature of the phase-space coordinates, we show that the momentum space dimensions should be counted as negative dimensions for SPT classification purposes. Therefore, the classification of phase-space Chern insulators (or, more generally fermionic SPT phases) always reduces to a $(0+1)$-dimensional problem, which can then be answered by the cobordism approach. In addition to the codimension-1 Fermi surface case, we also discuss the codimension-$p$ Fermi surface case briefly. We provide concrete examples to demonstrate the validity of our classification scheme, and make connections to the recent development of Fermi surface symmetric mass generation.

arXiv:2302.12731•Physical review. B./Physical review. B•Jan 12, 2024•11 citations

Global anomalies on the surface of fermionic symmetry-protected topological phases in (3+1) dimensions

Chang-Tse Hsieh, Gil Young Cho, Shinsei Ryu

Quantum anomalies, breakdown of classical symmetries by quantum effects, provide a sharp definition of symmetry protected topological phases. In particular, they can diagnose interaction effects on the noninteracting classification of fermionic symmetry protected topological phases. In this paper, we identify quantum anomalies in two kinds of (3+1)d fermionic symmetry protected topological phases: (i) topological insulators protected by CP (charge conjugation $\ifmmode\times\else\texttimes\fi{}$ reflection) and electromagnetic $\mathrm{U}(1)$ symmetries, and (ii) topological superconductors protected by reflection symmetry. For the first example, which is related to, by CPT-theorem, time-reversal symmetric topological insulators, we show that the CP-projected partition function of the surface theory is not invariant under large $\mathrm{U}(1)$ gauge transformations, but picks up an anomalous sign, signaling a ${\mathbb{Z}}_{2}$ topological classification. Similarly, for the second example, which is related to, by CPT-theorem, class DIII topological superconductors, we discuss the invariance/noninvariance of the partition function of the surface theory, defined on the three-torus and its descendants generated by the orientifold projection, under large diffeomorphisms (coordinate transformations). The connection to the collapse of the noninteracting classification by an integer ($\mathbb{Z}$) to ${\mathbb{Z}}_{16}$, in the presence of interactions, is discussed.

arXiv:1503.01411•Physical review. B./Physical review. B•Feb 18, 2016•74 citations
We study gapped systems with anomalous time-reversal symmetry and global gravitational anomaly in three and four spacetime dimensions. These systems describe topological order on the boundary of bosonic Symmetry Protected Topological (SPT) Phases. Our description of these phases is via the recent cobordism proposal for their classification. In particular, the behavior of these systems is determined by the geometry of Stiefel-Whitney classes. We discuss electric and magnetic operators defined by these classes, and new types of Wilson lines and surfaces that sit on their boundary. The lines describe fermionic particles, while the surfaces describe a sort of fermionic string. We show that QED with a fermionic monopole exhibits the 4d global gravitational anomaly and has a fermionic π-flux.
We present the correspondence between symmetry-protected topological (SPT) phases and their anomalous boundary states, based on examples in various spacetime dimensions. Through the study of the effect of interactions on these SPT phases, we discovered a new formalism of quantum anomalies, associated with discrete spacetime (such as time-reversal and spatial reflection) symmetries in particular, to classify distinct interacting topological phases. An example is the Z2 classification of the (2+1)d topological insulator protected by charge U(1) and time-reversal (or CP) symmetries, which can be deduced by the form of the global U(1) gauge anomalies on its edge theories defined on closed unorientable manifolds. In this case, the nontrivial phase (in free systems) is robust against electron interactions. Another example is the (3+1)d topological superconductor protected by only time-reversal or reflection symmetry. For this system, we identified the bulk phase by studying the global gravitational anomalies of the surface theories formulated on unorientable spacetime manifolds, and also discussed its connection to the collapse of the non-interacting classification by an integer Z to Z16, in the presence of interactions. We also revisit the problem of gauging a discrete internal symmetry in theories of chiral (Weyl) fermions in 3+1 dimensions – which have not been fully understood so far – from the perspective of fermionic SPT phases in 4+1 dimensions. Comparing with the previous results, we give a complete answer for the anomalies constraints on the discrete symmetry, as our approach is based on purely geometrical considerations, namely, our assumption is more fundamental and general. Furthermore, our result also provides an understanding of gapped states of fermions with anomalous discrete symmetries, and we present a model, based on weak coupling, for constructing these anomalous gapped states.
We study bosonic systems on a space-time lattice (with imaginary time) defined by path integrals of commuting fields. We introduce a concept of branch-independent bosonic (BIB) systems, whose path integral is independent of the branch structure of the space-time simplicial complex, even for a space-time with boundaries. In contrast, a generic lattice bosonic (GLB) system's path integral may depend on the branch structure. We find the invertible topological order characterized by the Stiefel-Whitney cocycle [such as (4+1)d(4+1)\mathrm{d}(4+1)d w2w3{\mathrm{w}}_{2}{\mathrm{w}}_{3}w2​w3​] to be nontrivial for branch-independent bosonic systems, but this topological order and a trivial gapped tensor product state belong to the same phase (via a smooth deformation without any phase transition) for generic lattice bosonic systems. This implies that the invertible topological orders in generic lattice bosonic systems on a space-time lattice are not classified by the oriented cobordism. The branch dependence on a lattice may be related to the orthonormal frame of smooth manifolds and the framing anomaly of continuum field theories. In general, the branch structure on a discretized lattice may be related to a frame structure on a smooth manifold that trivializes any Stiefel-Whitney classes. We construct branch-independent bosonic systems to realize the w2w3{\mathrm{w}}_{2}{\mathrm{w}}_{3}w2​w3​ topological order, and its (3+1)d(3+1)\mathrm{d}(3+1)d gapped or gapless boundaries. One of the gapped boundaries is a (3+1)d(3+1)\mathrm{d}(3+1)d Z2{\mathbb{Z}}_{2}Z2​ gauge theory with (1) fermionic Z2{\mathbb{Z}}_{2}Z2​ gauge charge particle which trivializes w2{\mathrm{w}}_{2}w2​ and (2) ``fermionic'' Z2{\mathbb{Z}}_{2}Z2​ gauge flux line trivializes w3{\mathrm{w}}_{3}w3​. In particular, if the flux loop's world sheet is unorientable, then the orientation-reversal one-dimensional world line must correspond to a fermion world line that does not carry the Z2{\mathbb{Z}}_{2}Z2​ gauge charge. We also explain why Spin and Spinc{}^{c}c structures trivialize the w2w3{\mathrm{w}}_{2}{\mathrm{w}}_{3}w2​w3​ nonperturbative global pure gravitational anomaly to zero [which helps to construct the anomalous (3+1)d(3+1)\mathrm{d}(3+1)d gapped Z2{\mathbb{Z}}_{2}Z2​ and gapless all-fermion U(1) gauge theories], but the Spinh{}^{h}h and Spin${\ifmmode\times\else\texttimes\fi{}}_{{\mathbb{Z}}_{2}}$Spin$(n\ensuremath{\ge}3)$ structures modify the w2w3{\mathrm{w}}_{2}{\mathrm{w}}_{3}w2​w3​ into a nonperturbative global mixed gauge-gravitational anomaly, which helps to constrain grand unifications (e.g., n=10,18n=10,18n=10,18) or construct new models.
We classify Bosonic Topological Insulators and Paramagnets in D<=4 spatial dimensions using the cobordism approach. For D<4 we confirm that the only such phase which does not fit into the group cohomology classification is the 3D Bosonic Topological Insulator protected by time-reversal symmetry whose surface admits an all-fermion topologically ordered state. For D=4 there is a unique "beyond group cohomology" phase. It is protected by gravitational anomalies of the boundary theory and is stable without any additional symmetry.
We propose that the Fermi surface anomaly of symmetry group GGG in any dimension is universally classified by GGG-symmetric interacting fermionic symmetry-protected topological (SPT) phases in (0+1)(0+1)(0+1)-dimensional spacetime. The argument is based on the perspective that the gapless fermions on the Fermi surface can be viewed as the topological boundary modes of Chern insulators in the phase space (position-momentum space). Given the noncommutative nature of the phase-space coordinates, we show that the momentum space dimensions should be counted as negative dimensions for SPT classification purposes. Therefore, the classification of phase-space Chern insulators (or, more generally fermionic SPT phases) always reduces to a (0+1)(0+1)(0+1)-dimensional problem, which can then be answered by the cobordism approach. In addition to the codimension-1 Fermi surface case, we also discuss the codimension-ppp Fermi surface case briefly. We provide concrete examples to demonstrate the validity of our classification scheme, and make connections to the recent development of Fermi surface symmetric mass generation.
Quantum anomalies, breakdown of classical symmetries by quantum effects, provide a sharp definition of symmetry protected topological phases. In particular, they can diagnose interaction effects on the noninteracting classification of fermionic symmetry protected topological phases. In this paper, we identify quantum anomalies in two kinds of (3+1)d fermionic symmetry protected topological phases: (i) topological insulators protected by CP (charge conjugation $\ifmmode\times\else\texttimes\fi{}$ reflection) and electromagnetic U(1)\mathrm{U}(1)U(1) symmetries, and (ii) topological superconductors protected by reflection symmetry. For the first example, which is related to, by CPT-theorem, time-reversal symmetric topological insulators, we show that the CP-projected partition function of the surface theory is not invariant under large U(1)\mathrm{U}(1)U(1) gauge transformations, but picks up an anomalous sign, signaling a Z2{\mathbb{Z}}_{2}Z2​ topological classification. Similarly, for the second example, which is related to, by CPT-theorem, class DIII topological superconductors, we discuss the invariance/noninvariance of the partition function of the surface theory, defined on the three-torus and its descendants generated by the orientifold projection, under large diffeomorphisms (coordinate transformations). The connection to the collapse of the noninteracting classification by an integer (Z\mathbb{Z}Z) to Z16{\mathbb{Z}}_{16}Z16​, in the presence of interactions, is discussed.