Braided fusion categories, gravitational anomalies, and the mathematical framework for topological orders in any dimensions

Kong, Liang; Wen, Xiao-Gang

doi:10.48550/arxiv.1405.5858

Cited by 103 publications

(265 citation statements)

References 99 publications

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“…Here we attempt to use a neutral terminology to avoid any confusion 8. We note that the definitions of gravitational anomalies in[17,54] and[18] are different. This leads to different opinions, between[17,54] and[18], whether SPT states allow non-perturbative global gravitational anomalies or not along their boundaries, especially for SPT states with time reversal symmetries.…”

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confidence: 99%

Symmetric Gapped Interfaces of SPT and SET States: Systematic Constructions

Wang,

Wen,

Witten

2017

Preprint

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Symmetry protected topological (SPT) states have boundary 't Hooft anomalies that obstruct an effective boundary theory realized in its own dimension with UV completion and with an on-site G-symmetry. In this work, yet we show that a certain anomalous non-on-site G symmetry along the boundary becomes on-site when viewed as an extended H symmetry, via a suitable group extension 1 → K → H → G → 1. Namely, a non-perturbative global (gauge/gravitational) anomaly in G becomes anomaly-free in H. This guides us to construct exactly soluble lattice path integral and Hamiltonian of symmetric gapped boundaries, always existent for any SPT state in any spacetime dimension d ≥ 2 of any finite symmetry group, including on-site unitary and anti-unitary time-reversal symmetries. The resulting symmetric gapped boundary can be described either by an H-symmetry extended boundary of bulk d ≥ 2, or more naturally by a topological emergent K-gauge theory with a global symmetry G on a 3+1D bulk or above. The excitations on such a symmetric topologically ordered boundary can carry fractional quantum numbers of the symmetry G, described by representations of H. (Apply our approach to a 1+1D boundary of 2+1D bulk, we find that a deconfined gauge boundary indeed has spontaneous symmetry breaking with long-range order. The deconfined symmetry-breaking phase crosses over smoothly to a confined phase without a phase transition.) In contrast to known gapped boundaries/interfaces obtained via symmetry breaking (either global symmetry breaking or Anderson-Higgs mechanism for gauge theory), our approach is based on symmetry extension. More generally, applying our approach to SPT states, topologically ordered gauge theories and symmetry enriched topologically ordered (SET) states, leads to generic boundaries/interfaces constructed with a mixture of symmetry breaking, symmetry extension, and dynamical gauging.

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Symmetric Gapped Interfaces of SPT and SET States: Systematic Constructions

Wang,

Wen,

Witten

2017

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“…Finally, we would like to stress that just like we use tensor category theory to describe Chern-Simons theory, the algebraic tensor 2-category theory [26] might also provide us an equivalent UV-complete description for such a topological quantum gravity theory in 3 + 1D.…”

Section: B Canonical Quantizationmentioning

confidence: 99%

Topological gravity in 3+1D and a possible origin of dark matte

Fang¹,

Gu²

2021

Preprint

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Dark matter is one of the deepest mystery of the universe. So far there is no natural explanation why the dark matter should exist and even dominate the universe. In this paper, we begin with a 3+1D topological gravity theory which is super renormalizable with vanishing beta functions, then we argue that Einstein gravity can emerge by condensing loop-like excitation from the underlying topological gravity theory. In the meanwhile, the uncondensed loop-like excitations serves as a natural candidate of dark matter and a generalized Einstein equation can be derived in the presence of loop-source(dark matter) background. Surprisingly, we find that such kind of dark matter will not contribute to scalar curvature, however, it will become a source of torsion. Finally, we derive the generalized Einstein equation in the presence of Dirac field. Very different from the usual Einstein-Carton theory, our theory further predicts that any type of normal matter, including Dirac field will not produce torsion. All these unique predictions can be tested by future experiments. Our framework suggests that topological invariant principle might play a more profound role than the well-known general covariance principle, especially towards understanding the nature of dark matter and quantum gravity in 3+1D.

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“…As pointed out by Ref. 26, for the invertible anomalies (described by single-component partition function), the properties of the partition function Z(M D , A) is directly related to the topological invariance 8,[27][28][29] of the SPT order and the invertible topological order in one higher dimension. Therefore, such an approach may allow us to compute all the 't Hooft anomaly of any symmetry, as well as the global gravitational anomaly.…”

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confidence: 90%

“…Thus the gravitational anomaly simply corresponds to the topological order in one higher dimension. 2,8 Recently, it was pointed out that the gravitational anomaly defined this way is more general than the usual gravitational anomaly defined via the obstruction to define a path integral (i.e. to have a diffeomorphism invariant path integral).…”

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A unified view on symmetry, anomalous symmetry and non-invertible gravitational anomaly

Ji¹,

Wen²

2021

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In this paper, using 1+1D models as examples, we study symmetries and anomalous symmetries via multi-component partition functions obtained through symmetry twists, and their transformations under the mapping class group of spacetime. This point of view allows us to treat symmetries and anomalous symmetries as non-invertible gravitational anomalies (which are also described by multi-component partition functions, transforming covariantly under the mapping group transformations). This allows us to directly see how symmetry and anomalous symmetry constraint the low energy dynamics of the systems, since the low energy dynamics is directly encoded in the partition functions. More generally, symmetries, anomalous symmetries, non-invertible gravitational anomalies, and their combinations, can all be viewed as constraints on low energy dynamics. In this paper, we demonstrate that they all can be viewed uniformally and systematically as pure (non-invertible) gravitational anomalies.

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Braided fusion categories, gravitational anomalies, and the mathematical framework for topological orders in any dimensions

Cited by 103 publications

References 99 publications

Symmetric Gapped Interfaces of SPT and SET States: Systematic Constructions

Symmetric Gapped Interfaces of SPT and SET States: Systematic Constructions

Topological gravity in 3+1D and a possible origin of dark matte

A unified view on symmetry, anomalous symmetry and non-invertible gravitational anomaly

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