• Corpus ID: 32799231

Symmetry Protected Topological Phases, Anomalies, and Cobordisms: Beyond Group Cohomology

@article{Kapustin2014SymmetryPT,
  title={Symmetry Protected Topological Phases, Anomalies, and Cobordisms: Beyond Group Cohomology},
  author={Anton Kapustin},
  journal={arXiv: Strongly Correlated Electrons},
  year={2014}
}
  • A. Kapustin
  • Published 6 March 2014
  • Mathematics
  • arXiv: Strongly Correlated Electrons
We propose that Symmetry Protected Topological Phases with a finite symmetry group G are classified by cobordism groups of the classifying space of G. This provides an explanation for the recent discovery of bosonic SPT phases which do not fit into the group cohomology classification. We discuss the connection of the cobordism classification of SPT phases to gauge and gravitational anomalies in various dimensions. 
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References

SHOWING 1-10 OF 18 REFERENCES
Anomalies of discrete symmetries in various dimensions and group cohomology
We study 't Hooft anomalies for discrete global symmetries in bosonic theories in 2, 3 and 4 dimensions. We show that such anomalies may arise in gauge theories with topological terms in the action,
Anomalous discrete symmetries in three dimensions and group cohomology.
TLDR
This work constructs examples of bosonic field theories in three dimensions with a nonvanishing 't Hooft anomaly for a discrete global symmetry G such that gauging G(1) necessarily breaks G(2) and vice versa.
Classifying gauge anomalies through symmetry-protected trivial orders and classifying gravitational anomalies through topological orders
In this paper, we systematically study gauge anomalies in bosonic and fermionic weak-coupling gauge theories with gauge group G (which can be continuous or discrete). We show a very close relation
Exactly soluble model of a three-dimensional symmetry-protected topological phase of bosons with surface topological order
We construct an exactly soluble Hamiltonian on the D=3 cubic lattice, whose ground state is a topological phase of bosons protected by time-reversal symmetry, i.e., a symmetry-protected topological
Exactly Soluble Model of a 3 D Symmetry Protected Topological Phase of Bosons with Surface Topological Order
We construct an exactly soluble Hamiltonian on the D=3 cubic lattice, whose ground state is a topological phase of bosons protected by time reversal symmetry, i.e a symmetry protected topological
Periodic table for topological insulators and superconductors
Gapped phases of noninteracting fermions, with and without charge conservation and time-reversal symmetry, are classified using Bott periodicity. The symmetry and spatial dimension determines a
Topological phases of fermions in one dimension
In this paper we show how the classification of topological phases in insulators and superconductors is changed by interactions, in the case of one-dimensional systems. We focus on the
Physics of three dimensional bosonic topological insulators: Surface Deconfined Criticality and Quantized Magnetoelectric Effect
We discuss physical properties of `integer' topological phases of bosons in D=3+1 dimensions, protected by internal symmetries like time reversal and/or charge conservation. These phases invoke
Classification of gapped symmetric phases in one-dimensional spin systems
Quantum many-body systems divide into a variety of phases with very different physical properties. The questions of what kinds of phases exist and how to identify them seem hard, especially for
Boson topological insulators: A window into highly entangled quantum phases
We study several aspects of the realization of global symmetries in highly entangled phases of quantum matter. Examples include gapped topological ordered phases, gapless quantum spin liquids and
...
...