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Title: Fermionic Finite-Group Gauge Theories and Interacting Symmetric/Crystalline Orders via Cobordisms

Abstract

We formulate a family of spin Topological Quantum Field Theories (spin-TQFTs) as fermionic generalization of bosonic Dijkgraaf–Witten TQFTs. They are obtained by gauging G-equivariant invertible spin-TQFTs, or, in physics language, gauging the interacting fermionic Symmetry Protected Topological states (SPTs) with a finite group G symmetry. We use the fact that the torsion part of the classification is given by Pontryagin duals to spin-bordism groups of the classifying space BG. We also consider unoriented analogues, that is G-equivariant invertible pin±-TQFTs (fermionic time-reversal-SPTs) and their gauging. We compute these groups for various examples of abelian G using Adams spectral sequence and describe all corresponding TQFTs via certain bordism invariants in 3, 4 and other dimensions. This gives explicit formulas for the partition functions of spin-TQFTs on closed manifolds with possible extended operators inserted. The results also provide explicit classification of ’t Hooft anomalies of fermionic QFTs with finite abelian group symmetries in one dimension lower. We construct new anomalous boundary spin-TQFTs (surface fermionic topological orders). Finally, we explore SPT and symmetry enriched topologically (SET) ordered states, and crystalline SPTs protected by space-group (e.g. translation Z) or point-group (e.g. reflection, inversion or rotation Cm) symmetries, via the layer-stacking construction.

Authors:
 [1];  [2]; ORCiD logo [3];  [4];  [5]
+ Show Author Affiliations
  1. Harvard Univ., Cambridge, MA (United States); Perimeter Inst. for Theoretical Physics, Waterloo, ON (Canada); Univ. of Toronto, ON (Canada)
  2. Institute for Advanced Study, Princeton, NJ (United States)
  3. Institute for Advanced Study, Princeton, NJ (United States); International Centre for Theoretical Physics (ICTP), Trieste (Italy)
  4. Univ. of Science and Technology of China, Hefei (China)
  5. Institute for Advanced Study, Princeton, NJ (United States); Harvard Univ., Cambridge, MA (United States)
Publication Date:
Research Org.:
Institute for Advanced Study, Princeton, NJ (United States)
Sponsoring Org.:
USDOE Office of Science (SC); National Science Foundation (NSF); National Natural Science Foundation of China (NSFC)
OSTI Identifier:
1801983
Grant/Contract Number:  
SC0009988; PHY-1606531; DMS-1607871; 11431010; 11571329; PHY-1748958
Resource Type:
Accepted Manuscript
Journal Name:
Communications in Mathematical Physics
Additional Journal Information:
Journal Volume: 376; Journal Issue: 2; Journal ID: ISSN 0010-3616
Publisher:
Springer
Country of Publication:
United States
Language:
English
Subject:
97 MATHEMATICS AND COMPUTING; Physics

Citation Formats

Guo, Meng, Ohmori, Kantaro, Putrov, Pavel, Wan, Zheyan, and Wang, Juven. Fermionic Finite-Group Gauge Theories and Interacting Symmetric/Crystalline Orders via Cobordisms. United States: N. p., 2020. Web. doi:10.1007/s00220-019-03671-6.
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Guo, Meng, Ohmori, Kantaro, Putrov, Pavel, Wan, Zheyan, & Wang, Juven. Fermionic Finite-Group Gauge Theories and Interacting Symmetric/Crystalline Orders via Cobordisms. United States. https://doi.org/10.1007/s00220-019-03671-6
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Guo, Meng, Ohmori, Kantaro, Putrov, Pavel, Wan, Zheyan, and Wang, Juven. Wed . "Fermionic Finite-Group Gauge Theories and Interacting Symmetric/Crystalline Orders via Cobordisms". United States. https://doi.org/10.1007/s00220-019-03671-6. https://www.osti.gov/servlets/purl/1801983.
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@article{osti_1801983,
title = {Fermionic Finite-Group Gauge Theories and Interacting Symmetric/Crystalline Orders via Cobordisms},
author = {Guo, Meng and Ohmori, Kantaro and Putrov, Pavel and Wan, Zheyan and Wang, Juven},
abstractNote = {We formulate a family of spin Topological Quantum Field Theories (spin-TQFTs) as fermionic generalization of bosonic Dijkgraaf–Witten TQFTs. They are obtained by gauging G-equivariant invertible spin-TQFTs, or, in physics language, gauging the interacting fermionic Symmetry Protected Topological states (SPTs) with a finite group G symmetry. We use the fact that the torsion part of the classification is given by Pontryagin duals to spin-bordism groups of the classifying space BG. We also consider unoriented analogues, that is G-equivariant invertible pin±-TQFTs (fermionic time-reversal-SPTs) and their gauging. We compute these groups for various examples of abelian G using Adams spectral sequence and describe all corresponding TQFTs via certain bordism invariants in 3, 4 and other dimensions. This gives explicit formulas for the partition functions of spin-TQFTs on closed manifolds with possible extended operators inserted. The results also provide explicit classification of ’t Hooft anomalies of fermionic QFTs with finite abelian group symmetries in one dimension lower. We construct new anomalous boundary spin-TQFTs (surface fermionic topological orders). Finally, we explore SPT and symmetry enriched topologically (SET) ordered states, and crystalline SPTs protected by space-group (e.g. translation Z) or point-group (e.g. reflection, inversion or rotation Cm) symmetries, via the layer-stacking construction.},
doi = {10.1007/s00220-019-03671-6},
journal = {Communications in Mathematical Physics},
number = 2,
volume = 376,
place = {United States},
year = {Wed Jan 22 00:00:00 EST 2020},
month = {Wed Jan 22 00:00:00 EST 2020}
}
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https://doi.org/10.1007/s00220-019-03671-6
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