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

DataBETA

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

Meng Guo
(
)
,
Kantaro Ohmori
(
- Princeton, Inst. Advanced Study
)
,
Pavel Putrov
(
- Princeton, Inst. Advanced Study and
- ICTP, Trieste
)
,
Zheyan Wan
(
- Tsinghua U., Beijing and
- USTC, Hefei
)
,
Juven Wang
(
- Princeton, Inst. Advanced Study and
- Harvard U., CMSA
)

Dec 31, 2018

82 pages

Published in:

Commun.Math.Phys. 376 (2020) 2, 1073-1154

e-Print:

1812.11959 [hep-th]

DOI:

10.1007/s00220-019-03671-6 (publication)

View in:

reference search95 citations

Citations per year

Abstract: (Springer)

We formulate a family of spin Topological Quantum Filed 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 latter are classified by Pontryagin duals to spin-bordism groups of the classifying space

BG

. We also consider unoriented analogues, that is

G

-equivariant invertible pin

^\pm

-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 dimensions 3, 4, and other. 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 deconfined spin-TQFTs (surface fermionic topological orders). We explore SPT and SET (symmetry enriched topologically ordered) states, and crystalline SPTs protected by space-group (e.g. translation

\mathbb{Z}

) or point-group (e.g. reflection, inversion or rotation

C_m

) symmetries, via the layer-stacking construction.

Note:

81 pages, 28 figures

field theory: topological
fermion: interaction
group: abelian
group: finite
partition function
deconfinement
crystal
anomaly
time reversal

References(74)

Figures(5)

[1]

: The structure of the spin cobordism ring

D.W. Anderson
,
E.H. Brown
,
F.P. Peterson

- Annals Math. 86 (1967) 271-298

[2]

Topological Crystalline Insulators and Topological Superconductors: From Concepts to Materials

- Ann.Rev.Condensed Matter Phys. 6 (2015) 361
•
e-Print:
- 1501.00531
•
DOI:
- 10.1146/annurev-conmatphys-031214-014501

[3]

: Topological quantum field theories. Publications Mathématiques de l’Institut des Hautes Études Scientifiques 68(1), 175-186

M. Atiyah

[4]

A Guide for Computing Stable Homotopy Groups

e-Print:
- 1801.07530

[5]

: Spin modular categories. arXiv preprint

A. Beliakova
,
C. Blanchet
,
E. Contreras

•
e-Print:
- 1411.4232

[6]

Classification of Abelian spin Chern-Simons theories

e-Print:
- hep-th/0505235

[7]

: Topological quantum field theories for surfaces with spin structure

C. Blanchet
,
G. Masbaum

- Duke Math.J. 82 (1996) 229-268

[8]

: Invariants on three-manifolds with spin structure

C. Blanchet

- Comment.Math.Helv. 67 (1992) 406-427

[9]

: The Pontrjagin dual of 3-dimensional spin bordism. ArXiv e-prints, December

G. Brumfiel
,
J. Morgan

[10]

: The Pontrjagin dual of 4-dimensional spin bordism. ArXiv e-prints, March

G. Brumfiel
,
J. Morgan

[11]

: Greenlees, JPC: Connective real K-theory of finite groups, vol. 169. American Mathematical Society, Providence

R.R. Bruner

[12]

Homotopy Theoretic Classification of Symmetry Protected Phases

Jonathan A. Campbell

e-Print:
- 1708.04264

[13]

Braiding with Borromean Rings in (3+1)-Dimensional Spacetime

AtMa P.O. Chan
(
- Illinois U., Urbana (main)
)
,
Peng Ye
(
- Illinois U., Urbana (main) and
- Zhongshan U.
)
,
Shinsei Ryu
(
- Chicago U.
)

- Phys.Rev.Lett. 121 (2018) 6, 061601
•
e-Print:
- 1703.01926
•
DOI:
- 10.1103/PhysRevLett.121.061601

[14]

Symmetry-protected topological phases from decorated domain walls

- Nature Commun. 5 (2014) 1, 3507
•
DOI:
- 10.1038/ncomms4507

[15]

Loop Braiding Statistics and Interacting Fermionic Symmetry-Protected Topological Phases in Three Dimensions

- Phys.Rev.X 8 (2018) 1, 011054
•
e-Print:
- 1705.08911
•
DOI:
- 10.1103/PhysRevX.8.011054

[16]

Rotation symmetry-protected topological phases of fermions

Meng Cheng
(
- Yale U.
)
,
Chenjie Wang
(
- Hong Kong U.
)

- Phys.Rev.B 105 (2022) 19, 195154
•
e-Print:
- 1810.12308
•
DOI:
- 10.1103/PhysRevB.105.195154

[17]

: On an invariant of link cobordism in dimension four

T.D. Cochran

- Topology Appl. 18 (1984) 97-108

[18]

The Arf-Brown TQFT of Pin $^-$ Surfaces

e-Print:
- 1803.11183
•
DOI:
- 10.1090/conm/718/14478

[19]

Topological Gauge Theories and Group Cohomology

- Commun.Math.Phys. 129 (1990) 393
•
DOI:
- 10.1007/BF02096988

[20]

The effects of interactions on the topological classification of free fermion systems

- Phys.Rev.B 81 (2010) 134509
•
e-Print:
- 0904.2197
•
DOI:
- 10.1103/PhysRevB.81.134509

[21]

: Surface topological order and a new ’t Hooft anomaly of interaction enabled 3+1D Fermion SPTs. ArXiv e-prints, April

L. Fidkowski
,
A. Vishwanath
,
M.A. Metlitski

[22]

Reflection positivity and invertible topological phases

- Geom.Topol. 25 (2021) 1165-1330
•
e-Print:
- 1604.06527
•
DOI:
- 10.2140/gt.2021.25.1165

[23]

Chern-Simons theory with finite gauge group

Daniel S. Freed
(
- Texas U. and
- Virginia Tech.
)
,
Frank Quinn
(
- Texas U. and
- Virginia Tech.
)

- Commun.Math.Phys. 156 (1993) 435-472
•
e-Print:
- hep-th/9111004
•
DOI:
- 10.1007/BF02096860

[24]

Fu, L.: Topological crystalline insulators

- Phys.Rev.Lett. 106 (2011) 106802

[25]

Symmetry Protected Topological phases and Generalized Cohomology

Davide Gaiotto
(
- Unlisted
)
,
Theo Johnson-Freyd
(
- Unlisted
)

- JHEP 05 (2019) 007
•
e-Print:
- 1712.07950
•
DOI:
- 10.1007/JHEP05(2019)007

1-25 of 74
1
2
3
25 / page