Equivariant Topological Quantum Field Theory and Symmetry Protected Topological Phases - INSPIRE

DataBETA

Equivariant Topological Quantum Field Theory and Symmetry Protected Topological Phases

Apr 8, 2015

26 pages

Published in:

JHEP 03 (2017) 006

Published: Mar 1, 2017

e-Print:

1504.01830 [cond-mat.str-el]

DOI:

10.1007/JHEP03(2017)006

View in:

reference search26 citations

Citations per year

Abstract: (Springer)

Short-Range Entangled topological phases of matter are closely related to Topological Quantum Field Theory. We use this connection to classify Symmetry Protected Topological phases in low dimensions, including the case when the symmetry involves time-reversal. To accomplish this, we generalize Turaev’s description of equivariant TQFT to the unoriented case. We show that invertible unoriented equivariant TQFTs in one or fewer spatial dimensions are classified by twisted group cohomology, in agreement with the proposal of Chen, Gu, Liu and Wen. We also show that invertible oriented equivariant TQFTs in spatial dimension two or fewer are classified by ordinary group cohomology.

Topological Field Theories
Discrete Symmetries
Topological States of Matter
phase: topological
field theory: topological
boson: symmetry
group: twist
dimension: 2
cohomology
time reversal

References(22)

Figures(0)

[1]

Homotopy-theoretic approach to SPT phases in action: ℤ (16) classification of three-dimensional superconductors, talk given at Symmetry and Topology in Quantum Matter, Institute for Pure and Applied Mathematics, Los Angeles, U.S.A 26-30 January

A. Kitaev

[2]

Local unitary transformation, long-range quantum entanglement, wave function renormalization, and topological order

Xie Chen
(
- MIT
)
,
Zheng Cheng Gu
(
- Santa Barbara, KITP
)
,
Xiao Gang Wen
(
- MIT
)

- Phys.Rev.B 82 (2010) 155138
•
e-Print:
- 1004.3835
•
DOI:
- 10.1103/PhysRevB.82.155138

[3]

Symmetry protected topological orders and the group cohomology of their symmetry group

Xie Chen
(
- MIT, LNS and
- UC, Berkeley (main)
)
,
Zheng-Cheng Gu
(
- Caltech
)
,
Zheng-Xin Liu
(
- Tsinghua U., Beijing and
- MIT, LNS
)
,
Xiao-Gang Wen
(
- MIT, LNS and
- Tsinghua U., Beijing and
- Perimeter Inst. Theor. Phys.
)

- Phys.Rev.B 87 (2013) 15, 155114
•
e-Print:
- 1106.4772
•
DOI:
- 10.1103/PhysRevB.87.155114

[4]

Physics of three dimensional bosonic topological insulators: Surface Deconfined Criticality and Quantized Magnetoelectric Effect

- Phys.Rev.X 3 (2013) 1, 011016
•
e-Print:
- 1209.3058
•
DOI:
- 10.1103/PhysRevX.3.011016

[5]

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

Anton Kapustin
(
- Caltech
)

e-Print:
- 1403.1467

[6]

Short-range entanglement and invertible field theories

Daniel S. Freed
(
- Texas U., Math Dept.
)

e-Print:
- 1406.7278

[7]

Classification of gapped symmetric phases in one-dimensional spin systems

- Phys.Rev.B 83 (2011) 3, 035107
•
e-Print:
- 1008.3745
•
DOI:
- 10.1103/PhysRevB.83.035107

[7]

Classification of gapped symmetric phases in one-dimensional spin systems

- Phys.Rev.B 83 (2011) 3, 035107
•
e-Print:
- 1008.3745
•
DOI:
- 10.1103/PhysRevB.83.035107

[8]

Topological phases of fermions in one dimension

- Phys.Rev.B 83 (2011) 7, 075103
•
e-Print:
- 1008.4138
•
DOI:
- 10.1103/PhysRevB.83.075103

[8]

Topological phases of fermions in one dimension

- Phys.Rev.B 83 (2011) 7, 075103
•
e-Print:
- 1008.4138
•
DOI:
- 10.1103/PhysRevB.83.075103

[9]

Topological quantum field theories

M. Atiyah

- Inst.Hautes Etudes Sci.Publ.Math. 68 (1989) 175-186
•
DOI:
- 10.1007/BF02698547

[10]

D-branes and K-theory in 2D topological field theory

Gregory W. Moore
(
- Rutgers U., Piscataway
)
,
Graeme Segal
(
- Oxford U.
)

e-Print:
- hep-th/0609042

[11]

Homotopy field theory in dimension two and group algebras

Vladimir Turaev
(
- Louis Pasteur U., Strasbourg I
)

e-Print:
- math/9910010

[12]

Taming the Conformal Zoo

- Phys.Lett.B 220 (1989) 422-430
•
DOI:
- 10.1016/0370-2693(89)90897-6

[13]

Lectures on Tensor Categories and Modular Functors, American Mathematical Society

B. Bakalov
,
A. Kirillov, Jr.

[14]

Homotopy quantum field theory, in EMS Tracts in Mathematics, volume 10, EMS Publishing House, Zürich Switzerland

V. Turaev

[15]

On 3-dimensional homotopy quantum field theory II: The surgery approach

Vladimir Turaev
(
- Indiana U.
)
,
Alexis Virelizier
(
- Lille U.
)

- Int.J.Math. 25 (2014) 1450027
•
DOI:
- 10.1142/S0129167X1450027X

[16]

On braided fusion categories I

e-Print:
- 0906.0620

[17]

Commutator algebra of a finite group of collineations

H. Weyl

- Duke Math.J. 3 (1937) 200

[18]

Link homology and unoriented topological quantum field theory

Vladimir Turaev
(
- IRMA, Strasbourg
)
,
Paul Turner
(
- Heriot-Watt U.
)

e-Print:
- math/0506229

[19]

Reflection positivity and invertible topological phases

D.S. Freed
,
M.J. Hopkins

e-Print:
- 1605.06527

[20]

Unoriented 3d TFTs

Lakshya Bhardwaj
(
- Perimeter Inst. Theor. Phys.
)

- JHEP 05 (2017) 048
•
e-Print:
- 1611.02728
•
DOI:
- 10.1007/JHEP05(2017)048