Abstract
We discuss a recipe to produce a lattice construction of fermionic phases of matter on unoriented manifolds. This is performed by extending the construction of spin TQFT via the Grassmann integral proposed by Gaiotto and Kapustin, to the unoriented pin± case. As an application, we construct gapped boundaries for time-reversal-invariant Gu-Wen fermionic SPT phases. In addition, we provide a lattice definition of (1+1)d pin_ invertible theory whose partition function is the Arf-Brown-Kervaire invariant, which generates the ℤ8 classification of (1+1)d topological superconductors. We also compute the indicator formula of ℤ16 valued time-reversal anomaly for (2+1)d pin+ TQFT based on our construction.
Article PDF
We’re sorry, something doesn't seem to be working properly.
Please try refreshing the page. If that doesn't work, please contact support so we can address the problem.
Avoid common mistakes on your manuscript.
References
D. Gaiotto and A. Kapustin, Spin TQFTs and fermionic phases of matter, Int. J. Mod. Phys.A 31 (2016) 1645044 [arXiv:1505.05856] [INSPIRE].
Z.-C. Gu and X.-G. Wen, Symmetry-protected topological orders for interacting fermions: Fermionic topological nonlinear σ models and a special group supercohomology theory, Phys. Rev.B 90 (2014) 115141 [arXiv:1201.2648] [INSPIRE].
C. Wang, C.-H. Lin and Z.-C. Gu, Interacting fermionic symmetry-protected topological phases in two dimensions, Phys. Rev.B 95 (2017) 195147 [arXiv:1610.08478v1] [INSPIRE].
E. Witten, Fermion Path Integrals And Topological Phases, Rev. Mod. Phys.88 (2016) 035001 [arXiv:1508.04715v2] [INSPIRE].
M.A. Metlitski, L. Fidkowski, X. Chen and A. Vishwanath, Interaction effects on 3D topological superconductors: surface topological order from vortex condensation, the 16 fold way and fermionic Kramers doublets, arXiv:1406.3032 [INSPIRE].
M. Cheng, Z. Bi, Y.-Z. You and Z.-C. Gu, Classification of symmetry-protected phases for interacting fermions in two dimensions, Phys. Rev.B 97 (2018) 205109 [arXiv:1501.01313v3] [INSPIRE].
Z.-C. Gu, Z. Wang and X.-G. Wen, Lattice Model for Fermionic Toric Code, Phys. Rev.B 90 (2014) 085140 [arXiv:1309.7032v3] [INSPIRE].
M. Guo, K. Ohmori, P. Putrov, Z. Wan and J. Wang, Fermionic Finite-Group Gauge Theories and Interacting Symmetric/Crystalline Orders via Cobordisms, arXiv:1812.11959 [INSPIRE].
L. Bhardwaj, D. Gaiotto and A. Kapustin, State sum constructions of spin-TFTs and string net constructions of fermionic phases of matter, JHEP04 (2017) 096 [arXiv:1605.01640v2] [INSPIRE].
T.D. Ellison and L. Fidkowski, Disentangling Interacting Symmetry-Protected Phases of Fermions in Two Dimensions, Phys. Rev.X 9 (2019) 011016 [arXiv:1806.09623v3] [INSPIRE].
A. Kapustin, R. Thorngren, A. Turzillo and Z. Wang, Fermionic Symmetry Protected Topological Phases and Cobordisms, JHEP12 (2015) 052 [arXiv:1406.7329] [INSPIRE].
E. Witten, The “Parity” Anomaly On An Unorientable Manifold, Phys. Rev.B 94 (2016) 195150 [arXiv:1605.02391] [INSPIRE].
L. Bhardwaj, Unoriented 3d TFTs, JHEP05 (2017) 048 [arXiv:1611.02728v3] [INSPIRE].
H. Shapourian, K. Shiozaki and S. Ryu, Many-Body Topological Invariants for Fermionic Symmetry-Protected Topological Phases, Phys. Rev. Lett.118 (2017) 216402 [arXiv:1607.03896v3] [INSPIRE].
A. Turzillo, Diagrammatic State Sums for 2D Pin-Minus TQFTs, arXiv:1811.12654 [INSPIRE].
L. Fidkowski and A. Kitaev, Topological phases of fermions in one dimension, Phys. Rev.B 83 (2011) 075103 [arXiv:1008.4138v2].
D.S. Freed and M.J. Hopkins, Reflection positivity and invertible topological phases, arXiv:1604.06527 [INSPIRE].
K. Yonekura, On the cobordism classification of symmetry protected topological phases, Commun. Math. Phys.368 (2019) 1121 [arXiv:1803.10796] [INSPIRE].
L. Fidkowski, X. Chen and A. Vishwanath, Non-Abelian Topological Order on the Surface of a 3D Topological Superconductor from an Exactly Solved Model, Phys. Rev.X 3 (2013) 041016 [arXiv:1305.5851v4] [INSPIRE].
C.-T. Hsieh, G.Y. Cho and S. Ryu, Global anomalies on the surface of fermionic symmetry-protected topological phases in (3 + 1) dimensions, Phys. Rev.B 93 (2016) 075135 [arXiv:1503.01411v4] [INSPIRE].
Y. Tachikawa and K. Yonekura, More on time-reversal anomaly of 2 + 1-dimensional topological phases, Phys. Rev. Lett.119 (2017) 111603 [arXiv:1611.01601v2] [INSPIRE].
E.H. Brown, Generalizations of the Kervaire invariant, Ann. Math.95 (1972) 368 and online pdf version at https://www3.nd.edu/∼taylor/papers/PSKT.pdf.
A. Debray and S. Gunningham, The Arf-Brown TQFT of Pin −Surfaces, arXiv:1803.11183v1 [INSPIRE].
R. Thorngren, Anomalies and Bosonization, arXiv:1810.04414 [INSPIRE].
C. Wang and M. Levin, Anomaly indicators for time-reversal symmetric topological orders, Phys. Rev. Lett.119 (2017) 136801 [arXiv:1610.04624v1] [INSPIRE].
K. Walker and Z. Wang, (3 + 1)-TQFTs and Topological Insulators, arXiv:1104.2632v2 [INSPIRE].
K. Knapp, Wu class, (2014) http://www.map.mpim-bonn.mpg.de/Wu class.
G. Brumfiel and J. Morgan, Quadratic Functions of Cocycles and Pin Structures, arXiv:1808.10484.
X. Chen, Z.-C. Gu, Z.-X. Liu and X.-G. Wen, Symmetry protected topological orders and the group cohomology of their symmetry group, Phys. Rev.B 87 (2013) 155114 [arXiv:1106.4772v6] [INSPIRE].
R. Kirby and L. Taylor, Pin Structure on Low-dimensional Manifolds, in Geometry of Low-Dimensional Manifolds: Symplectic Manifolds and Jones-Witten Theory , S. Donaldson and C. Thomas eds., London Mathematical Society Lecture Note Series, Cambridge University Press, Cambridge U.K. (1991), pp. 177–242 and online pdf version at https://www3.nd.edu/∼taylor/papers/PSKT.pdf.
J. Wang, X.-G. Wen and E. Witten, Symmetric Gapped Interfaces of SPT and SET States: Systematic Constructions, Phys. Rev.X 8 (2018) 031048 [arXiv:1705.06728] [INSPIRE].
Y. Tachikawa, On gauging finite subgroups, arXiv:1712.09542 [INSPIRE].
R. Kobayashi, K. Ohmori and Y. Tachikawa, On gapped boundaries for SPT phases beyond group cohomology, arXiv:1905.05391 [INSPIRE].
M. Barkeshli, P. Bonderson, C.-M. Jian, M. Cheng and K. Walker, Reflection and time reversal symmetry enriched topological phases of matter: path integrals, non-orientable manifolds and anomalies, arXiv:1612.07792 [INSPIRE].
K. Walker, TQFTs [early incomplete draft] version 1h, (2006) and online pdf version at http://canyon23.net/math/tc.pdf.
Open Access
This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited
Author information
Authors and Affiliations
Corresponding author
Additional information
ArXiv ePrint: 1905.05902
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made.
The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.
To view a copy of this licence, visit https://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Kobayashi, R. Pin TQFT and Grassmann integral. J. High Energ. Phys. 2019, 14 (2019). https://doi.org/10.1007/JHEP12(2019)014
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP12(2019)014