Abstract
This paper generalizes two facts about oriented 3d TFTs to the unoriented case. On one hand, it is known that oriented 3d TFTs having a topological boundary condition admit a state-sum construction known as the Turaev-Viro construction. This is related to the string-net construction of fermionic phases of matter. We show how Turaev-Viro construction can be generalized to unoriented 3d TFTs. On the other hand, it is known that the “fermionic” versions of oriented TFTs, known as Spin-TFTs, can be constructed in terms of “shadow” TFTs which are ordinary oriented TFTs with an anomalous ℤ 2 1-form symmetry. We generalize this correspondence to Pin+-TFTs by showing that they can be constructed in terms of ordinary unoriented TFTs with anomalous ℤ 2 1-form symmetry having a mixed anomaly with time-reversal symmetry. The corresponding Pin+-TFT does not have any anomaly for time-reversal symmetry however and hence it can be unambiguously defined on a non-orientable manifold. In case a Pin+-TFT admits a topological boundary condition, one can combine the above two statements to obtain a Turaev-Viro-like construction of Pin+-TFTs. As an application of these ideas, we construct a large class of Pin+-SPT phases.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Y. Tachikawa and K. Yonekura, On time-reversal anomaly of 2 + 1d topological phases, PTEP 2017 (2017) 033B04 [arXiv:1610.07010] [INSPIRE].
D. Gaiotto, A. Kapustin, N. Seiberg and B. Willett, Generalized global symmetries, JHEP 02 (2015) 172 [arXiv:1412.5148] [INSPIRE].
V.G. Turaev and O.Y. Viro, State sum invariants of 3-manifolds and quantum 6j-symbols, Topology 31 (1992) 865
A. Kirillov, Jr. and B. Balsam, Turaev-Viro invariants as an extended TQFT, arXiv:1004.1533 [INSPIRE].
L. Bhardwaj, D. Gaiotto and A. Kapustin, State sum constructions of spin-TFTs and string net constructions of fermionic phases of matter, JHEP 04 (2017) 096 [arXiv:1605.01640] [INSPIRE].
E. Witten, The “Parity” anomaly on an unorientable manifold, Phys. Rev. B 94 (2016) 195150 [arXiv:1605.02391] [INSPIRE].
A. Kapustin, R. Thorngren, A. Turzillo and Z. Wang, Fermionic symmetry protected topological phases and cobordisms, JHEP 12 (2015) 052 [arXiv:1406.7329] [INSPIRE].
J. Fuchs, C. Schweigert and A. Valentino, Bicategories for boundary conditions and for surface defects in 3 − D TFT, Commun. Math. Phys. 321 (2013) 543 [arXiv:1203.4568] [INSPIRE].
M. Cheng, Z.C. Gu, S. Jiang, and Y. Qi, Exactly solvable models for symmetry-enriched topological phases, arXiv:1606.08482.
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:1106.4772] [INSPIRE].
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].
M. Cheng, Z. Bi, Y.-Z. You and Z.-C. Gu, Towards a complete classification of symmetry-protected phases for interacting fermions in two dimensions, arXiv:1501.01313 [INSPIRE].
G. Moore and N. Seiberg, Classical and quantum conformal field theory, Commun. Math. Phys. 123 (1989) 177.
G. Moore and N. Seiberg, Lectures on RCFT, in Physics, geometry and topology, M. Nakahara ed., Springer, Germany (1990).
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].
C. Wang and M. Levin, Anomaly indicators for time-reversal symmetric topological orders, arXiv:1610.04624 [INSPIRE].
Y. Tachikawa and K. Yonekura, More on time-reversal anomaly of 2 + 1d topological phases, arXiv:1611.01601 [INSPIRE].
M.A. Levin and X.-G. Wen, String net condensation: a physical mechanism for topological phases, Phys. Rev. B 71 (2005) 045110 [cond-mat/0404617] [INSPIRE].
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: 1611.02728
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (https://creativecommons.org/licenses/by/4.0), which permits use, duplication, 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 license, and indicate if changes were made.
About this article
Cite this article
Bhardwaj, L. Unoriented 3d TFTs. J. High Energ. Phys. 2017, 48 (2017). https://doi.org/10.1007/JHEP05(2017)048
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP05(2017)048