[1]
[2]
[3]
[4]
[5]
[6]
[7]
[8]
[9]
[10]
[11]
[12]
[13]
[14]
[15]
[16]
[17]
[18]
Z.-C. Gu and X.-G. Wen, Phys. Rev. B 80, 155131 (2009).
X. Chen, Z.-C. Gu, and X.-G. Wen, Phys. Rev. B 82, 155138 (2010).
N. Schuch, D. Pérez-García, and I. Cirac, Phys. Rev. B 84, 165139 (2011).
X. Chen, Z.-C. Gu, Z.-X. Liu, and X.-G. Wen, Phys. Rev. B 87, 155114 (2013).
M. Levin and Z.-C. Gu, Phys. Rev. B 86, 115109 (2012).
F. Pollmann, A. M. Turner, E. Berg, and M. Oshikawa, Phys. Rev. B 81, 064439 (2010).
X. Chen, Z.-C. Gu, and X.-G. Wen, Phys. Rev. B 83, 035107 (2011).
Y.-M. Lu and A. Vishwanath, Phys. Rev. B 86, 125119 (2012).
Z.-C. Gu and X.-G. Wen, Phys. Rev. B 90, 115141 (2014).
A. Vishwanath and T. Senthil, Phys. Rev. X 3, 011016 (2013).
A. P. Schnyder, S. Ryu, A. Furusaki, and A. W. W. Ludwig, Phys. Rev. B 78, 195125 (2008).
A. Kitaev, AIP Conf. Proc. 1134, 22 (2009).
D. S. Freed and G. W. Moore, Ann. Henri Poincaré 14, 1927 (2013).
L. Fidkowski and A. Kitaev, Phys. Rev. B 83, 075103 (2011).
C. Wang, A. C. Potter, and T. Senthil, Science 343, 629 (2014).
A. Kapustin, arXiv:1403.1467 [cond-mat.str-el] [Search INSPIRE].
A. Kapustin, R. Thorngren, A. Turzillo, and Z. Wang, J. High Energy Phys. 2015, 52 (2015).
D. S. Freed and M. J. Hopkins, Geom. Topol. 25, 1165 (2021).
62/64
Downloaded from https://academic.oup.com/ptep/article/2023/8/083I01/7220033 by Kyoto University user on 18 December 2023
Corollary A.6. The following holds true for N > 0:
PTEP 2023, 083I01
[43]
[44]
[45]
[46]
[47]
[48]
[49]
[50]
[51]
[52]
[53]
[54]
[55]
[56]
[57]
[58]
[59]
[60]
[61]
[62]
[63]
[64]
[65]
E. Witten, Rev. Mod. Phys. 88, 035001 (2016).
F. J. Burnell, X. Chen, L. Fidkowski, and A. Vishwanath, Phys. Rev. B 90, 245122 (2014).
M. Cheng, Z. Bi, Y.-Z. You, and Z.-C. Gu, Phys. Rev. B 97, 205109 (2018).
R. Dijkgraaf and E. Witten, Commun. Math. Phys. 129, 393 (1990).
A. Kapustin and A. Turzillo, J. High Energy Phys. 2017, 6 (2017).
K. Yonekura, Commun. Math. Phys. 368, 1121 (2019).
J. C. Y. Teo, L. Fu, and C. L. Kane, Phys. Rev. B 78, 045426 (2008).
L. Fu, Phys. Rev. Lett. 106, 106802 (2011).
X. Chen, Z.-C. Gu, and X.-G. Wen, Phys. Rev. B 84, 235128 (2011).
C.-T. Hsieh, O. M. Sule, G. Y. Cho, S. Ryu, and R. G. Leigh, Phys. Rev. B 90, 165134 (2014).
C.-T. Hsieh, T. Morimoto, and S. Ryu, Phys. Rev. B 90, 245111 (2014).
Y.-Z. You and C. Xu, Phys. Rev. B 90, 245120 (2014).
Y. Fuji, F. Pollmann, and M. Oshikawa, Phys. Rev. Lett. 114, 177204 (2015).
G. Y. Cho, C.-T. Hsieh, T. Morimoto, and S. Ryu, Phys. Rev. B 91, 195142 (2015).
T. Yoshida, T. Morimoto, and A. Furusaki, Phys. Rev. B 92, 245122 (2015).
H. Song, S.-J. Huang, L. Fu, and M. Hermele, Phys. Rev. X 7, 011020 (2017).
S. Jiang and Y. Ran, Phys. Rev. B 95, 125107 (2017).
R. Thorngren and D. V. Else, Phys. Rev. X 8, 011040 (2018).
H. C. Po, H. Watanabe, C.-M. Jian, and M. P. Zaletel, Phys. Rev. Lett. 119, 127202 (2017).
S.-J. Huang, H. Song, Y.-P. Huang, and M. Hermele, Phys. Rev. B 96, 205106 (2017).
W. A. Benalcazar, B. A. Bernevig, and T. L. Hughes, Science 357, 61 (2017).
W. A. Benalcazar, B. A. Bernevig, and T. L. Hughes, Phys. Rev. B 96, 245115 (2017).
C. Fang and L. Fu, Sci. Adv. 5, eaat2374 (2019).
F. Schindler, A. M. Cook, M. G. Vergniory, Z. Wang, S. S. P. Parkin, B. A. Bernevig, and T. Neupert,
Sci. Adv. 4, eaat034 (2018).
Z. Song, Z. Fang, and C. Fang, Phys. Rev. Lett. 119, 246402 (2017).
J. Langbehn, Y. Peng, L. Trifunovic, F. von Oppen, and P. W. Brouwer, Phys. Rev. Lett. 119, 246401
(2017).
Z. Song, T. Zhang, Z. Fang, and C. Fang, Nat. Commun. 9, 1 (2018).
E. Khalaf, H. C. Po, A. Vishwanath, and H. Watanabe, Phys. Rev. X 8, 031070 (2018).
E. Khalaf, Phys. Rev. B 97, 205136 (2018).
M. Geier, L. Trifunovic, M. Hoskam, and P. W. Brouwer, Phys. Rev. B 97, 205135 (2018).
L. Trifunovic and P. W. Brouwer, Phys. Rev. X 9, 011012 (2019).
A. Kitaev, On the classification of short-range entangled states( available at: http://scgp.stonybrook.
edu/ video_portal/ video.php?id=2010). Talk at Simons Center, 20 June 2013.
A. Kitaev, Homotopy-theoretic approach to SPT phases in action: Z16 classification of threedimensional superconductors, in Symmetry and Topology in Quantum Matter Workshop (Los Angeles, CA, Institute for Pure and Applied Mathematics, University of California, January 28, 2015)
(available at: http://www.ipam.ucla.edu/abst ract /?tid=12389&pcode=STQ2015).
C. Z. Xiong, J. Phys. A: Math. Theor. 51, 445001 (2018).
C. Z. Xiong and A. Alexandradinata, Phys. Rev. B 97, 115153 (2018).
D. Gaiotto and T. Johnson-Freyd, J. High Energy Phys. 2019, 1 (2019).
M. F. Atiyah and F. Hirzebruch, Vector bundles and homogeneous spaces, in Topological Library Part
3: Spectral Sequences in Topology (World Scientific, Singapore, 2012), p. 423.
G. W. Whitehead, Trans. Am. Math. Soc. 102, 227 (1962).
A. K¯ono and D. Tamaki, Generalized Cohomology (American Mathematical Society, Providence,
2006).
G. E. Bredon, Bull. Am. Math. Soc. 73, 266 (1967).
J. P. May, L. G. Lewis, R. J. Piacenza, M. Cole, G. Comezana, S. Costenoble, A. D. Elmendorf, and
J. P. C. Greenlees, Am. Math. Soc. 91, (1996).
A. Hatcher, Algebraic Topology (Cambridge University Press, Cambridge, UK, 2002).
N. Tantivasadakarn, Phys. Rev. B 96, 195101 (2017).
S. C. Furuya and M. Oshikawa, Phys. Rev. Lett. 118, 021601 (2017).
T. H. Hsieh, G. B. Halász, and T. Grover, Phys. Rev. Lett. 117, 166802 (2016).
G. Y. Cho, C.-T. Hsieh, and S. Ryu, Phys. Rev. B 96, 195105 (2017).
M. A. Metlitski and R. Thorngren, Phys. Rev. B 98, 085140 (2018).
63/64
Downloaded from https://academic.oup.com/ptep/article/2023/8/083I01/7220033 by Kyoto University user on 18 December 2023
[19]
[20]
[21]
[22]
[23]
[24]
[25]
[26]
[27]
[28]
[29]
[30]
[31]
[32]
[33]
[34]
[35]
[36]
[37]
[38]
[39]
[40]
[41]
[42]
K. Shiozaki et al.
PTEP 2023, 083I01
[78]
[79]
[80]
[81]
[82]
[83]
[84]
[85]
[86]
[87]
[88]
[89]
[90]
[91]
[92]
[93]
[94]
[95]
[96]
[97]
[98]
[99]
[100]
[101]
[102]
[103]
[104]
[105]
M. Cheng, Phys. Rev. B 99, 075143 (2019).
R. Kobayashi, K. Shiozaki, Y. Kikuchi, and S. Ryu, Phys. Rev. B 99, 014402 (2019).
X. Yang, S. Jiang, A. Vishwanath, and Y. Ran, Phys. Rev. B 98, 125120 (2018).
Y.-M. Lu,arXiv:1705.04691 [cond-mat.str-el] [Search INSPIRE].
E. Lieb, T. Schultz, and D. Mattis, Ann. Phys. 16, 407 (1961).
M. Oshikawa, Phys. Rev. Lett. 84 1535 (2000).
M. B. Hastings, Europhys. Lett. 70, 824 (2005).
H. Watanabe, H. C. Po, A. Vishwanath, and M. Zaletel, Proc. Natl. Acad. Sci. 112, 14551 (2015).
H. C. Po, H. Watanabe, M. P. Zaletel, and A. Vishwanath, Sci. Adv. 2, e1501782 (2016).
A. Y. Kitaev, Phys.-Uspekhi 44, 131 (2001).
K. Shiozaki, H. Shapourian, and S. Ryu, Phys. Rev. B 95, 205139 (2017).
P. B. Gilkey, The Geometry of Spherical Space Form Groups (World Scientific, Singapore, 1989),
Series in Pure Mathematics, Vol. 7.
M. Guo, P. Putrov, and J. Wang, Ann. Phys. 394, 244 (2018).
Y.-M. Lu, Y. Ran, and M. Oshikawa, Ann. Phys. 413, 168060 (2020).
A. Matsugatani, Y. Ishiguro, K. Shiozaki, and H. Watanabe, Phys. Rev. Lett. 120, 096601 (2018).
X.-L. Qi and S.-C. Zhang, Phys. Rev. Lett. 101, 086802 (2008).
Y. Ran, A. Vishwanath, and D.-H. Lee, Phys. Rev. Lett. 101, 086801 (2008).
D. Gaiotto, A. Kapustin, N. Seiberg, and B. Willett, J. High Energy Phys. 2015, 172 (2015).
D. Gaiotto, A. Kapustin, Z. Komargodski, and N. Seiberg, J. High Energy Phys. 2017, 91 (2017).
A. Altland and M. R. Zirnbauer, Phys. Rev. B 55, 1142 (1997).
G. C. Thiang, Ann. Henri Poincaré 17, 757 (2016).
K. Gomi, arXiv:1705.09134 [math.KT] [Search INSPIRE].
K. Shiozaki, M. Sato, and K. Gomi, Phys. Rev. B 95, 235425 (2017).
K. Shiozaki, M. Sato, and K. Gomi, Phys. Rev. B 106, 165103 (2022) arXiv:1802.06694 [cond-mat.strel] [Search INSPIRE].
K. Gomi and G. C. Thiang, J. Geom. Phys. 139, 50 (2019).
J. Kruthoff, J. de Boer, J. van Wezel, C. L. Kane, and R.-J. Slager, Phys. Rev. X 7, 041069 (2017).
H. C. Po, A. Vishwanath, and H. Watanabe, Nat. Commun. 8, 50 (2017).
B. Bradlyn, L. Elcoro, J. Cano, M. G. Vergniory, Z. Wang, C. Felser, M. I. Aroyo, and B. A. Bernevig,
Nature 547, 298 (2017).
C.-K. Chiu, H. Yao, and S. Ryu, Phys. Rev. B 88, 075142 (2013).
T. Morimoto and A. Furusaki, Phys. Rev. B 88, 125129 (2013).
K. Shiozaki and M. Sato, Phys. Rev. B 90, 165114 (2014).
K. Shiozaki, M. Sato, and K. Gomi, Phys. Rev. B 93, 195413 (2016).
K. Shiozaki, H. Shapourian, K. Gomi, and S. Ryu, Phys. Rev. B 98, 035151 (2018).
A. G. Abanov and P. B. Wiegmann, Nucl. Phys. B 570, 685 (2000).
O. Dubinkin and T. L. Hughes, Phys. Rev. B 99, 235132 (2019).
Y. You, T. Devakul, F. J. Burnell, and T. Neupert, Phys. Rev. B 98, 235102 (2018).
A. Rasmussen and Y.-M. Lu, Phys. Rev. B 101, 085137 (2020).
R. Bott and L. W. Tu, Differential Forms in Algebraic Topology (Springer, Berlin, 2013), Graduate
Texts in Mathematics, Vol. 82.
J.-P. Serre, Comment. Math. Helv. 27, 198 (1953).
A. Kapustin and R. Thorngren, Higher symmetry and gapped phases of gauge theories, in Algebra,
Geometry, and Physics in the 21st Century (Springer, Cham, 2017), p. 177.
© The Author(s) 2023. Published by Oxford University Press on behalf of the Physical Society of Japan. This is an Open Access article distributed under the terms of the Creative Commons
Attribution License (https://cr eativecommons.or g/licenses/by/4.0/), which permits unrestricted reuse, distribution, and reproduction in any medium, provided the original work is properly cited.
64/64
Downloaded from https://academic.oup.com/ptep/article/2023/8/083I01/7220033 by Kyoto University user on 18 December 2023
[66]
[67]
[68]
[69]
[70]
[71]
[72]
[73]
[74]
[75]
[76]
[77]
K. Shiozaki et al.
...