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Reflection positivity and invertible topological phases

Daniel S Freed and Michael J Hopkins

Geometry & Topology 25 (2021) 1165–1330
1 J F Adams, Prerequisites (on equivariant stable homotopy) for Carlsson’s lecture, from: "Algebraic topology" (editors I Madsen, B Oliver), Lecture Notes in Math. 1051, Springer (1984) 483 MR764596
2 J F Adams, H R Margolis, Modules over the Steenrod algebra, Topology 10 (1971) 271 MR294450
3 A Altland, M R Zirnbauer, Nonstandard symmetry classes in mesoscopic normal-superconducting hybrid structures, Phys. Rev. B 55 (1997) 1142
4 D W Anderson, E H Brown Jr., F P Peterson, The structure of the Spin cobordism ring, Ann. of Math. 86 (1967) 271 MR219077
5 D W Anderson, E H Brown Jr., F P Peterson, Pin cobordism and related topics, Comment. Math. Helv. 44 (1969) 462 MR261613
6 M F Atiyah, K–theory and reality, Q. J. Math. 17 (1966) 367 MR206940
7 M Atiyah, Topological quantum field theories, Inst. Hautes Études Sci. Publ. Math. 68 (1988) 175 MR1001453
8 M F Atiyah, R Bott, A Shapiro, Clifford modules, Topology 3 (1964) 3 MR167985
9 M F Atiyah, V K Patodi, I M Singer, Spectral asymmetry and Riemannian geometry, I, Math. Proc. Cambridge Philos. Soc. 77 (1975) 43 MR397797
10 M F Atiyah, I M Singer, Index theory for skew-adjoint Fredholm operators, Inst. Hautes Études Sci. Publ. Math. 37 (1969) 5 MR285033
11 D Ayala, J Francis, The cobordism hypothesis, preprint (2017) arXiv:1705.02240
12 J C Baez, J Dolan, Higher-dimensional algebra and topological quantum field theory, J. Math. Phys. 36 (1995) 6073 MR1355899
13 A Bahri, P Gilkey, The eta invariant, Pinc bordism, and equivariant Spinc bordism for cyclic 2–groups, Pacific J. Math. 128 (1987) 1 MR883375
14 C Barwick, C Schommer-Pries, On the unicity of the homotopy theory of higher categories, preprint (2011) arXiv:1112.0040
15 A Beaudry, J A Campbell, A guide for computing stable homotopy groups, from: "Topology and quantum theory in interaction" (editors D Ayala, D S Freed, R E Grady), Contemp. Math. 718, Amer. Math. Soc. (2018) 89 MR3869642
16 M Berg, C DeWitt-Morette, S Gwo, E Kramer, The Pin groups in physics : C, P and T, Rev. Math. Phys. 13 (2001) 953 MR1845915
17 G Birkhoff, M K Bennett, Felix Klein and his “Erlanger Programm”, from: "History and philosophy of modern mathematics" (editors W Aspray, P Kitcher), Minnesota Stud. Philos. Sci. 11, Univ. Minnesota Press (1988) 145 MR945470
18 M Bökstedt, I Madsen, The cobordism category and Waldhausen’s K–theory, from: "An alpine expedition through algebraic topology" (editors C Ausoni, K Hess, B Johnson, W Lück, J Scherer), Contemp. Math. 617, Amer. Math. Soc. (2014) 39 MR3243393
19 E H Brown Jr., M Comenetz, Pontrjagin duality for generalized homology and cohomology theories, Amer. J. Math. 98 (1976) 1 MR405403
20 G Brumfiel, J Morgan, The Pontrjagin dual of 3–dimensional spin bordism, preprint (2016) arXiv:1612.02860
21 U Bunke, Transgression of the index gerbe, Manuscripta Math. 109 (2002) 263 MR1948015
22 U Bunke, T Schick, Smooth K–theory, from: "From probability to geometry, II" (editors X Dai, R Léandre, X Ma, W Zhang), Astérisque 328, Soc. Math. France (2009) 45 MR2664467
23 D Calaque, C Scheimbauer, A note on the (,n)–category of cobordisms, Algebr. Geom. Topol. 19 (2019) 533 MR3924174
24 J A Campbell, Homotopy theoretic classification of symmetry protected phases, preprint (2017) arXiv:1708.04264
25 F Catanese, On the moduli spaces of surfaces of general type, J. Differential Geom. 19 (1984) 483 MR755236
26 X Chen, Z C Gu, X G Wen, Local unitary transformation, long-range quantum entanglement, wave function renormalization, and topological order, Phys. Rev. B 82 (2010)
27 S Coleman, J Mandula, All possible symmetries of the S matrix, Phys. Rev. 159 (1967) 1251
28 C Cordova, D S Freed, H T Lam, N Seiberg, Anomalies in the space of coupling constants and their dynamical applications, I, SciPost Phys. 8 (2020)
29 X Dai, D S Freed, η–invariants and determinant lines, C. R. Acad. Sci. Paris Sér. I Math. 320 (1995) 585 MR1322342
30 P Deligne, Notes on spinors, from: "Quantum fields and strings: a course for mathematicians, I" (editors P Deligne, P Etingof, D S Freed, L C Jeffrey, D Kazhdan, J W Morgan, D R Morrison, E Witten), Amer. Math. Soc. (1999) 99 MR1701598
31 P Deligne, J W Morgan, Notes on supersymmetry (following Joseph Bernstein), from: "Quantum fields and strings: a course for mathematicians, I" (editors P Deligne, P Etingof, D S Freed, L C Jeffrey, D Kazhdan, J W Morgan, D R Morrison, E Witten), Amer. Math. Soc. (1999) 41 MR1701597
32 T tom Dieck, Transformation groups and representation theory, 766, Springer (1979) MR551743
33 F J Dyson, The threefold way: algebraic structure of symmetry groups and ensembles in quantum mechanics, J. Math. Phys. 3 (1962) 1199 MR177643
34 J Ebert, A vanishing theorem for characteristic classes of odd-dimensional manifold bundles, J. Reine Angew. Math. 684 (2013) 1 MR3181555
35 L Fidkowski, X Chen, A Vishwanath, Non-abelian topological order on the surface of a 3D topological superconductor from an exactly solved model, Phys. Rev. X 3 (2013)
36 L Fidkowski, A Kitaev, Effects of interactions on the topological classification of free fermion systems, Phys. Rev. B 81 (2010)
37 L Fidkowski, A Kitaev, Topological phases of fermions in one dimension, Phys. Rev. B 83 (2011)
38 D S Freed, Higher algebraic structures and quantization, Comm. Math. Phys. 159 (1994) 343 MR1256993
39 D S Freed, The cobordism hypothesis, Bull. Amer. Math. Soc. 50 (2013) 57 MR2994995
40 D S Freed, Anomalies and invertible field theories, from: "String-Math 2013" (editors R Donagi, M R Douglas, L Kamenova, M Roček), Proc. Sympos. Pure Math. 88, Amer. Math. Soc. (2014) 25 MR3330283
41 D S Freed, Lectures on Field theory and topology, 133, Amer. Math. Soc. (2019) MR3969923
42 D S Freed, M J Hopkins, Chern–Weil forms and abstract homotopy theory, Bull. Amer. Math. Soc. 50 (2013) 431 MR3049871
43 D S Freed, M J Hopkins, Consistency of M–theory on unorientable manifolds, preprint (2019) arXiv:1908.09916
44 D S Freed, M J Hopkins, C Teleman, Consistent orientation of moduli spaces, from: "The many facets of geometry" (editors O García-Prada, J P Bourguignon, S Salamon), Oxford Univ. Press (2010) 395 MR2681705
45 D S Freed, M J Hopkins, C Teleman, Loop groups and twisted K–theory, III, Ann. of Math. 174 (2011) 947 MR2831111
46 D S Freed, J Lott, An index theorem in differential K–theory, Geom. Topol. 14 (2010) 903 MR2602854
47 D S Freed, G W Moore, Setting the quantum integrand of M–theory, Comm. Math. Phys. 263 (2006) 89 MR2207325
48 D S Freed, G W Moore, Twisted equivariant matter, Ann. Henri Poincaré 14 (2013) 1927 MR3119923
49 L Fu, C L Kane, E J Mele, Topological insulators in three dimensions, Phys. Rev. Lett. 98 (2007)
50 D Gaiotto, A Kapustin, Spin TQFTs and fermionic phases of matter, Int. J. Mod. Phys. A 31 (2016)
51 D Gaiotto, A Kapustin, Z Komargodski, N Seiberg, Theta, time reversal and temperature, J. High Energy Phys. 2017 (2017) MR3662840
52 S Galatius, U Tillmann, I Madsen, M Weiss, The homotopy type of the cobordism category, Acta Math. 202 (2009) 195 MR2506750
53 J Glimm, A Jaffe, Quantum physics: a functional integral point of view, Springer (1987) MR887102
54 H Greaves, T Thomas, On the CPT theorem, Stud. Hist. Philos. Sci. B Stud. Hist. Philos. Modern Phys. 45 (2014) 46 MR3168090
55 J P C Greenlees, J P May, Equivariant stable homotopy theory, from: "Handbook of algebraic topology" (editor I M James), North-Holland (1995) 277 MR1361893
56 Z C Gu, 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)
57 M Guo, P Putrov, J Wang, Time reversal, SU(N) Yang–Mills and cobordisms : interacting topological superconductors/insulators and quantum spin liquids in 3 + 1D, Ann. Physics 394 (2018) 244 MR3812704
58 P Heinzner, A Huckleberry, M R Zirnbauer, Symmetry classes of disordered fermions, Comm. Math. Phys. 257 (2005) 725 MR2164950
59 M A Hill, M J Hopkins, D C Ravenel, On the nonexistence of elements of Kervaire invariant one, Ann. of Math. 184 (2016) 1 MR3505179
60 M J Hopkins, Algebraic topology and modular forms, from: "Proceedings of the International Congress of Mathematicians, I" (editor T Li), Higher Ed. Press (2002) 291 MR1989190
61 M J Hopkins, I M Singer, Quadratic functions in geometry, topology, and M–theory, J. Differential Geom. 70 (2005) 329 MR2192936
62 T Johnson-Freyd, Spin, statistics, orientations, unitarity, Algebr. Geom. Topol. 17 (2017) 917 MR3623677
63 R Jost, Eine Bemerkung zum CTP theorem, Helv. Phys. Acta 30 (1957) 409 MR89720
64 C L Kane, E J Mele, 2 topological order and the quantum spin Hall effect, Phys. Rev. Lett. 95 (2005)
65 M Kapranov, Supergeometry in mathematics and physics, preprint (2015) arXiv:1512.07042
66 A Kapustin, Topological field theory, higher categories, and their applications, from: "Proceedings of the International Congress of Mathematicians, III" (editors R Bhatia, A Pal, G Rangarajan, V Srinivas, M Vanninathan, P Gastesi), Hindustan (2010) 2021 MR2827874
67 A Kapustin, Symmetry protected topological phases, anomalies, and cobordisms: beyond group cohomology, preprint (2014) arXiv:1403.1467
68 A Kapustin, R Thorngren, A Turzillo, Z Wang, Fermionic symmetry protected topological phases and cobordisms, J. High Energy Phys. 2015 (2015) MR3464750
69 D Kazhdan, Introduction to QFT, from: "Quantum fields and strings: a course for mathematicians, I" (editors P Deligne, P Etingof, D S Freed, L C Jeffrey, D Kazhdan, J W Morgan, D R Morrison, E Witten), Amer. Math. Soc. (1999) 377 MR1701603
70 R Kennedy, M R Zirnbauer, Bott periodicity for 2 symmetric ground states of gapped free-fermion systems, Comm. Math. Phys. 342 (2016) 909 MR3465435
71 R C Kirby, L R Taylor, A calculation of Pin+ bordism groups, Comment. Math. Helv. 65 (1990) 434 MR1069818
72 R C Kirby, L R Taylor, Pin structures on low-dimensional manifolds, from: "Geometry of low-dimensional manifolds, II" (editors S K Donaldson, C B Thomas), Lond. Math. Soc. Lect. Note Ser. 151, Cambridge Univ. Press (1990) 177 MR1171915
73 A Kitaev, Anyons in an exactly solved model and beyond, Ann. Physics 321 (2006) 2 MR2200691
74 A Kitaev, Periodic table for topological insulators and superconductors, AIP Conf. Proc. 1134 (2009) 22
75 A Kitaev, Toward topological classification of phases with short-range entanglement, video lecture (2011)
76 A Kitaev, On the classification of short-range entangled states, video lecture (2013)
77 A Kitaev, Short range entangled quantum states, lecture (2014)
78 A Kitaev, Homotopy-theoretic approach to SPT phases in action : 16 classification of three-dimensional superconductors, video lecture (2015)
79 K R Klonoff, An index theorem in differential K–theory, PhD thesis, University of Texas at Austin (2008)
80 M Kontsevich, G B Segal, Wick rotation and the positivity of energy in quantum field theory, in preparation
81 R J Lawrence, Triangulations, categories and extended topological field theories, from: "Quantum topology" (editors L H Kauffman, R A Baadhio), Ser. Knots Everything 3, World Sci. (1993) 191 MR1273575
82 H B Lawson Jr., M L Michelsohn, Spin geometry, 38, Princeton Univ. Press (1989) MR1031992
83 J Lott, Higher-degree analogs of the determinant line bundle, Comm. Math. Phys. 230 (2002) 41 MR1930571
84 Y M Lu, A Vishwanath, Theory and classification of interacting integer topological phases in two dimensions : a Chern–Simons approach, Phys. Rev. B 86 (2012)
85 J Lurie, On the classification of topological field theories, from: "Current developments in mathematics" (editors D Jerison, B Mazur, T Mrowka, W Schmid, R Stanley, S T Yau), International (2009) 129 MR2555928
86 H R Margolis, Spectra and the Steenrod algebra : modules over the Steenrod algebra and the stable homotopy category, 29, North-Holland (1983) MR738973
87 M A Metlitski, S–duality of u(1) gauge theory with 𝜃 = π on non-orientable manifolds : applications to topological insulators and superconductors, preprint (2015) arXiv:1510.05663
88 M A Metlitski, L Fidkowski, X Chen, A Vishwanath, Interaction effects on 3D topological superconductors : surface topological order from vortex condensation, the 16 fold way and fermionic Kramers doublets, preprint (2014) arXiv:1406.3032
89 G W Moore, G B Segal, D–branes and K–theory in 2D topological field theory, from: "Dirichlet branes and mirror symmetry" (editors P S Aspinwall, T Bridgeland, A Craw, M R Douglas, M Gross, A Kapustin, G W Moore, G Segal, B Szendrői, P M H Wilson), Clay Math. Monogr. 4, Amer. Math. Soc. (2009) 27 MR2567952
90 S Morrison, K Walker, Blob homology, Geom. Topol. 16 (2012) 1481 MR2978449
91 R M Nandkishore, M Hermele, Fractons, Ann. Rev. Condensed Matter Phys. 10 (2019) 295
92 H K Nguyen, Higher bordism categories, Master’s thesis, Universität Bonn (2014)
93 M L Ortiz, Differential equivariant K–theory, PhD thesis, University of Texas at Austin (2009)
94 K Osterwalder, R Schrader, Axioms for Euclidean Green’s functions, II, Comm. Math. Phys. 42 (1975) 281 MR376002
95 X L Qi, T L Hughes, S C Zhang, Topological field theory of time-reversal invariant insulators, Phys. Rev. B 78 (2008)
96 M Reid, The moduli space of 3–folds with K = 0 may nevertheless be irreducible, Math. Ann. 278 (1987) 329 MR909231
97 B L Reinhart, Cobordism and the Euler number, Topology 2 (1963) 173 MR153021
98 S Ryu, A P Schnyder, A Furusaki, A W W Ludwig, Topological insulators and superconductors: tenfold way and dimensional hierarchy, New J. Phys. 12 (2010)
99 C Schommer-Pries, Invertible topological field theories, preprint (2017) arXiv:1712.08029
100 S Schwede, Lecture notes on equivariant stable homotopy theory, (2020)
101 G Segal, Classifying spaces and spectral sequences, Inst. Hautes Études Sci. Publ. Math. 34 (1968) 105 MR232393
102 G Segal, The definition of conformal field theory, from: "Topology, geometry and quantum field theory" (editor U Tillmann), Lond. Math. Soc. Lect. Note Ser. 308, Cambridge Univ. Press (2004) 421 MR2079383
103 G Segal, Three roles of quantum field theory, I–VI, video lectures (2011)
104 N Seiberg, E Witten, Gapped boundary phases of topological insulators via weak coupling, Prog. Theor. Exp. Phys. 12 (2016) MR3628684
105 S Sternberg, Lectures on differential geometry, Prentice-Hall (1964) MR0193578
106 S Stolz, P Teichner, Supersymmetric field theories and generalized cohomology, from: "Mathematical foundations of quantum field theory and perturbative string theory" (editors H Sati, U Schreiber), Proc. Sympos. Pure Math. 83, Amer. Math. Soc. (2011) 279 MR2742432
107 R E Stong, Notes on cobordism theory, Princeton Univ. Press (1968) MR0248858
108 R F Streater, A S Wightman, PCT, spin and statistics, and all that, Benjamin (1964) MR0161603
109 R Thom, Quelques propriétés globales des variétés différentiables, Comment. Math. Helv. 28 (1954) 17 MR61823
110 A M Turner, F Pollmann, E Berg, Topological phases of one-dimensional fermions: an entanglement point of view, Phys. Rev. B 83 (2011)
111 C Wang, A C Potter, T Senthil, Classification of interacting electronic topological insulators in three dimensions, Science 343 (2014) 629
112 C Wang, T Senthil, Interacting fermionic topological insulators/superconductors in three dimensions, Phys. Rev. B 89 (2014)
113 X G Wen, SPT order and algebraic topology, lecture notes (2015)
114 E Witten, Quantum field theory and the Jones polynomial, Comm. Math. Phys. 121 (1989) 351 MR990772
115 E Witten, What one can hope to prove about three-dimensional gauge theory, video lecture (2012)
116 E Witten, Fermion path integrals and topological phases, Rev. Mod. Phys. 88 (2016)
117 Y Z You, Z Wang, J Oon, C Xu, Topological number and fermion Green’s function for strongly interacting topological superconductors, Phys. Rev. B 90 (2014)