The Ko-Valued Spectral Flow for Skew-Adjoint Fredholm Operators

[ABS64] M. F. ATIYAH, R. BOTT, and A. SHAPIRO, Clifford modules, Topology 3 (1964), no. suppl. 1, 3–38. MR 0167985

[ALZI97] A. ALTLAND and M. R. ZIRNBAUER, Nonstandard symmetry classes in mesoscopic normal-/superconducting hybrid structures, Phys. Rev. B 55 (1997), 1142–1162.

[AMZ19] A. ALLDRIDGE, C. MAX, and M. R. ZIRNBAUER, Bulk-boundary correspondence for disordered free-fermion topological phases, Comm. Math. Phys., online first (2019). https:dx.doi. org/10.1007/s00220-019-03581-7

[APS76] M. F. ATIYAH, V. K. PATODI, and I. M. SINGER, Spectral asymmetry and Riemannian geometry. III, Math. Proc. Cambridge Philos. Soc. 79 (1976), no. 1, 71–99. MR 0397799 (53 #1655c)

[ASS94A] J. AVRON, R. SEILER, and B. SIMON, The index of a pair of projections, J. Funct. Anal. 120 (1994), no. 1, 220–237. MR 1262254 (95b:47012)

[ASS94B] J. AVRON, R. SEILER, and B. SIMON, Charge deficiency, charge transport and comparison of dimensions, Comm. Math. Phys. 159 (1994), 399–422. MR 1256994

[ATI66] M. F. ATIYAH, K-theory and reality, Quart. J. Math. Oxford Ser. (2) 17 (1966), 367–386. MR 0206940

[ATSI69] M. F. ATIYAH and I. M. SINGER, Index theory for skew-adjoint Fredholm operators, Inst. Hautes E´ tudes Sci. Publ. Math. (1969), no. 37, 5–26. MR 0285033

[AZWA11] S. AZZALI and C. WAHL, Spectral flow, index and the signature operator, J. Topol. Anal. 3 (2011), no. 1, 37–67. MR 2784763

[BLA98] B. BLACKADAR, K-theory for operator algebras, second ed., Mathematical Sciences Re- search Institute Publications, vol. 5, Cambridge University Press, Cambridge, 1998. MR 1656031 (99g:46104)

[BBLP05] B. BOOSS-BAVNBEK, M. LESCH, and J. PHILLIPS, Unbounded Fredholm operators and spectral flow, Canad. J. Math. 57 (2005), no. 2, 225–250. MR 2124916

[BBWO93] B. BOOß-BAVNBEK and K. P. WOJCIECHOWSKI, Elliptic boundary problems for Dirac opera- tors, Mathematics: Theory & Applications, Birkha¨user Boston Inc., Boston, MA, 1993. MR 1233386 (94h:58168)

[BCR16] C. BOURNE, A. L. CAREY, and A. RENNIE, A non-commutative framework for topological insulators, Rev. Math. Phys. 28 (2016), no. 2, 1650004, 51. MR 3484317

[BKR17] C. BOURNE, J. KELLENDONK, and A. RENNIE, The K-theoretic bulk-edge correspondence for topological insulators, Ann. Henri Poincare´ 18 (2017), no. 5, 1833–1866. MR 3635969

[BOSB20] C. BOURNE and H. SCHULZ-BALDES, On Z2-indices for ground states of fermionic chains, Rev. Math. Phys. 32 (2020), 2050028.

[BDF73] L. G. BROWN, R. G. DOUGLAS, and P. A. FILLMORE, Unitary equivalence modulo the com- pact operators and extensions of C∗-algebras, 58–128. Lecture Notes in Math., Vol. 345. MR 0380478

[BRLE01] J. BRU¨ NING and M. LESCH, On boundary value problems for Dirac type operators. I. Regularity and self-adjointness, J. Funct. Anal. 185 (2001), no. 1, 1–62. arXiv:9905181 [math.FA], MR 1853751 (2002g:58034)

[CPSB19] A. L. CAREY, J. PHILLIPS, and H. SCHULZ-BALDES, Spectral flow for skew-adjoint Fredholm operators, J. Spectr. Theory 9 (2019), no. 1, 137–170. MR 3900782

[CASB19] A. L. CAREY and H. SCHULZ-BALDES, Spectral flow of monopole insertion in topological insu- lators, Comm. Math. Phys., 370 (2019), no. 3, 895–923. MR 3995923

[DNSB16] G. DE NITTIS and H. SCHULZ-BALDES, Spectral flows associated to flux tubes, Ann. Henri Poincare´, 17 (2016), 1–35. MR 3437823

[DSBW19] N. DOLL, H. SCHULZ-BALDES, and N. WATERSTRAAT, Parity as Z2-valued spectral flow, Bull. London Math. Soc., 51 (2019), 836–852 (2019). MR 4022430

[VDD19] K. VAN DEN DUNGEN, The index of generalised Dirac-Schro¨dinger operators, J. Spectr. Theory 9 (2019), no. 4, 1459–1506. MR 4033528

[FRMO13] D. S. FREED and G. W. MOORE, Twisted equivariant matter, Ann. Henri Poincare´ 14 (2013), no. 8, 1927–2023. MR 3119923

[GRSB16] J. GROßMANN and H. SCHULZ-BALDES, Index pairings in presence of symmetries with applica- tions to topological insulators, Comm. Math. Phys. 343 (2016), no. 2, 477–513. MR 3477345

[HHZ05] P. HEINZNER, A. HUCKLEBERRY, and M. R. ZIRNBAUER, Symmetry classes of disordered fermions, Comm. Math. Phys. 257 (2005), no. 3, 725–771. MR 2164950

[KALE13] J. KAAD and M. LESCH, Spectral flow and the unbounded Kasparov product, Adv. Math. 248 (2013), 495–530. arXiv:1110.1472 [math.OA], MR 3107519

[KAR70] M. KAROUBI, Espaces classifiants en K-the´orie, Trans. Amer. Math. Soc. 147 (1970), 75–115. MR 0261592

[KAS80] G. G. KASPAROV, The operator K-functor and extensions of C∗-algebras, Izv. Akad. Nauk SSSR Ser. Mat. 44 (1980), no. 3, 571–636, 719. MR 582160

[KAT95] T. KATO, Perturbation theory for linear operators, Classics in Mathematics, Springer-Verlag, Berlin, 1995, Reprint of the 1980 edition. MR 1335452 (96a:47025)

[KAKO18] H. KATSURA and T. KOMA, The noncommutative index theorem and the periodic table for disordered topological insulators and superconductors, J. Math. Phys. 59 (2018) 031903.MR 3774825

[KEL17] J. KELLENDONK, On the C∗-algebraic approach to topological phases for insulators, Ann. Henri Poincare´ 18 (2017), no. 7, 2251–2300. MR 3665214

[KEL19] , Cyclic cohomology for graded C∗,r-algebras and its pairings with van Daele K-theory, Comm. Math. Phys. 368 (2019), no. 2, 467–518. MR 3949717

[KEZI16] R. KENNEDY and M. R. ZIRNBAUER, Bott periodicity for Z2 symmetric ground states of gapped free-fermion systems, Comm. Math. Phys. 342 (2016), no. 3, 909–963. MR 3465435

[KILE04] P. KIRK and M. LESCH, The η-invariant, Maslov index, and spectral flow for Dirac-type opera- tors on manifolds with boundary, Forum Math. 16 (2004), no. 4, 553–629. arXiv:0012123 [math.DG], MR 2044028 (2005b:58029)

[KIT09] A. KITAEV, Periodic table for topological insulators and superconductors, Proceedings of the L.D.Landau Memorial Conference ‘Advances in Theoretical Physics’ (V. LEBEDEV and M. FEIGEL’MAN, eds.), vol. 1134, American Institute of Physics Conference Series, 2009, pp. 22–30.

[LAMI89] H. B. LAWSON, JR. and M.-L. MICHELSOHN, Spin geometry, Princeton Mathematical Series, vol. 38, Princeton University Press, Princeton, NJ, 1989. MR 1031992

[LES05] M. LESCH, The uniqueness of the spectral flow on spaces of unbounded self-adjoint Fredholm operators, Spectral geometry of manifolds with boundary and decomposition of mani- folds, Contemp. Math., vol. 366, Amer. Math. Soc., Providence, RI, 2005, pp. 193–224. MR 2114489

[LIMO19] Z. LI and R. MONG Local formula for the Z2-invariant of topological insulators, Phys. Rev. B 100 (2019) 205101.

[LOT88] J. LOTT, Real anomalies, J. Math. Phys. 29 (1988), no. 6, 1455–1464. MR 944463

[MIL63] J. MILNOR, Morse theory, Based on lecture notes by M. Spivak and R. Wells. An- nals of Mathematics Studies, No. 51, Princeton University Press, Princeton, N.J., 1963. MR 0163331

[PHI96] J. PHILLIPS, Self-adjoint Fredholm operators and spectral flow, Canad. Math. Bull. 39 (1996), no. 4, 460–467. MR 1426691

[ROSA95] J. ROBBIN and D. SALAMON, The spectral flow and the Maslov index, Bull. London Math. Soc. 27 (1995), no. 1, 1–33. MR 1331677 (96d:58021)

[RSFL10] S. RYU, A. P. SCHNYDER, A. FURUSAKI and A. W. W. LUDWIG, Topological insulators and superconductors: ten-fold way and dimensional hierarchy, New J. Phys. 12 (2010), 065010.

[SCH93] H. SCHRO¨ DER, K-theory for real C∗-algebras and applications, Pitman Research Notes in Mathematics Series, vol. 290, Longman Scientific & Technical, Harlow; copublished in the United States with John Wiley & Sons, Inc., New York, 1993. MR 1267059

[THI16] G. C. THIANG, On the K-theoretic classification of topological phases of matter, Ann. Henri Poincare´ 17 (2016), no. 4, 757–794. MR 3472623

[WIT82] E. WITTEN, An SU(2) anomaly, Phys. Lett. B 117 (1982), no. 5, 324–328. MR 678541