Twists on the Torus Equivariant under the $2$-Dimensional Crystallographic Point Groups

Kiyonori Gomi

Symmetry, integrability and geometry: methods and applications, Tome 13 (2017)

Résumé

A twist is a datum playing a role of a local system for topological $K$-theory. In equivariant setting, twists are classified into four types according to how they are realized geometrically. This paper lists the possible types of twists for the torus with the actions of the point groups of all the $2$-dimensional space groups (crystallographic groups), or equivalently, the torus with the actions of all the possible finite subgroups in its mapping class group. This is carried out by computing Borel's equivariant cohomology and the Leray–Serre spectral sequence. As a byproduct, the equivariant cohomology up to degree three is determined in all cases. The equivariant cohomology with certain local coefficients is also considered in relation to the twists of the Freed–Moore $K$-theory.

Détail

Keywords: twist; Borel equivariant cohomology; crystallographic group; topological insulator.

Bibliographie

[1] Adem A., Duman A. N., Gómez J. M., “Cohomology of toroidal orbifold quotients”, J. Algebra, 344 (2011), 114–136, arXiv: 1003.0435 | DOI | MR | Zbl

[2] Adem A., Ge J., Pan J., Petrosyan N., “Compatible actions and cohomology of crystallographic groups”, J. Algebra, 320 (2008), 341–353, arXiv: 0704.1823 | DOI | MR | Zbl

[3] Adem A., Pan J., “Toroidal orbifolds, Gerbes and group cohomology”, Trans. Amer. Math. Soc., 358 (2006), 3969–3983, arXiv: math.AT/0406130 | DOI | MR | Zbl

[4] Bott R., Tu L. W., Differential forms in algebraic topology, Graduate Texts in Mathematics, 82, Springer-Verlag, New York–Berlin, 1982 | DOI | MR | Zbl

[5] Donovan P., Karoubi M., “Graded Brauer groups and $K$-theory with local coefficients”, Inst. Hautes Études Sci. Publ. Math., 1970, 5–25 | DOI | MR | Zbl

[6] Dupont J. L., Curvature and characteristic classes, Lecture Notes in Math., 640, Springer-Verlag, Berlin–New York, 1978 | DOI | MR | Zbl

[7] Freed D. S., Hopkins M. J., Teleman C., “Loop groups and twisted $K$-theory I”, J. Topol., 4 (2011), 737–798, arXiv: 0711.1906 | DOI | MR | Zbl

[8] Freed D. S., Moore G. W., “Twisted equivariant matter”, Ann. Henri Poincaré, 14 (2013), 1927–2023, arXiv: 1208.5055 | DOI | MR | Zbl

[9] Gomi K., “Equivariant smooth Deligne cohomology”, Osaka J. Math., 42 (2005), 309–337, arXiv: math.DG/0307373 | MR | Zbl

[10] Gomi K., “A variant of $K$-theory and topological T-duality for real circle bundles”, Comm. Math. Phys., 334 (2015), 923–975, arXiv: 1310.8446 | DOI | MR | Zbl

[11] Hatcher A., Algebraic topology, Cambridge University Press, Cambridge, 2002 | MR | Zbl

[12] Hiller H., “Crystallography and cohomology of groups”, Amer. Math. Monthly, 93 (1986), 765–779 | DOI | MR | Zbl

[13] Karoubi M., $K$-theory. An introduction, Grundlehren der Mathematischen Wissenschaften, 226, Springer-Verlag, Berlin–New York, 1978 | DOI | MR | Zbl

[14] Karpilovsky G., Projective representations of finite groups, Monographs and Textbooks in Pure and Applied Mathematics, 94, Marcel Dekker, Inc., New York, 1985 | MR | Zbl

[15] Kaufmann R. M., Khlebnikov S., Wehefritz-Kaufmann B., “Projective representations from quantum enhanced graph symmetries”, J. Phys. Conf. Ser., 597 (2015), 012048, 16 pp. | DOI

[16] Kaufmann R. M., Khlebnikov S., Wehefritz-Kaufmann B., “Re-gauging groupoid, symmetries and degeneracies for graph Hamiltonians and applications to the gyroid wire network”, Ann. Henri Poincaré, 17 (2016), 1383–1414, arXiv: 1208.3266 | DOI | MR | Zbl

[17] Kitaev A., “Periodic table for topological insulators and superconductors”, AIP Conf. Proc., 1134 (2009), 22–30, arXiv: 0901.2686 | DOI | Zbl

[18] Kubota Y., “Notes on twisted equivariant K-theory for $\rm C^*$-algebras”, Internat. J. Math., 27 (2016), 1650058, 28 pp., arXiv: 1511.05312 | DOI | MR | Zbl

[19] May J. P., Cole M., Comezana G. R., Costenoble S. R., Elmendorf A. D., Greenlees J. P., Lewis L. G., Piacenza R. J., Triantafillou G., Waner S., Equivariant homotopy and cohomology theory, CBMS Regional Conference Series in Mathematics, 91, Amer. Math. Soc., Providence, RI, 1996 | DOI | MR | Zbl

[20] Newman M., Integral matrices, Pure and Applied Mathematics, 45, Academic Press, New York–London, 1972 | MR | Zbl

[21] Rolfsen D., Knots and links, Mathematics Lecture Series, 7, Publish or Perish, Inc., Houston, TX, 1990 | MR | Zbl

[22] Rosenberg J., “Continuous-trace algebras from the bundle theoretic point of view”, J. Austral. Math. Soc. Ser. A, 47 (1989), 368–381 | DOI | MR | Zbl

[23] Schattschneider D., “The plane symmetry groups: their recognition and notation”, Amer. Math. Monthly, 85 (1978), 439–450 | DOI | MR | Zbl

[24] Schwarzenberger R. L. E., “The $17$ plane symmetry groups”, Math. Gaz., 58 (1974), 123–131 | DOI | MR | Zbl

[25] Schwarzenberger R. L. E., “Colour symmetry”, Bull. London Math. Soc., 16 (1984), 209–240 | DOI | MR | Zbl

[26] Shiozaki K., Sato M., Gomi K., “$Z_2$-topology in nonsymmorphic crystalline insulators: Mobius twist in surface states”, Phys. Rev. B, 91 (2015), 155120, 9 pp., arXiv: 1502.03265 | DOI

[27] Shiozaki K., Sato M., Gomi K., “Topology of nonsymmorphic crystalline insulators and superconductors”, Phys. Rev. B, 93 (2016), 195413, 28 pp., arXiv: 1511.01463 | DOI

[28] Shiozaki K., Sato M., Gomi K., Topological crystalline materials — general formulation and wallpaper group classification, arXiv: 1701.08725

[29] Thiang G. C., “On the $K$-theoretic classification of topological phases of matter”, Ann. Henri Poincaré, 17 (2016), 757–794, arXiv: 1406.7366 | DOI | MR | Zbl