Twisted K-theory of differentiable stacks
Jean-Louis Tu; Ping Xu; Camille Laurent-Gengoux
Annales scientifiques de l'École Normale Supérieure (2004)
- Volume: 37, Issue: 6, page 841-910
- ISSN: 0012-9593
Access Full Article
topHow to cite
topTu, Jean-Louis, Xu, Ping, and Laurent-Gengoux, Camille. "Twisted K-theory of differentiable stacks." Annales scientifiques de l'École Normale Supérieure 37.6 (2004): 841-910. <http://eudml.org/doc/82648>.
@article{Tu2004,
author = {Tu, Jean-Louis, Xu, Ping, Laurent-Gengoux, Camille},
journal = {Annales scientifiques de l'École Normale Supérieure},
keywords = {-theory; differentiable stack; gerbe; groupoid; orbifold},
language = {eng},
number = {6},
pages = {841-910},
publisher = {Elsevier},
title = {Twisted K-theory of differentiable stacks},
url = {http://eudml.org/doc/82648},
volume = {37},
year = {2004},
}
TY - JOUR
AU - Tu, Jean-Louis
AU - Xu, Ping
AU - Laurent-Gengoux, Camille
TI - Twisted K-theory of differentiable stacks
JO - Annales scientifiques de l'École Normale Supérieure
PY - 2004
PB - Elsevier
VL - 37
IS - 6
SP - 841
EP - 910
LA - eng
KW - -theory; differentiable stack; gerbe; groupoid; orbifold
UR - http://eudml.org/doc/82648
ER -
References
top- [1] Adem A., Ruan Y., Twisted orbifold K-theory, Comm. Math. Phys.237 (2003) 533-556. Zbl1051.57022MR1993337
- [2] Artin M. et al. , Théorie des topos et cohomologie étale des schémas, in: Séminaire de géométrie algébrique, Lecture Notes in Mathematics, vols. 269, 270, 305, Springer, Berlin, 1972–1973.
- [3] Atiyah M., K-theory. Lecture notes by D.W. Anderson, 1967. MR224083
- [4] Atiyah M., K-theory past and present, math.KT/0012213. MR2091892
- [5] Atiyah M., Segal G., Twisted K-theory, math.KT/0407054.
- [6] Baum P., Connes A., Chern character for discrete groups, in: A fête of topology, Academic Press, New York, 1988, pp. 163-232. Zbl0656.55005MR928402
- [7] Baum P., Connes A., Higson N., Classifying space for proper actions and K-theory of group -algebras, in: -Algebras: 1943–1993 (San Antonio, TX, 1993), Contemp. Math., vol. 167, Amer. Math. Soc., Providence, RI, 1994, pp. 240-291. Zbl0830.46061MR1292018
- [8] Behrend K., Edidin B., Fantechi B., Fulton W., Gottsche L., Kresch K., Introduction to stacks, in preparation.
- [9] Behrend K., Xu P., -bundles and gerbes over differentiable stack, C. R. Acad. Sci. Paris Sér. I336 (2003) 163-168. Zbl1039.58016MR1969572
- [10] Behrend, K., Xu P., Differentiable stacks and gerbes, in preparation. Zbl1227.14007
- [11] Blackadar B., K-Theory for Operator Algebras, Mathematical Sciences Research Institute Publications, vol. 5, Cambridge University Press, Cambridge, 1998. Zbl0913.46054MR1656031
- [12] Blanchard E., Déformations de -algèbres de Hopf, Bull. Soc. Math. France124 (1996) 141-215. Zbl0851.46040MR1395009
- [13] Bost J.-B., Principe d'Oka, K-théorie et systèmes dynamiques non commutatifs, Invent. Math.101 (1990) 261-333. Zbl0719.46038MR1062964
- [14] Brylinski J.-L., Loop Spaces, Characteristic Classes and Geometric Quantization, Progress in Mathematics, vol. 107, Birkhäuser, Basel, 1993. Zbl0823.55002MR1197353
- [15] Bouwknegt P., Mathai V., D-branes, B-fields and twisted K-theory, J. High Energy Phys.3 (2000) 7-11. Zbl0959.81037MR1756434
- [16] Bouwknegt P., Carey A., Mathai V., Murray M., Stevenson D., Twisted K-theory and K-theory of bundle gerbes, Comm. Math. Phys.228 (2002) 17-45. Zbl1036.19005MR1911247
- [17] Connes A., Cyclic Cohomology and the Transverse Fundamental Class of a Foliation, in: Pitman Research Notes Math. Ser., vol. 123, Longman Sci., Harlow, 1986, pp. 52-144. Zbl0647.46054MR866491
- [18] Connes A., Skandalis G., The longitudinal index theorem for foliations, Publ. Res. Inst. Math. Sci. Kyoto Univ.20 (1984) 1139-1183. Zbl0575.58030MR775126
- [19] Crainic M., Differentiable and algebroid cohomology, van Est isomorphisms, and characteristic classes, Comment. Math. Helv.78 (2003) 681-721. Zbl1041.58007MR2016690
- [20] Crainic M., Moerdijk I., A homology theory for étale groupoids, J. Reine Angew. Math.521 (2000) 25-46. Zbl0954.22002MR1752294
- [21] Dixmier J., -Algebras, North Holland, Amsterdam, 1977. Zbl0372.46058MR458185
- [22] Dixmier J., Douady A., Champs continus d’espaces hilbertiens et de -algèbres, Bull. Soc. Math. France91 (1963) 227-284. Zbl0127.33102MR163182
- [23] Donovan P., Karoubi M., Graded Brauer groups and K-theory with local coefficients, Inst. Hautes Études Sci. Publ.38 (1970) 5-25. Zbl0207.22003MR282363
- [24] Dupont J., Curvature and Characteristic Classes, Lecture Notes in Mathematics, vol. 640, Springer, Berlin, 1978. Zbl0373.57009MR500997
- [25] Dupré M.J., Gillette R.M., Banach Bundles, Banach Modules and Automorphisms of -Algebras, Pitman Research Notes in Mathematics, vol. 92, 1983. Zbl0536.46048MR721812
- [26] Fell J., Doran R., Representations of -Algebras, Locally Compact Groups, and Banach ∗-Algebraic Bundles, vol. 2, Pure and Applied Mathematics, vol. 126, Academic Press, Boston, MA, 1988. Zbl0652.46051
- [27] Freed D., The Verlinde algebra is twisted equivariant K-theory, Turkish J. Math.25 (2001) 159-167. Zbl0971.55006MR1829086
- [28] Fulman I., Muhly P., Bimodules, spectra, and Fell bundles, Israel J. Math.108 (1998) 193-215. Zbl0917.46064MR1669380
- [29] Gabriel P., Zisman M., Calculus of Fractions and Homotopy Theory, Ergebnisse der Mathematik und ihrer Grenzgebiete, vol. 35, Springer, Berlin, 1967. Zbl0186.56802MR210125
- [30] Giraud J., Cohomologie non abélienne, Die Grundlehren der mathematischen Wissenschaften, vol. 179, Springer, Berlin, 1971. Zbl0226.14011MR344253
- [31] Grothendieck A., Le groupe de Brauer, Séminaire Bourbaki (1964/65), Collection Hors Série de la S.M.F.9 (1995) 199-219. Zbl0186.54702MR1608798
- [32] Guillemin V., Sternberg S., Supersymmetry and Equivariant de Rham Theory, Springer, Berlin, 1999. Zbl0934.55007MR1689252
- [33] Haefliger A., Groupoïdes d'holonomie et classifiants, Astérisque116 (1984) 70-97. Zbl0562.57012MR755163
- [34] Higson N., On a technical theorem of Kasparov, J. Funct. Anal.73 (1987) 107-112. Zbl0623.46035MR890657
- [35] Higson N., The Baum–Connes conjecture, in: Proceedings of the International Congress of Mathematicians, vol. II (Berlin, 1998), Doc. Math., Extra vol. II, 1998, pp. 637-646, (electronic). Zbl0911.46041MR1648112
- [36] Higson N., Lafforgue V., Skandalis G., Counterexamples to the Baum–Connes conjecture, Geom. Funct. Anal.12 (2) (2002) 330-354. Zbl1014.46043MR1911663
- [37] Higson N., Roe J., Yu G., A coarse Mayer–Vietoris principle, Math. Proc. Cambridge Philos. Soc.114 (1993) 85-97. Zbl0792.55001MR1219916
- [38] Hilsum M., Skandalis G., Morphismes K-orientés d'espaces de feuilles et fonctorialité en théorie de Kasparov (d'après une conjecture d'A. Connes), Ann. Sci. Éc. Norm. Sup.20 (1987) 325-390. Zbl0656.57015MR925720
- [39] Hitchin N., Lectures on special Lagrangian submanifolds, AMS/IP Stud. Adv. Math.23 (1999) 151-182. Zbl1079.14522MR1876068
- [40] Kellendonk J., Noncommutative geometry of tilings and gap labelling, Rev. Math. Phys.7 (7) (1995) 1133-1180. Zbl0847.52022MR1359991
- [41] Koosis P., An irreducible unitary representation of a compact group is finite dimensional, Proc. Amer. Math. Soc.8 (1957) 712-715. Zbl0079.32802MR87888
- [42] Kumjian A., Fell Bundles over groupoids, Proc. Amer. Math. Soc.128 (1998) 1115-1125. Zbl0891.46038MR1443836
- [43] Kumjian A., Muhly P., Renault J., Williams D., The Brauer group of a locally compact groupoid, Amer. J. Math.120 (1998) 901-954. Zbl0916.46050MR1646047
- [44] Le Gall P.-Y., Théorie de Kasparov équivariante et groupoïdes, 16 (1999) 361-390. Zbl0932.19004MR1686846
- [45] Lupercio E., Uribe B., Gerbes over orbifolds and twisted K-theory, Comm. Math. Phys.245 (2004) 449-489. Zbl1068.53034MR2045679
- [46] Mathai V., Stevenson D., Chern character in twisted K-theory, equivariant and holomorphic cases, Comm. Math. Phys.236 (2003) 161-186. Zbl1030.19004MR1977885
- [47] Meinrenken E., The basic gerbe over a compact simple Lie group, Enseign. Math. (2)49 (2003) 307-333. Zbl1061.53034MR2026898
- [48] Mickelsson J., Gerbes, (twisted) K-theory, and the supersymmetric WZW model, hep-th/0206139. Zbl1058.81067
- [49] Minasian R., Moore G., K-theory and Ramond–Ramond charge, J. High Energy Phys.11 (1997) 2-7. Zbl0949.81511MR1606278
- [50] Monthubert B., Groupoids of manifolds with corners and index theory, in: Groupoids in Analysis, Geometry, and Physics (Boulder, CO, 1999), Contemp. Math., vol. 282, Amer. Math. Soc., Providence, RI, 2001, pp. 147-157. Zbl1043.35148MR1855248
- [51] Mrčun J., Functoriality of the bimodule associated to a Hilsum–Skandalis map, 18 (1999) 235-253. Zbl0938.22002MR1722796
- [52] Moerdijk I., Orbifolds as groupoids: an introduction, in: Orbifolds in Mathematics and Physics (Madison, WI, 2001), Contemp. Math., vol. 310, 2002, pp. 205-222. Zbl1041.58009MR1950948
- [53] Moerdijk I., Introduction to the language of stacks and gerbes, math.AT/0212266.
- [54] Moerdijk I., Pronk D.A., Orbifolds, sheaves and groupoids, 12 (1997) 3-21. Zbl0883.22005MR1466622
- [55] Muhly P., Bundles over groupoids, in: Groupoids in Analysis, Geometry and Physics (Boulder, CO, 1999), Contemp. Math., vol. 282, Amer. Math. Soc., Providence, RI, 2000, pp. 67-82. Zbl1011.46052MR1855243
- [56] Muhly P., Renault J., Williams D., Equivalence and isomorphism for groupoid -algebras, J. Operator Theory17 (1987) 3-22. Zbl0645.46040MR873460
- [57] Paterson A., The analytic index for proper, Lie groupoid actions, in: Groupoids in Analysis, Geometry, and Physics (Boulder, CO, 1999), Contemp. Math., vol. 282, Amer. Math. Soc., Providence, RI, 2001, pp. 115-135. Zbl0995.19005MR1855246
- [58] Pedersen G., -Algebras and their Automorphism Groups, London Mathematical Society Monographs, vol. 14, Academic Press, London, 1979. Zbl0416.46043MR548006
- [59] Pressley A., Segal G., Loop Groups, Oxford Mathematical Monographs, Oxford University Press, New York, 1986. Zbl0618.22011MR900587
- [60] Raeburn I., Taylor J., Continuous trace -algebras with given Dixmier–Douady class, J. Austral. Math. Soc. Ser. A38 (1985) 394-407. Zbl0627.46068MR779202
- [61] Renault J., A Groupoid Approach to -Algebras, Lecture Notes in Mathematics, vol. 793, Springer, Berlin, 1980. Zbl0433.46049MR584266
- [62] Renault J., Représentation des produits croisés d'algèbres de groupoïdes, J. Operator Theory18 (1987) 67-97. Zbl0659.46058MR912813
- [63] Rosenberg J., Continuous-trace algebras from the bundle theoretic point of view, J. Austral. Math. Soc. Ser. A47 (1989) 368-381. Zbl0695.46031MR1018964
- [64] Schweitzer L.B., A short proof that is local ifA is local and Fréchet, Internat. J. Math.3 (4) (1992) 581-589. Zbl0804.46054MR1168361
- [65] Segal G., Equivariant K-theory, Inst. Hautes Études Sci. Publ. Math.34 (1968) 129-151. Zbl0199.26202MR234452
- [66] Segal G., Fredholm complexes, Quart. J. Math. Oxford Ser. (2)21 (1970) 385-402. Zbl0213.25403MR271930
- [67] Skandalis G., Tu J.L., Yu G., The coarse Baum–Connes conjecture and groupoids, Topology41 (4) (2002) 807-834. Zbl1033.19003MR1905840
- [68] Tu J.-L., La conjecture de Novikov pour les feuilletages hyperboliques, 16 (1999) 129-184. Zbl0932.19005MR1671260
- [69] Tu J.-L., La conjecture de Baum–Connes pour les feuilletages moyennables, 17 (3) (1999) 215-264. Zbl0939.19001MR1703305
- [70] Tuynman G.M., Wiegerinck G.M., Central extensions and physics, J. Geom. Phys.4 (1987) 207-258. Zbl0649.58014MR948561
- [71] Yamagami S., On primitive ideal spaces of -algebras over certain locally compact groupoids, in: Mappings of Operator Algebras (Philadelphia, PA, 1988), Progress in Math., vol. 84, Birkhäuser Boston, Boston, MA, 1990, pp. 199-204. Zbl0733.46035MR1103378
- [72] Wegge-Olsen N.E., K-theory and -algebras, A friendly approach, Oxford Science Publications, The Clarendon Press, Oxford University Press, New York, 1993. Zbl0780.46038MR1222415
- [73] Weil A., Sur les théorèmes de De Rham, Comment. Math. Helv.26 (1959) 119-145. Zbl0047.16702MR50280
- [74] Weinstein A., Xu P., Extensions of symplectic groupoids and quantization, J. Reine Angew. Math.417 (1991) 159-189. Zbl0722.58021MR1103911
- [75] Witten E., D-branes and K-theory, J. High Energy Phys.12 (1998) 19-44. Zbl0959.81070MR1674715
- [76] Witten E., Overview of K-theory applied to strings, Internat. J. Modern Phys. A16 (2001) 693-706. Zbl0980.81049MR1827946
- [77] Xu P., Morita equivalent symplectic groupoids, Math. Sci. Res. Inst. Publ.20 (1991) 291-311. Zbl0733.58013MR1104935
NotesEmbed ?
topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.