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An index for gauge-invariant operators and the Dixmier-Douady invariant

Authors: Victor Nistor and Evgenij Troitsky
Journal: Trans. Amer. Math. Soc. 356 (2004), 185-218
MSC (2000): Primary 46L80
Published electronically: August 25, 2003
MathSciNet review: 2020029
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Abstract: Let $\mathcal{G}\to B$ be a bundle of compact Lie groups acting on a fiber bundle $Y \to B$. In this paper we introduce and study gauge-equivariant $K$-theory groups $K_\mathcal{G}^i(Y)$. These groups satisfy the usual properties of the equivariant $K$-theory groups, but also some new phenomena arise due to the topological non-triviality of the bundle $\mathcal{G}\to B$. As an application, we define a gauge-equivariant index for a family of elliptic operators $(P_b)_{b \in B}$ invariant with respect to the action of $\mathcal{G}\to B$, which, in this approach, is an element of $K_\mathcal{G}^0(B)$. We then give another definition of the gauge-equivariant index as an element of $K_0(C^*(\mathcal{G}))$, the $K$-theory group of the Banach algebra $C^*(\mathcal{G})$. We prove that $K_0(C^*(\mathcal{G})) \simeq K^0_\mathcal{G}(\mathcal{G})$ and that the two definitions of the gauge-equivariant index are equivalent. The algebra $C^*(\mathcal{G})$ is the algebra of continuous sections of a certain field of $C^*$-algebras with non-trivial Dixmier-Douady invariant. The gauge-equivariant $K$-theory groups are thus examples of twisted $K$-theory groups, which have recently turned out to be useful in the study of Ramond-Ramond fields.

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  • 1. Orlando Alvarez, I. M. Singer, and Bruno Zumino, Gravitational anomalies and the family’s index theorem, Comm. Math. Phys. 96 (1984), no. 3, 409–417. MR 769356
  • 2. Luis Alvarez-Gaumé, An introduction to anomalies, Fundamental problems of gauge field theory (Erice, 1985) NATO Adv. Sci. Inst. Ser. B Phys., vol. 141, Plenum, New York, 1986, pp. 93–206. MR 878768
  • 3. M. F. Atiyah, 𝐾-theory, Lecture notes by D. W. Anderson, W. A. Benjamin, Inc., New York-Amsterdam, 1967. MR 0224083
  • 4. M. F. Atiyah and R. Bott, The Yang-Mills equations over Riemann surfaces, Philos. Trans. Roy. Soc. London Ser. A 308 (1983), no. 1505, 523–615. MR 702806, 10.1098/rsta.1983.0017
  • 5. M. F. Atiyah and I. M. Singer, The index of elliptic operators. IV, Ann. of Math. (2) 93 (1971), 119–138. MR 0279833
  • 6. Thierry Aubin, Espaces de Sobolev sur les variétés riemanniennes, Bull. Sci. Math. (2) 100 (1976), no. 2, 149–173 (French). MR 0488125
  • 7. A. A. Belavin and V. G. Knizhnik, Algebraic geometry and the geometry of quantum strings, Phys. Lett. B 168 (1986), no. 3, 201–206. MR 830618, 10.1016/0370-2693(86)90963-9
  • 8. Jean-Michel Bismut and Daniel S. Freed, The analysis of elliptic families. II. Dirac operators, eta invariants, and the holonomy theorem, Comm. Math. Phys. 107 (1986), no. 1, 103–163. MR 861886
  • 9. Bruce Blackadar, 𝐾-theory for operator algebras, Mathematical Sciences Research Institute Publications, vol. 5, Springer-Verlag, New York, 1986. MR 859867
  • 10. P. Bouwknegt, A. Carey, V. Mathai, M. Murray, and D. Stevenson, Twisted $K$-theory and $K$-theory of bundle gerbes, E-print, hep-th/0106194.
  • 11. Theodor Bröcker and Tammo tom Dieck, Representations of compact Lie groups, Graduate Texts in Mathematics, vol. 98, Springer-Verlag, New York, 1985. MR 781344
  • 12. Jean-Luc Brylinski, Loop spaces, characteristic classes and geometric quantization, Progress in Mathematics, vol. 107, Birkhäuser Boston, Inc., Boston, MA, 1993. MR 1197353
  • 13. Alain Connes, Noncommutative geometry, Academic Press, Inc., San Diego, CA, 1994. MR 1303779
  • 14. Marius Dădărlat, A suspension theorem for continuous trace 𝐶*-algebras, Proc. Amer. Math. Soc. 120 (1994), no. 3, 761–769. MR 1166354, 10.1090/S0002-9939-1994-1166354-3
  • 15. Pierre Deligne and Daniel S. Freed, Classical field theory, Quantum fields and strings: a course for mathematicians, Vol. 1, 2 (Princeton, NJ, 1996/1997) Amer. Math. Soc., Providence, RI, 1999, pp. 137–225. MR 1701599
  • 16. Jacques Dixmier, Les 𝐶*-algèbres et leurs représentations, Deuxième édition. Cahiers Scientifiques, Fasc. XXIX, Gauthier-Villars Éditeur, Paris, 1969 (French). MR 0246136
  • 17. P. Donovan and M. Karoubi, Graded Brauer groups and 𝐾-theory with local coefficients, Inst. Hautes Études Sci. Publ. Math. 38 (1970), 5–25. MR 0282363
  • 18. Daniel S. Freed and Michael Hopkins, On Ramond-Ramond fields and 𝐾-theory, J. High Energy Phys. 5 (2000), Paper 44, 14. MR 1769477, 10.1088/1126-6708/2000/05/044
  • 19. D. Freed and E. Witten, Anomalies in string theory with $D$-branes, E-print, hep-th/9907189.
  • 20. James Glimm and Arthur Jaffe, Quantum physics, 2nd ed., Springer-Verlag, New York, 1987. A functional integral point of view. MR 887102
  • 21. Michel Hilsum and Georges Skandalis, Morphismes 𝐾-orientés d’espaces de feuilles et fonctorialité en théorie de Kasparov (d’après une conjecture d’A. Connes), Ann. Sci. École Norm. Sup. (4) 20 (1987), no. 3, 325–390 (French, with English summary). MR 925720
  • 22. Lars Hörmander, The analysis of linear partial differential operators. III, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 274, Springer-Verlag, Berlin, 1994. Pseudo-differential operators; Corrected reprint of the 1985 original. MR 1313500
  • 23. J. E. Humphreys, Introduction to Lie algebras and representation theory, Graduate texts in Math. 9, Springer-Verlag, Berlin - Heidelberg - New York - Tokyo, 1972.
  • 24. James E. Humphreys, Introduction to Lie algebras and representation theory, Springer-Verlag, New York-Berlin, 1972. Graduate Texts in Mathematics, Vol. 9. MR 0323842
  • 25. Max Karoubi, Algèbres de Clifford et 𝐾-théorie, Ann. Sci. École Norm. Sup. (4) 1 (1968), 161–270 (French). MR 0238927
  • 26. Max Karoubi, 𝐾-theory, Springer-Verlag, Berlin-New York, 1978. An introduction; Grundlehren der Mathematischen Wissenschaften, Band 226. MR 0488029
  • 27. 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 (Russian). MR 582160
  • 28. Nicolaas H. Kuiper, The homotopy type of the unitary group of Hilbert space, Topology 3 (1965), 19–30. MR 0179792
  • 29. Robert Lauter, Bertrand Monthubert, and Victor Nistor, Pseudodifferential analysis on continuous family groupoids, Doc. Math. 5 (2000), 625–655 (electronic). MR 1800315
  • 30. R. Lauter and V. Nistor, Analysis of geometric operators on open manifolds: a groupoid approach, to appear.
  • 31. V. Mathai, R. B. Melrose, and I. M. Singer, The index of projective families of elliptic operators, math. DG/0206002.
  • 32. Gregory Moore and Philip Nelson, The ætiology of sigma model anomalies, Comm. Math. Phys. 100 (1985), no. 1, 83–132. MR 796163
  • 33. G. Moore and E. Witten, Self-duality, Ramond-Ramond fields, and $K$-theory, E-print, hep-th/9912086.
  • 34. V. Nistor, An index theorem for gauge-invariant families: The case of solvable groups, submitted for publication.
  • 35. Victor Nistor, Higher McKean-Singer index formulae and noncommutative geometry, Representation theory of groups and algebras, Contemp. Math., vol. 145, Amer. Math. Soc., Providence, RI, 1993, pp. 439–452. MR 1216202, 10.1090/conm/145/1216202
  • 36. V. Nistor and E. Troitsky, An index theorem for gauge-invariant families: The case of compact groups (tentative title, work in progress).
  • 37. D. Kvillen, Determinants of Cauchy-Riemann operators on Riemann surfaces, Funktsional. Anal. i Prilozhen. 19 (1985), no. 1, 37–41, 96 (Russian). MR 783704
  • 38. John Roe, An index theorem on open manifolds. I, II, J. Differential Geom. 27 (1988), no. 1, 87–113, 115–136. MR 918459
  • 39. Jonathan Rosenberg, The role of 𝐾-theory in noncommutative algebraic topology, Operator algebras and 𝐾-theory (San Francisco, Calif., 1981) Contemp. Math., vol. 10, Amer. Math. Soc., Providence, R.I., 1982, pp. 155–182. MR 658514
  • 40. Jonathan Rosenberg, 𝐾-theory of group 𝐶*-algebras, foliation 𝐶*-algebras, and crossed products, Index theory of elliptic operators, foliations, and operator algebras (New Orleans, LA/Indianapolis, IN, 1986) Contemp. Math., vol. 70, Amer. Math. Soc., Providence, RI, 1988, pp. 251–301. MR 948696, 10.1090/conm/070/948696
  • 41. Jonathan Rosenberg, Continuous-trace algebras from the bundle theoretic point of view, J. Austral. Math. Soc. Ser. A 47 (1989), no. 3, 368–381. MR 1018964
  • 42. Shôichirô Sakai, 𝐶*-algebras and 𝑊*-algebras, Classics in Mathematics, Springer-Verlag, Berlin, 1998. Reprint of the 1971 edition. MR 1490835
  • 43. Georges Skandalis, Kasparov’s bivariant 𝐾-theory and applications, Exposition. Math. 9 (1991), no. 3, 193–250. MR 1121156
  • 44. Evgenij V. Troitsky, ``Twice'' equivariant $C\sp *$-index theorem and the index theorem for families, Acta Appl. Math. 68 (2001), no. 1-3, 39-70, Noncommutative geometry and operator $K$-theory.
  • 45. Edward Witten, Two-dimensional gauge theories revisited, J. Geom. Phys. 9 (1992), no. 4, 303–368. MR 1185834, 10.1016/0393-0440(92)90034-X
  • 46. Edward Witten, D-branes and 𝐾-theory, J. High Energy Phys. 12 (1998), Paper 19, 41 pp. (electronic). MR 1674715, 10.1088/1126-6708/1998/12/019
  • 47. Edward Witten, Overview of 𝐾-theory applied to strings, Strings 2000. Proceedings of the International Superstrings Conference (Ann Arbor, MI), 2001, pp. 693–706. MR 1827946, 10.1142/S0217751X01003820

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Additional Information

Victor Nistor
Affiliation: Department of Mathematics, Pennsylvania State University, University Park, Pennsylvania 16802

Evgenij Troitsky
Affiliation: Department of Mechanics and Mathematics, Moscow State University, 119992 Moscow, Russia

Received by editor(s): April 22, 2002
Published electronically: August 25, 2003
Additional Notes: The first author was partially supported by NSF Young Investigator Award DMS-9457859 and NSF Grants DMS 991981 and 0200808
The second author was partially supported by RFFI Grant 99-01-01202 and Presidential Grant 00-15-99263.
Article copyright: © Copyright 2003 American Mathematical Society