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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48.

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An index for gauge-invariant operators and the Dixmier-Douady invariant
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by Victor Nistor and Evgenij Troitsky PDF
Trans. Amer. Math. Soc. 356 (2004), 185-218 Request permission


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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Additional Information
  • Victor Nistor
  • Affiliation: Department of Mathematics, Pennsylvania State University, University Park, Pennsylvania 16802
  • Email:
  • Evgenij Troitsky
  • Affiliation: Department of Mechanics and Mathematics, Moscow State University, 119992 Moscow, Russia
  • Email:
  • 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.
  • © Copyright 2003 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 356 (2004), 185-218
  • MSC (2000): Primary 46L80
  • DOI:
  • MathSciNet review: 2020029