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A variant of \(K\)-theory: \(K_\pm\). (English) Zbl 1090.19004
Tillmann, Ulrike (ed.), Topology, geometry and quantum field theory. Proceedings of the 2002 Oxford symposium in honour of the 60th birthday of Graeme Segal, Oxford, UK, June 24–29, 2002. Cambridge: Cambridge University Press (ISBN 0-521-54049-6/pbk). London Mathematical Society Lecture Note Series 308, 5-17 (2004).
E. Witten [J. High Energy Phys. 1998, No. 12, Paper 19, 41 p. (1998; Zbl 0959.81070)] introduced a variant \(K_\pm(X)\) of \(K\)-theory for manifolds \(X\) with involution \(\tau\) having a fixed sub-manifold \(Y\) such that \(\tau\) interchanges pairs of complex vector bundles \((E^+,E^-)\). In physics \(X\) is a 10-dimensional Lorentzian manifold and maps \(\Sigma \to X\) of surface \(\Sigma\) describe the world-sheet of strings. Using their previously constructed K-theory [M. F. Atiyah and D. W. Anderson, “K-theory. With reprints of M. F. Atiyah: Power operations in K-theory.” New York-Amsterdam: W.A. Benjamin, Inc., 166 p. (1967; Zbl 0159.53302)] the authors defined \(K^*_\pm\) through relative \(\mathbb Z_2\)-equivariant K-theory \[ K^*_\pm(X) = K^*_{\mathbb Z_2}(X\times I, X\times\partial I). \] If the involution \(\tau\) is trivial, then \(K^0_\pm(X) \cong K^1(X).\) This K-theory fits to the requirements of the physics which involves a switch from type IIA to type IIB string theory. Another definition of \(K_\pm\) is from operator version \[ \mathcal K_\pm(X) := [X,\mathcal F]^s_*, \] where \(\mathcal F\) is the space of Fredholm operators, \([.,.]\) the set of homotopy classes of \(\mathbb Z_2\)-maps, \(*\) means use \(\mathbb Z_2\)-maps compatible with \(*\) and \(s\) means use stable homotopy equivalence. In §§4-6 the authors show that there is a natural isomorphism \(\mathcal K_\pm(X) \cong K_\pm(X)\) and discuss also the real versions.
For the entire collection see [Zbl 1076.19001].

MSC:
19L64 Geometric applications of topological \(K\)-theory
19L47 Equivariant \(K\)-theory
55N91 Equivariant homology and cohomology in algebraic topology
81T30 String and superstring theories; other extended objects (e.g., branes) in quantum field theory
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