nLab
odd Chern character

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Cohomology

cohomology

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Differential cohomology

Contents

Idea

Where the Chern character of a map to BU\mathbf{B}U (for UU the stable unitary group) is a sum of characteristic classes corresponding to invariant polynomials, the odd Chern character ch(g)ch(g) of a map to g:XUg \colon X \to U is a sum of the corresponding WZW term curvatures

ch(g)k=0(1) kk!(2k+1)!tr[(g 1dg) 2k+1], ch(g) \coloneqq \underoverset{k= 0}{\infty}{\sum} (-1)^k \frac{k!}{(2k+1)!} tr[(g^{-1} d g)^{2k+1}] \,,

where g 1dgg *θg^{-1}d g \coloneqq g^\ast \theta is the pullback of differential forms along gg of the Maurer-Cartan form on UU.

The odd Chern character appears in the index theory for Toeplitz operators and the eta invariant for even-dimensional manifolds (Dai-Zhang 12).

References

  • Paul Baum, R. G. Douglas, K-homology and index theory, in Proc. Sympos. Pure

    and Appl. Math., Vol. 38, pp. 117-173, Amer. Math. Soc. Providence, 1982.

  • Ezra Getzler, The odd Chern character in cyclic homology and spectral

    flow_, Topology, 32(3):489–507, 1993.

Discussion of the relation to index theory of Toeplitz operators and eta invariants is in

Discussion of odd differential K-theory via the odd Chern character is in

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