nLab odd Chern character

Idea

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

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

where $g^{-1}d g \coloneqq g^\ast \theta$ is the pullback of differential forms along $g$ of the Maurer-Cartan form on $U$.

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

Related concepts

• K-theory, K-homology, index theory

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

• Xianzhe Dai, Weiping Zhang, Eta invariant and holonomy, the even dimensional case (arXiv:1205.0562)

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

• Thomas Tradler, Scott Wilson, Mahmoud Zeinalian, An Elementary Differential Extension of Odd K-theory, J. of K-theory, K-theory and its Applications to Algebra, Geometry, Analysis

  and Topology, (arXiv:1211.4477)

• Scott Wilson, A loop group extension of the odd Chern character (arXiv:1311.6393)