# nLab odd Chern character

cohomology

### Theorems

#### Differential cohomology

differential cohomology

# Contents

## 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).

## 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