A loop group extension of the odd Chern character

We show that the universal odd Chern form, defined on the stable unitary group UU, extends to the loop group LULU as an equivariantly closed differential form. This provides an odd analogue to the Bismut–Chern form that appears in supersymmetric field theories. We also describe the associated transgression form, the so-called Bismut–Chern–Simons form, and explicate some properties it inherits as a differential form on the space of maps of a cylinder into the stable unitary group.As one corollary, we show that in a precise sense the spectral flow of a loop of self adjoint Fredholm operators equals the lowest degree component of the Bismut–Chern–Simons form, and the latter, when restricted to cylinders which are tori, is an equivariantly closed extension of spectral flow. As another corollary, we construct the Chern character homomorphism from odd KK-theory to the periodic cohomology of the free loop space, represented geometrically on the level of differential forms.