Odd characteristic classes in entire cyclic homology and equivariant loop space homology

Given a compact manifold $M$ and $g\in C^{\infty}(M,U(l;\mathbb{C}))$ we construct Chern character $\mathrm{Ch}^-(g)$ which lives in the odd part of equivariant (entire) cyclic Chen-normalized bar complex $\underline{\mathscr{C}}(\Omega_{\mathbb{T}}(M\times \mathbb{T}))$ $M$, is mapped to Bismut-Chern under Chen integral map. It also shown that assignment $g\mapsto \mathrm{Ch}^-(g)$ induces well-defined group homomorphism from $K^{-1}$ theory homology