A cut-off tubular geometry of loop space

Motivated by the computation of loop space quantum mechanics as indicated in [7], here we seek a better understanding of the tubular geometry of loop space ${\cal L}{\cal M}$ corresponding to a Riemannian manifold ${\cal M}$ around the submanifold of vanishing loops. Our approach is to first compute the tubular metric of $({\cal M}^{2N+1})_{C}$ around the diagonal submanifold, where $({\cal M}^N)_{C}$ is the Cartesian product of $N$ copies of ${\cal M}$ with a cyclic ordering. This gives an infinite sequence of tubular metrics such that the one relevant to ${\cal L}{\cal M}$ can be obtained by taking the limit $N\to \infty$. Such metrics are computed by adopting an indirect method where the general tubular expansion theorem of [12] is crucially used. We discuss how the complete reparametrization isometry of loop space arises in the large-$N$ limit and verify that the corresponding Killing equation is satisfied to all orders in tubular expansion. These tubular metrics can alternatively be interpreted as some natural Riemannian metrics on certain bundles of tangent spaces of ${\cal M}$ which, for ${\cal M} \times {\cal M}$, is the tangent bundle $T{\cal M}$.