On the geometry of free loop spaces
Let \(M\) be a finite-dimensional path-connected Riemannian manifold. The infinite-dimensional space of all smooth maps from the circle group \(S^1\) to \(M\) is the Fréchet manifold \(\Lambda M\) called the free loop space. The points in \(\Lambda M\) can be described as free loops in \(M\). \textit{M. F. Atiyah} [Colloq. Honneur L. Schwartz, Ec. Polytech. 1983, Vol. 1, Astérisque 131, 43-59 (1985; Zbl 0578.58039)] showed that there is a fundamental closed 2-form \(\omega\) on \(\Lambda M\) which, unlike the finite-dimensional case, can be degenerate at some points. In this paper, the author proves that the set of all points where \(\omega\) is nondegenerate is open. In modern physical theories such as string theory, the idea of representing particles not as points but rather as loops on some manifold \(M\) has resulted in the formulation of spinors on the loop space \(\Lambda M\) of \(M\). A string structure is defined as a lifting of the structural group to an \(S^1\)-central extension of the loop group. If \(\widetilde G\to \widetilde P\to X\) is a lifting of a principal Fréchet bundle \(G\to P\to X\) over a Fréchet manifold \(X\) and if \(S^1\to \widetilde G\to G\) is an \(S^1\)-central extension of \(G\), then the author shows that every connection on the principal bundle \(G\to P\to X\) together with a \(\widetilde G\)-invariant connection on \(S^1\to \widetilde P\to P\) defines a connection on \(\widetilde G\to \widetilde P\to X\), and, as a consequence of this result, that there exist connections on the string structure of \(\Lambda M\). Finally, the author defines Christoffel symbols on \(\Lambda M\) which are described in terms of the Christoffel symbols on \(M\) and compute the corresponding curvature.
scientific article; zbMATH DE number 1779095