Spin structures on loop spaces that characterize string manifolds (Q276567)

Let \(M\) be a spin manifold of dimension \(3\) or of dimension bigger than \(4\). Using bundle \(2\)-gerbes (in particular, considering string structures as trivializations of the Chern-Simons \(2\)-gerbe; see [\textit{A. L. Carey} et al., Commun. Math. Phys. 259, No. 3, 577--613 (2005; Zbl 1088.58018)]), as well as a functor that serves to transgress string structures, the author proves that the existence of a fusion spin structure (i.e. a spin structure on the infinite dimensional loop manifold \(LM\) together with a fusion product on the associated principal Fréchet \(U(1)\)-bundle over \(LFM\), the loop space of the total space of the frame bundle of \(M\)) is equivalent to the existence of a string structure on \(M\), that is, to vanishing of the first fractional Pontryagin class \(\frac{1}{2}p_1(M)\in H^4(M; \mathbb Z)\).