Equivalence of string and fusion loop-spin structures

The importance of the fusion relation of loops was recognized in the context of spin structures on the loop space by Stolz and Teichner and further developed by Waldorf. On a spin manifold M the equivalence classes of fusive' spin structures on the loop space LM, incorporating the fusion property, strong regularity and reparameterization-invariance, are shown to be in 1-1 correspondence with equivalence classes of string structures on M. The identification is through the affine space of string’ cohomology classes considered by Redden.