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The importance of the fusion relation of loops was recognized in the contextof spin structures on the loop space by Stolz and Teichner and furtherdeveloped by Waldorf. On a spin manifold M the equivalence classes of `fusive'spin structures on the loop space LM, incorporating the fusion property, strongregularity and reparameterization-invariance, are shown to be in 1-1correspondence with equivalence classes of string structures on M. Theidentification is through the affine space of `string' cohomology classesconsidered by Redden.

Equivalence of string and fusion loop-spin structures