String structures and canonical 3-forms

Details String structures and canonical 3-forms String structures and canonical 3-forms

2009-12-10

arxiv.org/abs/0912.2086v2

Pacific J. Math., 249(2):447-484, 2011

Mathematics

Differential Geometry

32 pages; v.2 minor typos corrected

Scientific paper

10.2140/pjm.2011.249.447

Using basic homotopy constructions, we show that isomorphism classes of string structures on spin bundles are naturally given by certain degree 3 cohomology classes, which we call string classes, on the total space of the bundle. Using a Hodge isomorphism, we then show that the harmonic representative of a string class gives rise to a canonical 3-form on the base space, refining the associated differential character. We explicitly calculate this 3-form for homogeneous metrics on 3-spheres, and we discuss how the cohomology theory tmf could potentially encode obstructions to positive Ricci curvature metrics.

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