Canonical Metric Connections Associated to String Structures

In this thesis, we look at principal $Spin(n)$-bundles $P o M$ whose Pontryagin class $phalf (P)=0 in H4(M;R)$. We say that such a bundle admits a string structure, and a choice of string structure is given by particular elements in $H3(P;)$. This provides an analogue to the idea of a spin structure, where the topological group $String(n)$ is, up to homotopy, the unique 3-connected cover of $Spin(n)$ $(ngeq 5)$. Choosing a Riemannian metric on the base $M$ and a connection on $P$ determines a 1-parameter family of metrics on $P$. We prove that in a scaling limit known as the adiabatic limit, the harmonic representative of a string structure is equal to the Chern–Simons 3-form on $P$ minus a 3-form on $M$, denoted $Hin Omega3(M)$. This 3-form $H$ is closely related to the Chern–Simons form, and can be thought of as a reduction of the Chern–Simons form on $P$ to a form on $M$. The exterior derivative of $H$ is the $phalf$-form; the integral of $H$ on any 3-cycle is equal, modulo $$, to the integral of the Chern–Simons 3-form pulled back via a global section on the same 3-cycle. Finally, we note that for the $Spin(n)$-frame bundle $Spin(M)$ of a Riemannian spin manifold, this 3-form $H$ determines a canonical metric connection on $M$. This canonical metric connection depends on both the metric and string structure, and its torsion is determined by $H$.