The families analytic index for $1|1$-dimensional Euclidean field theories

Abstract

We construct a $\mathrm{KO}$-valued families index for a class of $1|1$-dimensional Euclidean field theories. This realizes a conjectured cocycle map in the Stolz--Teichner program. We further show that a bundle of spin manifolds leads to a family of partially-defined $1|1$-Euclidean field theories, yielding a cocycle refinement of the families analytic index. The methods are chosen with the goal of generalizing to $2|1$-dimensional field theories, where analogous structures are expected to provide an analytic index valued in topological modular forms.