The Index of M-theory – I

Motivated by M-theory physics we will discuss a conjectural curve-counting theory in Calabi–Yau 5-folds and its relation to K-theoretic Donaldson–Thomas invariants of smooth 3-folds. Specifically, if a Calabi–Yau 5-fold Z admits an automorphism $q$ with 3-dimensinal fixed locus $X$, then the trace of $q$ on K-theoretic curve counts in $Z$ is written in terms of DT invariants of $X$ with boxcounting parameter $q$.