Abstract. Geometric engineering in M-theory predicts a correspondence between the index of certain 5d SCFTs on Taub-NUT spaces with n centers, and the index of 7d U(n) gauge theories on certain Calabi Yau three-folds, also known as higher equivariant Donaldson-Thomas theory. In this talk we will discuss some aspects of such correspondence as well as some possible subtleties and generalisations to other spaces and dimensions of the theories involved.