Playing With the Index of M-Theory

Motivated by M-theory, we study rank n K-theoretic Donaldson–Thomas theory on a toric threefold X. In the presence of compact four-cycles, we discuss how to include the contribution of D4-branes wrapping them. Combining this with a simple assumption on the (in)dependence on Coulomb moduli in the 7d theory, we show that the partition function factorizes and, when X is Calabi–Yau and it admits an ADE ruling, it reproduces the 5d master formula for the geometrically engineered theory on A_n-1 ALE space, thus extending the usual geometric engineering dictionary to n> 1. We finally speculate about implications for instanton counting on Taub-NUT.