Playing with the index of M-theory

TL;DR

This paper explores the extension of Donaldson-Thomas theory in M-theory contexts, demonstrating how partition functions factorize and relate to 5d theories on ALE spaces, with implications for instanton counting.

Contribution

It introduces a rank n K-theoretic Donaldson-Thomas framework on toric threefolds, extending geometric engineering to higher ranks and including D4-brane contributions.

Findings

Partition function factorization under certain assumptions

Reproduction of 5d master formula for ADE-ruling Calabi-Yau threefolds

Implications for instanton counting on Taub-NUT spaces

Abstract

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.