Playing With the Index of M-Theory

Michele Del Zotto; Nikita Nekrasov; Nicolò Piazzalunga; Maxim Zabzine

doi:10.1007/s00220-022-04479-7

Playing With the Index of M-Theory

Commun Math Phys. 2022;396(2):817-865. doi: 10.1007/s00220-022-04479-7. Epub 2022 Aug 23.

Authors

Michele Del Zotto^{1

2}, Nikita Nekrasov^{3

4

5}, Nicolò Piazzalunga², Maxim Zabzine²

Affiliations

¹ Mathematics Institute, Uppsala University, Box 480, SE-75106 Uppsala, Sweden.
² Department of Physics and Astronomy, Uppsala University, Box 516, SE-75120 Uppsala, Sweden.
³ Simons Center for Geometry and Physics, Stony Brook University, Stony Brook, NY 11794-3636 USA.
⁴ Center for Advanced Studies, Skoltech, Moscow, Russia.
⁵ Kharkevich Institute for Information Transmission Problems, Moscow, Russia.

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.