Global magni$4$icence, or: 4G Networks
Nikita Nekrasov, Nicolo Piazzalunga
TL;DR
The work extends DT4/K-theoretic counting to toric Calabi–Yau fourfolds by constructing a global, four-valent vertex formalism that localizes the 8+1D gauge theory, counting D-brane bound states via equivariant localization. It introduces solid/plane/4D partitions, Euler-characteristic-type indices, and square-root prescriptions to define a covariant vertex, then couples these with edge and face contributions to form a full partition function. The main contributions are the explicit vertex construction, the derivation of D0–D8 (and related) charges from local geometry, and the demonstration of how perturbative factors separate from instanton data, paving the way for a complete global DT4 framework and potential connections to M-theory and Gromov–Witten/DT-type correspondences. The results provide a combinatorial, residue-based path to compute twisted Witten indices of high-dimensional gauge theories and offer a foundation for verifying M-theory predictions in higher-dimensional Calabi–Yau backgrounds.
Abstract
The global magnificent four theory is the homological version of a maximally supersymmetric $(8+1)$-dimensional gauge theory on a Calabi-Yau fourfold fibered over a circle. In the case of a toric fourfold we conjecture the formula for its twisted Witten index. String-theoretically we count the BPS states of a system of $D0$-$D2$-$D4$-$D6$-$D8$-branes on the Calabi-Yau fourfold in the presence of a large Neveu-Schwarz $B$-field. Mathematically, we develop the equivariant $K$-theoretic DT4 theory, by constructing the four-valent vertex with generic plane partition asymptotics. Physically, the vertex is a supersymmetric localization of a non-commutative gauge theory in $8+1$ dimensions.