Global magni$4$icence, or: 4G Networks

TL;DR

This paper explores the mathematical and physical aspects of a homological gauge theory on Calabi-Yau fourfolds, proposing formulas for the twisted Witten index and developing equivariant K-theoretic DT4 theory with a four-valent vertex.

Contribution

It introduces a conjecture for the twisted Witten index on toric fourfolds and constructs a four-valent vertex in equivariant K-theoretic DT4 theory, linking geometry and string theory.

Findings

Conjectured formula for the twisted Witten index on toric fourfolds.

Construction of the four-valent vertex with plane partition asymptotics.

Development of equivariant K-theoretic DT4 theory and its physical interpretation.

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.