From 0-Branes to Torons

TL;DR

The paper investigates the moduli space of torons in $U(n)$ Super Yang-Mills theory on $T^4$, predicting it from the moduli space of $p$ zero-branes via U-duality. It identifies the 0-brane count with the greatest common divisor $p = \gcd(n, \nu, n_{\mu\nu})$ of the rank, the instanton number $\\nu$, and the magnetic fluxes $n_{\\mu\\nu}$, and provides explicit checks through analysis of $U(n)$ bundles with nonzero first and second Chern classes. The authors show that, for fixed flux data, the toron moduli space is the symmetric product $(T^4)^p / S_p$, and they extend the framework to configurations preserving eight supersymmetries and to $T^2$ and $T^3$, with dualities interpreted as Lorentz boosts in M-theory along the eleventh dimension. They discuss implications for the spectrum of BPS bound states and connections to Matrix theory. Overall, the work clarifies how brane-bound-state physics constrains toron moduli spaces across dimensions.

Abstract

The moduli space of 0-branes on $T^4$ gives a prediction for the moduli space of torons in U(n) Super Yang Mills theory which preserve 16 supersymmetries. The zero brane number corresponds to the greatest common denominator of the rank $n$, magnetic fluxes and the instanton number. This prediction is derived using U-duality. We explicitly check this prediction by analyzing U(n) bundles with non-zero first as well as second Chern classes. The argument is extended to deduce the moduli space of torons which preserve 8 supersymmetries. Parts of the discussion extend naturally to $T^2$ and $T^3$. Some of the U-dualities involved are related to Lorentz boosts along the eleventh direction in M theory.