Notes on Soliton Bound-State Problems in Gauge Theory and String Theory

TL;DR

This work presents a unified $L^2$-index framework to count threshold BPS soliton bound-states predicted by string/gauge dualities across IIA/M and IIB theories. By decomposing the index into a bulk instanton contribution and a boundary term from asymptotic moduli-space regions, it relates gauge theory solitons, ADHM/Nahm instanton moduli spaces, and D-brane instanton calculus to exact counting results; duality then fixes the exact values of the index in each case. Across the IIA, IIB', IIA', and IIB examples, the bulk contributions reproduce instanton/Wilsonian actions, while the boundary terms subtract the spurious multi-winding sectors, yielding a single threshold bound-state for coprime charge sectors, in agreement with Sen’s conjecture and M/M-duality expectations. The analysis connects nonperturbative spectra to geometric properties of moduli spaces (e.g., Euler characteristics) and to exact higher-derivative actions in compactified theories, providing a cohesive nonperturbative picture of BPS bound-states in string theory and its gauge theory realizations.

Abstract

We review four basic examples where string theory and/or field theory dualities predict the existence of soliton bound-states. These include the existence of threshold bound-states of D0 branes required by IIA/M duality and the closely-related bound-states of instantons in the maximally supersymmetric five dimensional gauge theory. In the IIB theory we discuss (p,q)-strings as bound-states of D and F strings, as well as the corresponding bound-states of monopoles and dyons in N=4 supersymmetric Yang-Mills theory whose existence was predicted by Sen. In particular we consider the L^2-index theory relevant for counting these states. In each case we show that the bulk contribution to the index can be evaluated by relating it to an instanton effect in the corresponding theory with a compact Euclidean time dimension. The boundary contribution to the index can be determined by considering the asymptotic regions of the relevant moduli space.