Classifying bions in Grassmann sigma models and non-Abelian gauge theories by D-branes

Abstract

We classify bions in the Grassmann $Gr_{N_{\rm F},N_{\rm C}}$ sigma model (including the ${\mathbb C}P^{N_{\rm F}-1}$ model) on ${\mathbb R}^{1}\times S^{1}$ with twisted boundary conditions. We formulate these models as $U(N_{\rm C})$ gauge theories with $N_{\rm F}$ flavors in the fundamental representations. These theories can be promoted to supersymmetric gauge theories and further can be embedded into D-brane configurations in type II superstring theories. We focus on specific configurations composed of multiple fractional instantons, termed neutral bions and charged bions, which are identified as perturbative infrared renormalons by Ünsal and his collaborators. We show that D-brane configurations as well as the moduli matrix offer a very useful tool to classify all possible bion configurations in these models. Contrary to the ${\mathbb C}P^{N_{\rm F}-1}$ model, there exist Bogomol'nyi-Prasad-Sommerfield (BPS) fractional instantons with topological charge greater than unity (of order $N_{\rm C}$) that cannot be reduced to a composite of an instanton and fractional instantons. As a consequence, we find that the Grassmann sigma model admits neutral bions made of BPS and anti-BPS fractional instantons each of which has topological charge greater (less) than one (minus one), that are not decomposable into instanton anti-instanton and the rests. The ${\mathbb C}P^{N_{\rm F}-1}$ model is found to have no charged bions. In contrast, we find that the Grassmann sigma model admits charged bions, for which we construct exact non-BPS solutions of the field equations.