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 [G. V. Dunne and M. Ünsal, J. High Energy Phys. 1211, 170 (2012); G. V. Dunne and M. Ünsal, Phys. Rev. D 87, 025015 (2013)]. 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. In contrast to the |${\mathbb C}P^{N_{\rm F}-1}$| model, there exist Bogomol’nyi–Prasad–Sommerfield (BPS) fractional instantons with topological charges 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 a topological charge greater (less) than one (minus one), that are not decomposable into an instanton–anti-instanton pair and the rest. 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.