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

TL;DR

The paper investigates topologically trivial configurations, namely bions, in Grassmann sigma models on ${\mathbb R}^{1}\times S^{1}$ with ${\mathbb Z}_{N_{ m F}}$ twists, aiming to clarify their role in resurgence and infrared renormalons.By formulating these models as $U(N_{ m C})$ gauge theories and employing the moduli-matrix method, supplemented by string-theoretic D-brane realizations, the authors classify fractional instantons and their bound states in both ${\mathbb C}P^{N_{ m F}-1}$ and Grassmann settings.A key result is that Grassmann models admit neutral bions built from BPS and anti-BPS fractional instantons with charges that can exceed unity and resist decomposition, and importantly, Grassmann models also admit charged bions, unlike ${\mathbb C}P^{N_{ m F}-1}$, with exact non-BPS charged-bion solutions constructed.The work further computes energy and topological-charge densities for representative configurations, analyzes inter-bion interactions showing exponential separation dependence, and demonstrates that brane reconnection and pair-creation pictures capture the full spectrum of irreducible bions.Together, these findings deepen understanding of nonperturbative dynamics and offer a concrete bridge between gauge theories, sigma models, and string-theoretic brane constructions, with potential implications for resurgence-based approaches to quantum field theories.

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.

Figures (36)

• Figure 1: Brane configuration for $k$ vortices. As for separation of the two NS5-branes, $\Delta x^3$ corresponds to $1/g^2$ and $(\Delta x^4,\Delta x^5,\Delta x^6)$ correspond to the triplet of the FI parameters.
• Figure 2: vacuum configuration $(k=0)$
• Figure 3: T-dualized configurations for $k=1$: the $\rm D2'$ brane is assumed to be located at $x^1=x_{0}$.
• Figure 5: T-dual picture for $N_{\rm F}=2, N_{\rm C}=1, k=1$.
• Figure 6: Brane picture of the Seiberg dual for kinks, where the presence and absence of color branes is exchanged. Here we give an example of the duality between $Gr_{3,2}$ and $Gr_{3,1} \simeq {\mathbb C}P^2$. The horizontal and vertical directions are $x_1$ and $x_2$, respectively.