Non-BPS exact solutions and their relation to bions in ${\mathbb C}P^{N-1}$ models

TL;DR

This work characterizes non-BPS exact solutions in the ${\mathbb C}P^{N-1}$ model on ${\mathbb R}^1\times S^1$ with ${\mathbb Z}_{N}$ twists, using the Din–Zakrzewski projection to relate them to multi-bion configurations that drive resurgence. It demonstrates that non-BPS solutions are contained as subspaces of a general two-bion ansatz, with balance of many-body forces governed by relative phases that yield three essential properties: opposite left-right contributions, reflection symmetry, and flipping-partner transitions. A diagrammatic construction reveals a generic pattern for flipping partners across higher N, while detailed stability analyses uncover local and global negative modes and the role of deformations in connecting to other saddles. The paper also shows how similar non-BPS structures arise in Grassmann sigma models, signaling broader applicability and potential links to non-BPS sectors in Yang–Mills via vortex correspondences, thereby enriching the non-perturbative landscape in low-dimensional gauge theories.

Abstract

We investigate non-BPS exact solutions in ${\mathbb C}P^{N-1}$ sigma models on ${\mathbb R}^1 \times S^{1}$ with twisted boundary conditions, by using the Din-Zakrzewski projection method. We focus on the relation of the non-BPS solutions to the ansatz of multi-instanton (bion) configurations and discuss their significance in the context of the resurgence theory. We find that the transition between seemingly distinct configurations of multi-instantons occur as moduli changes in the non-BPS solutions, and the simplest non-BPS exact solution corresponds to multi-bion configurations with fully-compressed double fractional instantons in the middle. It indicates that the non-BPS solutions make small but nonzero contribution to the resurgent trans-series as special cases of the multi-bion configurations. We observe a generic pattern of transitions between distinct multi-bion configurations (flipping partners), leading to the three essential properties of the non-BPS exact solution: (i) opposite sign for terms corresponding to the left and right infinities, (ii) symmetric location of fractional instantons, and (iii) the transition between distinct bion configurations. By studying the balance of forces, we show that the relative phases between the instanton constituents play decisive roles in stability and instability of the muli-instanton configurations. We discuss local and global instabilities of the solutions such as negative modes and the flow to the other saddle points, by considering the deformations of the non-BPS exact solutions within our multi-instanton ansatz. We also briefly discuss some classes of the non-BPS exact solutions in Grassmann sigma models.

Figures (34)

• Figure 1: An example of BPS fractional instantons in the ${\mathbb C}P^{N-1}$ model. Black lines stand for some of $N$ discrete vacua while a kinky red line indicates a transition between different vacua.
• Figure 2: An example of bion configurations in the ${\mathbb C}P^{N-1}$ model. For simplicity, we depict only four vacua lines among $N$ here.
• Figure 3: Generic two-bion configurations in Eq. (\ref{['eq:2bion']}).
• Figure 4: BPS solution $\omega_{\cal I I}$ of two fractional instantons in Eq. (\ref{['bps']}) in ${\mathbb C}P^{2}$ model.
• Figure 5: The non-BPS exact solution in ${\mathbb C}P^{2}$ model makes transition between these two configurations (flipping parteners) as a moduli $2l_{1}/l_{2}^2$ varies.