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.
Highlights
In order to reach deeper understanding on bions and the related physics, it is of great importance to study examples in the field theory models in low-dimensions such as CP N−1 models [9, 10, 19,20,21, 23, 24], principal chiral models [13,14,15] and quantum mechanics [15,16,17, 21]
We investigate non-BPS exact solutions in CP N−1 sigma models on R1 × S1 with twisted boundary conditions, by using the Din-Zakrzewski projection method
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
Summary
Action density s(x) and topological charge density q(x) of the CP N−1 model in euclidean 2dimensions are given in terms of a normalized row vector n(x) of an N -component complex scalar fields ω(x) as s(x). The simplest BPS solution with the topological charge Q = 1/N is the fractional instanton given by the following unnormalized N -component fields ω(x). Before discussing the non-BPS exact solutions in CP N−1 models with the ZN -twisted boundary conditions (2.4), we introduce an ansatz [19, 23, 24] for multi fractional instanton configurations that reduces to solutions of field equations for asymptotically large separations of constituent fractional instantons, and carries the correct number of moduli for each individual constituent: its phase and position in x1. We note that action density s(x) and topological charge density q(x) depend on the (relative) phase θb of two terms in a single component, but are independent of the phase θa. If we wish to consider the diagram with ordering IIIIin figure 3(b), we need to have a term f zin the third component
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