Clustering and decomposition for non BPS solutions of the $\mathbb{CP}^{N-1}$ models

TL;DR

The paper studies non-BPS solutions of the CP^{N-1} sigma model on $\mathbb{R}\times S^1$ with twisted boundary conditions by leveraging a conformal map to $\mathbb{R}^2$, enabling a decomposition of non-holomorphic solutions into constituent partons. It introduces and analyzes the nonholomorphic generator $P_+$, clarifying how solutions transform and how the action decomposes into $R_1$ and $R_2$ terms, with detailed CP$^2$-specific structures. Through explicit two-, three-, and four-soliton examples under both periodic and twisted twists, the work reveals severe clustering pathologies and nonlocal effects of $P_+$, leading to discontinuities in the decompactification limit and a lack of a one-to-one cylinder-to-plane correspondence for non-BPS sectors. The results caution against naive radius extrapolations in non-BPS CP^{N-1} theories and motivate refined notions of convergence across radii in semi-classical analyses.

Abstract

We look at solutions (both BPS and non-BPS) of the $\mathbb{CP}^{N-1}$ model on $\mathbb{R} \times S^1$ (with twisted boundary conditions), in particular by using a conformal mapping technique, and we show how to interpret these solutions by decomposing them into expressions describing constituent solitons. We point out the problems that may arise (for non-BPS solutions) when one naively looks at the clustering properties of these solutions. This could lead to misunderstandings when studying extrapolations between small and large compactification radii.

Figures (5)

• Figure 1: Toric diagram for $\mathbb{CP}^2$.
• Figure 2: Conformal map for $\mathbb{CP}^2$ with twisted boundary conditions.
• Figure 3: Action density for $P_+ f$ with $f$ given by the two-soliton example (\ref{['examplez2']}). $P_+ f$ is a mixture of two anti-solitons located at $x_+=\pm 1$ and of two solitons located at $x_+=\pm i$. The three plots correspond, respectively, to $(\alpha,\lambda) = (.4,.5), (.01,.5), (.4,.005)$. The second plot corresponds to the $\alpha$ small case, thus very close to the embedding limit (\ref{['embeddinglimit']}). The third one corresponds to the $\lambda$ small case thus very close to the clustering limit (\ref{['clusteringlimit']}).
• Figure 4: Action density for $P_+ f$ for $f$ given by the three-soliton example in (\ref{['examplez3']}). $P_+ f$ is a mixture of three anti-solitons located at $1,\omega, \omega^2$ and three solitons located at $\rho,\rho \omega, \rho \omega^2$. The three plots correspond respectively to $(\alpha,\lambda) = (.8,.5),\, (.015,.5),\, (.8,.01)$. The second plot corresponds to the $\alpha$ small case; the third one to the $\lambda$ small case.
• Figure 5: The first line is the action density for $P_+ f$ of the $\mathbb{Z}_2$ symmetric configuration with $4$ anti-solitons plus $6$ solitons (\ref{['examplez24']}) for the values $(\alpha,\lambda) = (.2,.25),\, (.2,.025)$ and $\epsilon = 1.6 \lambda$. The second line is obtained by zooming into the right-hand cluster for the same values of $(\alpha,\lambda)$.