A Study of Wall-Crossing: Flavored Kinks in D=2 QED

TL;DR

This work analyzes the BPS spectrum and wall-crossing of a 2D $ abla$N$ abla$=(2,2) QED with $N+1$ massive chiral multiplets, focusing on flavored kinks in the IR CP$^N$ model with twisted masses. It develops a comprehensive framework—via moduli-space dynamics, the CFIV index, and explicit BPS equations—to construct simple and composite flavored kinks, count their degeneracies, and track wall-crossing in the weak-coupling regime. The authors compare the 2D flavored-kink spectrum to the 4D ${ m SU}(N+1)$ Seiberg-Witten theory with flavors, finding that while individual dyons differ, summing over compatible dyons reproduces the 2D flavored-kink index, hinting at a deeper 2D–4D correspondence. The results illuminate how flavor charges organize BPS spectra and provide a controlled setting to study wall-crossing phenomena and their connections to open-string web analogies and tt$^*$-GMN structures.

Abstract

We study spectrum of D=2 N=(2,2) QED with N+1 massive charged chiral multiplets, with care given to precise supermultiplet countings. In the infrared the theory flows to CP^N model with twisted masses, where we construct generic flavored kink solitons for the large mass regime, and study their quantum degeneracies. These kinks are qualitatively different and far more numerous than those of small mass regime, with features reminiscent of multi-pronged (p,q) string web, complete with the wall-crossing behavior. It has been also conjectured that spectrum of this theory is equivalent to the hypermultiplet spectrum of a certain D=4 Seiberg-Witten theory. We find that the correspondence actually extends beyond hypermultiplets in D=4, and that many of the relevant indices match. However, a D=2 BPS state is typically mapped to several different kind of dyons whose individual supermultiplets are rather complicated; the match of index comes about only after summing over indices of these different dyons. We note general wall-crossing behavior of flavored BPS kink states, and compare it to those of D=4 dyons.

Figures (5)

• Figure 2.1: Configuration of the GLSM field $\sigma$. It implies that the system is placed in $\sigma=m_1$ vacuum at ${\bf x}^3 = - \infty$, and in $\sigma=m_2$ vacuum at ${\bf x}^3 = + \infty$. The size of the plateau near $m_1$ is determined by how far 10-kink and 21-kink are separated, which is in turn determined by certain ratio between $\zeta^1$ and $\zeta^2$.
• Figure 3.1: For a simple flavor kink, the mass parameter $\vec{m}_{10}$ can be decomposed into arbitrary two orthogonal vectors ${\vec{m}}_\text{M}$ and ${\vec{m}}_\text{E}$. For (a), $m_\text{M}$ lies on the right hand side of $m_{10}$ while for (b) $m_\text{M}$ lies on the left hand side of $m_{10}$.
• Figure 3.2: (a) Schematic diagram for decomposition of the mass parameter $\vec{m}_{10}$. Let us denote the relative angle between two mass parameters $m_{10}$ and $m_{20}$ by $\theta$. By definition, $\vec{m}_{\text{M}}$ is parallel to $\vec{m}_{20}$. We are considering cases where $|\vec{m}_{\text{M}}|<| m_{20}|$. (b) Each node denotes the vacuum of the theory, i.e., grey for $\sigma=m_0$, red for $\sigma=m_1$ and green for $\sigma=m_2$. The solid lines schematically describe the GLSM $\sigma$ field. It somehow parallels with the four-dimensional picture of pronged strings where each node represents the D3-brane and solid line denotes the (p,q)-string. In section 5, the parallel between $D=2$ sigma models and $D=4$ gauge theories will be discussed in more details.
• Figure 4.1: The profiles of attractive scalar potential in the moduli space dynamics of two-kinks system, induced by tension of composite kinks.
• Figure B.1: The profiles of the effective potentials (a) $V^{(1)}_\pm({\bf x}^3)$ and (b) $V^{(2)}({\bf x}^3)$ in the case of $|m_{20}|>|m_{10}|$.