Marginal deformations of SU(2)$_k$ WZW model boundary states in open string field theory

TL;DR

This work investigates the moduli space of Cardy boundary states in SU(2)$_k$ WZW models by constructing marginally deformed OSFT solutions within level truncation. The approach blends marginal and relevant deformations by turning on a marginal coefficient $\lambda_S$ for the $J^3$ current while partially fixing SU(2) symmetry, yielding continuous families of solutions labeled by the boundary angle $\theta$ via the BCFT parameter $\lambda_B$. Across $k=2,3,4$, the authors identify branches that describe deformed boundary states but find they do not cover the entire moduli space; each branch terminates at finite $\lambda_S$ with inconsistent behavior beyond, and the relation between $\lambda_B$ and $\lambda_S$ shows signs of universality within solution groups. The results indicate that level truncation OSFT can capture substantial but incomplete segments of the boundary-state moduli space, with the coverage decreasing as $k$ grows and seed choices constrain the accessible region. The findings also reveal intriguing connections between marginal OSFT data and BCFT deformations, including a roughly universal linear response for small $\lambda_S$ and proportionality relations between 0-brane and B-brane sectors, offering guidance for future investigations of D-brane moduli via OSFT.

Abstract

We attempt to describe the moduli space of boundary states in the SU(2)$_k$ WZW model by constructing marginally deformed solutions in open string field theory in the level truncation approximation. In contrast with other approaches to marginal deformations, our solutions exhibit a $g$-function different from that of the background (typically lower). Thus, our method effectively combines features of both marginal and relevant deformations. After partially fixing an SU(2) symmetry of the equations of motion, we find families of solutions parameterized by the coefficient of the marginal field associated with the $J^3$ current, and we identify them as Cardy boundary states with varying angle $θ$. However, it turns out that these solutions become inconsistent once the marginal parameter exceeds a certain value, implying that they cover only a part of the moduli space. Finally, we also compare the relation between the marginal parameter and the angle $θ$ for different solutions and we find evidence suggesting that this relation is universal for certain classes of solutions.

Figures (17)

• Figure 1.1: Visualizations of moduli space coverage by consistent 0-brane solutions in SU(2)$_k$ WZW models with $k=2$ (top left), $k=3$ (top right) and $k=4$ (bottom). The initial D-branes are denoted by black lines, the seed 0-brane solutions by red dots and the covered parts of moduli space by red arcs. The lengths of the arcs are based on the results from sections \ref{['sec:k=2']}, \ref{['sec:k=3']} and \ref{['sec:k=4']}. We were able to determine the maximal values of $\lambda_{\rm B}$ only with a limited precision, so the covered area may be somewhat larger or smaller. In case of one of the $k=4$ solutions (denoted by the dotted curves), we do not have a good estimate of maximal $\lambda_{\rm B}$, so this part of the figure is only schematic.
• Figure 3.1: Plots of $\lambda_{\rm S}$-dependence of the energy (computed independently as $E_{tot}$, $E_{kin}$ and $E_{0,0}$) and the out-of-Siegel equations of the 0-brane branch of solutions in the $k=2$ model. Levels are distinguished by colors following the rainbow spectrum from red to purple and infinite level extrapolations are denoted by black lines.
• Figure 3.2: Plots of $\lambda_{\rm S}$-dependence of invariants $E_{1,0}$ and $J_{33}$ and absolute values of invariants $E_{1/2,1/2}$ and $E_{1,1}$ of the 0-brane solution in the $k=2$ model. The figures have the same style as in figure \ref{['fig:k=2 energy']}. These observables should be constant and their expected values are given by positions of the horizontal axes.
• Figure 3.3: Plot of the ratio $\sigma/\sigma_{est}$ for several observables of the marginal branch solutions in the free boson theory (based on data from KudrnaThesis). The four curves represent $E_{tot}$ (red), $E_{kin}$ (green), $E_{0}$ invariant (blue) and $\Delta_S$ (magenta). The black vertical line denotes the estimated point where the branch goes off-shell $\lambda_{\rm S}^\ast=0.392$.
• Figure 3.5: Plots of $\lambda_{\rm S}$-dependence of real and imaginary parts of invariants $E_{1/2,1/2}$, $E_{1,1}$ and $J_{+-}$ of the 0-brane solution in the $k=2$ model. The figures have the same style as in figure \ref{['fig:k=2 energy']}, the expected behavior (based on a fit of the phase of $E_{1/2,1/2}$ invariant) is denoted by dashed black lines.