String Webs and 1/4 BPS Monopoles

TL;DR

Bergman and Kol construct 1/4 BPS states in $N=4$ $SU(N_c)$ SYM from supersymmetric string webs ending on D3-branes, showing web masses match the BPS bound and identifying marginal stability curves where these states decay to BPS constituents. They systematically count bosonic and fermionic zero modes to infer degeneracy and spin multiplets, revealing towers of high-spin states that become massless at the conformal point. The work derives central charges and mass formulas from the brane geometry, classifies 1/4 BPS states via $SL(2,\mathbb{Z})$ invariants, and analyzes how moduli variation can render states long BPS or non-BPS, including extremal stable non-BPS configurations. The results imply rich non-perturbative spectra in 4d $N=4$ SYM with geometric and duality-inspired structure, and suggest broader geometric analogs in string/M-theory realizations.

Abstract

We argue for the existence of many new 1/4 BPS states in N=4 SU(N_c) Super-Yang-Mills theory with N_c>=3, by constructing them from supersymmetric string webs whose external strings terminate on parallel D3-branes. The masses of the string webs are shown to agree with the BPS bound for the corresponding states in SYM. We identify the curves of marginal stability, at which these states decay into other BPS states. We find the bosonic and fermionic zero modes of the string webs, and thereby the degeneracy and spin content of some of the BPS states. States of arbitrarily high spin are predicted in this manner, all of which become massless at the conformal point. For N_c>=4 we find BPS states which transform in long multiplets, and are therefore not protected against becoming stable non-BPS states as moduli are varied. The mass of these extremal non-BPS states is constrained as they are connected to BPS states. Analogous geometric phenomena are anticipated.

Figures (5)

• Figure 1: Examples of simple string webs: (a) a 3-string junction, (b) a web with an internal string, (c) a web with an internal face, (d) another web with an internal string.
• Figure 2: Grid diagrams dual to the webs in figure 1. Note that (b) is a quadrangle.
• Figure 3: Grid diagrams with $E_{ext}=3, q=3$ and varying shapes. The cases $p=0,2$ are excluded as they contain new edge points, and therefore really have $E_{ext}>3$.
• Figure 4: 1/4 BPS states from string webs: (a) completely degenerate web, (b) web without internal faces, (c) generic web.
• Figure 5: Curves of marginal stability and decay of 1/4 BPS states. (a) 3-string junction decays into (b) two strings; (c) string web of fig. 1d decays first into (b) a string and a 3-string junction, which proceeds to deacy into (c) two more strings.