Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

A Geometric Derivation of the Dyon Wall-Crossing Group

Miranda C. N. Cheng, Lotte Hollands

TL;DR

This work geometrically derives the dyon wall-crossing group for ${\cal N}=4$, $d=4$ compactifications by modeling 1/4-BPS dyons as a supersymmetric network of $(p,q)$ strings and K3-wrapped five-branes, and then lifting to Euclidean M-theory where the dyons correspond to a genus-two Riemann surface embedded in $T^4$. The authors show that crossing walls of marginal stability corresponds to degenerations of this surface, which induce basis changes in the homology that act as Weyl reflections, thereby generating a non-compact hyperbolic reflection group $W$; the full symmetry is captured by the semi-direct product $W\rtimes D_3 = PGL(2,\mathbb Z)$. A key result is that the moduli space can be interpreted as the dual graph of the five-brane network, with each region labeled by an effective charge pair $(Q_v,P_v)$ and a corresponding BPS index $D_v$, all computed from the same partition function. The framework provides a geometric foundation for the observed Weyl-group structure and its connection to a Borcherds–Kac–Moody algebra, while outlining limitations and avenues for extending to CHL models and attractor-flow interpretations.

Abstract

Recently, using supergravity analysis, a hyperbolic reflection group was found to underlie the structure of wall-crossing, or the discontinuous moduli dependence of the supersymmetric index due to the presence of walls of marginal stability, of the BPS dyons in the N=4, d=4 compactification. In this paper we work in the regime where four-dimensional gravity decouples and we show how the presence of such a group structure can be easily understood as a consequence of the supersymmetry of a system of (p,q) five-brane network, or equivalently the holomorphicity of the Riemann surface wrapped by the appropriate M5 branes in the Euclidean M-theory frame.

A Geometric Derivation of the Dyon Wall-Crossing Group

TL;DR

This work geometrically derives the dyon wall-crossing group for , compactifications by modeling 1/4-BPS dyons as a supersymmetric network of strings and K3-wrapped five-branes, and then lifting to Euclidean M-theory where the dyons correspond to a genus-two Riemann surface embedded in . The authors show that crossing walls of marginal stability corresponds to degenerations of this surface, which induce basis changes in the homology that act as Weyl reflections, thereby generating a non-compact hyperbolic reflection group ; the full symmetry is captured by the semi-direct product . A key result is that the moduli space can be interpreted as the dual graph of the five-brane network, with each region labeled by an effective charge pair and a corresponding BPS index , all computed from the same partition function. The framework provides a geometric foundation for the observed Weyl-group structure and its connection to a Borcherds–Kac–Moody algebra, while outlining limitations and avenues for extending to CHL models and attractor-flow interpretations.

Abstract

Recently, using supergravity analysis, a hyperbolic reflection group was found to underlie the structure of wall-crossing, or the discontinuous moduli dependence of the supersymmetric index due to the presence of walls of marginal stability, of the BPS dyons in the N=4, d=4 compactification. In this paper we work in the regime where four-dimensional gravity decouples and we show how the presence of such a group structure can be easily understood as a consequence of the supersymmetry of a system of (p,q) five-brane network, or equivalently the holomorphicity of the Riemann surface wrapped by the appropriate M5 branes in the Euclidean M-theory frame.

Paper Structure

This paper contains 8 sections, 83 equations, 5 figures.

Table of Contents

  1. Introduction
  2. The Five-brane Network
  3. The Riemann Surface
  4. Deriving the Group of Discrete Attractor Flow
  5. The First Degeneration
  6. The Symmetry of the Weyl Chamber
  7. Moduli Space as the Dual Graph
  8. Discussion

Figures (5)

  • Figure 1: Two examples, described in (\ref{['network_1']}) and (\ref{['network_2']}), of the effective string network with $(q_1,q_2)=(1,0)$ and $(p_1,p_2)=(0,1)$. As discussed in (\ref{['length_general']}), depending on the moduli, these networks may or may not be realized.
  • Figure 2: The degeneration of the genus two surface described in (\ref{['degenerate_area']}).
  • Figure 3: In section \ref{['The First Degeneration']} we study the change of the Riemann surface $\varSigma$ when its period matrix changes as (\ref{['change_1']}) following the above path, where $\epsilon\to 0_+$ and $\rho$ and $\sigma$ are held fixed at values satisfying $\mathrm{Im}\rho\,\mathrm{Im}\sigma\gg (\mathrm{Im}\nu_0)^2$ .
  • Figure 4: Hyperelliptic representation of the genus two surface $\varSigma$ together with a choice of its $A_i$ and $B_i$-cycles. A degeneration corresponding to the one shown in Fig \ref{['genus2degenerating']} corresponds to coalescing the branch points $b_1$, $b_2$ and $b_3$. Note that when we set the background two-form fields $B$ and $C$ along the timelike direction to zero, so that $\mathrm{Re}\Omega=0$ (\ref{['re_omega']}), all branch points are colinear.
  • Figure 5: (i) Different possible ways of compactifying the periodic network on a torus. (ii) The moduli space as the dual graph of the five-brane network.