Genus Two Surface and Quarter BPS Dyons: The Contour Prescription

TL;DR

This work addresses the spectrum of quarter BPS dyons in ${\cal N}=4$ string theories by casting dyons as string networks and linking their degeneracies to a genus-two partition function. Through a chain of dualities, the authors connect complex chemical potentials $(\rho,\sigma,v)$ to the period matrix of a genus-two surface embedded in $T^4$, with the imaginary parts fixed by a chamber-contour prescription that governs marginal stability walls. Although an explicit computation of the full partition function is not completed, the paper clarifies how the inverse Igusa cusp form $1/\Phi_{10}$ arises and how the contour rules emerge from genus-two geometry, offering a physical origin for the moduli-dependent integration contours. The framework provides a path to extend the analysis to other BPS sectors and highlights genus-two data as central to encoding dyon spectra in ${\cal N}=4$ theories.

Abstract

Following the suggestion of hep-th/0506249 and hep-th/0612011, we represent quarter BPS dyons in N=4 supersymmetric string theories as string network configuration and explore the role of genus two surfaces in determining the spectrum of such dyons. Our analysis leads to the correct contour prescription for integrating the partition function to determine the spectrum in different domains of the moduli space separated by the walls of marginal stability.

Figures (1)

• Figure 1: The string network configuration.