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Walls of Marginal Stability and Dyon Spectrum in N=4 Supersymmetric String Theories

Ashoke Sen

TL;DR

The paper addresses how the exact degeneracies $d(Q,P)$ of quarter-BPS dyons in $N=4$ string theories change when crossing walls of marginal stability. It adopts a contour-integral approach on the Siegel upper half-plane with a modular-form denominator and analyzes how dualities map between moduli regions, revealing that spectrum changes across walls arise solely from contour deformations rather than changes in the integrand. A key finding is that, at large charges, wall-crossing contributions are exponentially suppressed, consistent with the attractor mechanism governing black hole entropy. The work also provides concrete tests of $S$-duality invariance by demonstrating duality-consistent contour mapping for certain $N$ and discusses limitations for larger $N$, thereby reinforcing the duality structure of the dyon spectrum in these theories.

Abstract

The spectrum of quarter BPS dyons in N=4 supersymmetric string theories can change as the asymptotic moduli cross walls of marginal stability on which the dyon can break apart into a pair of half BPS states. In this paper we classify these marginal stability walls and examine this phenomenon in the context of exact dyon spectrum found in a class of N=4 supersymmetric string theories. We argue that the dyon partition functions in different domains separated by marginal stability walls are the same, but the choice of integration contour needed for extracting the degeneracies from the partition function differ in these different regions. We also find that in the limit of large charges the change in the degeneracy is exponentially suppressed compared to the leading contribution. This is consistent with the fact that in the computation of black hole entropy we do not encounter any change as the asymptotic moduli fields move across the walls of marginal stability. Finally we carry out some tests of S-duality invariance in the theory.

Walls of Marginal Stability and Dyon Spectrum in N=4 Supersymmetric String Theories

TL;DR

The paper addresses how the exact degeneracies of quarter-BPS dyons in string theories change when crossing walls of marginal stability. It adopts a contour-integral approach on the Siegel upper half-plane with a modular-form denominator and analyzes how dualities map between moduli regions, revealing that spectrum changes across walls arise solely from contour deformations rather than changes in the integrand. A key finding is that, at large charges, wall-crossing contributions are exponentially suppressed, consistent with the attractor mechanism governing black hole entropy. The work also provides concrete tests of -duality invariance by demonstrating duality-consistent contour mapping for certain and discusses limitations for larger , thereby reinforcing the duality structure of the dyon spectrum in these theories.

Abstract

The spectrum of quarter BPS dyons in N=4 supersymmetric string theories can change as the asymptotic moduli cross walls of marginal stability on which the dyon can break apart into a pair of half BPS states. In this paper we classify these marginal stability walls and examine this phenomenon in the context of exact dyon spectrum found in a class of N=4 supersymmetric string theories. We argue that the dyon partition functions in different domains separated by marginal stability walls are the same, but the choice of integration contour needed for extracting the degeneracies from the partition function differ in these different regions. We also find that in the limit of large charges the change in the degeneracy is exponentially suppressed compared to the leading contribution. This is consistent with the fact that in the computation of black hole entropy we do not encounter any change as the asymptotic moduli fields move across the walls of marginal stability. Finally we carry out some tests of S-duality invariance in the theory.

Paper Structure

This paper contains 6 sections, 125 equations, 1 figure.

Table of Contents

  1. Introduction and Summary
  2. Dyon Spectrum in a Class of ${\cal N}=4$ Supersymmetric Models
  3. Walls of Marginal Stability
  4. Duality Transformation of the Degeneracy Formula
  5. The Large Charge Limit
  6. Test of S-duality Invariance

Figures (1)

  • Figure 1: A schematic diagram representing the domain ${\cal R}$ in the upper half $\tau_\infty$ plane, bounded by the walls of marginal stability, for ${\hbox{Z Z}}_1$, ${\hbox{Z Z}}_2$ and ${\hbox{Z Z}}_3$ orbifolds. The shapes of the circles and the slopes of the straight lines bordering the domain depend on the charges and other asymptotic moduli, but the vertices are universal.