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Spectrum of CHL Dyons from Genus-Two Partition Function

Atish Dabholkar, Davide Gaiotto

TL;DR

The paper provides a genus-two, left-moving twisted partition-function computation for the Z2 CHL orbifold, showing it matches the inverse Siegel modular form Φ′6 and thus yields the exact dyon degeneracies at weak coupling via a string-web/M-theory lifting interpretation. By expressing the twisted determinants in terms of untwisted ones and demonstrating a Prym-period cancellation against the odd-charge lattice sum, it derives a concrete partition-function form that reproduces Fourier coefficients of Φ′6 and fixes normalization. The work then outlines a consistent generalization to general ZN CHL orbifolds, deriving the weight k = 24/(N+1) − 2 and arguing for theta-function representations of Φk, with caveats for certain N (e.g., N = 7). Altogether, the results connect microscopic CHL CFT data to the Siegel modular structure governing dyon degeneracies and provide a framework for theta-based realizations of CHL dyon partition functions, with implications for black hole entropy comparisons and duality-invariant counting. $k = \frac{24}{N+1}-2$ and the inverse Siegel form $Φ′_k$ encode the dyon spectrum, while Prym-period cancellation ensures modular-consistent, q-series degeneracies across CHL sectors.

Abstract

We compute the genus-two chiral partition function of the left-moving heterotic string for a $\mathbb{Z}_2$ CHL orbifold. The required twisted determinants can be evaluated explicitly in terms of the untwisted determinants and theta functions using orbifold techniques. The dependence on Prym periods cancels neatly once summation over odd charges is properly taken into account. The resulting partition function is a Siegel modular form of level two and precisely equals recently proposed dyon partition function for this model. This result provides an independent weak coupling derivation of the dyon partition function using the M-theory lift of string webs representing the dyons. We discuss generalization of this technique to general $\mathbb{Z}_N$ orbifolds.

Spectrum of CHL Dyons from Genus-Two Partition Function

TL;DR

The paper provides a genus-two, left-moving twisted partition-function computation for the Z2 CHL orbifold, showing it matches the inverse Siegel modular form Φ′6 and thus yields the exact dyon degeneracies at weak coupling via a string-web/M-theory lifting interpretation. By expressing the twisted determinants in terms of untwisted ones and demonstrating a Prym-period cancellation against the odd-charge lattice sum, it derives a concrete partition-function form that reproduces Fourier coefficients of Φ′6 and fixes normalization. The work then outlines a consistent generalization to general ZN CHL orbifolds, deriving the weight k = 24/(N+1) − 2 and arguing for theta-function representations of Φk, with caveats for certain N (e.g., N = 7). Altogether, the results connect microscopic CHL CFT data to the Siegel modular structure governing dyon degeneracies and provide a framework for theta-based realizations of CHL dyon partition functions, with implications for black hole entropy comparisons and duality-invariant counting. and the inverse Siegel form encode the dyon spectrum, while Prym-period cancellation ensures modular-consistent, q-series degeneracies across CHL sectors.

Abstract

We compute the genus-two chiral partition function of the left-moving heterotic string for a CHL orbifold. The required twisted determinants can be evaluated explicitly in terms of the untwisted determinants and theta functions using orbifold techniques. The dependence on Prym periods cancels neatly once summation over odd charges is properly taken into account. The resulting partition function is a Siegel modular form of level two and precisely equals recently proposed dyon partition function for this model. This result provides an independent weak coupling derivation of the dyon partition function using the M-theory lift of string webs representing the dyons. We discuss generalization of this technique to general orbifolds.

Paper Structure

This paper contains 8 sections, 71 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. Dyon Partition Function in CHL Orbifolds
  3. Dyon Partition Function and Genus-Two Riemann Surfaces
  4. Computation of the Twisted Chiral Partition Function
  5. Generalization to $\mathbb{Z}_N$ Orbifolds
  6. Comments
  7. Computation of the Classical Piece of the Partition Function
  8. An Outline of An Alternative Derivation

Figures (2)

  • Figure 1: A dyon can be represented as a string web on a torus which in M-theory looks like a genus two Riemann surface.
  • Figure 2: A double cover of a genus-two Riemann surface is a genus-three surface that admits an involution i.