Communications in Number Theory and Physics

Volume 14 (2020)

Number 4

CHL Calabi–Yau threefolds: curve counting, Mathieu moonshine and Siegel modular forms

Pages: 785 – 862



Jim Bryan (Department of Mathematics, University of British Columbia, Vancouver, B.C., Canada)

Georg Oberdieck (Mathematisches Institut, Universität Bonn, Germany)


A CHL model is the quotient of $\mathrm{K}3 \times E$ by an order $N$ automorphism which acts symplectically on the $\mathrm{K}3$ surface and acts by shifting by an $N$-torsion point on the elliptic curve $E$. We conjecture that the primitive Donaldson–Thomas partition function of elliptic CHL models is a Siegel modular form, namely the Borcherds lift of the corresponding twisted-twined elliptic genera which appear in Mathieu moonshine. The conjecture matches predictions of string theory by David, Jatkar and Sen. We use the topological vertex to prove several base cases of the conjecture. Via a degeneration to $\mathrm{K}3 \times \mathbb{P}^1$ we also express the DT partition functions as a twisted trace of an operator on Fock space. This yields further computational evidence. An extension of the conjecture to nongeometric CHL models is discussed.

We consider CHL models of order $N = 2$ in detail.We conjecture a formula for the Donaldson–Thomas invariants of all order two CHL models in all curve classes. The conjecture is formulated in terms of two Siegel modular forms. One of them, a Siegel form for the Iwahori subgroup, has to our knowledge not yet appeared in physics. This discrepancy is discussed in an appendix with Sheldon Katz.


Donaldson–Thomas theory, moonshine, $\mathrm{K}3$ surfaces, Siegel modular forms

2010 Mathematics Subject Classification

Primary 14N35. Secondary 11F46.

S. Katz was supported by NSF grants DMS-1502170 and DMS-1802242, together with DMS-1440140 while at MSRI during Spring 2018.

Received 5 January 2019

Accepted 7 April 2020

Published 2 October 2020