Twisted elliptic genus for K3 and Borcherds product

TL;DR

This paper explores the connection between the twisted elliptic genus of K3 surfaces and the Mathieu group M24, revealing simple eta-product representations linked to Borcherds products and Siegel modular forms.

Contribution

It introduces a simple eta-product formula for twisted elliptic genera of K3 surfaces associated with M24 conjugacy classes, based on Borcherds product identities.

Findings

Twisted elliptic genera can be expressed via eta-products.

The eta-product formula follows from Borcherds and additive lift identities.

The work links modular forms, Mathieu group M24, and K3 surface invariants.

Abstract

We further discuss the relation between the elliptic genus of K3 surface and the Mathieu group M24. We find that some of the twisted elliptic genera for K3 surface, defined for conjugacy classes of the Mathieu group M24, can be represented in a very simple manner in terms of the eta-product of the corresponding conjugacy classes. It is shown that our formula is a consequence of the identity between the Borcherds product and additive lift of some Siegel modular forms.