Eta Products, BPS States and K3 Surfaces

TL;DR

This paper explores the connection between eta-product generating functions, BPS states in heterotic orbifolds, and elliptic K3 surfaces with automorphisms, revealing deep links via string duality and relations to sporadic groups.

Contribution

It demonstrates the equivalence of certain eta-product functions with K3 surfaces admitting Nikulin automorphisms through string duality.

Findings

Identifies eta-products as generating functions for BPS states in heterotic orbifolds.

Shows these eta-products correspond to elliptic K3 surfaces with automorphisms.

Derives identities from q-expansions and links to Mathieu group M24.

Abstract

Inspired by the multiplicative nature of the Ramanujan modular discriminant, Delta, we consider physical realizations of certain multiplicative products over the Dedekind eta-function in two parallel directions: the generating function of BPS states in certain heterotic orbifolds and elliptic K3 surfaces associated to congruence subgroups of the modular group. We show that they are, after string duality to type II, the same K3 surfaces admitting Nikulin automorphisms. In due course, we will present some identities arising from q-expansions as well as relations to the sporadic Mathieu group M24.