Recounting Special Lagrangian Cycles in Twistor Families of K3 Surfaces. Or: How I Learned to Stop Worrying and Count BPS States

TL;DR

This paper explores the asymptotic counting of BPS states in M-theory on K3 surfaces, linking physical state counts to mathematical problems involving special Lagrangian fibrations and billiard dynamics, offering new proofs and insights.

Contribution

It provides an alternative proof of Filip's results on special Lagrangian fibrations in K3 surfaces and connects BPS state counts to mathematical counting problems in billiard dynamics.

Findings

Established asymptotic formulas for BPS state counts.

Linked BPS counts to special Lagrangian fibrations in K3 surfaces.

Provided simplified proofs of existing mathematical results.

Abstract

We consider asymptotics of certain BPS state counts in M-theory compactified on a K3 surface. Our investigation is parallel to (and was inspired by) recent work in the mathematics literature by Filip, who studied the asymptotic count of special Lagrangian fibrations of a marked K3 surface, with fibers of volume at most $V_*$, in a generic twistor family of K3 surfaces. We provide an alternate proof of Filip's results by adapting tools that Douglas and collaborators have used to count flux vacua and attractor black holes. We similarly relate BPS state counts in 4d ${\cal N}=2$ supersymmetric gauge theories to certain counting problems in billiard dynamics and provide a simple proof of an old result in this field.