Advances in Theoretical and Mathematical Physics

Volume 24 (2020)

Number 7

The first of two special issues in honor of Cumrun Vafa’s 60th birthday

Recounting special Lagrangian cycles in twistor families of K3 surfaces (or: How I learned to stop worrying and count BPS states)

Pages: 1917 – 1930

DOI: https://dx.doi.org/10.4310/ATMP.2020.v24.n7.a5

Authors

Shamit Kachru (Stanford Institute for Theoretical Physics, Stanford University, Stanford, California, U.S.A.)

Arnav Tripathy (Department of Mathematics, Harvard University, Cambridge, Massachusetts, U.S.A.)

Max Zimet (Stanford Institute for Theoretical Physics, Stanford University, Stanford, California, U.S.A.)

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 [1], who studied the asymptotic count of special Lagrangian fibrations of a marked K3 surface, with fibers of volume at most $V_\ast$, 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 [2–7] to count flux vacua and attractor black holes. We similarly relate BPS state counts in 4d $\mathcal{N} = 2$ supersymmetric gauge theories to certain counting problems in billiard dynamics and provide a simple proof of an old result in this field.

Published 8 September 2021