BPS States and Minimal Surfaces

TL;DR

This work presents a geometric realization of BPS states in 4D N=2 SU(2) gauge theory via M-theory, identifying BPS states with minimal-area membranes ending on a fivebrane and deriving their spectra from holomorphic membrane configurations. The approach unifies monopole, dyon, and vector multiplet states through membrane topology (disk vs cylinder) and their deformations across moduli, including complex structure variations and marginal stability. It extends the framework to include matter in mixed and fundamental representations via spikes and Taub-NUT reductions, and develops a robust deformation theory for membranes and a membrane worldsheet SUSY description that reproduces the correct multiplet structures. The results connect Seiberg-Witten data with M-theory geometry, provide tools for analyzing BPS spectra in broader theories, and reveal deep links to geodesic structures and Calabi-Yau-type calibrations in string compactifications.

Abstract

It was observed recently, that the low energy effective action of the four-dimensional supersymmetric theories may be obtained as a certain limit of M Theory. From this point of view, the BPS states correspond to the minimal area membranes ending on the M Theory fivebrane. We prove that for the configuration, corresponding to the SU(2) Super Yang-Mills theory, the BPS spectrum is correctly reproduced, and develop techniques for analyzing the BPS spectrum in more general cases. We show that the type of the supermultiplet is related to the topology of the membrane: disks correspond to hypermultiplets, and cylinders to vector multiplets. We explain the relation between minimal surfaces and geodesic lines, which shows that our description of BPS states is closely related to one arising in Type II string compactification on Calabi-Yau threefolds.