BPS Spectrum of 5 Dimensional Field Theories, (p,q) Webs and Curve Counting

TL;DR

This paper investigates the BPS spectrum of 5D supersymmetric field theories using string web representations, providing a diagrammatic method to determine the spectrum of curves on toric surfaces and their moduli.

Contribution

It introduces a novel diagrammatic approach to compute BPS spectra and moduli spaces of curves in toric geometries, linking string webs to membrane configurations in M-theory.

Findings

States correspond to irreducible string webs with specific charges.

Fermionic zero modes depend on the web's faces and boundaries.

A diagrammatic method for curve spectrum and moduli is developed.

Abstract

We study the BPS spectrum of supersymmetric 5 dimensional field theories and their representations as string webs. It is found that a state of given charges exists when it has a representation as an irreducible string web. Its spin is determined by the string web. The number of fermionic zero modes is 8g+4b, where g is the number of internal faces and b is the number of boundaries. In the lift to M theory of 4d field theories such states are described by membranes ending on the 5-brane, breaking SUSY from 8 to 4, and g becomes the genus of the membrane. Mathematically, we obtain a diagrammatic method to find the spectrum of curves on a toric complex surface, and the number of their moduli.