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

Barak Kol, J. Rahmfeld

TL;DR

This work develops a coherent framework to compute the BPS spectrum of 5d N=1 field theories via (p,q) webs and their M-theory lifts to membranes ending on M5-branes. By mapping webs to toric Calabi–Yau geometries, the authors derive zero-mode counts and a diagrammatic method to count holomorphic curves and their moduli, linking irreducible webs to marginal bound states and reading spin from web topology. The approach yields explicit spectra for toric examples (E0, E1, tilde E1) and clarifies how Seiberg–Witten data and curve counting emerge from the toric/Web dictionary, with fermionic zero-mode counts refined in a subsequent update. Overall, the paper provides a practical, geometrically grounded toolkit to extract BPS data and prepotentials from toric 5d theories via brane webs.

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.

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

TL;DR

This work develops a coherent framework to compute the BPS spectrum of 5d N=1 field theories via (p,q) webs and their M-theory lifts to membranes ending on M5-branes. By mapping webs to toric Calabi–Yau geometries, the authors derive zero-mode counts and a diagrammatic method to count holomorphic curves and their moduli, linking irreducible webs to marginal bound states and reading spin from web topology. The approach yields explicit spectra for toric examples (E0, E1, tilde E1) and clarifies how Seiberg–Witten data and curve counting emerge from the toric/Web dictionary, with fermionic zero-mode counts refined in a subsequent update. Overall, the paper provides a practical, geometrically grounded toolkit to extract BPS data and prepotentials from toric 5d theories via brane webs.

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.

Paper Structure

This paper contains 8 sections, 11 equations, 9 figures.

Table of Contents

  1. Introduction
  2. Review of $(p,q)$ webs
  3. Webs and toric geometry
  4. BPS spectrum and Curves
  5. Examples
  6. $E_0$ or ${\bf P}^2$
  7. $E_1$ or ${\bf P}^1 \times {\bf P}^1$
  8. $\tilde{E}_1$ or ${\bf F}_1$

Figures (9)

  • Figure 1: A flop transition.
  • Figure 2: An enhanced gauge symmetry singularity.
  • Figure 3: A conformal theory singularity.
  • Figure 4: The $E_0$ theory. (a) The BPS state $\Delta$. (b) A bound state of $n$$\Delta$'s.
  • Figure 5: The BPS state made out of 2 $\Delta$'s. (a) A reducible configuration. (b) A general irreducible configuration.
  • ...and 4 more figures