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

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.

20 pages, latex, 9 figures. Changes are described at the end of the introduction

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• Counting Yang-Mills Dyons with Index Theorems
• Intersecting brane solutions in string and M-theory
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• An Index for Ray Operators in 5d $E_n$ SCFTs
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• On the Spatial Structure of Monopoles
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• Equivalent String Networks and Uniqueness of BPS States
• Modular properties of 6d (DELL) systems
• The Condensate from Torus Knots
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• Deconstructing the E_0 SCFT to Solve the Orbifold Paradox of the Heterotic M Theory