  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.
