Five-branes, seven-branes and five-dimensional E-n field theories

We generalize the (p, q) 5-brane web construction of five-dimensional field theories by introducing (p, q) 7-branes, and apply this construction to th eories with a one-dimensional Coulomb branch. The 7-branes render the excep tional global symmetry of these theories manifest. Additionally, 7-branes a llow the construction of all E-n theories up to n = 8, previously not possi ble in 5-brane configurations. The exceptional global symmetry in the field theory is a subalgebra of an affine symmetry on the 7-branes, which is nec essary for the existence of the system. We explicitly determine the quantum numbers of the BPS states of all E-n theories using two simple geometrical constraints.