M-Theory Five-brane Wrapped on Curves for Exceptional Groups

We study the M-theory five-brane wrapped around the Seiberg-Witten curves for pure classical and exceptional groups given by an integrable system. Generically, the D4-branes arise as cuts that collapse to points after compactifying the eleventh dimension and going to the semiclassical limit, producing brane configurations of NS5and D4-branes with N = 2 gauge theories on the world volume of the four-branes. We study the symmetries of the different curves to see how orientifold planes are related to the involutions needed to obtain the distinguished Prym variety of the curve. This approach explains some subtleties encountered for the Sp(2n) and SO(2n + 1). Using this approach we investigate the curves for exceptional groups, especially G2 and E6, and show that unlike for classical groups taking the semiclassical ten dimensional limit does not reduce the cuts to D4-branes. For G2 we find a genus two quotient curve that contains the Prym and has the right properties to describe the G2 field theory, but the involutions are far more complicated than the ones for classical groups. To realize them in M-theory instead of an orientifold plane we would need another object, a kind of curved orientifold surface.