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

TL;DR

This paper explores the geometry of M-theory five-branes wrapped on Seiberg-Witten curves for various gauge groups, revealing new insights into exceptional groups and the role of orientifold-like objects.

Contribution

It extends the analysis of brane configurations to exceptional groups G_2 and E_6, uncovering complex involutions and proposing the need for curved orientifold surfaces.

Findings

Classical groups' cuts reduce to D4-branes in the semiclassical limit.

G_2 curve has genus two quotient with Prym properties matching G_2 theory.

Exceptional groups require novel objects beyond traditional orientifold planes.

Abstract

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 NS5- and 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 explains the subtleties encountered for the Sp(2n) and SO(2n +1). Using this approach we investigate the curves for exceptional groups, especially G_2 and E_6, and show that unlike for classical groups taking the semiclassical ten dimensional limit does not reduce the cuts to D4-branes. For G_2 we find a genus two quotient curve that contains the Prym and has the right properties to describe the G_2 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.