Script N = 1G2SYM theory and compactification to three dimensions

We study four dimensional N=2 G_2 supersymmetric gauge theory on R^3\timesS^1 deformed by a tree level superpotential. We will show that the exactsuperpotential can be obtained by making use of the Lax matrix of thecorresponding integrable model which is the periodic Toda lattice based on thedual of the affine G_2 Lie algebra. At extrema of the superpotential theSeiberg-Witten curve typically factorizes, and we study the algebraic equationsunderlying this factorization. For U(N) theories the factorization was closelyrelated to the geometrical engineering of such gauge theories and to matrixmodel descriptions, but here we will find that the geometrical interpretationis more mysterious. Along the way we give a method to compute the gauge theoryresolvent and a suitable set of one-forms on the Seiberg-Witten curve. We willalso find evidence that the low-energy dynamics of G_2 gauge theories caneffectively be described in terms of an auxiliary hyperelliptic curve.Comment: 27 pages, late