Exceptional SW Geometry from ALE Fibrations

We show that the genus 34 Seiberg-Witten curve underlying $N=2$ Yang-Mills theory with gauge group $E_6$ yields physically equivalent results to the manifold obtained by fibration of the $E_6$ ALE singularity. This reconciles a puzzle raised by $N=2$ string duality

The Refined Elliptic Genus and Coulomb Gas Formulations of N=2N=2 Superconformal Coset Models

We describe, in some detail, a number of different Coulomb gas formulations of $N=2$ superconformal coset models. We also give the mappings between these formulations. The ultimate purpose of this is to show how the Landau-Ginzburg structure of these models can be used to extract the $W$-generators, and to show how the computation of the elliptic genus can be refined so as to extract very detailed information about the characters of component parts of the model.Comment: 40 pages in harvmac, no figure

Quark Potentials in the Higgs Phase of Large N Supersymmetric Yang-Mills Theories

We compute, in the large N limit, the quark potential for ${\cal N}=4$ supersymmetric SU(N) Yang-Mills theory broken to $SU(N_1) \times SU(N_2)$. At short distances the quarks see only the unbroken gauge symmetry and have an attractive potential that falls off as 1/L. At longer distances the interquark interaction is sensitive to the symmetry breaking, and other QCD states appear. These states correspond to combinations of the quark-antiquark pair with some number of W-particles. If there is one or more W-particles then this state is unstable because of the coulomb interaction between the W-particles and between the W's and the quarks. As L is decreased the W-particles delocalize and these coulomb branches merge onto a branch with a linear potential. The quarks on this branch see the unbroken gauge group, but the flux tube is unstable to the production of W-particles.Comment: 23 pages, 7 figures, harvmac (b

Quartic Gauge Couplings from K3 Geometry

We show how certain F^4 couplings in eight dimensions can be computed using the mirror map and K3 data. They perfectly match with the corresponding heterotic one-loop couplings, and therefore this amounts to a successful test of the conjectured duality between the heterotic string on T^2 and F-theory on K3. The underlying quantum geometry appears to be a 5-fold, consisting of a hyperk"ahler 4-fold fibered over a IP^1 base. The natural candidate for this fiber is the symmetric product Sym^2(K3). We are lead to this structure by analyzing the implications of higher powers of E_2 in the relevant Borcherds counting functions, and in particular the appropriate generalizations of the Picard-Fuchs equations for the K3.Comment: 32 p, harvmac; One footnote on page 11 extended; results unchanged; Version subm. to ATM