The "Dual" Variables Of Yang-Mills Theory And Local Gauge Invariant Variables

After adding auxiliary fields and integrating out the original variables, the Yang-Mills action can be expressed in terms of local gauge invariant variables. This method reproduces the known solution of the two dimensional $SU(N)$ theory. In more than two dimensions the action splits into a topological part and a part proportional to $\alpha_s$. We demonstrate the procedure for $SU(2)$ in three dimensions where we reproduce a gravity-like theory. We discuss the four dimensional case as well. We use a cubic expression in the fields as a space-time metric to obtain a covariant Lagrangian. We also show how the four-dimensional $SU(2)$ theory can be expressed in terms of a local action with six degrees of freedom only.Comment: 34pp, LaTeX, Corrections in reference

Non-Critical Strings, Del Pezzo Singularities And Seiberg-Witten Curves

We study limits of four-dimensional type II Calabi-Yau compactifications with vanishing four-cycle singularities, which are dual to \IT^2 compactifications of the six-dimensional non-critical string with $E_8$ symmetry. We define proper subsectors of the full string theory, which can be consistently decoupled. In this way we obtain rigid effective theories that have an intrinsically stringy BPS spectrum. Geometrically the moduli spaces correspond to special geometry of certain non-compact Calabi-Yau spaces of an intriguing form. An equivalent description can be given in terms of Seiberg-Witten curves, given by the elliptic simple singularities together with a peculiar choice of meromorphic differentials. We speculate that the moduli spaces describe non-perturbative non-critical string theories.Comment: 29 pages, harvmac, 1 figure, minor correction