The Chiral Ring and the Periods of the Resolvent

The strongly coupled vacua of an N=1 supersymmetric gauge theory can be described by imposing quantization conditions on the periods of the gauge theory resolvent, or equivalently by imposing factorization conditions on the associated N=2 Seiberg-Witten curve (the so-called strong-coupling approach). We show that these conditions are equivalent to the existence of certain relations in the chiral ring, which themselves follow from the fact that the gauge group has a finite rank. This provides a conceptually very simple explanation of why and how the strongly coupled physics of N=1 theories, including fractional instanton effects, chiral symmetry breaking and confinement, can be derived from purely semi-classical calculations involving instantons only.Comment: 17 pages, 1 figure; v2: cosmetic change

Chiral Rings, Vacua and Gaugino Condensation of Supersymmetric Gauge Theories

We find the complete chiral ring relations of the supersymmetric U(N) gauge theories with matter in adjoint representation. We demonstrate exact correspondence between the solutions of the chiral ring and the supersymmetric vacua of the gauge theory. The chiral ring determines the expectation values of chiral operators and the low energy gauge group. All the vacua have nonzero gaugino condensation. We study the chiral ring relations obeyed by the gaugino condensate. These relations are generalizations of the formula $S^N=\Lambda^{3N}$ of the pure ${\cal N} =1$ gauge theory.Comment: 38 page

Extending the Veneziano-Yankielowicz Effective Theory

We extend the Veneziano Yankielowicz (VY) effective theory in order to account for ordinary glueball states. We propose a new form of the superpotential including a chiral superfield for the glueball degrees of freedom. When integrating it ``out'' we obtain the VY superpotential while the N vacua of the theory naturally emerge. This fact has a counterpart in the Dijkgraaf and Vafa geometric approach. We suggest a link of the new field with the underlying degrees of freedom which allows us to integrate it ``in'' the VY theory. We finally break supersymmetry by adding a gluino mass and show that the Kahler independent part of the ``potential'' has the same form of the ordinary Yang-Mills glueball effective potential.Comment: LaTeX, 20 page

On the Factorisation of the Connected Prescription for Yang-Mills Amplitudes

We examine factorisation in the connected prescription of Yang-Mills amplitudes. The multi-particle pole is interpreted as coming from representing delta functions as meromorphic functions. However, a naive evaluation does not give a correct result. We give a simple prescription for the integration contour which does give the correct result. We verify this prescription for a family of gauge-fixing conditions.Comment: 16 pages, 1 figur

Resultants and Gravity Amplitudes

Two very different formulations of the tree-level S-matrix of N=8 Einstein supergravity in terms of rational maps are known to exist. In both formulations, the computation of a scattering amplitude of n particles in the k R-charge sector involves an integral over the moduli space of certain holomorphic maps of degree d=k-1. In this paper we show that both formulations can be simplified when written in a manifestly parity invariant form as integrals over holomorphic maps of bi-degree (d,n-d-2). In one formulation the full integrand becomes directly the product of the resultants of each of the two maps defining the one of bi-degree (d,n-d-2). In the second formulation, a very different structure appears. The integrand contains the determinant of a (n-3)x(n-3) matrix and a 'Jacobian'. We prove that the determinant is a polynomial in the coefficients of the maps and contains the two resultants as factors.Comment: 21 page