On the Quantum Moduli Space of Vacua of N=2N=2 Supersymmetric SU(Nc)SU(N_c) Gauge Theories

We construct families of hyper-elliptic curves which describe the quantum moduli spaces of vacua of $N=2$ supersymmetric $SU(N_c)$ gauge theories coupled to $N_f$ flavors of quarks in the fundamental representation. The quantum moduli spaces for $N_f < N_c$ are determined completely by imposing $R$-symmetry, instanton corrections and the proper classical singularity structure. These curves are verified by residue and weak coupling monodromy calculations. The quantum moduli spaces for $N_f\geq N_c$ theories are parameterized and their general structure is worked out using residue calculations. Global symmetry considerations suggest a complete description of them. The results are supported by weak coupling monodromy calculations. The exact metrics on the quantum moduli spaces as well as the exact spectrum of stable massive states are derived. We find an example of a novel symmetry of a quantum moduli space: Invariance under the exchange of a moduli parameter and the bare mass. We apply our method for the construction of the quantum moduli spaces of vacua of $N=1$ supersymmetric theories in the coulomb phase.Comment: 35 pages, late

Mirror Symmetry in Three Dimensional Gauge Theories

We discuss non-trivial fixed points of the renormalization group with dual descriptions in $N=4$ gauge theories in three dimensions. This new duality acts as mirror symmetry, exchanging the Higgs and Coulomb branches of the theories. Quantum effects on the Coulomb branch arise classically on the Higgs branch of the dual theory. We present examples of dual theories whose Higgs/Coulomb branch are the ALE spaces and whose Coulomb/Higgs branches are the moduli space of instantons of the corresponding $ADE$ gauge group. In particular, we show that in three dimensions small $E_8$ instantons in string theory are described by a local quantum field theory.Comment: 13 pages, harvma

Non-trivial Fixed Points of The Renormalization Group in Six Dimensions

We start a systematic analysis of supersymmetric field theories in six dimensions. We find necessary conditions for the existence of non-trivial interacting fixed points. String theory provides us with examples of such theories. We conjecture that there are many other examples.Comment: 6 pages, uses harvma

Toroidal Compactification of Heterotic 6D Non-Critical Strings Down to Four Dimensions

The low-energy limit of the 6D non-critical string theory with $N=1$ SUSY and $E_8$ chiral current algebra compactified on $T^2$ is generically an $N=2$ $U(1)$ vector multiplet. We study the analog of the Seiberg-Witten solution for the low-energy effective action as a function of $E_8$ Wilson lines on the compactified torus and the complex modulus of that torus. The moduli space includes regions where the Seiberg-Witten curves for $SU(2)$ QCD are recovered as well as regions where the newly discovered 4D theories with enhanced $E_{6,7,8}$ global symmetries appear.Comment: 15pp LaTe

Duality and the Legendre Transform

We define a weak-strong coupling transformation based on the Legendre transformation of the effective action. In the case of N\es 2 supersymmetric Yang-Mills theory, this coincides with the duality transform on the low energy effective action considered by Seiberg and Witten. This Legendre transform interpretation of duality generalizes directly to the full effective action, and in principle to other theories.Comment: 6 pages, LaTe

On Duality in Supersymmetric Yang-Mills Theory

We discuss non-abelian $SU(N_c)$ gauge theory coupled to an adjoint chiral superfield $X$, and a number of fundamental chiral superfields $Q^i$. Using duality, we show that turning on a superpotential W(X)=\Tr\sum_{l=1}^k g_l X^{l+1} leads to non-trivial long distance dynamics, a large number of multicritical IR fixed points and vacua, connected to each other by varying the coefficients $g_l$.Comment: 11 pages, harvma

Remarks on Non-Abelian Duality in N=1 Supersymmetric Gauge Theories

Recently Seiberg has conjectured a duality symmetry connecting different theories of the supersymmetric QCD type. We provide support for this conjecture by analyzing a flat direction of the theory along which the two dual theories go over to the same theory in the IR.Comment: 17 pages, harvma

On the Quantum Moduli Space of N=2 Supersymmetric Gauge Theories

Families of hyper-elliptic curves which describe the quantum moduli spaces of vacua of $N=2$ supersymmetric $SO(N_c)$ gauge theories coupled to $N_f$ flavors of quarks in the vector representation are constructed. The quantum moduli spaces for $2N_f < N_c-1$ are determined completely by imposing $R$-symmetry, instanton corrections and the proper classical singularity structure. These curves are verified by residue calculations. The quantum moduli spaces for $2N_f\geq N_c-1$ theories are parameterized and their general structure is worked out using residue calculations. The exact metrics on the quantum moduli spaces as well as the exact spectrum of stable massive states are derived. The results presented here together with recent results of Martinec and Warner provide a natural conjecture for the form of the curves for the other gauge groups.Comment: 19 page

IR Dynamics on Branes and Space-Time Geometry

We consider the type I theory compactified on $T^3$. When the D5-brane wraps the $T^3$ it yields a D2-brane in seven dimensions. In the leading approximation the moduli space of vacua of the three dimensional field theory on the brane is T^4/\ZZ_2. The dual M theory description of this theory is a compactification on K3 and our 2-brane is the eleven dimensional 2-brane at a point in K3. We use this fact to conclude that strong coupling IR effects in the three dimensional theory on the brane turn its moduli space into a K3. This interpretation allows us to solve various strongly coupled gauge theories in three dimensions by identifying their Coulomb branch with a piece of a (sometime singular) K3.Comment: 8 pages, uses harvmac, one reference adde