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

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

Duality of Sp(2N) and SO(N) Supersymmetric Gauge Theories with Adjoint Matter

We discuss electric-magnetic duality in two new classes of supersymmetric Yang-Mills theories. The models have gauge group Sp(2\nc) or SO(\nc) with matter in both the adjoint and defining representations. By perturbing these theories with various superpotentials, we find a variety of new infrared fixed points with dual descriptions. This work is complementary to that of Kutasov and Schwimmer on SU(\nc) and of Intriligator on other models involving Sp(2\nc) and SO(\nc).Comment: 14 pages, no figures, uses harvma

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

Five Dimensional SUSY Field Theories, Non-trivial Fixed Points and String Dynamics

We study (non-renormalizable) five dimensional supersymmetric field theories. The theories are parametrized by quark masses and a gauge coupling. We derive the metric on the Coulomb branch exactly. We use stringy considerations to learn about new non-trivial interacting field theories with exceptional global symmetry $E_n$ ($E_8$, $E_7$, $E_6$, $E_5=Spin(10)$, $E_4=SU(5)$, $E_3=SU(3)\times SU(2)$, $E_2=SU(2)\times U(1)$ and $E_1=SU(2)$). Their Coulomb branch is ${\bf R}^+$ and their Higgs branch is isomorphic to the moduli space of $E_n$ instantons. One of the relevant operators of these theories leads to a flow to $SU(2)$ gauge theories with $N_f=n-1$ flavors. In terms of these $SU(2)$ IR theories this relevant parameter is the inverse gauge coupling constant. Other relevant operators (which become quark masses after flowing to the $SU(2)$ theories) lead to flows between them. Upon further compactifications to four and three dimensions we find new fixed points with exceptional symmetries.Comment: 13 pages, uses harvmac, one reference adde

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

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

Recently N. Seiberg has shown that certain N=1 supersymmetric Yang -- Mills theories have the property that their infrared physics can be equivalently described by {\it two different} gauge theories, with the strong coupling region of one corresponding to the weak coupling region of the other. This duality leads to interesting results about the infrared dynamics of these theories. We generalize Seiberg's ideas to a new class of models for which a similar duality allows one to obtain rather detailed dynamical information about the long distance physics.Comment: 9 pages, 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

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

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