Exploring the space of many-flavor QED's in 2 < d < 6

We study many-flavor Quantum Electrodynamics (QED) in 2&lt;6. In the first chapter we review and summarise our results. In the second chapter we consider QED's in three dimensions, with Nf fermionic or bosonic flavors, allowing for interactions that respect the global symmetry U(Nf/2)^2. There are four bosonic and four fermionic fixed points, which we analyze using the large Nf expansion. We systematically compute, at order O(1/Nf), the scaling dimensions of quadratic and quartic mesonic operators. We also consider three dimensional QED with minimal supersymmetry. In this case the large Nf scaling dimensions extrapolated at Nf=2, agree quite well with the scaling dimensions of a dual supersymmetric Wess-Zumino model. This provides a quantitiative check of the conjectured duality. In the third chapter, we analyze the fate of the non-supersymmetric QED's for small values of Nf. Large Nf arguments suggest that, lowering Nf, the fixed points collide pairwise, which leads the fixed points either to merge and to annihilate into the complex plane, or to pass through each other, exchanging their stability properties. In the bosonic QED's the merging happens around Nf ~ 9-11. In the fermionic QED's collision happens around Nf ~ 3-7. In the fermionic case, the fixed points with different symmetries are colliding. In the last chapter we consider the CP(Nf-1) Non-Linear-Sigma-Model in the dimension 4&lt;6. The critical behaviour of this model in the large Nf limit is reviewed. We propose a Higher Derivative Gauge (HDG) theory as an ultraviolet completion of the CP(Nf-1) NLSM. Tuning mass operators to zero, the HDG in the IR limit reaches to the critical CP(Nf-1). With partial tunings the HDG reaches either to the critical U(Nf)-Yukawa model or to the critical pure scalar QED (no Yukawa interactions). We renormalize the HDG in its critical dimension d=6. We study the fixed points of the HDG in d=6-2epsilon and we calculate the scaling dimensions of various observables finding a full agreement with the order O(1/Nf) predictions of the corresponding critical models.