Superconformal indices of 4d N=1 SYM theories with SU(N) and SP(2N) gauge groups are investigated for N_f=N and N_f=N+1 flavors, respectively. These indices vanish for generic values of the flavor fugacities. However, for a singular submanifold of fu
gacities they behave like the Dirac delta functions and describe the chiral symmetry breaking phenomenon. Similar picture holds for partition functions of 3d supersymmetric field theories with the chiral symmetry breaking.
Using the superconformal (SC) indices techniques, we construct Seiberg type dualities for $mathcal{N}=1$ supersymmetric field theories outside the conformal windows. These theories are physically distinguished by the presence of chiral superfields with small or negative $R$-charges.
Following a recent work of Dolan and Osborn, we consider superconformal indices of four dimensional ${mathcal N}=1$ supersymmetric field theories related by an electric-magnetic duality with the SP(2N) gauge group and fixed rank flavour groups. For t
he SP(2) (or SU(2)) case with 8 flavours, the electric theory has index described by an elliptic analogue of the Gauss hypergeometric function constructed earlier by the first author. Using the $E_7$-root system Weyl group transformations for this function, we build a number of dual magnetic theories. One of them was originally discovered by Seiberg, the second model was built by Intriligator and Pouliot, the third one was found by Csaki et al. We argue that there should be in total 72 theories dual to each other through the action of the coset group $W(E_7)/S_8$. For the general $SP(2N), N>1,$ gauge group, a similar multiple duality takes place for slightly more complicated flavour symmetry groups. Superconformal indices of the corresponding theories coincide due to the Rains identity for a multidimensional elliptic hypergeometric integral associated with the $BC_N$-root system.
We demonstrate how one can construct renormalizable perturbative expansion in formally nonrenormalizable higher dimensional scalar theories. It is based on 1/N-expansion and results in a logarithmically divergent perturbation theory in arbitrary high
odd space-time dimension. The resulting effective coupling is dimensionless and is running in accordance with the usual RG equations. The corresponding beta function is calculated in the leading order and is nonpolynomial in effective coupling. It exhibits either UV asymptotically free or IR free behaviour depending on the dimension of space-time.
