We give a full list of known N = 1 supersymmetric quantum field theories related by the Seiberg electric-magnetic duality conjectures for SU (N ), SP (2N ) and G 2 gauge groups. Many of the presented dualities are new, not considered earlier in the literature. For all these theories we construct superconformal indices and express them in terms of elliptic hypergeometric integrals. This gives a systematic extension of the related Römelsberger and Dolan-Osborn results. Equality of indices in dual theories leads to various identities for elliptic hypergeometric integrals. About half of them were proven earlier, and another half represents new challenging conjectures. In particular, we conjecture a dozen new elliptic beta integrals on root systems extending the univariate elliptic beta integral discovered by the first author.

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Recently, Kim and Imamura and Yokoyama derived an exact formula for superconformal indices in three-dimensional field theories. Using their results, we prove analytically the equality of superconformal indices in some U(1)-gauge group theories related by the mirror symmetry. The proofs are based on the well known identities of the theory of $q$-special functions. We also suggest the general index formula taking into account the $U(1)_J$ global symmetry present for abelian theories

We revisit the definition of the 6j symbols from the modular double of U q (sl(2, R)), referred to as b-6j symbols. Our new results are (i) the identification of particularly natural normalization conditions, and (ii) new integral representations for this object. This is used to briefly discuss possible applications to quantum hyperbolic geometry, and to the study of certain supersymmetric gauge theories. We show, in particular, that the b-6j symbol has leading semiclassical asymptotics given by the volume of a non-ideal tetrahedron. We furthermore observe a close relation with the problem to quantize natural Darboux coordinates for moduli spaces of flat connections on Riemann surfaces related to the Fenchel-Nielsen coordinates. Our new integral representations finally indicate a possible interpretation of the b-6j symbols as partition functions of non-abelian three-dimensional N = 2 supersymmetric gauge theories. Racah-Wigner 6j symbols for the modular double2.1 Self-dual representations of U q (sl(2, R)) and the modular doubleWe will be considering the Hopf-algebra U q (sl(2, R)) which has generators E, F and K subject to the usual relations. This algebra has a one-parameter family of representations P α E α ≡ π α (E) := e +πbx cosh πb(p − s) sin πb 2 e +πbx , F α ≡ π α (F ) := e −πbx cosh πb(p + s) sin πb 2 e −πbx , K α ≡ π α (K) := e −πbp , (2.1)

In this paper we study the form factors for the half-BPS operators O (n) I and the N = 4 stress tensor supermultiplet current T AB up to the second order of perturbation theory and for the Konishi operator K at first order of perturbation theory in the N = 4 SYM theory at weak coupling. For all the objects we observe the exponentiation of the IR divergences with two anomalous dimensions: the cusp anomalous dimension and the collinear anomalous dimension. For the IR finite parts we obtain a similar situation as for the gluon scattering amplitudes, namely, apart from the case of T AB and K the finite part has some remainder function which we calculate up to the second order. It involves the generalized Goncharov polylogarithms of several variables. All the answers are expressed in terms of the integrals related to the dual conformal invariant ones which might be a signal of integrable structure standing behind the form factors.Keywords: N = 4 Super Yang-Mills Theory, form factors, N = 1 superspace.