We consider 4d N=1 gauge theories with R-symmetry on a hemisphere times a torus. We apply localization techniques to evaluate the exact partition function through a cohomological reformulation of the supersymmetry transformations. Our results represent the natural elliptic lifts of the lower dimensional analogs as well as a field theoretic derivation of the conjectured 4d holomorphic blocks, from which partition functions of compact spaces with diverse topology can be recovered through gluing. We also analyze the different boundary conditions which can naturally be imposed on the chiral multiplets, which turn out to be either Dirichlet or Robin-like. We show that different boundary conditions are related to each other by coupling the bulk to 3d N=1 degrees of freedom on the boundary three-torus, for which we derive explicit 1-loop determinants.

Dual superconformal symmetry is a remarkable, hidden feature of N=4 SYM in 4 dimensions. Via AdS/CFT, such a symmetry corresponds to the invariance of the AdS(5) x S(5) superstring under specific combinations of bosonic and fermionic T-dualities. We show that AdS(d) x S(d) x S(d) superstrings with D(2,1;\alpha) isometry supergroup are T-self-dual if additional T-dualities along complexified S(d) directions are performed. This implies that CFTs dual to AdS(d) x S(d) x S(d) x T(10-3d) superstrings enjoy a new type of dual superconformal symmetry.