New N=1 superconformal field theories in four dimensions

We construct a large new family of four-dimensional N = 1 superconformal field theories by coupling Gaiotto's T-N theories to N = 1 vector multiplets. In particular, we consider theories in which various T-N blocks are linked together via bifundamental and adjoint chiral superfields, with no superpotential. We find that while some of these constructions appear to give new strongly coupled SCFTs and can be realized via RG flows, others lead to violations of the a-theorem.