  Using Majorana fermions and elementary mesons we find new massless quantum
field theories with weakly interacting ultraviolet fixed points. We also find
new classes of large N equivalences amongst SU, SO and Sp gauge theories with
different types of matter fields and Yukawa interactions. Results include a
triality of asymptotically safe theories and dualities between asymptotically
free matter-gauge theories with identical fixed points, phase diagrams, and
scaling exponents. Implications for conformal field theory and orbifold
reductions are indicated.
