Extremal Transitions and Five-Dimensional Supersymmetric Field Theories

We study five-dimensional supersymmetric field theories with one-dimensional Coulomb branch. We extend a previous analysis which led to non-trivial fixed points with $E_n$ symmetry ($E_8$, $E_7$, $E_6$, $E_5=Spin(10)$, $E_4=SU(5)$, $E_3=SU(3)\times SU(2)$, $E_2=SU(2)\times U(1)$ and $E_1=SU(2)$) by finding two new theories: $\tilde E_1$ with $U(1)$ symmetry and $E_0$ with no symmetry. The latter is a non-trivial theory with no relevant operators preserving the super-Poincar\'e symmetry. In terms of string theory these new field theories enable us to describe compactifications of the type I$'$ theory on $\bS^1/\IZ_2$ with 16, 17 or 18 background D8-branes. These theories also play a crucial role in compactifications of M-theory on Calabi--Yau spaces, providing physical models for the contractions of del Pezzo surfaces to points (thereby completing the classification of singularities which can occur at codimension one in K\"ahler moduli). The structure of the Higgs branch yields a prediction which unifies the known mathematical facts about del Pezzo transitions in a quite remarkable way.