Enhanced gauge symmetry and braid group actions

Enhanced gauge symmetry appears in Type II string theory (as well as F- and M-theory) compactified on Calabi--Yau manifolds containing exceptional divisors meeting in Dynkin configurations. It is shown that in many such cases, at enhanced symmetry points in moduli a braid group acts on the derived category of sheaves of the variety. This braid group covers the Weyl group of the enhanced symmetry algebra, which itself acts on the deformation space of the variety in a compatible fashion. Extensions of this result are given for nontrivial $B$-fields on K3 surfaces, explaining physical restrictions on the $B$-field, as well as for elliptic fibrations. The present point of view also gives new evidence for the enhanced gauge symmetry content in the case of a local $A_{2n}$-configuration in a threefold having global $\Z/2$ monodromy.