On the fate of tachyonic quivers

We study gauge theories on the world-volume of D3-branes probingsingularities. Seiberg duality can be realized as a sequence ofPicard-Lefschetz monodromies on 3-cycles in the mirror manifold. In previouswork, the precise meaning of gauge theories obtained by monodromies that do notcorrespond to Seiberg duality was unclear. Recently, it was pointed out thatthese theories contain tachyons, suggesting that the collection of marginallybound branes at the singularity is unstable. We address this problem using(p,q) web techniques. It is shown that theories with tachyons appear wheneverthe (p,q) web contains crossing legs. A recent study of these theories withtachyons using exceptional collections proposed the notion of "well splitcondition.'' We show the equivalence between the well split condition and theabsence of crossing legs in the (p,q) web. The (p,q) web has a naturalresolution of crossing legs which was first studied in the construction of fivedimensional fixed points using branes. Exploiting this result, we propose ageneric procedure which determines the quiver that corresponds to the stablebound state of D-branes that live on the singularity after the monodromy. Thisset is generically larger than the original set, meaning that there are extramassless gauge fields and matter fields in the quiver. Alternatively, one canargue that since these gauge and matter fields are initially assumed to beabsent, the theory exhibits tachyonic excitations. We illustrate our ideas inan explicit example for D3-branes on a complex cone over dP1, computing boththe quiver and the superpotential.Comment: 22 pages, 13 figures. v2.: genus 2 quiver corrected, comments and a reference adde