On the fate of tachyonic quivers

TL;DR

This paper investigates the stability of gauge theories on D3-branes probing singularities, linking tachyonic instabilities to crossing legs in (p,q) webs and proposing a resolution method to identify stable bound states.

Contribution

It establishes the equivalence between the well split condition and the absence of crossing legs in (p,q) webs, and introduces a procedure to determine stable quivers after monodromy.

Findings

Tachyons appear when (p,q) web legs cross.

Crossing legs can be resolved to find stable bound states.

Explicit example on D3-branes over dP1 confirms the approach.

Abstract

We study gauge theories on the world-volume of D3-branes probing singularities. Seiberg duality can be realized as a sequence of Picard-Lefschetz monodromies on 3-cycles in the mirror manifold. In previous work, the precise meaning of gauge theories obtained by monodromies that do not correspond to Seiberg duality was unclear. Recently, it was pointed out that these theories contain tachyons, suggesting that the collection of marginally bound branes at the singularity is unstable. We address this problem using (p,q) web techniques. It is shown that theories with tachyons appear whenever the (p,q) web contains crossing legs. A recent study of these theories with tachyons using exceptional collections proposed the notion of "well split condition.'' We show the equivalence between the well split condition and the absence of crossing legs in the (p,q) web. The (p,q) web has a natural resolution of crossing legs which was first studied in the construction of five dimensional fixed points using branes. Exploiting this result, we propose a generic procedure which determines the quiver that corresponds to the stable bound state of D-branes that live on the singularity after the monodromy. This set is generically larger than the original set, meaning that there are extra massless gauge fields and matter fields in the quiver. Alternatively, one can argue that since these gauge and matter fields are initially assumed to be absent, the theory exhibits tachyonic excitations. We illustrate our ideas in an explicit example for D3-branes on a complex cone over dP1, computing both the quiver and the superpotential.