We construct supergravity solutions describing a stack of D3-branes localized at a point on a blown-up cycle of a resolved $L^{a,b,c}$ cone. The geometry flows from ${\mathrm{AdS}}_5\times L^{a,b,c}$ to ${\mathrm{AdS}}_5\times S^5/{\mathbb{Z}}_k$. The corresponding quiver gauge theory undergoes a renormalization group flow between two superconformal fixed points, which leads to semi-infinite chains of flows between the various $L^{a,b,c}$ fixed points. The general system is described by a triplet of Heun equations, which can each be solved by an expansion with a three-term recursion relation, though there are closed-form solutions for certain cases. This enables us to read off the operators that acquire nonzero vacuum expectation values as the quiver gauge theory flows away from a fixed point.

• Received 26 February 2008

DOI:https://doi.org/10.1103/PhysRevD.77.126003