Gravitational and Yang-Mills instantons in holographic RG flows

Abstract

We study various holographic RG flow solutions involving warped asymptotically locally Euclidean (ALE) spaces of $A_{N-1}$ type. A two-dimensional RG flow from a UV (2,0) CFT to a (4,0) CFT in the IR is found in the context of (1,0) six dimensional supergravity, interpolating between $AdS_3\times S^3/\mathbb{Z}_N$ and $AdS_3\times S^3$ geometries. We also find solutions involving non trivial gauge fields in the form of SU(2) Yang-Mills instantons on ALE spaces. Both flows are of vev type, driven by a vacuum expectation value of a marginal operator. RG flows in four dimensional field theories are studied in the type IIB and type I$'$ context. In type IIB theory, the flow interpolates between $AdS_5\times S^5/\mathbb{Z}_N$ and $AdS_5\times S^5$ geometries. The field theory interpretation is that of an N=2 $SU(n)^N$ quiver gauge theory flowing to N=4 SU(n) gauge theory. In type I$'$ theory the solution describes an RG flow from N=2 quiver gauge theory with a product gauge group to N=2 gauge theory in the IR, with gauge group $USp(n)$. The corresponding geometries are $AdS_5\times S^5/(\mathbb{Z}_N\times \mathbb{Z}_2)$ and $AdS_5\times S^5/\mathbb{Z}_2$, respectively. We also explore more general RG flows, in which both the UV and IR CFTs are N=2 quiver gauge theories and the corresponding geometries are $AdS_5\times S^5/(\mathbb{Z}_N\times \mathbb{Z}_2)$ and $AdS_5\times S^5/(\mathbb{Z}_M\times \mathbb{Z}_2)$. Finally, we discuss the matching between the geometric and field theoretic pictures of the flows.