M-theory on `toric' G₂ cones and its type II reduction

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We analyze a class of conical G₂ metrics admitting two commuting isometries, together with a certain one-parameter family of G₂ deformations which preserves these symmetries. Upon using recent results of Calderbank and Pedersen, we write down the explicit G₂ metric for the most general member of this family and extract the IIA reduction of M-theory on such backgrounds, as well as its type IIB dual. By studying the asymptotics of type II fields around the relevant loci, we confirm the interpretation of such backgrounds in terms of localized IIA 6-branes and delocalized IIB 5-branes. In particular, we find explicit, general expressions for the string coupling and R-R/NS-NS forms in the vicinity of these objects. Our solutions contain and generalize the field configurations relevant for certain models considered in recent work of Acharya and Witten.