M2-branes and AdS/CFT

Abstract

We obtain a simple explicit expression for the hyper-Kähler Calabi metric on the co-tangent bundle of CP n+1 , for all n, in which it is constructed as a metric of cohomogeneity one with SU (n + 2)/U (n) principal orbits. These results enable us to obtain explicit expressions for an L 2-normalisable harmonic 4-form in D = 8, and an L 2-normalisable harmonic 6-form in D = 12. We use the former in order to obtain an explicit resolved M2-brane solution, and we show that this solution is invariant under all three of the supersymmetries associated with the covariantly-constant spinors in the 8-dimensional Calabi metric. We give some discussion of the corresponding dual N = 3 three-dimensional field theory. Various other topics are also addressed, including superpotentials for the Calabi metrics and the metrics of exceptional G 2 and Spin(7) holonomy in D = 7 and D = 8. We also present complex and quaternionic conifold constructions, associated with the cone metrics whose resolutions are provided by the Stenzel T * S n+1 and Calabi T * CP n+1 metrics. In the latter case we relate the construction to the hyper-Kähler quotient. We then use the hyper-Kähler quotient to give a quaternionic rederivation of the Calabi metrics.

Double-trace flows and the swampland

“…(For more detailed reviews, see e.g. [40][41][42].) The ABJM theory [18] is a parity-invariant, U (N ) k × U (N ) −k Chern-Simons theory coupled to bifundamental matter.…”

$ \mathcal{N} = 8 $ superspace constraints for three-dimensional gauge theories

“…Also the restrictions due the N = 8 superconformal symmetry as discussed in superspace in [11] might be helpful for further investigations. A big challenge in the AdS 4 /CF T 3 correspondence remains the understanding of the scaling of degrees of freedom with N 3/2 for the strongly coupled theory describing N M 2-branes [12,13].…”

“…However, the dimensional reduction of the the ten-dimensional SYM superspace constraints to six dimensions, describing six-dimensional N = 2 SYM, can be reformulated as a linear auxiliary system. 13 In three dimensions a twistorial description of SYM has been given in N = 6 superspace [41]. However, it is an interesting question if there exists an auxiliary linear system and an associated twistor space description for the solution (4.4) of the W -constraints which eventually would give rise to a linear system and associated twistor space formulation underlying the equations of motion of the conformal BLG model.…”

Quantum phase transitions with dynamical flavors

“…When N and k are large the theory admits a supergravity description in the ten-dimensional type IIA theory. The corresponding geometry is of the form AdS 4 × CP 3 with fluxes (see [21][22][23][24] for reviews of several aspects of the ABJM model).…”