From weak to strong coupling in ABJM theory
The world-volume theory of coincident M2 branes discovered by \textit{O. Aharony, O. Bergman, D. L. Jafferis} and \textit{J. Maldacena} [J. High Energy Phys. 2008, No. 10, 091, 37 p. (2008; Zbl 1245.81130)] provides a new interacting field theory with well defined weak and strong coupling expansions. These two limits are \(\mathcal{N} = 6\) supersymmetric Chern-Simons matter on \({\mathbb{S}}^3\) and type IIA string theory on \(AdS_4 \times {\mathbb{CP}}^3\) (both descriptions are much more involved than their famous D3-brane analogs, 4d \(\mathcal{N} = 4\) supersymmetric Yang-Mills theory and type IIB string theory on \(Ad\,S_5\times\mathbb S^5\)). An important development was the reduction of the calculation of the partition function and certain Wilson loop observables in the gauge theory on \({\mathbb{S}}^3\) to finite dimensional integrals in a zero dimensional super-matrix model by \textit{A. Kapustin, B. Willett} and \textit{I. Yaakov} [``Exact results for Wilson loops in superconformal Chern-Simons theories with matter'', J. High Energy Phys. 2010, No. 10, 089, 21 p. (2010; \url{doi:10.1007/JHEP03(2010)089})]. The paper explores these recent observations further and extracts more exact results in ABJM theory from the matrix model, in particular the free energy which matches at strong coupling the classical supergravity action and gives the correct \(N^{3/2}\) scaling for the number of degrees of freedom of the M2 brane theory. Non-planar corrections are also calculated. The same matrix model appears also in the study of topological strings on a toric Calabi-Yau manifold, establishing a connection between the space of couplings of the planar ABJM theory and the moduli space of the latter. In particular it suggests that in addition to the usual perturbative and strong coupling (AdS) expansions a third expansion locus is the line where one of the two 't Hooft couplings vanishes and the other is finite. This is the conifold locus of the Calabi-Yau manifold and leads to an expansion around topological Chern-Simons theory, in which again some explicit results are obtained for the partition function and Wilson loop observables.
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