ABJM theory as a Fermi gas

Marcos Marino, Pavel Putrov

The partition function on the three-sphere of many supersymmetric
Chern-Simons-matter theories reduces, by localization, to a matrix model. We
develop a new method to study these models in the M-theory limit, but at all
orders in the 1/N expansion. The method is based on reformulating the matrix
model as the partition function of an ideal Fermi gas with a non-trivial,
one-particle quantum Hamiltonian. This new approach leads to a completely
elementary derivation of the N^{3/2} behavior for ABJM theory and N=3 quiver
Chern-Simons-matter theories. In addition, the full series of 1/N corrections
to the original matrix integral can be simply determined by a next-to-leading
calculation in the WKB or semiclassical expansion of the quantum gas, and we
show that, for several quiver Chern-Simons-matter theories, it is given by an
Airy function. This generalizes a recent result of Fuji, Hirano and Moriyama
for ABJM theory. It turns out that the semiclassical expansion of the Fermi gas
corresponds to a strong coupling expansion in type IIA theory, and it is dual
to the genus expansion. This allows us to calculate explicitly non-perturbative
effects due to D2-brane instantons in the AdS background.