I have read recently that the partition function / half-BPS wilson vev (w/ NG probe) of a Chern-Simons matter theory with N=6 U(N)k x U(N)-k super-conformal symmetry (ABJM) on S3 is proportional to pure gravity on mini-superspace -- ignoring radiative / higher-curvature corrections, membrane / worldsheet instantons (in the case of holographic dual type 2a AdS4 x CP3), winding and higher string probes (in the case of half-BPS wilson vev), etc.
The statement is made using airy, which I can last recall from Kontsevich's proof of the Witten conjecture (interestingly enough, motivated by equivalent partition functions / intersection counting of moduli stack on curves of, in that case two models of two dimensional gravity).
Can one write the partition function of a level k=1 (or general k) Chern-Simons matter theory, or pure Chern-Simons / Chern-Simons matter with one loop renormalization in terms of ABJM partition function / half-BPS wilson vev? How are the partition functions of ABJM and Chern-Simons matter / Chern-Simons pure related? Can Chern-Simons matter / Chern-Simons pure partition function be written in terms of Airy?
Any comment / suggestion to any of the above questions is greatly appreciated.
Edit: I realize maximal Chern-Simons / ABJM may be obviously related to pure, but a reading that has detailed this I have yet to come across.