Ensembles in M-theory and Holography

We discuss that the string/M-theory partition function requires a choice of ensembles, depending on which background fields are held fixed. The background fields correspond to worldvolume couplings in the effective action approach to the superstring, which we extrapolate to the M2-brane. One natural ensemble in this context, which we call the M2-ensemble, corresponds to fixing the value of the M-theory three-form potential. In holographic setups the choice of ensemble is important when comparing to observables in the dual field theory. Indeed, in AdS$_4$ holography the M2-ensemble does not map gravitational observables directly to field theory observables at a fixed rank $N$, but rather to observables in the grand canonical ensemble. We remark that many M2-brane partition functions take a simple form in this ensemble hinting at one-loop exactness. We also discuss how in AdS$_7$ holography, the M2-ensemble does correspond to the canonical ensemble in the field theory, i.e. the (2,0) theory at fixed rank $N$.