  The partition function for a canonical ensemble of 2D Coulomb charges in a
background potential (the Dyson gas) is realized as a vacuum expectation value
of a group-like element constructed in terms of free fermionic operators. This
representation provides an explicit identification of the partition function
with a tau-function of the 2D Toda lattice hierarchy. Its dispersionless
(quasiclassical) limit yields the tau-function for analytic curves encoding the
integrable structure of the inverse potential problem and parametric conformal
maps. A similar fermionic realization of partition functions for grand
canonical ensembles of 2D Coulomb charges in the presence of an ideal conductor
is also suggested. Their representation as Fredholm determinants is given and
their relation to integrable hierarchies, growth problems and conformal maps is
discussed.
