Invariances of conformal field theories (CFTs) would seem to suggest that correlation functions behave as power laws. However, logarithms also exhibit conformal invariance. When logarithms are permitted in two-dimensional CFTs, the corresponding state spaces characteristically involve reducible but indecomposable Virasoro modules with non-diagonalisable algebra action. Notably, such state spaces no longer naturally admit a grading into energy eigenspaces. Despite this non-diagonalisable energy operator, one finds significant physical motivation for the study of such representations, with many interesting statistical mechanical models exhibiting this behaviour: percolation, dilute polymers, self-avoiding walks, and more. A vast amount of effort has been made in the study of these two-dimensional logarithmic CFTs, both their internal structure and their fusion rules. However, it would be fair to say that logarithmic CFTs are still less well understood than their non-logarithmic counterparts. In recent years, the relevance of free-field oscillator algebras to the study of such representations has become more and more apparent. Many of the module structures in question might more appropriately be considered as Fock-type spaces.
In this thesis we develop free field realisations of logarithmic CFTs. We analyse some general features, examining staggered modules of the Virasoro algebra in particular, before providing a construction for staggered modules consisting of Fock spaces considered as Virasoro modules. We derive an explicit formula for a module invariant of staggered Fock modules, verifying that the given construction agrees with those seen to date in the literature. We then turn to more conjectural areas, examining how the non-diagonalisaiblity of the Virasoro representation can be reproduced by the inclusion of additional modes into the underlying oscillator algebras, and how the states created by these modes correspond to the vacuum evaluations of logarithmic fields. We take these as our motivating examples for a subsequent working definition of logarithmic vertex operator algebras, in the hope that not only do their state spaces correspond to the staggered structures developed to this point, but that they provide an additional avenue of approach in the construction and study of logarithmic conformal field theories
