Bose-Fermi identities and Bailey flows in statistical mechanics and conformal field theory

The Rogers-Ramanujan identities express the duality between bosonic and fermionic bases in two-dimensional conformal field theories. Bosonic bases are usually obtained from the Feigin and Fuchs construction, whereas fermionic bases are suitable for studying massive integrable perturbations of conformal field theories or problems in condensed matter physics which utilize quasiparticle descriptions. Equating the partition functions (or more precisely the characters or branching functions) of these theories in the different bases, yields Rogers-Ramanujan type or Bose-Fermi identities. In this dissertation many new identities are found for A{dollar}sb1sp{lcub}(1){rcub}{dollar} coset conformal field theories with one integer and one fractional level, and for {dollar}N=1{dollar} and {dollar}N=2{dollar} superconformal models. Some of these identities involve new q-functions such as q-multinomial and q-supernomial coefficients. Two methods of proof are used, the L-difference method and the Bailey method. The Bailey method is particularly interesting because it allows conformal field theory characters or branching functions to be derived from polynomial identities of simpler conformal field theories. This construction is referred to as the Bailey flow. A new higher-level version of Bailey's lemma is presented which yields a Bailey flow from the minimal models {dollar}M(p,pspprime){dollar} to the A{dollar}sb1sp{lcub}(1){rcub}{dollar} cosets with one integer and one fractional level.