In this study, we examine effective field theories of superconducting phases with topological order, making connection to proposed realizations of exotic topological phases(including those hosting Ising and Fibonacci anyons) in superconductor-quantum
Hall heterostructures. Our effective field theories for the non-Abelian superconducting states are non-Abelian Chern-Simons theories in which the condensation of vortex-quasiparticle composites lead to the associated Abelian quantum Hall states. This Chern-Simons-Higgs condensation process is dual to the emergence of superconducting non-Abelian topological phases in coupled chain constructions. In such transitions, the chiral central charge of the system generally changes, so they fall outside the description of bosonic condensation transitions put forth by Bais and Slingerland (though the two approaches agree when the described transitions coincide). Our condensation process may be generalized to Chern-Simons theories based on arbitrary Lie groups, always describing a transition from a Lie Algebra to its Cartan subalgebra. We include several instructive examples of such transitions.
