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In this study, we examine effective field theories of superconducting phaseswith topological order, making connection to proposed realizations of exotictopological phases(including those hosting Ising and Fibonacci anyons) insuperconductor-quantum Hall heterostructures. Our effective field theories forthe non-Abelian superconducting states are non-Abelian Chern-Simons theories inwhich the condensation of vortex-quasiparticle composites lead to theassociated Abelian quantum Hall states. This Chern-Simons-Higgs condensationprocess is dual to the emergence of superconducting non-Abelian topologicalphases in coupled chain constructions. In such transitions, the chiral centralcharge of the system generally changes, so they fall outside the description ofbosonic condensation transitions put forth by Bais and Slingerland (though thetwo approaches agree when the described transitions coincide). Our condensationprocess may be generalized to Chern-Simons theories based on arbitrary Liegroups, always describing a transition from a Lie Algebra to its Cartansubalgebra. We include several instructive examples of such transitions.

Chern-Simons-Higgs transitions out of Topological Superconducting Phases