2D superconductivity: Classification of universality classes by infinite symmetry

I consider superconducting condensates which become incompressible in the infinite gap limit. Classical 2D incompressible fluids possess the dynamical symmetry of area-preserving diffeomorphisms. I show that the corresponding infinite dynamical symmetry of 2D superconducting fluids is the coset W1+∞⊗W¯1+∞U(1)diagonal, with W1+∞ the chiral algebra of quantum area-preserving diffeomorphisms and I derive its minimal models. These define a discrete set of 2D superconductivity universality classes which fall into two main categories: conventional superconductors with their vortex excitations and unconventional superconductors. These are characterized by a broken U(1)vector⊗U(1)axial symmetry and are labeled by an integer level m. They possess neutral spinon excitations of fractional spin and statistics S=θ2π=m−12m, which carry also an SU(m) isospin quantum number; this hidden SU(m) symmetry implies that these anyon excitations are non-Abelian. The simplest unconventional superconductor is realized for m=2: in this case the spinon excitations are semions (half-fermions). My results show that spin-charge separation in 2D superconductivity is a universal consequence of the infinite symmetry of the ground state. This infinite symmetry and its superselection rules realize a quantum protectorate in which the neutral spinons can survive even as soft modes on a rigid, spinless charge condensate.