  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
${{W_{1+\infty} \otimes \bar W_{1+\infty}} \over U(1)_{\rm diagonal}}$, with
$W_{1+\infty}$ 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 $U(1)_{\rm vector} \otimes
U(1)_{\rm axial}$ symmetry and are labeled by an integer level $m$. They
possess neutral spinon excitations of fractional spin and statistics $S =
{\theta \over 2\pi} = {{m-1} \over 2m}$ 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.
