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It is widely recognized that the main difficulty in designing devices whichcould process information using quantum states is due to the decoherence oflocal excitations about a ground state. A solution to this problem wassuggested in \cite{Kitaev}, relying on (non-local) topological excitations,structurally protected against local noise. However, a practical implementationof this proposal using special Landau levels in fractional quantum Hall effectsystems (FQHE) \cite{QHE} has proven elusive, while accessible FQHE states aretheoretically not optimal because their representations in the Hilbert space ofstates are not dense. We propose using a different physical system (cold Fermiatoms), whose semiclassical dynamics is described by a hyperelliptic functionin the Sklyanin formalism. The homological structure of the complex curvecorresponds to representations of the braid group, with the action of Heckeoperators leading to singularities detectable in the semiclassicaloscillations. We argue that, for a fixed genus of the hyperelliptic curve, theRichardson-Gaudin pairing Hamiltonian problem is the singular limit $m \to\infty$ of level-$k$ $\widehat{sl}(2)$, with $k+2 = \frac{4}{8m+1}\to 0$, sothat the level $k$ is admissible in the sense of Kac and Kazhdan \cite{KK}, butthe corresponding Hecke algebra is a $q-$deformation of the symmetric groupwith fixed $q = e^{i\pi/4}$, as $m \to \infty$.

Braid group representations and cold Fermi gases in the fast pairing regime