  We study the structure of the Kauffman algebra of a surface with parameter
equal to sqrt(-1). We obtain an interpretation of this algebra as an algebra of
parallel transport operators acting on sections of a line bundle over the
moduli space of flat connections in a trivial SU(2)-bundle over the surface. We
analyse the asymptotics of traces of curve-operators in TQFT in non standard
regimes where the root of unity parametrizing the TQFT accumulates to a root of
unity. We interpret the case of sqrt(-1) in terms of parallel transport
operators.
