We use a quite concrete and simple realization of $\slq$ involving finite difference operators. We interpret them as derivations (in the non-commutative sense) on a suitable graded algebra, which gives rise to the double of the projective line as the non commutative version of the standard homogeneous space.

1 aFranco, Davide1 aReina, Cesare uhttp://hdl.handle.net/1963/353801654nas a2200121 4500008004300000245006200043210006000105260001300165520127600178100002401454700001801478856003601496 1992 en_Ud 00aTopological "observables" in semiclassical field theories0 aTopological observables in semiclassical field theories bElsevier3 aWe give a geometrical set up for the semiclassical approximation to euclidean field theories having families of minima (instantons) parametrized by suitable moduli spaces ${\mathcal{M}}$. The standard examples are of course Yang-Mills theory and non-linear $\sigma$-models. The relevant space here is a family of measure spaces $\tilde{\mathcal{N}} \rightarrow \mathcal{M}$, with standard fibre a distribution space, given by a suitable extension of the normal bundle to $\mathcal{M}$ in the space of smooth fields. Over $\tilde{\mathcal{N}}$ there is a probability measure $d\mu$ given by the twisted product of the (normalized) volume element on $\mathcal{M}$ and the family of gaussian measures with covariance given by the tree propagator $C_\phi$ in the background of an instanton $\phi \in \mathcal{M}$. The space of "observables", i.e. measurable functions on ($\tilde{\mathcal{N}},\, d\mu$), is studied and it is shown to contain a topological sector, corresponding to the intersection theory on $\mathcal{M}$. The expectation value of these topological "observables" does not depend on the covariance; it is therefore exact at all orders in perturbation theory and can moreover be computed in the topological regime by setting the covariance to zero.

1 aNolasco, Margherita1 aReina, Cesare uhttp://hdl.handle.net/1963/354100855nas a2200121 4500008004100000245005400041210005300095260003400148520047700182100001800659700002100677856003500698 1990 en d00aN=2 super Riemann surfaces and algebraic geometry0 aN2 super Riemann surfaces and algebraic geometry bAmerican Institute of Physics3 aThe geometric framework for N=2 superconformal field theories are described by studying susy2 curves-a nickname for N=2 super Riemann surfaces. It is proved that \\\"single\\\'\\\' susy2 curves are actually split supermanifolds, and their local model is a Serre self-dual locally free sheaf of rank two over a smooth algebraic curve. Superconformal structures on these sheaves are then examined by setting up deformation theory as a first step in studying moduli problems.1 aReina, Cesare1 aFalqui, Gregorio uhttp://hdl.handle.net/1963/80700355nas a2200109 4500008004100000245005700041210005500098260001800153100001800171700002100189856003500210 1990 en d00aA note on the global structure of supermoduli spaces0 anote on the global structure of supermoduli spaces bSISSA Library1 aReina, Cesare1 aFalqui, Gregorio uhttp://hdl.handle.net/1963/80600306nas a2200109 4500008004100000245003200041210003100073260001800104100002100122700001800143856003500161 1988 en d00aSusy-curves and supermoduli0 aSusycurves and supermoduli bSISSA Library1 aFalqui, Gregorio1 aReina, Cesare uhttp://hdl.handle.net/1963/761