The noncommutative geometry of elliptic difference equations

Abstract: We give a new construction of noncommutative surfaces via elliptic difference operators, attaching a 1-parameter noncommutative deformation to any projective rational surface with smooth anticanonical curve. The construction agrees with one implicit in work of Van den Bergh (iterated blowups of noncommutative Hirzebruch surfaces), but the construction enables one to prove a number of new facts about these surfaces. We show that they are noncommutative smooth proper surfaces in the sense of Chan and Nyman, with projective Quot schemes, that moduli spaces of simple sheaves are Poisson and that moduli spaces classifying semistable sheaves of rank 0 or 1 are projective. We further show that the action of SL_2(Z) as derived autoequivalences of rational elliptic surfaces extends to an action as derived equivalences of surfaces in our family with K^2=0. We also discuss applications to the theory of special functions arising by interpreting moduli spaces of 1-dimensional sheaves as moduli spaces of difference equations. When the moduli space is a single point, the equation is rigid, and we give an integral representation for the solutions. More generally, twisting by line bundles corresponds to isomonodromy deformations, so this gives rise to Lax pairs. When the moduli space is 2-dimensional, one obtains Lax pairs for the elliptic Painlev\'e equation; this associates a Lax pair to any rational number, of order twice the denominator. There is also an elliptic analogue of the Riemann-Hilbert correspondence: an analytic equivalence between categories of elliptic difference equations, swapping the role of the shift of the equation and the nome of the curve.