Large-small dualities between periodic collapsing/expanding branes and brane funnels

We consider space and time dependent fuzzy spheres S2p arising in D1-D(2p+1) intersections in IIB string theory and collapsing D(2p)-branes in IIA string theory. In the case of S2, where the periodic space and time-dependent solutions can be described by Jacobi elliptic functions, there is a duality of the form r to 1/r which relates the space and time dependent solutions. This duality is related to complex multiplication properties of the Jacobi elliptic functions. For S4 funnels, the description of the periodic space and time dependent solutions involves the Jacobi inversion problem on a hyper-elliptic Riemann surface of genus 3. Special symmetries of the Riemann surface allow the reduction of the problem to one involving a product of genus one surfaces. The symmetries also allow a generalisation of the r to 1/r duality. Some of these considerations extend to the case of the fuzzy S6.