Large-small Dualities between Periodic Collapsing/expanding Branes and Brane Funnels

We consider space and time dependent fuzzy spheres S 2p arising in D1 − D(2p + 1) intersections in IIB string theory and collapsing D(2p)-branes in IIA string theory. In the case of S 2 , 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 S 4 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 S