Abstract
We consider space and time dependent fuzzy spheres S 2 p arising in D 1 – D ( 2 p + 1 ) intersections in IIB string theory and collapsing D ( 2 p ) -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 6 .
Highlights
Fuzzy spheres of two, four, six dimensions arise in a variety of related contexts
It is of interest to consider using the constraint to solve u1 in terms of u2 and describe x1 as a function of u2. The reason for this is that the automorphisms of the Riemann surface allow us to relate the large r behaviour of the spatial problem described in terms of the u1 variable, to the small r behaviour of the time dependent problem described in terms of the u2 variable and vice versa
The space and time dependence of fuzzy spheres S2, S4 and S6 are governed by equations which follow from the DBI action of D-branes
Summary
Four, six dimensions arise in a variety of related contexts. On the one hand they describe the cross-sections of fuzzy funnels appearing at the intersection of. In the case of the fuzzy 2-sphere, there are purely spatial and purely time-dependent solutions described in terms of Jacobi elliptic functions. For solutions described in terms of elliptic functions, the inversion symmetry is related to the property of complex multiplication u1 → iu. Using the conservation laws of the spatial or time evolution, the time elapsed or distance along the 1-brane can be expressed in terms of an integral of a holomorphic differential on the genus 3 hyper-elliptic curve. The solution r(t, σ) can still be related to a constrained Jacobi inversion problem, which can be solved in terms of genus 3 Riemann theta functions. Appendix C describes the derivation of the Jacobi-Cn solution for the fuzzy S2 by steps using Weierstrass ℘ functions, since the discussion of section 5 on the fuzzy S6 is expressed in terms of higher genus generalisations of the ℘ functions
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