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

TL;DR

This paper explores dualities between space and time-dependent fuzzy spheres in string theory, revealing mathematical structures involving elliptic functions and Riemann surfaces that relate collapsing and expanding brane solutions.

Contribution

It introduces a duality between space and time solutions for fuzzy spheres, connecting elliptic functions and Riemann surface symmetries in brane configurations.

Findings

Duality r to 1/r relates space and time solutions for S^2.

Solutions involve Jacobi elliptic functions and hyper-elliptic Riemann surfaces.

Symmetries enable reduction to simpler mathematical problems.

Abstract

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 ${1 \over r}$ 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 \over r} $ duality. Some of these considerations extend to the case of the fuzzy $S^6$.