On membrane interactions and a three-dimensional analog of Riemann surfaces

TL;DR

This paper introduces a novel continuum approximation method for constructing BPS instanton solutions in the pp-wave matrix model, modeling membrane splitting and joining in M-theory through solutions of a 3D Laplace equation on a Riemann surface analog.

Contribution

It presents a new approach to explicitly construct instanton solutions for membrane interactions using a continuum approximation and a 3D Laplace equation mapping.

Findings

Explicit analytic solutions for instantons are provided.

The continuum approximation simplifies the analysis of membrane interactions.

Membrane splitting/joining processes are described via solutions on a 3D Riemann surface analog.

Abstract

Membranes in M-theory are expected to interact via splitting and joining processes. We study these effects in the pp-wave matrix model, in which they are associated with transitions between states in sectors built on vacua with different numbers of membranes. Transition amplitudes between such states receive contributions from BPS instanton configurations interpolating between the different vacua. Various properties of the moduli space of BPS instantons are known, but there are very few known examples of explicit solutions. We present a new approach to the construction of instanton solutions interpolating between states containing arbitrary numbers of membranes, based on a continuum approximation valid for matrices of large size. The proposed scheme uses functions on a two-dimensional space to approximate matrices and it relies on the same ideas behind the matrix regularisation of membrane degrees of freedom in M-theory. We show that the BPS instanton equations have a continuum counterpart which can be mapped to the three-dimensional Laplace equation through a sequence of changes of variables. A description of configurations corresponding to membrane splitting/joining processes can be given in terms of solutions to the Laplace equation in a three-dimensional analog of a Riemann surface, consisting of multiple copies of R^3 connected via a generalisation of branch cuts. We discuss various general features of our proposal and we also present explicit analytic solutions.