Non-Collapsing Membrane Instantons in Higher Dimensions

TL;DR

This paper explores non-collapsing membrane instantons in higher dimensions, revealing solutions with conserved charges that prevent collapse and exhibit periodic motion, extending complex Toda equations.

Contribution

It introduces new non-collapsing membrane instanton solutions in seven dimensions with SU(3) symmetry and extends complex Toda equations to complex space.

Findings

Membranes exhibit non-zero conserved charges preventing collapse.

Solutions include periodic, bouncing membrane instantons.

Extensions of Toda equations to complex space are derived.

Abstract

We introduce a particular embedding of seven dimensional self-duality membrane equations in C^3\times R which breaks G_2 invariance down to SU(3). The world-volume membrane instantons define SU(3) special lagrangian submanifolds of C^3. We discuss in detail solutions for spherical and toroidal topologies assuming factorization of time. We show that the extra dimensions manifest themselves in the solutions through the appearance of a non-zero conserved charge which prevents the collapse of the membrane. We find non-collapsing rotating membrane instantons which contract from infinite size to a finite one and then they bounce to infinity in finite time. Their motion is periodic. These generalized complex Nahm equations, in the axially symmetric case, lead to extensions of the continuous Toda equation to complex space.