Non-Collapsing Membrane Instantons in Higher Dimensions.pdf

a r X i v : h e p - t h / 0 2 0 8 1 5 1 v 1 2 1 A u g 2 0 0 2 CERN-TH/02-198 hep-th/0208151 Non-Collapsing Membrane Instantons in Higher Dimensions E.G. Floratos a,b,d and G.K. Leontaris c,d aInstitute of Nuclear Physics, NRCS Demokritos, Athens, Greece b Physics Department, University of Athens, Athens, Greece cTheoretical Physics Division, Ioannina University, GR-45110 Ioannina, Greece d CERN, Theory Division, 1211 Geneva 23, Switzerland Abstract We introduce a particular embedding of seven dimensional self-duality membrane equations in C3×R which breaks G2 invariance down to SU(3). The world-volume membrane instantons define SU(3) spe- cial lagrangian submanifolds of C3. 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. CERN-TH/02-198 August 2002 –2– 1. Introduction Some years ago, we introduced the notion of the self-duality for supermembrane in 4 + 1-dimensions and in the light-cone gauge. The corresponding self-duality (s-d) equations proved to be an integrable system with an infinite number of conservation laws and particular solutions were found [1, 2] which were collapsing configurations of membrane instantons to point-like or string-like objects. Similar covariant self-duality equations have been introduced before for 2 + 1-dimensions [3] and later generalized to 6 + 1 dimensions in [4]. These objects, represent world-volume instantons of the supermembrane. In the light cone gauge, the world-volume time and the target tim