Wrapped M-branes and Three-dimensional Topologies

TL;DR

This paper explores how three-dimensional topologies of M-theory membranes can be constructed via Dehn surgery along knots, linking topology, gauge theory, and phase transitions in a novel way.

Contribution

It introduces a model connecting M-theory membrane topologies with knot theory and gauge fields, proposing a phase transition related to topology fluctuations.

Findings

Topologies are constructed via Dehn surgery along knot lines.

A phase transition occurs at a critical string coupling.

Topology fluctuations are irrelevant in one phase and condense in another.

Abstract

The three-dimensional topologies of the membrane of M-theory can be constructed by performing Dehn surgery along knot lines. We investigate membranes wrapped around a circle and the correponding subset of topologies (Seifert manifolds). The knot lines are interpreted as magnetic flux tubes in an XY model coupled to Maxwell theory. In this model the eleventh dimension of M-theory gets ``eaten'' by the world-brane metric. There is argued to be a second-order phase transition at a critical value of the string coupling constant. The topology fluctuations that correspond to the knot lines are irrelevant in one phase while they condense in the other phase.