Critical Points and Phase Transitions in 5D Compactifications of M-Theory

TL;DR

This paper analyzes critical points and phase transitions in five-dimensional M-theory compactifications on Calabi-Yau three-folds, revealing simple stabilization equations, the nature of phase transitions, and the behavior of black hole and string potentials across flop transitions.

Contribution

It provides a simplified formulation of stabilization equations in 5D, explores phase transitions in the extended K"ahler cone, and characterizes the continuity properties of potentials during topological changes.

Findings

Black hole and string potentials exhibit specific continuity properties across flop transitions.

Supersymmetric stabilization occurs at most once within the extended K"ahler cone.

Electric states and magnetic strings become massless or tensionless at the cone boundaries.

Abstract

We study critical points of the BPS mass $Z$, the BPS string tension $Z_m$, the black hole potential $V$ and the gauged central charge potential $P$ for M-theory compactified on Calabi-Yau three-folds. We first show that the stabilization equations for $Z$ (determining the black hole entropy) take an extremely simple form in five dimensions as opposed to four dimensions. The stabilization equations for $Z_m$ are also very simple and determine the size of the infinite $adS_3$-throat of the string. The black hole potential in general exhibits two classes of critical points: supersymmetric critical points which coincide with those of the central charge and non-supersymmetric critical points. We then generalize the discussion to the entire extended K\"ahler cone encompassing topologically different but birationally equivalent Calabi-Yau three-folds that are connected via flop transitions. We examine behavior of the four potentials to probe the nature of these phase transitions. We find that $V$ and $P$ are continuous but not smooth across the flop transition, while $Z$ and its first two derivatives, as well as $Z_m$ and its first derivative, are continuous. This in turn implies that supersymmetric stabilization of $Z$ and $Z_m$ for a given configuration takes place in at most one point throughout the entire extended K\"ahler cone. The corresponding black holes (or string states) interpolate between different Calabi-Yau three-folds. At the boundaries of the extended K\"ahler cone we observe that electric states become massless and/or magnetic strings become tensionless.