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

TL;DR

This work analyzes critical points of the BPS central charge $Z$, its magnetic counterpart $Z_m$, and the potentials $V$ (black-hole) and $P$ (gauged) for M-theory compactified on Calabi–Yau three-folds to five dimensions. By exploiting very special geometry, it derives simple stabilization relations such as $q_I = Z t_I$ for electric charges and $t_I = {C_{IJK} m^J m^K \over Z_m^2}$ for magnetic charges, and shows that extrema of $Z$ are minima. Across flop transitions, $Z$ and $Z_m$ remain continuous (and their relevant derivatives), while $V$ and $P$ remain continuous but develop a kink, implying the existence and uniqueness of supersymmetric stabilization throughout the extended Kähler cone. The paper also provides explicit CY examples illustrating how potentials behave near conifold-like boundaries, including the appearance of tensionless magnetic strings and massless electric states at the cone boundaries, and demonstrates that five-dimensional dynamics interpolate between different CY geometries. Overall, the results illuminate how topology-changing transitions in the internal space manifest in the moduli dynamics and the spectra of BPS states in five-dimensional M-theory compactifications.

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 Kähler 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 Kähler cone. The corresponding black holes (or string states) interpolate between different Calabi-Yau three-folds. At the boundaries of the extended Kähler cone we observe that electric states become massless and/or magnetic strings become tensionless.

Figures (7)

• Figure 1: Contour plots of $Z^2$, $V$ and $P$ for charge configurations $(q_S,q_T,q_U)=(1,1,1),(1,-1,1),(1,-1,-1)$
• Figure 2: The extended moduli space
• Figure 3: $Z^2$, $V$ and $P$ for $(q_U,q_W)=(4,2)$ (left) and $(q_U,q_T)= (2,2)$ (right)
• Figure 4: $Z^2,V,P$ for $q_U=3, \ q_T=2$
• Figure 5: Potential $P$ with 4 critical points