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Rolling among G_2 vacua

H. Partouche, B. Pioline

TL;DR

The work develops a framework to study topology-changing transitions between $G_2$ manifolds constructed as quotients $(\mathcal{C}\times S^1)/\sigma$ of CICYs by antiholomorphic involutions, focusing on fixed-point-free cases to enable controlled analysis. By translating CY conifold transitions into the $G_2$ setting, the authors connect geometric transitions to low-energy ${\cal N}=1$ dynamics, illustrating Higgsing and branch changes in the effective theory, and they provide explicit examples with involutions on $\mathbb{C}P^7$ factors that yield different Betti-number patterns. The paper also builds a web of connected $G_2$ manifolds via determinantal contractions, showing how fixed-point data can alter the spectrum and exploring the possibility of mirror-like pairs with constant $b_2+b_3$. Finally, it discusses fixed-point cases at conifold points, highlighting new phenomena and signaling further work on Joyce-type fixed loci and their phase structure. The results advance understanding of the landscape of $G_2$ vacua and their non-perturbative transitions in M-theory.

Abstract

We consider topology-changing transitions between 7-manifolds of holonomy G_2 constructed as a quotient of CY x S^1 by an antiholomorphic involution. We classify involutions for Complete Intersection CY threefolds, focussing primarily on cases without fixed points. The ordinary conifold transition between CY threefolds descends to a transition between G_2 manifolds, corresponding in the N=1 effective theory incorporating the light black hole states either to a change of branch in the scalar potential or to a Higgs mechanism. A simple example of conifold transition with a fixed nodal point is also discussed. As a spin-off, we obtain examples of G_2 manifolds with the same value for the sum of Betti numbers b_2+b_3, and hence potential candidates for mirror manifolds.

Rolling among G_2 vacua

TL;DR

The work develops a framework to study topology-changing transitions between manifolds constructed as quotients of CICYs by antiholomorphic involutions, focusing on fixed-point-free cases to enable controlled analysis. By translating CY conifold transitions into the setting, the authors connect geometric transitions to low-energy dynamics, illustrating Higgsing and branch changes in the effective theory, and they provide explicit examples with involutions on factors that yield different Betti-number patterns. The paper also builds a web of connected manifolds via determinantal contractions, showing how fixed-point data can alter the spectrum and exploring the possibility of mirror-like pairs with constant . Finally, it discusses fixed-point cases at conifold points, highlighting new phenomena and signaling further work on Joyce-type fixed loci and their phase structure. The results advance understanding of the landscape of vacua and their non-perturbative transitions in M-theory.

Abstract

We consider topology-changing transitions between 7-manifolds of holonomy G_2 constructed as a quotient of CY x S^1 by an antiholomorphic involution. We classify involutions for Complete Intersection CY threefolds, focussing primarily on cases without fixed points. The ordinary conifold transition between CY threefolds descends to a transition between G_2 manifolds, corresponding in the N=1 effective theory incorporating the light black hole states either to a change of branch in the scalar potential or to a Higgs mechanism. A simple example of conifold transition with a fixed nodal point is also discussed. As a spin-off, we obtain examples of G_2 manifolds with the same value for the sum of Betti numbers b_2+b_3, and hence potential candidates for mirror manifolds.

Paper Structure

This paper contains 20 sections, 62 equations, 2 figures.

Table of Contents

  1. Introduction
  2. $G_2$ manifolds as Calabi-Yau orbifolds
  3. General facts about $G_2$ manifolds
  4. Joyce's construction of $G_2$ manifold
  5. Antiholomorphic involutions of projective spaces
  6. Antiholomorphic involutions of CICY
  7. A review of conifold transition between CICY's
  8. An example
  9. The web of complete intersection CY's
  10. Black hole resolution of the conifold singularity
  11. Conifold transitions between G$_2$ manifolds
  12. An example
  13. Involution $B\times B$ on $\mathbb{C}\mathbb{P}^7 \times \mathbb{C}\mathbb{P}^1$
  14. Involution $B\times A$ on $\mathbb{C}\mathbb{P}^7 \times \mathbb{C}\mathbb{P}^1$
  15. Involution $B\times C$ on $\mathbb{C}\mathbb{P}^7 \times \mathbb{C}\mathbb{P}^1\times \mathbb{C}\mathbb{P}^1$
  16. ...and 5 more sections

Figures (2)

  • Figure 1: Sequences of transitions relating the 9 $G_2$ configuration matrices descending from ${\cal M}^{(1111)}$. Only the first column of the $G_2$ configuration matrices is shown, the other ones being identical to those appearing in the CY configuration matrices of Eq. (\ref{['seq']}). Dotted arrows denote transitions to non-freely acting orbifolds.
  • Figure 2: Sequences of transitions descending from the CICY sequence \ref{["seq'"]}. Only the first column of the matrices is shown, the other are as in (\ref{["seq'"]}).