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The heterotic G$_2$ system with reducible characteristic holonomy

Mateo Galdeano, Leander Stecker

TL;DR

This work addresses constructing supersymmetric heterotic vacua with $G_2$-structure on seven-manifolds by exploiting almost contact geometries with reduced characteristic holonomy. It introduces a one-parameter family of $G_2$-connections $\nabla^{\lambda}$ and analyzes both exact and approximate solutions within the $3$-$(\alpha,\delta)$-Sasaki and $(\alpha,\delta)$-Sasaki frameworks, including a new exact solution in the degenerate case $\delta=0$ when $\alpha^2=1/(12\alpha')$. For the $\mathrm{Sp}(1)\mathrm{Sp}(1)$ branch, exact solutions are obtained in the degenerate setting while Bianchi-identity constraints yield interesting algebraic relations for $A$ and $\Theta$; for the $\mathrm{SU}(3)$ branch, $G_2$-instantons occur for specific $\lambda$ but exact Bianchi satisfaction is not achieved, though approximate solutions exist in special scaling limits. The results connect spinorial descriptions of $G_2$-structures with submersion geometry and highlight the role of $\alpha'$ in shaping the AdS$_3$ vacua and potential supersymmetry enhancements, offering a foundation for further moduli and Swampland investigations.

Abstract

We construct solutions to the heterotic G$_2$ system on almost contact metric manifolds with reduced characteristic holonomy. We focus on $3$-$(α,δ)$-Sasaki manifolds and $(α,δ)$-Sasaki manifolds, the latter being a convenient reformulation of spin $η$-Einstein $α$-Sasaki manifolds. Investigating a $1$-parameter family of G$_2$-connections on the tangent bundle, we obtain several approximate solutions as well as one new class of exact solutions on degenerate $3$-$(α,δ)$-Sasaki manifolds.

The heterotic G$_2$ system with reducible characteristic holonomy

TL;DR

This work addresses constructing supersymmetric heterotic vacua with -structure on seven-manifolds by exploiting almost contact geometries with reduced characteristic holonomy. It introduces a one-parameter family of -connections and analyzes both exact and approximate solutions within the --Sasaki and -Sasaki frameworks, including a new exact solution in the degenerate case when . For the branch, exact solutions are obtained in the degenerate setting while Bianchi-identity constraints yield interesting algebraic relations for and ; for the branch, -instantons occur for specific but exact Bianchi satisfaction is not achieved, though approximate solutions exist in special scaling limits. The results connect spinorial descriptions of -structures with submersion geometry and highlight the role of in shaping the AdS vacua and potential supersymmetry enhancements, offering a foundation for further moduli and Swampland investigations.

Abstract

We construct solutions to the heterotic G system on almost contact metric manifolds with reduced characteristic holonomy. We focus on --Sasaki manifolds and -Sasaki manifolds, the latter being a convenient reformulation of spin -Einstein -Sasaki manifolds. Investigating a -parameter family of G-connections on the tangent bundle, we obtain several approximate solutions as well as one new class of exact solutions on degenerate --Sasaki manifolds.
Paper Structure (16 sections, 44 theorems, 180 equations, 2 figures)

This paper contains 16 sections, 44 theorems, 180 equations, 2 figures.

Table of Contents

  1. Introduction
  2. The heterotic G2 system
  3. Contact Geometry
  4. Characteristic holonomy Sp(1)Sp(1)
  5. 3-$(\alpha,\delta)$-Sasaki manifolds
  6. The family of connections $\nabla^\lambda$
  7. Solving the heterotic G$_2$ system
  8. Characteristic holonomy SU(3)
  9. $(\alpha,\delta)$-Sasaki manifolds
  10. The family of connections $\nabla^\lambda$
  11. Solving the heterotic G$_2$ system
  12. Conclusion and outlook
  13. Spinor conventions
  14. $\mathrm{G}_2$-structures in terms of spinors
  15. SU(m)-structures in terms of spinors
  16. ...and 1 more sections

Key Result

Theorem 1.1

Let $\alpha'>0$ and let $(M,g,\xi_i,\eta_i,\phi_i)_{i=1,2,3}$ be a degenerate $7$-dimensional $3$-$(\alpha,\delta)$-Sasaki manifold with its canonical G$_2$-structure $\varphi$ and canonical connection $\nabla$ with torsion $T$. If $\alpha^2=\frac{1}{12\alpha'}$, then where $\beta=2(\delta-2\alpha)$, is a solution to the heterotic G$_2$ system.

Figures (2)

  • Figure 1: Diagram of groups.
  • Figure 2: Distinguished values of $(\alpha,\delta)$ for an $(\alpha,\delta)$-Sasaki manifold.

Theorems & Definitions (118)

  • Theorem 1.1
  • Definition 2.1
  • Remark 2.2
  • Proposition 2.3
  • Definition 2.4
  • Definition 2.5
  • Definition 2.6
  • Proposition 2.7: Friedrich:2001nh
  • Definition 2.8
  • Definition 2.9
  • ...and 108 more