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Singular 7-manifolds with G_2 holonomy and intersecting 6-branes

Klaus Behrndt

TL;DR

The paper presents a constructive framework for 7-manifolds with $G_2$ holonomy, realized as $\mathbb{R}^3$ bundles over singular quaternionic bases whose metric depends on three parameters, including an interpolation between $S^4$ and $\mathbb{CP}^2$. By solving the Killing spinor equations and enforcing a closed/co-closed 3-form, explicit first-order relations for the metric functions are obtained, yielding a concrete $G_2$ metric. The base spaces are explored in homogeneous and four-isometry (non-homogeneous) cases, revealing an interpolating geometry that contains curvature singularities and multiple fixed points of isometries. In the M-theory reduction to IIA, the fixed points of Killing vectors correspond to D6-branes, with the possibility of constructing intersecting brane configurations and discussions of brane charges, dilaton behavior, and potential resolutions of singularities via membrane instantons or fluxes. The work also outlines prospects for extending to Spin$(7)$ holonomy and emphasizes the role of fluxes and instantons in smoothing the geometry, highlighting potential practical implications for brane world scenarios and flux compactifications.

Abstract

A 7-manifold with G_2 holonomy can be constructed as a R^3 bundle over a quaternionic space. We consider a quaternionic base space which is singular and its metric depends on three parameters, where one of them corresponds to an interpolation between S^4 and CP^2 or its non-compact analogs. This 4-d Einstein space has four isometries and the fixed point set of a generic Killing vector is discussed. When embedded into M-theory the compactification over a given Killing vector gives intersecting 6-branes as IIA configuration and we argue that membrane instantons may resolve the curvature singularity.

Singular 7-manifolds with G_2 holonomy and intersecting 6-branes

TL;DR

The paper presents a constructive framework for 7-manifolds with holonomy, realized as bundles over singular quaternionic bases whose metric depends on three parameters, including an interpolation between and . By solving the Killing spinor equations and enforcing a closed/co-closed 3-form, explicit first-order relations for the metric functions are obtained, yielding a concrete metric. The base spaces are explored in homogeneous and four-isometry (non-homogeneous) cases, revealing an interpolating geometry that contains curvature singularities and multiple fixed points of isometries. In the M-theory reduction to IIA, the fixed points of Killing vectors correspond to D6-branes, with the possibility of constructing intersecting brane configurations and discussions of brane charges, dilaton behavior, and potential resolutions of singularities via membrane instantons or fluxes. The work also outlines prospects for extending to Spin holonomy and emphasizes the role of fluxes and instantons in smoothing the geometry, highlighting potential practical implications for brane world scenarios and flux compactifications.

Abstract

A 7-manifold with G_2 holonomy can be constructed as a R^3 bundle over a quaternionic space. We consider a quaternionic base space which is singular and its metric depends on three parameters, where one of them corresponds to an interpolation between S^4 and CP^2 or its non-compact analogs. This 4-d Einstein space has four isometries and the fixed point set of a generic Killing vector is discussed. When embedded into M-theory the compactification over a given Killing vector gives intersecting 6-branes as IIA configuration and we argue that membrane instantons may resolve the curvature singularity.

Paper Structure

This paper contains 12 sections, 56 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Constructing metrics with $G_2$ holonomy
  3. The metric ansatz
  4. Solving the Killing spinor equations
  5. The closed and co-closed 3-form
  6. Explicit metric of the 7-manifold
  7. Homogeneous quaternionic spaces
  8. Quaternionic spaces with four isometries
  9. Isometries and IIA description
  10. Isometries
  11. Fixed point set
  12. Outlook: Membrane instantons and new spin(7) metric

Figures (2)

  • Figure 1: The multiplication table of imaginary components of octonians: $i_1 i_2 = i_3$, $i_6 i_2 = i_4$, $i_4 i_7 = i_1$, ... can be obtained from this figure by following the lines in the direction of the arrows.
  • Figure 2: A positive definite metric requires: $V>0$ as well as $\rho^2 > n^2$. Hence, there are two allowed coordinate regions, as we have indicated by the arrows.