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The canonical symmetry reduction of string backgrounds

Aaron Kennon, Jeffrey Streets

TL;DR

The paper develops a canonical symmetry reduction framework for string backgrounds—metric connections with skew-symmetric torsion and reduced holonomy—via a Bismut-parallel vector field $V$ (with ∇V = 0). It shows that, under dimension reduction, the transverse geometry satisfies gradient string generalized Ricci soliton equations and is conformally co-closed with the conformal factor given by the soliton potential, across almost Hermitian, almost contact, SU(3), G2, and Spin(7) structures. The results unify and extend Hull–Strominger-type systems, establishing rigidity when $V = 0$ and a robust dimension-reduction mechanism when $V eq 0$, with explicit descriptions for SU(3), G2, and Spin(7) reductions and concrete homogeneous-space examples. This framework clarifies how torsion, holonomy, and Lee forms interact to produce self-similar solitons and codifies the structure of reduced transverse geometries as string solitons, enabling systematic construction and classification of string backgrounds with reduced holonomy. The findings have potential implications for string compactifications and geometric flows by providing a unifying, structure-preserving reduction principle across multiple geometries.

Abstract

String backgrounds, defined here as metric connections with skew-symmetric torsion and reduced holonomy, yield generalized Ricci solitons relative to the Lee vector field. By a variational argument using the string action, they are also gradient generalized Ricci solitons relative to a potential function. These two observations combine to yield a canonical symmetry, and in this work we derive fundamental features of the transverse geometry, and rigidity phenomena. We prove in a unified conceptual fashion that the transverse geometry satisfies the string generalized Ricci soliton equations (a simplified Hull-Strominger system) in many settings including almost Hermitian, almost contact, $SU(3)$, $G_2$, and $\mathrm{Spin}(7)$ geometry. We also show that the transverse geometry is always conformally co-closed, with the conformal factor given by the associated soliton potential.

The canonical symmetry reduction of string backgrounds

TL;DR

The paper develops a canonical symmetry reduction framework for string backgrounds—metric connections with skew-symmetric torsion and reduced holonomy—via a Bismut-parallel vector field (with ∇V = 0). It shows that, under dimension reduction, the transverse geometry satisfies gradient string generalized Ricci soliton equations and is conformally co-closed with the conformal factor given by the soliton potential, across almost Hermitian, almost contact, SU(3), G2, and Spin(7) structures. The results unify and extend Hull–Strominger-type systems, establishing rigidity when and a robust dimension-reduction mechanism when , with explicit descriptions for SU(3), G2, and Spin(7) reductions and concrete homogeneous-space examples. This framework clarifies how torsion, holonomy, and Lee forms interact to produce self-similar solitons and codifies the structure of reduced transverse geometries as string solitons, enabling systematic construction and classification of string backgrounds with reduced holonomy. The findings have potential implications for string compactifications and geometric flows by providing a unifying, structure-preserving reduction principle across multiple geometries.

Abstract

String backgrounds, defined here as metric connections with skew-symmetric torsion and reduced holonomy, yield generalized Ricci solitons relative to the Lee vector field. By a variational argument using the string action, they are also gradient generalized Ricci solitons relative to a potential function. These two observations combine to yield a canonical symmetry, and in this work we derive fundamental features of the transverse geometry, and rigidity phenomena. We prove in a unified conceptual fashion that the transverse geometry satisfies the string generalized Ricci soliton equations (a simplified Hull-Strominger system) in many settings including almost Hermitian, almost contact, , , and geometry. We also show that the transverse geometry is always conformally co-closed, with the conformal factor given by the associated soliton potential.

Paper Structure

This paper contains 19 sections, 24 theorems, 123 equations.

Table of Contents

  1. Introduction
  2. Dimension reduction, Bismut connections, and generalized Ricci solitons
  3. Strong torsion almost-Hermitian structures
  4. Definitions and structure theorems
  5. Dimension Reduction
  6. Strong torsion HKT structures
  7. Definitions and structure theorems
  8. Dimension Reduction
  9. Strong torsion almost contact structures
  10. Definitions and structure theorems
  11. Dimension Reduction
  12. Strong torsion SU(3) structures
  13. Definitions and structure theorems
  14. Strong torsion G2 structures
  15. Definitions and structure theorems
  16. ...and 4 more sections

Key Result

Lemma 2.2

Suppose $(M^n, g, H)$ is a manifold with closed three form endowed a vector field $V$ such that $\nabla V = 0$, $\left| V \right| = 1$, where $\nabla = D + \tfrac{1}{2} H$ is the Bismut connection associated to $(g, H)$. Then, defining $\mu = V^{\flat}$, one has

Theorems & Definitions (81)

  • Remark 1.1
  • Remark 2.1
  • Lemma 2.2
  • proof
  • Remark 2.3
  • Definition 2.4
  • Definition 2.5
  • Proposition 2.6
  • proof
  • Proposition 2.7
  • ...and 71 more