The canonical symmetry reduction of string backgrounds

TL;DR

This paper explores the geometric structures of string backgrounds with special holonomy, revealing their nature as generalized Ricci solitons and establishing a canonical symmetry that governs their transverse geometry.

Contribution

It introduces a unified framework for understanding string backgrounds as gradient generalized Ricci solitons and derives fundamental geometric properties and rigidity phenomena across various special holonomy settings.

Findings

Transverse geometry satisfies string generalized Ricci soliton equations.

Transverse geometry is conformally co-closed with a conformal factor from the soliton potential.

The work unifies several geometric structures under a common Ricci soliton framework.

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.