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Derivations, holonomy groups and heterotic geometry

G. Papadopoulos

TL;DR

The paper develops a derivation-based framework for differential forms on manifolds with reduced holonomy, connecting the algebra of derivations to heterotic geometries admitting Killing spinors and to holonomy symmetries in sigma models. It introduces a Lie bracket on the space of fundamental forms for non-compact holonomy and a null-form extension bar-curlywedge, then analyzes both compact and non-compact holonomy cases, including heterotic-inspired geometries in Euclidean 8D and Lorentzian 10D. The work links the commutator of holonomy-symmetries to generalized Nijenhuis tensors and Noether currents, providing explicit algebraic structures for fundamental forms and highlighting the role of Gray-Hervella classifications for closure in compact cases. It lays out a roadmap for extending these algebraic structures to broader reduced-structure geometries and sigma-model settings, with implications for supersymmetric backgrounds and holonomy-based geometry.

Abstract

We investigate the superalgebra of derivations generated by the fundamental forms on manifolds with reduced structure group. In particular, we point out a relation between the algebra of derivations of heterotic geometries that admit Killing spinors and the commutator algebra of holonomy symmetries in sigma models. We use this to propose a Lie bracket on the space of fundamental forms of all heterotic geometries with a non-compact holonomy group and present the associated derivation algebras. We also explore the extension of these results to heterotic geometries with compact holonomy groups and, more generally, to manifolds with reduced structure group. A brief review of the classification of heterotic geometries that admit Killing spinors and an extension of this classification to some heterotic inspired geometries are also included.

Derivations, holonomy groups and heterotic geometry

TL;DR

The paper develops a derivation-based framework for differential forms on manifolds with reduced holonomy, connecting the algebra of derivations to heterotic geometries admitting Killing spinors and to holonomy symmetries in sigma models. It introduces a Lie bracket on the space of fundamental forms for non-compact holonomy and a null-form extension bar-curlywedge, then analyzes both compact and non-compact holonomy cases, including heterotic-inspired geometries in Euclidean 8D and Lorentzian 10D. The work links the commutator of holonomy-symmetries to generalized Nijenhuis tensors and Noether currents, providing explicit algebraic structures for fundamental forms and highlighting the role of Gray-Hervella classifications for closure in compact cases. It lays out a roadmap for extending these algebraic structures to broader reduced-structure geometries and sigma-model settings, with implications for supersymmetric backgrounds and holonomy-based geometry.

Abstract

We investigate the superalgebra of derivations generated by the fundamental forms on manifolds with reduced structure group. In particular, we point out a relation between the algebra of derivations of heterotic geometries that admit Killing spinors and the commutator algebra of holonomy symmetries in sigma models. We use this to propose a Lie bracket on the space of fundamental forms of all heterotic geometries with a non-compact holonomy group and present the associated derivation algebras. We also explore the extension of these results to heterotic geometries with compact holonomy groups and, more generally, to manifolds with reduced structure group. A brief review of the classification of heterotic geometries that admit Killing spinors and an extension of this classification to some heterotic inspired geometries are also included.
Paper Structure (18 sections, 14 theorems, 40 equations, 7 tables)

This paper contains 18 sections, 14 theorems, 40 equations, 7 tables.

Table of Contents

  1. Introduction
  2. Derivations
  3. Derivations and superalgebras
  4. Inner derivations for null forms
  5. Heterotic geometry
  6. Preliminaries
  7. Killing spinors and geometry
  8. Killing spinors
  9. Geometry
  10. Compact holonomy groups
  11. Non-compact holonomy groups
  12. Heterotic inspired geometries
  13. Examples with Euclidean signature
  14. Examples with Lorentzian signature
  15. Algebraic structures on fundamental forms
  16. ...and 3 more sections

Key Result

Theorem 2.1

Let $\lambda, \phi\in \Omega^*_\kappa({\mathcal{M}}^n)$. Define on $U_\alpha$ where $\breve \lambda$ and $\breve \phi$ are defined as in (tphi), respectively, and we have suppress the open set labelling. Then, both $\lambda\bar{\curlywedge} \phi$ and $\lambda\curlywedge \phi$ are globally defined forms on ${\mathcal{M}}^n$. Moreover, if $\kappa$, $\lambda$ and $\phi$ are cova

Theorems & Definitions (44)

  • Definition 2.1
  • Remark
  • Definition 2.2
  • Remark
  • Theorem 2.1
  • proof
  • Remark
  • Remark
  • Theorem 3.1
  • proof
  • ...and 34 more