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The $\barν$-Invariant of $G_2$-Structures on Aloff-Wallach Spaces

Artem Aleshin

Abstract

We compute the $\barν$-invariant of homogeneous nearly-parallel $G_2$-structures on Aloff--Wallach spaces $N_{k,l} = SU(3)/S^1_{k,l}$. Using Goette's formulas for the $η$-invariants of homogeneous spaces, we derive an explicit expression for $\barν$ in terms of representation-theoretic data and show that for the two homogeneous nearly-parallel structures $\varphi^\pm$ on $N_{k,l}$ one has \[\barν(\varphi^\pm) = \mp 41.\] Additionally, we compare the $\barν$-invariants of the nearly-parallel $G_2$-structures arising from the 3-Sasakian structure.

The $\barν$-Invariant of $G_2$-Structures on Aloff-Wallach Spaces

Abstract

We compute the -invariant of homogeneous nearly-parallel -structures on Aloff--Wallach spaces . Using Goette's formulas for the -invariants of homogeneous spaces, we derive an explicit expression for in terms of representation-theoretic data and show that for the two homogeneous nearly-parallel structures on one has Additionally, we compare the -invariants of the nearly-parallel -structures arising from the 3-Sasakian structure.

Paper Structure

This paper contains 22 sections, 15 theorems, 98 equations.

Table of Contents

  1. Introduction and main results
  2. Invariants of G2-structures
  3. nu-invariant of G2-structures
  4. Mathai-Quillen current
  5. Homogeneous G2-structures
  6. Reductive connection
  7. eta-invariants of homogeneous spaces
  8. nu-invariant in the homogeneous case
  9. Aloff-Wallach spaces N(k,l)
  10. Geometry of Aloff-Wallach spaces
  11. Homogeneous G2-structures on Aloff-Wallach spaces
  12. Scalar curvature
  13. Computations of the invariants
  14. Computing first terms
  15. Spectra of deformed Dirac operators
  16. ...and 7 more sections

Key Result

Theorem 1

Let $\varphi_\pm$ denote the two non-equivalent nearly-parallel homogeneous structures on $N_{k,l}$. Then $\blacktriangleleft$$\blacktriangleleft$

Theorems & Definitions (35)

  • Theorem 1
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3: Crowley2025
  • Lemma 2.1: Bismut1992
  • Lemma 2.2: Crowley2025
  • Lemma 3.1
  • proof : Proof of Lemma \ref{['lem:redpar']}
  • Remark 3.1
  • Lemma 3.2
  • ...and 25 more