[Paper Review] Quaternionic Kähler and hyperKähler manifolds with torsion and twistor spaces

Stefan Ivanov, Ivan Minchev|ArXiv.org|Dec 15, 2001

Black Holes and Theoretical Physics20 references3 citations

TL;DR

This paper establishes a geometric correspondence between Quaternionic Kähler manifolds with torsion (QKT) and HyperKähler manifolds with torsion (HKT) via their twistor spaces. It proves that the Swann bundle of a QKT manifold admits an HKT structure with special symmetry if and only if its twistor space carries an almost Hermitian structure with totally skew-symmetric Nijenhuis tensor, linking quantum field theory geometries from (4,0) supersymmetric sigma models with Wess-Zumino terms.

ABSTRACT

The target space of a (4,0) supersymmetric two-dimensional sigma model with Wess-Zumino term has a connection with totally skew-symmetric torsion and holonomy contained in Sp(n)Sp(1) (resp. Sp(n)), QKT (resp. HKT)-spaces. We study the geometry of QKT, HKT manifold and their twistor spaces. We show that the Swann bundle of a QKT manifold admits a HKT structure with special symmetry if and only if the twistor space of the QKT manifold admits an almost hermitian structure with totally skew-symmetric Nijenhuis tensor, thus connecting two structures arising from quantum field theories and supersymmetric sigma models with Wess-Zumino term.

Motivation & Objective

• To establish a geometric correspondence between QKT and HKT manifolds through their twistor spaces.
• To clarify the role of torsion in (4,0) supersymmetric sigma models and their target space geometry.
• To investigate the conditions under which the Swann bundle of a QKT manifold inherits an HKT structure with special symmetry.
• To analyze the twistor space of QKT and HKT manifolds and characterize their almost Hermitian structures.
• To connect quantum field theory geometries with differential geometric structures via holonomy and torsion.

Proposed method

• The paper studies QKT and HKT manifolds as target spaces of (4,0) supersymmetric two-dimensional sigma models with Wess-Zumino terms, which admit metric connections with totally skew-symmetric torsion.
• It uses the twistor construction to relate the geometry of QKT manifolds to that of their twistor spaces, focusing on almost Hermitian structures with totally skew-symmetric Nijenhuis tensors.
• The Swann bundle construction is applied to QKT manifolds to analyze whether it inherits an HKT structure with special symmetry.
• The authors derive curvature and Ricci tensor formulas on the twistor space using the Bismut and Chern connections, comparing their (2,0)+(0,2) components.
• They employ the Lee form and torsion 1-form equality θ = t on HKT manifolds to analyze the Ricci tensor and its symmetry.
• The analysis includes the use of the (1,1)-type curvature of the Chern connection and the vanishing curvature of the Bismut connection to deduce properties of dθ and d(Jαθ).

Experimental results

Research questions

• RQ1 Under what conditions does the Swann bundle of a QKT manifold admit an HKT structure with special symmetry?
• RQ2 When does the twistor space of a QKT manifold carry an almost Hermitian structure with totally skew-symmetric Nijenhuis tensor?
• RQ3 How are the geometric properties of HKT and QKT manifolds related through their twistor spaces?
• RQ4 What is the role of the Lee form and torsion 1-form in characterizing Ricci tensor symmetry on HKT manifolds?
• RQ5 What are the curvature and Ricci tensor formulas on the twistor space of a 4n-dimensional HKT manifold with n > 1?

Key findings

• The Swann bundle of a QKT manifold admits an HKT structure with special symmetry if and only if its twistor space admits an almost Hermitian structure with totally skew-symmetric Nijenhuis tensor.
• On a HKT manifold, the Lee form θ equals the torsion 1-form t, and dθ is of type (1,1) with respect to each Jα.
• The *-Ricci tensors on the twistor space of a HKT manifold are symmetric and Ii-invariant for i=1,2.
• If the HKT manifold is *-Einstein with positive scalar curvature, then its twistor space admits a *-Einstein almost Hermitian structure with c² = 4/Scal_Q.
• On a compact 4n-dimensional HKT manifold with n > 1, the integral of (Scal^g - Scal^g_Q) dV ≥ 0, with equality iff the structure is balanced.
• On a balanced HKT manifold with n ≥ 2, the Ricci tensor is Jα-invariant and symmetric, which implies the torsion 3-form is coclosed (δT = 0).