[Paper Review] Hyperkaehler manifolds with torsion, supersymmetry and Hodge theory

Misha Verbitsky|ArXiv.org|Dec 20, 2001

Advanced Topics in Algebra12 citations

TL;DR

This paper establishes a hyperkähler-de Rham superalgebra structure on HKT-manifolds, extending the classical Kähler superalgebra to non-Kähler settings. It proves an SL(2)-action on harmonic spinors via a normalized Dolbeault complex, yielding a canonical isomorphism $ H^i(K^{1/2}) \cong H^i(K^{1/2})^* $ and a Hard Lefschetz theorem, generalizing Hodge theory to hypercomplex Hermitian manifolds with torsion.

ABSTRACT

Let M be a hypercomplex Hermitian manifold, (M,I) the same manifold considered as a complex Hermitian with a complex structure I induced by the quaternions. The standard linear-algebraic construction produces a canonical nowhere degenerate (2,0)-form on (M,I). It is well known that M is hyperkaehler if and only if the form Ωis closed with respect to the de Rham differential. The M is called HKT (hyperkaehler with torsion) if this form is closed with respect to the Dolbeault differential (this condition is weaker. Conjecturally, all compact hypercomplex manifolds admit an HKT-metrics. We exploit a remarkable analogy between the de Rham DG-algebra of a Kaehler manifold and the Dolbeault DG-algebra of an HKT-manifold. The supersymmetry of a Kaehler manifold is given by an action of an 8-dimensional Lie superalgebra on its de Rham algebra, containing the Lefschetz SL(2)-triple, the Laplacian and the de Rham differential. We establish the action of this superalgebra on the Dolbeault DG-algebra of an HKT-manifold. This is used to construct a canonical Lefschetz-type SL(2)-action on the space of harmonic spinors of M.

Motivation & Objective

• To extend the classical Kähler superalgebra and Hodge theory to non-Kähler hypercomplex Hermitian manifolds with torsion (HKT-manifolds).
• To construct a canonical $ \mathfrak{sl}(2) $-action on the space of harmonic spinors using the normalized Dolbeault complex.
• To establish a Hard Lefschetz isomorphism and Serre duality for cohomology of the canonical bundle square root $ K^{1/2} $ on compact HKT-manifolds.
• To demonstrate that the harmonic spinor space is independent of the choice of complex structure in the quaternionic triple $ I,J,K $.
• To provide a geometric realization of the supersymmetry algebra on HKT-manifolds, analogous to the Kähler case.

Proposed method

• Define the canonical $ (2,0) $-form $ \Omega $ on $ (M,I) $, induced by the hypercomplex Hermitian structure.
• Introduce the normalized Dolbeault differential $ {}^n\partial $ on $ \Lambda^{p,0}(M) \otimes K^{1/2} $, forming the complex (10.5).
• Construct the normalized Laplacian $ {}^n\Delta = \{{}^n\partial, \partial^*\} $, whose cohomology identifies with harmonic spinors.
• Establish an $ \mathfrak{sl}(2) $-triple $ \langle L_\Omega, \Lambda_\Omega, H_\Omega \rangle $ acting on the cohomology, commuting with $ {}^n\Delta $.
• Use the $ \mathfrak{sl}(2) $-action to prove the Hard Lefschetz isomorphism $ L_\Omega^{n-i}: H^i(K^{1/2}) \to H^{2n-i}(K^{1/2}) $.
• Define the pairing $ \langle \eta, \eta' \rangle = L_\Omega^{n-i}(\eta \wedge J_c(\eta')) \in \mathbb{C} $, realizing the canonical duality.

Experimental results

Research questions

• RQ1 Can the classical Kähler superalgebra and Hodge theory be generalized to non-Kähler HKT-manifolds?
• RQ2 Does a canonical $ \mathfrak{sl}(2) $-action exist on the cohomology of $ K^{1/2} $ for compact HKT-manifolds?
• RQ3 Is there a Hard Lefschetz isomorphism for the cohomology of the canonical bundle square root on HKT-manifolds?
• RQ4 How does the harmonic spinor space on HKT-manifolds relate to the choice of complex structure in the quaternionic triple?
• RQ5 Can the supersymmetry algebra of Kähler manifolds be extended to HKT-geometry via a Lie superalgebra action?

Key findings

• The cohomology $ H^*(K^{1/2}) $ of the normalized Dolbeault complex is canonically isomorphic to the space of harmonic spinors on the HKT-manifold.
• The $ \mathfrak{sl}(2) $-triple $ \langle L_\Omega, \Lambda_\Omega, H_\Omega \rangle $ commutes with the normalized Laplacian $ {}^n\Delta $, enabling the Hard Lefschetz isomorphism.
• The Hard Lefschetz theorem holds: $ L_\Omega^{n-i}: H^i(K^{1/2}) \to H^{2n-i}(K^{1/2}) $ is an isomorphism for all $ i \leq n $.
• Serre duality is realized via the canonical isomorphism $ H^i(K^{1/2}) \cong H^i(K^{1/2})^* $, induced by the $ \mathfrak{sl}(2) $-action.
• The pairing on $ H^i(K^{1/2}) \times H^i(K^{1/2}) $ is given explicitly by $ \langle \eta, \eta' \rangle = L_\Omega^{n-i}(\eta \wedge J_c(\eta')) \in \mathbb{C} $.
• The space $ H^*(K^{1/2}) $ is independent of the choice of complex structure $ I $ in the quaternionic triple $ I,J,K $, and thus is intrinsically defined.