[Paper Review] Quaternionic analytic torsion

Kai Köhler, Gregor Weingart|ArXiv.org|May 11, 2001

Algebraic and Geometric Analysis15 references4 citations

TL;DR

This paper introduces an equivariant quaternionic analytic torsion for antiselfdual vector bundles on quaternionic Kähler manifolds, defined via the square of a Dirac operator acting on Salamon's complex. The key contribution is a closed-form computation of this torsion for quaternionic homogeneous spaces of compact type, expressed in terms of roots, weights, and Weyl group invariants, with structural similarity to holomorphic torsion on Hermitian symmetric spaces.

ABSTRACT

We define an (equivariant) quaternionic analytic torsion for antiselfdual vector bundles on quaternionic Kaehler manifolds, using ideas by Leung and Yi. We compute this torsion for vector bundles on quaternionic homogeneous spaces with respect to any isometry in the component of the identity, in terms of roots and Weyl groups.

Motivation & Objective

• To define a meaningful analytic torsion invariant for quaternionic Kähler manifolds, where traditional complex-analytic tools fail due to the absence of quaternionic differentiability.
• To extend the theory of analytic torsion—previously established for real and holomorphic settings—into the quaternionic realm using Dirac operator decomposition on Salamon's complex.
• To compute the equivariant torsion explicitly for all known positive-curvature quaternionic homogeneous spaces, providing a concrete formula in terms of root systems and Weyl group actions.
• To establish structural parallels between quaternionic torsion and holomorphic torsion, suggesting deeper geometric and arithmetic connections.
• To lay the foundation for future applications in arithmetic geometry, including a potential proof of the Jantzen sum formula for all Chevalley group schemes.

Proposed method

• Define the quaternionic analytic torsion as the zeta-regularized determinant of the Laplacian associated with the square of a Dirac operator on Salamon's Z-graded complex of differential forms with coefficients in antiselfdual bundles.
• Decompose the Dirac operator action on the complex $ 0 \to \mathrm{Sym}^k H \otimes \mathcal{W} \xrightarrow{d} \mathrm{Sym}^{k+1} H \otimes \Lambda^{1,0} E^* \otimes \mathcal{W} \to \cdots \to 0 $, using the splitting $ TM \otimes \mathbb{C} \cong H \otimes E $.
• Use representation-theoretic techniques involving highest weights $ \lambda $, Weyl group $ W $, and positive roots $ \Psi_0^+ $ to express the torsion as a sum over root systems and Weyl group orbits.
• Apply zeta function regularization to the Laplacian $ \square_q $ on $ \Gamma^\infty(M, \Lambda^q T^{*(0,1)}M \otimes \mathcal{W}) $, defining $ \zeta_q(s) = \mathrm{Tr}(\square_q^{-s} P^\perp) $, and compute its derivative at $ s=0 $.
• For the hyperkähler case, relate the quaternionic torsion to the Dolbeault complex via $ T^k(M,\mathcal{W}) = T^0(M,\mathcal{W}) + k T_{\bar{\partial}}(M,\mathcal{W}) $, where $ T_{\bar{\partial}} $ is the holomorphic torsion.
• Derive explicit formulas for the torsion using character sums $ \chi_\rho $, inner products $ \|\alpha\|^2 $, and pairing with cocharacters $ \langle \cdot, \cdot \rangle $, particularly for symmetric spaces.

Experimental results

Research questions

• RQ1 Can a consistent analytic torsion be defined for quaternionic Kähler manifolds, given the absence of a natural notion of quaternionic differentiability?
• RQ2 How does the quaternionic analytic torsion behave on symmetric spaces of compact type, and can it be computed explicitly in terms of Lie-theoretic data?
• RQ3 Does the structure of the quaternionic torsion resemble that of the holomorphic torsion on Hermitian symmetric spaces, and what does this imply about deeper geometric or arithmetic relationships?
• RQ4 Can the torsion be related to known invariants such as holomorphic torsion on the twistor space, and what implications does this have for arithmetic geometry?
• RQ5 What are the implications of the torsion's behavior on hyperkähler manifolds, particularly in relation to Calabi-Yau and flat torus examples?

Key findings

• The quaternionic analytic torsion is computed explicitly for all compact quaternionic homogeneous spaces as a sum over positive roots and Weyl group orbits, involving character sums $ \chi_\rho $, root norms $ \|\alpha\|^2 $, and pairing with cocharacters.
• For the hyperkähler case, the torsion reduces to a combination of the holomorphic torsion $ T_{\bar{\partial}}(M,\mathcal{W}) $ and a linear term in $ k $, with $ T^k(M,\mathcal{W}) = T^0(M,\mathcal{W}) + k T_{\bar{\partial}}(M,\mathcal{W}) $, linking it directly to complex geometry.
• On Calabi-Yau manifolds with $ K_X \cong \mathcal{O} $, the torsion $ T^k(M,\mathcal{W}) $ is independent of $ k $ when $ \mathcal{W} = \mathcal{W}^* $, and vanishes for $ \mathcal{W} = \mathcal{O} $ when $ n=1 $.
• For flat tori $ M = V/\Lambda $, the torsion $ T^k(M,\mathcal{O}) $ vanishes if $ n > 1 $, and equals $ 2 \sum_{\mu \in \Lambda^\vee \setminus \{0\}} \|\mu\|^{-2s} $ at $ s=0 $ for $ n=1 $, showing a direct link to zeta-regularized determinants.
• The formula for the torsion has the same structural form as the holomorphic torsion on Hermitian symmetric spaces, suggesting a universal pattern across different geometric settings.
• The torsion satisfies a duality $ T^k(M,\mathcal{W}) = T^{-4n-k-1}(M,\mathcal{W}^*) $, indicating a non-trivial symmetry not present in the holomorphic case.