Canonical metric on the space of symplectic invariant tensors and its applications

2014/04/13 by Shigeyuki Morita, Morita, Shigeyuki
Mathematics · Physics and Astronomy · #32G15 #55R40 #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Black Holes and Theoretical Physics #Cohomology #Combinatorics #Degenerate energy levels #FOS: Mathematics #Genus #Geometric Topology (math.GT) #Homotopy and Cohomology in Algebraic Topology #Invariant (physics) #Mathematical physics #Mathematics #Moduli space #Physics #Primary 20C30 #Pure mathematics #Quantum mechanics #Representation Theory (math.RT) #Secondary 20J06 #Symplectic geometry #Vector bundle #math.AG #math.GT #math.RT #msc:20C30 #msc:20J06 #msc:32G15 #msc:55R40

paper · pdf · doi:10.48550/arxiv.1404.3354

28 pages. References added

openalex publication_date 2014/04/13 · arxiv created 2015/05/17 · arxiv updated 2015/05/19 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/28

Abstract

Let Σ_g be a closed oriented surface of genus g and let H_ℚ denote H₁(Σ_g;ℚ) which we understand to be the standard symplectic vector space over ℚ of dimension 2g. We introduce a canonical metric on the space (H_ℚ^{⊗ 2k})^Sp of symplectic invariant tensors by analyzing the structure of the vector space ℚD^ℓ(2k) generated by linear chord diagrams with 2k vertices. This space, equipped with a certain inner product, serves as a universal model for (H^{⊗ 2k})^Sp for any g. We decompose ℚD^ℓ(2k) as an orthogonal direct sum of eigenspaces E_λ where λ is indexed by the set of all the Young diagrams with k boxes. We give a formula for the eigenvalue μ_λ of E_λ and thereby we obtain a complete description of how the spaces (H_ℚ^{⊗ 2k})^Sp degenerate according as the genus decreases from the stable range g≥ k to the last case g=1 with the largest eigenvalue 2g(2g+1) ⋯ (2g+k-1). As an application of our canonical metric, we obtain certain relations among the Mumford-Morita-Miller tautological classes, in a systematic way, which hold in the tautological algebra in cohomology of the moduli space of curves. We also indicate other possible applications such as characteristic classes of transversely symplectic foliations and a project with T. Sakasai and M. Suzuki where we study the structure of the symplectic derivation Lie algebra.

Citations

• Relations in the tautological ring of the moduli space of curves
• Relations in the tautological ring
• The Gelfand-Kalinin-Fuks class and characteristic classes of transversely symplectic foliations
• On the tautological rings of M_{g, 1} and its universal Jacobian
• Relations among characteristic classes of manifold bundles

Related

• Symplectic aspects of the first eigenvalue
• Symplectic Symmetric Spaces
• Characteristic Classes in Symplectic Topology
• Realizing the Teichmüller space as a symplectic quotient
• A canonical connection on bundles on Riemann surfaces and Quillen connection on the theta bundle