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

Abstract

Let $\Sigma_g$ be a closed oriented surface of genus g and let $H_\mathbb{Q}$ denote $H_1(\Sigma_g;\mathbb{Q})$ which we understand to be the standard symplectic vector space over $\mathbb{Q}$ of dimension $2g$. We introduce a canonical metric on the space $(H_\mathbb{Q}^{\otimes 2k})^{\mathrm{Sp}}$ of symplectic invariant tensors by analyzing the structure of the vector space $\mathbb{Q}\mathcal{D}^{\ell}(2k)$ generated by linear chord diagrams with $2k$ vertices. This space, equipped with a certain inner product, serves as a universal model for $(H^{\otimes 2k})^{\mathrm{Sp}}$ for any $g$. We decompose $\mathbb{Q}\mathcal{D}^\ell(2k)$ as an orthogonal direct sum of eigenspaces $E_\lambda$ where $\lambda$ is indexed by the set of all the Young diagrams with $k$ boxes. We give a formula for the eigenvalue $\mu_\lambda$ of $E_\lambda$ and thereby we obtain a complete description of how the spaces $(H_\mathbb{Q}^{\otimes 2k})^{\mathrm{Sp}}$ degenerate according as the genus decreases from the stable range $g\geq k$ to the last case $g=1$ with the largest eigenvalue $2g(2g+1) \cdots (2g+k-1)$. 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.