Universal Index Theorem on $Mob(S^1)\Diff_+(S^1)$

TL;DR

This paper establishes a universal index theorem on the complex Kähler manifold formed by the quotient of the Möbius group and orientation-preserving diffeomorphisms of the circle, linking holomorphic differentials, Grunsky matrices, and Weil-Petersson geometry.

Contribution

It generalizes Faber polynomials and Grunsky matrices to higher differentials, deriving identities and determinants that connect to the Weil-Petersson form on the moduli space.

Findings

Defined canonical bases of holomorphic differentials using generalized Faber polynomials.

Established identities among generalized Grunsky matrices reminiscent of classical Grunsky equality.

Proved the determinant relation et N_n = (et N_1)^{6n^2-6n+1} and its connection to Weil-Petersson geometry.

Abstract

By conformal welding, there is a pair of univalent functions $(f,g)$ associated to every point of the complex K\"ahler manifold $\Mob(S^1)\bk\Diff_+(S^1)$. For every integer $n\geq 1$, we generalize the definition of Faber polynomials to define some canonical bases of holomorphic $1-n$ and $n$ differentials associated to the pair $(f,g)$. Using these bases, we generalize the definition of Grunsky matrices to define matrices whose columns are the coefficients of the differentials with respect to standard bases of differentials on the unit disc and the exterior unit disc. We derive some identities among these matrices which are reminiscent of the Grunsky equality. By using these identities, we showed that we can define the Fredholm determinants of the period matrices of holomorphic $n$ differentials $N_n$, which are the Gram matrices of the canonical bases of holomorphic $n$-differentials with respect to the inner product given by the hyperbolic metric. Finally we proved that $\det N_n =(\det N_1)^{6n^2-6n+1}$ and $\pa\bar{\pa}\log\det N_n$ is $-(6n^2-6n+1)/(6\pi i)$ of the Weil-Petersson symplectic form.