On the law of the index of Brownian loops related to the Hopf and anti-de Sitter fibrations

TL;DR

This paper derives explicit formulas and asymptotics for the distribution of the index of Brownian loops across various geometric spaces, utilizing the geometry of Hopf and anti-de Sitter fibrations and their relation to winding and area forms.

Contribution

It provides new explicit formulas and asymptotic behaviors for Brownian loop indices in complex and hyperbolic geometries based on Hopf and anti-de Sitter fibrations.

Findings

Explicit formulas for Brownian loop index distributions.

Asymptotic behaviors in various geometric settings.

Connections between winding, area forms, and fibrations.

Abstract

We give explicit formulas and asymptotics for the distribution of the index of the Brownian loop in the following geometrical settings: the complex projective line from which two points have been removed; the complex hyperbolic line from which one point has been removed; the odd dimensional spheres from which a great hypersphere has been removed; and the complex anti-de Sitter spaces. Our analysis is based on the geometry of the Hopf and anti-de Sitter fibrations, and on the relationship between winding and area forms.