A Geometric transition from hyperbolic to anti de Sitter geometry

TL;DR

This paper introduces a geometric transition method connecting hyperbolic and anti de Sitter geometries via a new transitional geometry, enabling the continuation of collapsing hyperbolic structures into AdS structures with singularities.

Contribution

It presents a novel geometric transition framework and the concept of half-pipe geometry to connect hyperbolic and AdS structures, especially in collapsing scenarios.

Findings

Constructed a transition from hyperbolic to AdS geometries.

Demonstrated the transition on unit tangent bundles of (2,m,m) orbifolds.

Identified tachyon singularities in the resulting AdS manifolds.

Abstract

We introduce a geometric transition between two homogeneous three-dimensional geometries: hyperbolic geometry and anti de Sitter (AdS) geometry. Given a path of three-dimensional hyperbolic structures that collapse down onto a hyperbolic plane, we describe a method for constructing a natural continuation of this path into AdS structures. In particular, when hyperbolic cone manifolds collapse, the AdS manifolds generated on the "other side" of the transition have tachyon singularities. The method involves the study of a new transitional geometry called half-pipe geometry. We demonstrate these methods in the case when the manifold is the unit tangent bundle of the (2,m,m) triangle orbifold for m at least 5.