Caustics and Maxwell sets of world sheets in anti-de Sitter space

TL;DR

This paper develops a mathematical framework for analyzing caustics and Maxwell sets of world sheets in anti-de Sitter space, linking geometric singularity theory with concepts from theoretical physics.

Contribution

It introduces a rigorous mathematical approach to caustics of world sheets using graph-like Legendrian unfoldings, extending previous physics-based notions.

Findings

Established a new geometric framework for caustics in anti-de Sitter space

Connected singularity theory with physical concepts of lightlike hypersurfaces

Provided tools for further mathematical and physical analysis of world sheets

Abstract

A world sheet in anti-de Sitter space is a timelike submanifold consisting of a one-parameter family of spacelike submanifolds. We consider the family of lightlike hypersurfaces along spacelike submanifolds in the world sheet. The locus of the singularities of lightlike hypersurfaces along spacelike submanifolds forms the caustic of the world sheet. This notion is originally introduced by Bousso and Randall in theoretical physics. In this paper we give a mathematical framework for the caustics of world sheets as an application of the theory of graph-like Legendrian unfoldings.