Projective Coordinates and Projective Space Limit

TL;DR

This paper introduces a novel group contraction called the projective lightcone limit, which reduces spacetime dimensions and results in a projective space with linear fractional isometry, with applications to Hopf reduction.

Contribution

It generalizes the projective lightcone limit and applies it to Hopf reduction, deriving lower-dimensional complex projective spaces while preserving symmetry.

Findings

The projective lightcone limit preserves the isometry group G.

It reduces the spacetime dimension via a new group contraction.

Application to Hopf reduction yields complex projective spaces.

Abstract

The "projective lightcone limit" has been proposed as an alternative holographic dual of an AdS space. It is a new type of group contraction for a coset G/H preserving the isometry group G but changing H. In contrast to the usual group contraction, which changes G preserving the spacetime dimension, it reduces the dimensions of the spacetime on which G is realized. The obtained space is a projective space on which the isometry is realized as a linear fractional transformation. We generalize and apply this limiting procedure to the "Hopf reduction" and obtain (n-1)-dimensional complex projective space from (2n-1)-dimensional sphere preserving SU(n) symmetry.