Algebraic Geometry Approach in Theories with Extra Dimensions I. Application of Lobachevsky Geometry

TL;DR

This paper explores how Lobachevsky geometry and algebraic geometry can be applied to theories with extra dimensions, revealing new insights into orbifold compactification, mass generation, and black hole solutions.

Contribution

It introduces a novel application of Lobachevsky geometry to orbifold compactification and derives algebraic equations for black holes in higher-dimensional brane worlds.

Findings

Orbifold periodic identification enabled by algebraic solutions.

Corrections to extradimensional volume due to Lobachevsky space.

Multiple physical masses associated with a single Higgs mass.

Abstract

This present paper has the purpose to find certain physical appications of Lobachevsky geometry and of the algebraic geometry approach in theories with extra dimensions. It has been shown how the periodic properties of the uniformization functions-solutions of cubic algebraic equations in gravity theory enable the orbifold periodic identification of the points pr{c} and -pr{c} under compactification. It has been speculated that corrections to the extradimensional volume in theories with extra dimensions should be taken into account due to the non-euclidean nature of the Lobachevsky space. It has been demonstrated that in the Higgs mass generation model with two branes (a "hidden" and a "visible" one), to any mass on the visible brane there could correspond a number of physical masses. Algebraic equations for 4D Schwarzschild Black Holes in higher dimensional brane worlds have been obtained.