Algebraic Geometry Approach in Gravity Theory and New Relations between the Parameters in Type I Low-Energy String Theory Action in Theories with Extra Dimensions

TL;DR

This paper introduces a novel algebraic geometric approach to gravity theories with extra dimensions, deriving complex multivariable cubic equations and establishing new parameter relations in string theory actions.

Contribution

It develops a new multivariable cubic algebraic equation framework for gravity, parametrized with complex functions, and uncovers novel parameter relations in string theory with extra dimensions.

Findings

Derived a new multivariable cubic algebraic equation for gravitational invariance.

Parametrized the equation using complex non-elliptic functions involving Weierstrass functions.

Established new inequalities between string theory parameters not accessible in standard models.

Abstract

On the base of the distinction between covariant and contravariant metric tensor components, a new (multivariable) cubic algebraic equation for reparametrization invariance of the gravitational Lagrangian has been derived and parametrized with complicated non - elliptic functions, depending on the (elliptic) Weierstrass function and its derivative. This is different from standard algebraic geometry, where only two-dimensional cubic equations are parametrized with elliptic functions and not multivariable ones. Physical applications of the approach have been considered in reference to theories with extra dimensions. The s.c. "length function" l(x) has been introduced and found as a solution of quasilinear differential equations in partial derivatives for two different cases of "compactification + rescaling" and "rescaling + compactification". New physically important relations (inequalities) between the parameters in the action are established, which cannot be derived in the case $l=1$ of the standard gravitational theory, but should be fulfilled also for that case.