On the relationship of gravitational constants in KK reduction

TL;DR

This paper clarifies how the gravitational constant in lower-dimensional theories derived from Kaluza-Klein reduction relates to the higher-dimensional constant, focusing on M-theory and type IIA string theory.

Contribution

It provides a re-derivation of the lower-dimensional gravitational constant in M-theory using the metric relations with type IIA string theory, clarifying a common confusion.

Findings

Derived the lower-dimensional gravitational constant in M-theory in terms of string coupling and alpha'

Clarified the relation between higher- and lower-dimensional gravitational constants in KK reduction

Resolved a common confusion in relating gravitational constants across dimensions

Abstract

In this short note, we try to clarify a seemly trivial but often confusing question in relating a higher-dimensional physical gravitational constant to its lower-dimensional correspondence in Kaluza-Klein reduction. In particular, we re-derive the low-energy M-theory gravitational constant in terms of type IIA string coupling $g_s$ and constant $\alpha'$ through the metric relation between the two theories.