A gap between two approaches of dimensional reduction for a six-dimensional Kaluza-Klein theory

Abstract

Inspired by the five-dimensional Kaluza-Klein theory, we would like to study the dimensional reduction issue of six-dimensional Kaluza-Klein extension in this paper. In particular, we will examine two possible approaches of dimensional reduction from six-dimensional spacetimes to four-dimensional ones. The first one is a direct dimensional reduction, i.e., from six-dimensional spacetimes directly to four-dimensional ones, via a $T^2\equiv S^1\times S^1$ compactification, while the second one is an indirect dimensional reduction, i.e., from six-dimensional spacetimes to five-dimensional ones then four-dimensional ones, via two separated $S^1$ compactifications. Interestingly, we show that these two approaches lead to different four-dimensional effective actions although using the same six-dimensional metric. It could therefore address an important question of which approach is more reliable than the other.

16 pages. Accepted for publication in The European Physical Journal Plus (EPJ Plus). Comments are welcome