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

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.