In this talk I shall discuss the effect of composite connection in 10 and 8 dimensional supergravity theories (N=2 and 1), especially the anomaly generated by such composite connections. The counter-term for such anomaly provides us with an $alpha'^4$ correction in supergravity theories. I shall also discuss that this counter-term is also reproduced in 1-loop string calculations. Finally, we shall consider certain subtleties related to the gauge group of the 8-d N=1 supergravity theories which shall provide us with a classification for possible geometric and non-geometric compactifications towards D=6, N=(1,0) theories.

IPhT