There is no Mitzvah to light Chanuka candles in Shul.
Lighting Chanukah candles in Shul is not mentioned in
Bavli, Yerushalmi or Rambam. Even the Tur & Shulchan
Aruch don't say there is Chiyuv to light in Shul. They do
mention it indirectly. (where to place it and to light
it before Havdalah)
The Minhag of lighting in Shul started in the times of the
This Minhag is one of the rare Minhagim
where we make a Bracha אשר קדשנו במצותיו on a Minhag.