A homological field theory (or HFT) is an algebraic structure governed by surfaces and their mapping class groups. Although the name is inspired by theoretical physics, HFTs really belong to topology: an HFT is simply a functor on an appropriate category of cobordisms.

After reviewing these definitions I will explain how to obtain a HFT from a compact Lie group G. This recovers some classic algebraic structures on the homology of G and BG, and much new data besides. If time permits then I will explain the connection with Chern-Simons theory and the "parameterised umkehr maps" used to build the HFT.

This is joint work with Anssi Lahtinen.