Abstract: The purpose of the talk is to give some example of "partition functions" appearing in supersymmetric quantum field theory and string theory (using purely mathematical language) and exhibit their (still unexplained) connections with representation theory.

In particular, we shall introduce the notion of Gromov-Witten invariants and Gromov-Witten potential and discuss mostly the Okounkov-Pandharipande calculation of the Gromov-Witten potential of P^1 and quantum cohomology of flag varieties.

If time permits, we shall also mention the partition function of 4-dimensional gauge theory.