What do the different "levels" in a Kac-Moody algebra tell us physically?

Question

I'm reading a book on conformal field theory and I am on a section which derives the current algebra: $$[j_m^i,j_n^j]=\sum_l f^{ijl}j_{m+n}^l+k \;m \delta^{ij}\delta_{m,-n} $$ We can prove this result assuming we are working with chiral fields of conformal dimension 1 using some rather abstract arguments, but I would like to relate some on these things back to something more tangible. In particular this factor of $k$ which refers to the level of a conformal field.

This $k$ comes about by diagonalising a structure constant $d_{ij}=k\delta_{ij}$. The constant can be realized by re-scaling fields. What does this represent physically? I'd guess it has to do with the scale we elect to describe our theory with, but I am not sure. Any intuition would be nice!

Answer 1

The scale of the algebra is always chosen so that the length of the longest root in the underlying finite Lie algebra is 2. This choice simplifies a number of formulae, and in particular ensures that the level $k$ is an integer.