Intuitively I feel that if you compactified open bosonic strings on a product of $n$ circles such that each radius is fine-tuned to the self-dual point then the CFT of these $n$ world-sheet fields would "see" a Kac-Moody algebra for some rank "n" Lie group.
True? If yes then what is a good proof of this?
Now how does one construct the conserved $(1,0)$ currents in terms of these $n$ CFT fields such that they would reproduce the structure constants of the rank $n$ Lie group that one wants?
Said otherwise, by choosing the number of compactification directions one could only choose the rank but how does one tune the required structure constants/Lie algebra as well?
(..Polchinski just writes down the answer for SU(2) in page 243 vol 1..even in that "simple" case its not quite clear as to how he managed to get the structure constants right..as in till he writes down the $j$s and shows that the commuatation gives $SU(2)$ it wasn't obvious that it is $SU(2)$..or was there an apriori reason for it?..)
- Now if one didn't start from open bosonic strings but were just doing CFT of $n$ fields on a product of $n$ circles then (1) what would remain of this effect of tuning to a self-dual radius? (2) how would one fine tune the required affine algebra? (3) how would one write down the $CFT$ Lagrangian depending on the wanted affine Lie algebra?