This is motivated by the answers to this question http://mathoverflow.net/questions/36183/linear-non-linear-sigma-model .The answers particularly by José Figueroa-O'Farrill generated a new interest in me about supersymmetric sigma models. These sigma models relate to Complex algebraic geometric in a beautiful way. I am working on some aspects of Mirror Symmetry and have already seen some of such connections.
This post is an invitation to discuss mathematics inspired by supersymmetric sigma models in QFT/String Theory. The precise questions I am interested in are :
How do we construct the $\sigma$-models, the terms in it. What is the mathematics behind this construction.
What is the relation of the the supersymmetric $\sigma$-models to Ricci flow (Jose mentioned this in other post). I am looking for some elaboration.
What is the current research on $\sigma$-models centred at.
Please provide some references from where one can learn the basics of these theories,preferably mathematical.