# conformal blocks for beginners

I have given now a couple of talks that involve conformal blocks bundles on the moduli stack $\overline{\mathcal{M}}_{g,n}$, in front of a public of algebraic geometers but not specialists of the field. I have always encountered the same difficulty. The definition itself of bundle of conformal blocks is pretty elaborated and I think that it freightens the audience.

What can one do to give the flavour of it without killing the talk? Maybe just introduce the parabolic theta functions on the smooth locus and say that the associated bundle degenerates? Or any other brilliant idea coming from physics?

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Conformal blocks can of course be approached in lots of different ways which are more or less intelligible to different groups of people. Could you tell us more about your background or the background of the audience you'd like to present to? –  stankewicz Dec 3 '12 at 15:44
I approached CB mainly as vector bundles on the moduli of curves, as spaces of generalized theta functions on the moduli of bundles on curves, but the only definition working also on the boundary that I know is the one as sheaves of covacua. –  IMeasy Dec 3 '12 at 15:49
Ah, read Beauville! I found this paper math.unice.fr/~beauvill/pubs/Hirz65.pdf to be most helpful, but this may just be my own preference towards Lie algebras. He also has several papers which give a "generalized theta function" approach. –  stankewicz Dec 3 '12 at 16:50
I am sorry I did not mention: the hypothesis is "public of algebraic geometers, with some knowledge of moduli spaces, but zero knowledge of CB". this is the enviromment I find the most difficult. –  IMeasy Dec 3 '12 at 17:13
Have you tried starting with the "obvious" introduction/motivation: the moduli of vector bundles of fixed det,coprime rank & deg (and n=0) when the coarse moduli space is smooth, with $Pic\simeq \mathbb{Z}$. You can then discuss the Verlinde bundle over Teichmueller space. If you want to be really concrete, you can mention e.g. rank 2 bundles of degree zero and fixed determinant for $g=2$, when the coarse moduli space is $\mathbb{P}^3$. And then you can say that you want to upgrade this to a fancier version, living on $\overline{\mathcal{M}}_{g,n}$? Or is this too trivial for your audiences? –  Peter Dalakov Dec 4 '12 at 11:19