Reimundo Helluani
IMPA
On a construction of Deligne in the context of vertex algebras.
Abstract: To any Riemman Surface S and two nonvanishing functions f,g on S, Deligne associates a Line bundle L_(f,g) endowed with a connection. This construction is bimultiplicative, namely we have natural isomorphisms
*) L_{(fg,h)} \simeq L_{(f,h)} \otimes L_{(g,h)}. L_{(f,gh)} \simeq L_{(f,g)} \otimes L_{(f,h)}
We will construct a Hilbert space H with the following structure. To each n vectors in H we will construct a flat section of a suitable bundle L_{(f,g)} as above, together with a factorization property compatible with the isomorphisms *).
This structure generalizes that of a vertex algebra in that in the latter case, all the line bundles L_{(f,g)} are trivial.