Global vertex operators on Riemann surfaces
The author introduces a possible notion of global vertex operators and associated spaces of vacua on higher genus Riemann surfaces with ``punctures''. First he recalls the (classical) definition of vertex operator algebras and their representations. In the following his basic set-up is the data \(\widetilde {\Sigma}\) consisting of a Riemann surface, \(N\) points on this surface and local coordinates at these points. To every marked point a representation of a fixed vertex operator algebra \(V\) is assigned. Now it is possible to assign to every marked point for every primary field \(a\) of degree \(n\) and every meromorphic \((-n+1)\)-differential \(f\), which is holomorphic outside the marked points, a local vertex operator of the corresponding representation at this point. This is possible via the local coordinates there. The author calls the sum over the residues of all these local operators a global vertex operator \(a(f, \widetilde {\Sigma})\). The vertex operator depends in a well defined manner on the local coordinates chosen. A similar assignment is possible for the special homogeneous element of degree 2 in the vertex operator algebra which is used to define the associated Virasoro algebra. For this and some of the following results it is necessary to choose a projective structure on the Riemann surface with punctures (i.e. a special system of local coordinates). The space of vacua is defined to be the subspace of the dual of the tensor products of the involved local representations which is annihilated by all global vertex operators assigned to the primary fields and to \(\omega\). The author introduces the notion of quasi-global vertex operators which correspond to quasi-primary fields. This set contains the set of global vertex operators and is close to a Lie algebra. It is shown that under certain conditions on the vertex operator algebra the vacuum still is annihilated by them. The space of vacua is unchanged when a new point together with the vertex operator algebra \(V\) at this point is added. With the help of this result it is possible to define \(n\)-point ``correlation'' functions associated to an element of the vacuum. They are global meromorphic sections of a suitable line bundle with a certain structure of the singularities. Special examples are studied. In the special case where the vertex operator algebra is associated to an affine Lie algebra the vacuum defined by \textit{A. Tsuchiya}, \textit{K. Ueno} and \textit{Y. Yamada} [Adv. Stud. Pure Math. 19, 459-566 (1989; Zbl 0696.17010)] is isomorphic to the vacuum introduced in this article. The article closes with a conjecture on the construction of the vacua \(N(\widetilde {\Sigma})\) by gluing Riemann surfaces. This would imply that the assignment \(\widetilde {\Sigma}\to N (\widetilde {\Sigma})\) is a moduli functor in the sense of Segal.
scientific article; zbMATH DE number 694966