Nuclear Physics B322 (1989) 317-372 North-Holland, Amsterdam N-POINT g-LOOP VERTEX FOR A FREE BOSONIC THEORY WITH VACUUM CHARGE Q P. DI VECCHIA and F. PEZZELLA l NORDITA, Blegdam+~,¢] 17, DK 2100 Copenhagen O. Denmark M. FRAU INFN, Se-ione di Torino, Cso M. D'Azeglio 46, 1 10125 Torino, Italy K. HORNFECK Lahoratoire d'A nnecv-le- Vieux de Physique des Particules, Chemin de Bellet,ue BP 110, F 74941 A nneqv-le- Vieux Cedex. France A. LERDA2 Center for Theoretical Pto'si~w and Laboratory .['or Nuclear Science, Massachusetts Institute of Technology, Cambridge, MA 02139, USA S. SCIUTO 3 Dipartimento di Science Fisiche dell'Unit,ersitgt di Napoli and INFN, Seeione di Napoli, Mostra d'Oltremare Pad. 19, 1-80125 Napoli. ltah' Received 20 December 1988 Starting from the N-point vertex on the sphere and using the sewing procedure we construct the N-point g-loop vertex for a free bosonic theory, with vacuum charge Q. We then show that, when this vertex is saturated with N highest weight states, it gives their correlation function on an arbitrary Ricmann surface of genus g. We also extend our formalism to the case of a free scalar field compactified on a circle, which is related to the Coulomb gas description of minimal models. 1. Introduction A lot of effort has been recently put in the calculation of multiloop diagrams in string theories (ST) using many different approaches. They include the path integral 1 INFN Fellow, Sezione di Napoli, Italy. : Work partially supported by the US Department of Energy, under grant DE-AC02-76R03069. ) Work partially supported by the Italian Ministero della Pubblica Istruzione. 0550-3213/89/$03.50~',Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)