Volume 206, number 4 PHYSICS LETTERS B 2 June 1988 N-STRING, g-LOOP VERTEX FOR THE BOSONIC STRING P. DI VECCHIA, K. HORNFECK l Nordita, Blegdamsvej 17, DK-2100 Copenhagen O, Denmark M. FRAU INFN, Sezione di Torino, Corso M. D'Azeglio 46, 1-10125 Turin, Italy and Niels Bohr Institute, Blegdamsvej 17, DK- 2100 Copenhagen 0, Denmark A. LERDA 2 Institute for Theoretical Physics, State University of New York at Stony Brook, Stony Brook, NY 11794, USA and S. SCIUTO 3 Dipartmento di Scienze Fisiche dell'Universit& di Napoli and INFN, Sezione di Napoli, Mostra d'Oltremare Pad 19, 1-80125 Naples, Italy Received 9 March 1988 We construct the N-string,g-loop vertex V~v.g for the orbital degrees of freedom of the bosonic string in terms of the first abelian differentials, the period matrix and the prime form. We also build the Ig) vacuum recently discussed by many people in the framework of an operator formalism on an arbitrary Riemann surface; our expression also contains the measure that takes into account the ghost contribution. The old operator formalism has been recently revived in different forms [ 1-6 ] ~ for computing the multiloop amplitudes in string theories. Following in particular the approach of ref. [ 1 ] the basic ingredients for construct- ing loop diagrams are the BRST invariant of the N-string vertex VN and the twisted propagator T. VN is con- structed in ref. [ 1 ] by sewing together BRST invariant three-string vertices [4,8 ]. As it has been shown in refs. [9,1,2] VN provides an off-shell extrapolation of the string scattering amplitudes, that on one hand reproduces the dual amplitudes at the tree level for on shell physical states and on the other hand can be used for computing multiloop diagrams. This approach allowed to write an explicit and simple expression [ 10,2 ] for the multiloop partition function and N-tachyon scattering amplitude ~2. At one loop it reproduces the well-known result. A detailed comparison of the two- and three-loop partition functions with known explicit result derived using geometrical methods [ 11 ] has been carried out by Petersen, Roland and Sidenius [ 12 ] finding complete agreement to "high" order in some moduli and exactly in others. A new operator formalism on an arbitrary Riemann surface has been recently introduced by many authors Also at University of Wuppertal, GauBstraBe20, D-5600 Wuppertal 1, Fed. Rep. Germany. 2 A Della Riccia fellow. 3 Work partially supported by the Italian Ministero dellla Pubblica Istruzione. ~ Seealso ref. [7]. ~2 It is not difficult to prove that eq. (29) of ref. [ 10 ] actually holds for any number of loops g and not only for g= 2. 0370-2693/88/$ 03.50 © Elsevier Science Publishers B.V. ( North-Holland Physics Publishing Division ) 643