Operator approach to bosonic string: Multiloop calculation and<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi>b</mml:mi></mml:math>-ghost insertion

Abstract

We construct the $g$-loop $N$-string vertex using the Feynman-like rules in the operator formalism with bosonized $\mathrm{bc}$ ghosts. The measure factor is analyzed in detail, which consists of the ghost $g$ vacuum and the contour integration associated with the $b$-ghost insertion. The general form of the $g$ vacuum is derived using the handle operators. The handle operator is formulated by the geometrical quantities: period matrix, first Abelian integral, prime form, \textonehalf{} differential, and Riemann constant. Explicit results are given up to two-loop level. The contour integrations associated with the ghost insertions can be performed explicitly in the one-loop case, in which we reproduce the well-known formula of the one-loop $N$-tachyon amplitude.