This lecture derives the one-loop (torus) vacuum amplitude of the bosonic string by counting the torus's moduli and conformal Killing vectors (one complex modulus tau, no CKVs, unlike the sphere's three CKVs, zero moduli), showing that ghost zero-mode insertions can be written symmetrically so every vertex operator is effectively integrated with a universal 1/Im(tau) measure factor. It then computes the torus partition function as a trace over the matter (X) and ghost (bc) CFT Hilbert spaces using Dedekind-eta-function characters, and verifies modular invariance (tau -> -1/tau and SL(2,Z)) as the key consistency check. Domain: bosonic string one-loop perturbation theory / CFT on the torus.
This lecture derives the one-loop (torus) vacuum amplitude of the bosonic string by counting the torus's moduli and conformal Killing vectors (one complex modulus tau, no CKVs, unlike the sphere's th…