Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 306 / 2 / i / Solution

The term ${\hat{p}}^{,2}/2$ is the target-space zero-mode kinetic energy. The constant $-1/12$ is the zero-point energy of the left- and right-moving oscillators: each chiral boson contributes $-1/24$. Finally,count the energies of the two independent sets of string oscillators; one excitation of mode $n$ has energy $n$.