Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 306 / 1 / d / Solution

Let $M$ be the mass measured in the uncompactified 25-dimensional spacetime. The quantum zero-mode constraints areThe string level operatorshave nonnegative integer eigenvalues. Adding the constraints and using $(p_L^2+p_R^2)/2=n^2/R^2+w^2{R}^2/{\alpha }^{\prime 2}$ givesTheir difference gives closed-string level matching, $N-\tilde{N}+nw=0$ with the present left-right convention. The normal-ordering constant of a string $a$ is the regularized zero-point energy generated when oscillator products are normal ordered.