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

The Euler-Lagrange equation of the gauge-fixed Polyakov action is the two-dimensional wave equationIts general solution is a sum of left- and right-moving functions. Periodicity in $\sigma$ and a Fourier expansion give the closed-string mode expansionThe zero modes satisfy ${\alpha }_0^{\mu }={\tilde{\alpha }}_0^{\mu }=\sqrt{{\alpha }^{\prime }/2},p^{\mu }$. Here $x^{\mu }$ and $p^{\mu }$ are the center-of-mass position and momentum, while the nonzero string oscillators describe shape excitations.

A classical string must also satisfy the equations obtained by varying the worldsheet metric before imposing conformal gauge. Vanishing of the worldsheet stress-energy tensor givesIn modes these are the Virasoro constraintsfor every integer $m$. Reality of $X^{\mu }$ also requiresTogether with periodicity and the wave equation, these conditions remove the unphysical longitudinal worldsheet excitations.

For a target circle of radius $R$, maps may wind, so the boundary condition isSingle-valued target-space wavefunctions quantize the center-of-mass momentum as $p_Y=n/R$, with $n\in \mathbb{Z}$. The compact boson expansion becomesEquivalently, its left- and right-moving zero-mode momenta areThe integers $n$ and $w$ are the momentum and winding modes.

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.