Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 306 / 4 / Solution

Insert the given closed-string mode expansion into the stress tensor and compare $T(z)={\sum }_n{L}_n{z}^{-n-2}$. Normal ordering gives

The inverse of the worldsheet Laplacian gives the free-field operator-product expansionHenceExtracting the Laurent modes by contour integration yields the string oscillator algebra

The vanishing of the worldsheet stress tensor becomes the Virasoro constraintsSince $L_0={\alpha }^{\prime }{k}^2/4+N$ and the bosonic-string normal-ordering constant is $a=1$, these zero-mode constraints are the target-space string mass-shell conditionThereforeThe positive-mode constraints further requireand null string states identify polarizations that differ by momentum-longitudinal terms. The symmetric trace-free, antisymmetric, and trace sectors of ${ϵ}_{\mu \nu }$ describe the graviton, Kalb–Ramond field, and dilaton, respectively.

Finally, the state–operator correspondence maps ${\alpha }_{-1}^{\mu }{\bar{\alpha }}_{-1}^{\nu }|k⟩$ toAdding its integrated massless closed-string vertex operator to the string nonlinear sigma model changes the background coupling byThus a Fourier mode $\delta G_{\mu \nu }(X)={ϵ}_{(\mu \nu )}{e}^{ik\cdot X}$ is precisely a linearized deformation of the target-space metric. Its transversality, mass-shell condition, and polarization gauge redundancy are the worldsheet statements that the deformation is marginal and is defined up to a linearized spacetime diffeomorphism.