Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 306 (String Theory)

The holomorphic part of the free-boson worldsheet propagator givesApply Wick theorem to $T(z)T(w)$. The two double contractions give $D/[2(z-w{)}^4]$, while the single contractions reconstruct $T$ and its derivative. Thus the stress-tensor operator-product expansion isand the $D$ embedding coordinates have central charge $\boxed{c=D}$.

The holomorphic stress-energy tensor generates an infinitesimal conformal transformation throughTaking the three residues givesThe third derivative is the anomalous term that prevents $T$ from transforming as an ordinary weight-two Virasoro primary operator.

With Virasoro algebra modes $L_n=(2\pi i{)}^{-1}\oint dz{z}^{n+1}T(z)$, a second contour calculation givesso $A(m)=c(m^3-m)/12=D(m^3-m)/12$. This Virasoro central extension is the quantum conformal anomaly. In string theory the matter and ghost contributions must cancel; $c_{\mathrm{matter}}+c_{\mathrm{ghost}}=D-26=0$ gives the critical dimension of string theory $D=26$ and makes the BRST operator nilpotent.

Split the string embedding map into its constant worldsheet zero mode and orthogonal fluctuations, $X^{\mu }=x^{\mu }+X^{\prime \mu }$. The kinetic operator has no inverse on the constant mode, while its inverse on $X^{\prime }$ is the worldsheet Green function $G$. Completing the square in the Gaussian functional integral givesThe remaining ordinary integral iswhich proves (1), up to a source-independent functional determinant.

To insert $n$ tachyon vertex operators, chooseThe zero-mode integral produces target-space momentum conservation,In the nonzero-mode integral, discard the coincident self-contractions by normal ordering. With $G(z,w)=-({\alpha }^{\prime }/2)\log |z-w{|}^2$, the remaining pairwise contractions produce the Koba-Nielsen factor, so

For three closed-string tachyons on the sphere, $k_i^2=4/{\alpha }^{\prime }$ and ${\sum }_i{k}_i=0$, hence ${\alpha }^{\prime }{k}_i\cdot k_j=-2$ for $i\ne j$. The matter correlator is therefore $|z_{12}{z}_{13}{z}_{23}{|}^{-2}$. Gauge fixing the sphere's Möbius transformation group fixes three insertion points and supplies the bc ghost system correlator $|z_{12}{z}_{13}{z}_{23}{|}^2$, exactly cancelling this position dependence. ThusHere each vertex contributes one closed-string coupling $g_c$, $C_{S^2}$ contains the sphere vacuum normalization and conventions, the worldsheet zero mode gives the momentum delta function, the nonzero modes give the Koba-Nielsen factor, and the ghost determinant divides by the conformal Killing group. Since $C_{S^2}\propto g_s^{-2}$ and $g_c\propto g_s$, the net string genus expansion dependence is $g_s$.

The BRST operator $Q_B$ is Grassmann odd and represents the gauge symmetry on the gauge-fixed state space. Requiring two successive BRST transformations to vanish meansThis nilpotence makes physical states a BRST cohomology. If $[Q_B,S_0]=0$ and $\Psi$ is the gauge-fixing fermion, the graded Jacobi identity givesso $S=S_0+\{Q_B,\Psi \}$ is BRST invariant.

A holomorphic field of conformal weight $h$ has the Laurent expansionUnder the state–operator correspondence, $\varphi (z)|0⟩$ must be regular at the origin. Terms with $n>-h$ have negative powers, soFor the anticommuting bc system,Separating creation and annihilation modes and summing the geometric series for $|z|>|w|$ gives the bc ghost operator-product expansion

The BRST current built from the matter and ghost stress tensors has an operator-product expansion with $c$ whose residue givesFor a matter Virasoro primary operator $\Phi (z,\bar{z})$ of weights $(1,1)$, the two standard string vertex operators areThe first is the local, unintegrated vertex. Using $Q_{B}c=c\partial c$, $Q_B\bar{c}=\bar{c}\bar{\partial }\bar{c}$, and the weight-$(1,1)$ transformation of $\Phi$, the terms cancel pairwise and $Q_{B}U=0$. For the integrated vertex,so $\{Q_B,V\}=0$ on a closed worldsheet because the variation is a total derivative. Both vertices therefore represent the same BRST cohomology class.

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.