Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 306 / 2

The universal stress-tensor operator-product expansion in a two-dimensional conformal field theory isIts fourth-order coefficient defines the central charge $c$.

Differentiating $X(z)X(w)\sim -({\alpha }^{\prime }/2)\log (z-w)$ gives the contractions needed for Wick theorem. The free-boson part of $T$ supplies $1/[2(z-w{)}^4]$, corresponding to central charge one. The cross-contractions between $:\partial X\partial X:$ and ${\partial }^{2}X$ produce the required lower poles but no fourth-order scalar term. Finally,so the product of the two improvement terms contributes $3{\alpha }^{\prime }{q}^2/(z-w{)}^4$. Matching this with $c/[2(z-w{)}^4]$ in the stress-tensor operator-product expansion gives the linear dilaton conformal field theory

The first term of the action is the free boson conformal field theory. For the curvature coupling, insert the supplied first variation of $\sqrt{g}{R}^{(2)}$, integrate by parts twice and discard the boundary term. Its metric variation is the stress-tensor improvementin the conventions of the question. In a locally flat complex coordinate, the holomorphic component of the complete stress-energy tensor is thereforeas required. Thus the coupling $q\int \sqrt{g}X{R}^{(2)}$ makes $X$ a background-charge scalar field, equivalently a worldsheet coordinate in a linear dilaton conformal field theory.

Take $D$ embedding coordinates and give one coordinate the background charge $q$. The remaining $D-1$ free bosons contribute $D-1$, while the distinguished coordinate contributes $1+6{\alpha }^{\prime }{q}^2$, so the matter central charge isThe worldsheet ghosts contributes $-26$. Cancellation of the worldsheet Weyl anomaly therefore requireswhich is real for $D<26$ and produces a noncritical bosonic string.

In target-space language the dilaton is linear in this coordinate, so the local string coupling $g_s=e^{\Phi }$ changes exponentially. One end of the target direction is weakly coupled and the other is strongly coupled. Consequently string perturbation theory is trustworthy only in the weak-coupling region, rather than throughout the full background.