Past exam of the mathematics course of the University of Cambridge / 2017 / iii / Paper 306 / 2

The conformal condition is a quantum worldsheet Weyl anomaly cancellation, not the classical metric equation of the Polyakov action. Assume a smooth target metric, no antisymmetric background field, and curvature/gradient scales large compared with $\sqrt{{\alpha }^{\prime }}$. Use $[{\nabla }_a,{\nabla }_b]v^c=R^c{}_{dab}{v}^d$ and $R_{ab}=R^c{}_{acb}$, consistent with the Ricci-scalar Weyl transformation in the hint.

A sign convention in the printed action must be made explicit. With worldsheet signature $(-,+)$, continuation $\tau =-i{\tau }_E$ and $e^{i{S}_L}=e^{-S_E}$ sends its negative kinetic term to a positive Euclidean kinetic term, while its positive Lorentzian curvature coupling becomes a negative Euclidean curvature coupling. Define the conventional Euclidean dilaton by $\phi =-\Phi$ for this literal action. Then $S_{E,\mathrm{dil}}=(4\pi {)}^{-1}\int \sqrt{h}{R}^{(2)}\phi$. The Lorentzian dilaton coupling sign convention is therefore important: a convention with a negative Lorentzian curvature coupling would instead use $\phi =\Phi$ and reverse every term linear in the printed $\Phi$ below.

For completeness, the metric part of the one-loop calculation can be obtained by a geodesic background field expansion of a string sigma model. Write $X={\exp }_{\bar{X}}Y$. Its quadratic Euclidean action containsHere $D_{\mu }{Y}^a={\partial }_{\mu }{Y}^a+{\Gamma }^a{}_{bc}{\partial }_{\mu }{\bar{X}}^b{Y}^c$. In dimension $2-\epsilon$, the ultraviolet coincident contraction has pole $⟨Y^a{Y}^b{⟩}_{\mathrm{div}}={\alpha }^{\prime }{g}^{ab}/\epsilon$ with an infrared regulator. Contracting the curvature term gives a divergence $-(4\pi \epsilon {)}^{-1}\int \sqrt{h}{R}_{cd}\partial {\bar{X}}^c\partial {\bar{X}}^d$. It is cancelled by the metric counterterm $\delta g_{ab}={\alpha }^{\prime }{R}_{ab}/\epsilon$, giving the one-loop sigma-model beta function ${\beta }_{ab}^g={\alpha }^{\prime }{R}_{ab}+O({\alpha }^{\prime 2})$ before the dilaton improvement. Equivalently its local Euclidean Weyl variation is $-(4\pi {)}^{-1}\int \sqrt{h}\omega R_{ab}\partial X^a\partial X^b$, with terms proportional to the embedding equations understood as field redefinitions.

Set $\Omega =e^{\omega }$ in the supplied curvature transformation. On a closed string worldsheet, integration by parts givesThe chain rule and leading string embedding map equation implyThe kinetic embedding equation suffices for extracting the leading metric coefficient. More precisely, the full Euclidean embedding equation makes the parenthesis $({\alpha }^{\prime }/2)R^{(2)}{\nabla }^a\phi$, giving a curvature contribution $-({\alpha }^{\prime }/4\pi )\int \sqrt{h}\omega R^{(2)}|\nabla \phi {|}^2$ to the Weyl variation. This contributes to the scalar coefficient below rather than to the leading metric tensor coefficient. Combining the curvature counterterm anomaly with this variation gives the leading metric-dilaton Weyl condition:This is the gravitational equation in the string frame, rather than an ordinary Einstein equation with a minimally coupled scalar. In terms of the field and signs literally printed in this Lorentzian action it readsThe frequently used $R_{ab}+2{\nabla }_a{\nabla }_b\Phi =0$ is obtained if the printed $\Phi$ is identified with the conventional Euclidean dilaton, which requires the opposite Lorentzian curvature-coupling sign. Both conventions describe the same mathematics after $\Phi \mapsto -\Phi$, but one cannot change only the field equation silently. Also, vanishing of this tensor coefficient is the metric part of Weyl invariance; the dilaton curvature coefficient and central-charge condition remain to be checked.

Now derive the scalar consequence without presupposing its integration constant. The contracted Bianchi identity gives ${\nabla }^a{R}_{ab}=\frac{1}{2}{\nabla }_{b}R$. The Ricci identity applied to the gradient of a scalar givesDiverging the metric equation and then substituting $R_{ab}=-2{\nabla }_a{\nabla }_b\phi$ yieldsThus the bracket is constant on each connected target component. Its trace equation is $R+2{\square }_g\phi =0$, so the dilaton equation from contracted Bianchi identity becomesThe second equation uses the literal printed sign convention. These are nonlinear scalar wave equations analogous to the Klein-Gordon equation. For the exponentiated dilaton wave equation, put $F=e^{-2\phi }=e^{2\Phi }$; differentiating twice givesHere ${\square }_g={\nabla }^a{\nabla }_a$ is the Lorentzian Laplace-Beltrami operator, and $|\nabla \phi {|}^2=g^{ab}{\partial }_a\phi {\partial }_b\phi$ is a Lorentzian contraction, not necessarily nonnegative.

The Bianchi and Ricci identities alone do not imply $C=0$. The linear dilaton counterexample to zero integration constant is flat target space with $\phi =q_a{X}^a$: the tensor equation holds for every constant $q$, while $C=-2q^2$. A non-null $q$ disproves any deduction of the zero-constant scalar equation from that tensor equation alone.

The leading dilaton Weyl anomaly coefficient is obtained as follows. The matter and reparameterization-ghost central charges give the constant $(D-26)/6$. Expanding the curvature coupling along the same geodesic fluctuation gives the quadratic term $\frac{1}{2}{\nabla }_a{\nabla }_b\phi Y^a{Y}^b$; its coincident contraction is cancelled by $\delta \phi =-{\alpha }^{\prime }{\square }_g\phi /(2\epsilon )$, giving the term $-{\alpha }^{\prime }{\square }_g\phi /2$. The full embedding-equation contribution identified above supplies $+{\alpha }^{\prime }|\nabla \phi {|}^2$. Therefore full leading-order bosonic-string Weyl invariance also requiresIt fixes $C=(D-26)/(3{\alpha }^{\prime })$. Hence in the critical $D=26$ theory, or under boundary conditions making the constant vanish,This explains precisely the extra input needed for a zero-mass Klein-Gordon-type equation. In noncritical dimension the corresponding equation has the central-charge-deficit constant, subject to the usual controlled-background assumptions.