 Solution

Use the standard Euclidean convention $e^{-S_E}$ and restore the curvature normalization explicitly. Let $\phi$ be the conventional dimensionless dilaton, whose worldsheet term isFor constant ${\phi }_0$, the Gauss-Bonnet theorem gives $S_{\phi }={\phi }_0\chi (\Sigma )$. A closed surface of genus $g$ has Euler characteristic $\chi =2-2g$, so its contribution is weighted byThusThis is the dilaton normalization and Euler-characteristic weighting of string perturbation theory. If the unit curvature coefficient in the printed expression is retained literally in this Euclidean convention, then $\phi =4\pi \Phi$ and $g_s=e^{4\pi {\Phi }_0}$. If $\Phi$ denotes the conventional dilaton, the usual omitted $1/(4\pi )$ is understood and $g_s=e^{{\Phi }_0}$. One must specify that normalization before exponentiating the field. Lorentzian overall signs are fixed consistently by Wick rotation.

For the sigma-model beta function, usewith vanishing two-form and dilaton. Perform the background field expansion of a string sigma model covariantly by setting $X={\exp }_{\bar{X}}\eta$. In Riemann normal coordinates, the quadratic fluctuation action isThe covariant derivative contains the pulled-back target connection. This expansion accounts for the connection vertices as well as the normal-coordinate metric expansion; keeping only a single metric tadpole would not give a covariant answer.

Integrating the Gaussian fluctuations gives $\frac{1}{2}\mathrm{Tr}\log \Delta$, whereContracting the curvature vertex with the short-distance propagator traces $R^a{}_{cad}$ to the Ricci tensor $R_{cd}$. In dimensional regularization at $d=2-ϵ$, the logarithmic integral is $\int d^{d}p/(2\pi {)}^d(p^2+{\mu }^2{)}^{-1}=1/(2\pi ϵ)+O(1)$. With the stated curvature convention the divergent effective action isIt is cancelled by the metric counterterm $\delta g_{ab}={\alpha }^{\prime }{R}_{ab}/ϵ$. Equivalently, write $g^{\mathrm{bare}}={\mu }^{-ϵ}(g+{\alpha }^{\prime }R/ϵ+\cdots )$. Differentiating the bare coupling at fixed value gives the one-loop metric beta function of a string sigma modelThis loop is a loop of two-dimensional fluctuation fields: its expansion is in $\alpha \prime$, not a change of worldsheet genus or an extra power of $g_s$. The sign here defines the renormalization scale to increase toward the ultraviolet. For a round target sphere, positive Ricci curvature makes its squared radius increase with that scale, consistent with the usual asymptotic freedom of the inverse-radius sigma-model coupling. With the printed kinetic coefficient $1/2$ taken literally, $2\pi {\alpha }^{\prime }=1$ and the leading coefficient is $R_{ab}/(2\pi )$. Requiring the metric beta function to vanish gives $R_{ab}=0$ at this order. Full bosonic-string Weyl consistency also requires the other anomaly coefficients, including the critical-dimension condition when the dilaton vanishes.

For T-duality, choose coordinates $(y,x^i)$ adapted to the isometry and use ${\alpha }^{\prime }=1$ units for the local Buscher rules. Assume $g_{yy}\ne 0$, with a spacelike isometry for the usual unitary circle duality. In light-cone worldsheet coordinates take the convention that $E=g+B$ multiplies ${\partial }_{+}{X}^a{\partial }_{-}{X}^b$. Gauge translations of $y$, replace ${\partial }_{\pm }y$ by $A_{\pm }$, gauge fix $y=0$, and impose flatness with a multiplier $\tilde{y}$. The relevant first-order Lagrangian isIntegrating out $\tilde{y}$ gives a locally pure-gauge connection and hence the original model. Instead integrate the last term by parts and solve the algebraic equations for $A_{\pm }$:Substitution givesReading its symmetric and antisymmetric parts yields the Buscher dual of a metric with vanishing two-form:The spectator metric is the Schur complement of $g_{yy}$. Reversing the orientation of $\tilde{y}$ reverses the displayed mixed two-form signs; our multiplier and $E=g+B$ conventions fix them.

The metric rules arise already from the classical algebraic elimination. For quantum equivalence the regulated Gaussian measure also produces the Buscher dilaton shiftFor a compact isometry the multiplier periodicity and flat-connection sectors implement the exchange of momentum and winding. On a circle, restoring dimensions gives $R^{\prime }={\alpha }^{\prime }/R$. Thus the calculation derives the local dual background, while the measure and global sectors explain how it becomes a string-theory duality.