Sigma-model beta function

A sigma-model beta function measures the failure of a worldsheet background coupling to remain scale invariant. Vanishing metric, B-field and dilaton beta functions gives the spacetime field equations order by order in ${\alpha }^{\prime }$.

For the Euclidean kinetic normalization $(4\pi {\alpha }^{\prime }{)}^{-1}\int g_{ab}(X)\partial X^a\partial X^b$, vanishing two-form and constant dilaton, the leading beta function is ${\beta }_{ab}^g={\alpha }^{\prime }{R}_{ab}$. The covariant Gaussian fluctuation operator contains a curvature vertex whose one-loop trace contracts to the Ricci tensor. Its logarithmic divergence renormalizes the metric. The scale convention is $\beta =\mu dg/d\mu$; Weyl consistency requires vanishing of this beta function as well as the other anomaly coefficients.

For the conventional positive Euclidean dilaton curvature coupling, with no antisymmetric background field, the leading scalar Weyl coefficient is the displayed expression. The matter/ghost central charge gives $(D-26)/6$. In a geodesic background field expansion of a string sigma model, contracting $\frac{1}{2}{\nabla }_a{\nabla }_b\phi Y^a{Y}^b$ in the curvature coupling produces the counterterm $\delta \phi =-{\alpha }^{\prime }\square \phi /(2\epsilon )$ and hence the Laplacian term. The full embedding equation ${\square }_h{X}^a+{\Gamma }^a{}_{bc}\partial X^b\partial X^c=({\alpha }^{\prime }/2)R^{(2)}{\nabla }^a\phi$ supplies the gradient-square improvement in the dilaton Weyl variation. Setting this coefficient to zero fixes the constant in the dilaton equation from contracted Bianchi identity; the metric equation alone does not. The Lorentzian dilaton coupling sign convention determines the named-field translation.

The noncritical coefficient is also given in Tong's string theory notes, Section 7.4.4.

For the conventional Euclidean dilaton coupling $+(4\pi {)}^{-1}\int \sqrt{h}R\phi$ and no antisymmetric background, the metric worldsheet Weyl anomaly vanishes at first order in ${\alpha }^{\prime }$ when $R_{ab}+2{\nabla }_a{\nabla }_b\phi =0$. The one-loop Ricci term follows from the background field expansion of a string sigma model. Under a Weyl transformation, the dilaton coupling varies as $-(2\pi {)}^{-1}\int \sqrt{h}\omega {\square }_h\phi$; the embedding equations replace this by the target Hessian contribution. Full Weyl invariance also requires the dilaton curvature coefficient, including the central-charge deficit, to vanish. See the Lorentzian dilaton coupling sign convention when translating an action with a specified Lorentzian sign.

The conventional coefficient agrees with the background equations in Tong's string theory notes, Section 7.2.3.

Diverging $R_{ab}+2{\nabla }_a{\nabla }_b\phi =0$ and using the contracted Bianchi identity and Ricci identity gives ${\nabla }_b(R+4{\square }_g\phi -4|\nabla \phi {|}^2)=0$. The trace equation $R+2{\square }_g\phi =0$ then gives ${\square }_g\phi -2|\nabla \phi {|}^2=C$, with $C$ constant on each connected component. Zero does not follow from the tensor equation alone. The additional dilaton anomaly coefficient fixes $C=(D-26)/(3{\alpha }^{\prime })$ at leading order for the bosonic string without an antisymmetric background. In critical dimension it is zero; noncritical backgrounds have a central-charge-deficit term.

In flat target space, $\phi =q_a{X}^a$ has zero Hessian and therefore solves the leading metric-dilaton Weyl condition for every constant covector $q$. But $\square \phi -2|\nabla \phi {|}^2=-2q^2$, which need not vanish. This demonstrates the integration constant left by the contracted Bianchi identity. Full quantum Weyl invariance fixes the allowed $q^2$ through the central charge, as in a linear dilaton conformal field theory; the metric equation alone does not.

For the dilaton equation from contracted Bianchi identity, put $F=e^{-2\phi }$. The chain rule gives ${\square }_{g}F=F(-2{\square }_g\phi +4|\nabla \phi {|}^2)=-2CF$. Thus the exponential obeys a linear Klein-Gordon equation on the given background, although the original dilaton equation is nonlinear and the string-frame metric remains coupled to the dilaton. With the opposite named-field convention $\phi =-\Phi$, this variable is $F=e^{2\Phi }$.

A worldsheet Weyl anomaly is the quantum failure of the gauge-fixed string path integral to remain invariant under a Weyl transformation. Its local coefficients are the sigma-model beta functions, while its constant part is controlled by the total central charge.

To first order in ${\alpha }^{\prime }$, the conventional Euclidean dilaton coupling gives ${\overline{\beta }}_{\mu \nu }^G={\alpha }^{\prime }[R_{\mu \nu }-H_{\mu \rho \sigma }{H}_{\nu }{}^{\rho \sigma }/4+2{\nabla }_{\mu }{\nabla }_{\nu }\Phi ]$ and ${\overline{\beta }}_{\mu \nu }^B={\alpha }^{\prime }[-{\nabla }^{\rho }{H}_{\rho \mu \nu }/2+({\nabla }^{\rho }\Phi )H_{\rho \mu \nu }]$. The scalar coefficient is $(d-26)/6+{\alpha }^{\prime }[(\nabla \Phi {)}^2-{\nabla }^2\Phi /2-H^2/24]$. Here $H$ is the Kalb-Ramond field strength. Vanishing of these coefficients, including higher-order corrections, is the background-field condition for quantum Weyl transformations to remain a gauge symmetry. The scalar coefficient includes the central charge deficit and is not implied by the metric equation alone.