String nonlinear sigma model

The string nonlinear sigma model couples the worldsheet embedding fields to target-space backgrounds such as a metric, a Kalb–Ramond field and a dilaton. Quantum Weyl invariance requires its beta functions to vanish.

The leading massless-sector spacetime action is proportional to $\int \sqrt{-G}{e}^{-2\Phi }[R+4(\nabla \Phi {)}^2-H^2/12]$, with $H=dB$. The backgrounds are the string-frame metric, dilaton, and Kalb–Ramond field. Massive string exchange generates higher-derivative corrections ordered by ${\alpha }^{\prime }$. This perturbative bosonic massless-sector truncation does not remove the tachyon instability.

The string-frame metric is the target metric appearing directly in the kinetic term of the string nonlinear sigma model. With conventional Euclidean dilaton $\phi$, its leading spacetime gravitational action contains $\sqrt{-g^{(s)}}{e}^{-2\phi }(R+4|\nabla \phi {|}^2)$. The dilaton-dependent Ricci coefficient distinguishes it from the Einstein-frame metric. Frame changes are local field redefinitions where the conformal factor is finite and nonzero; the Lorentzian dilaton coupling sign convention fixes which named field is $\phi$.

For $D>2$, rescale the string-frame metric by $g^{(E)}=e^{-4\phi /(D-2)}{g}^{(s)}$. The Weyl transformation of the Ricci scalar makes the coefficient of the Einstein-frame Ricci term independent of the dilaton. Indeed $g^{(s)}=e^{4\phi /(D-2)}{g}^{(E)}$ contributes factors $e^{2D\phi /(D-2)}$ from the volume, $e^{-2\phi }$ from the original coupling and $e^{-4\phi /(D-2)}$ from the Ricci term; their product is one. Derivative terms supply the transformed dilaton kinetic term. The two-dimensional case is excluded from this particular rescaling.

Expand the string embedding map along target geodesics as $X={\exp }_{\bar{X}}Y$. The quadratic Euclidean fluctuation action has a covariant derivative term and a curvature term $-R_{acbd}{Y}^a{Y}^b\partial {\bar{X}}^c\partial {\bar{X}}^d$. The logarithmically divergent coincident contraction is proportional to ${\alpha }^{\prime }{g}^{ab}$; its curvature contraction is the Ricci tensor. A compensating metric counterterm gives ${\beta }_{ab}^g={\alpha }^{\prime }{R}_{ab}+O({\alpha }^{\prime 2})$. This is why the target gravitational equation arises from quantum worldsheet Weyl anomaly cancellation, rather than from the classical worldsheet metric variation. Infrared regulation and removal of equation-of-motion terms by field redefinitions specify the usual one-loop scheme.

The Kalb–Ramond field is an antisymmetric two-form background coupled to the oriented worldsheet area element. Its gauge-invariant field strength is the three-form $H=dB$.

When ${\partial }_{-}$ is invertible, choose $B_{-m}=0$. Residual gauge parameters are exact one-forms and act trivially on $B$. The remaining independent modes are antisymmetric transverse components $B_{IJ}$, satisfying $\square B_{IJ}=0$; longitudinal components obey $B_{+I}={\partial }_{-}^{-1}{\partial }^J{B}_{IJ}$. There are $(D-2)(D-3)/2$ physical polarizations.

The three-form field strength of the Kalb–Ramond field is $H=dB$. Its components are $H_{mnp}={\partial }_m{B}_{np}+{\partial }_n{B}_{pm}+{\partial }_p{B}_{mn}$. In flat spacetime the free equation is ${\partial }^m{H}_{mnp}=0$.

The H-field is the three-form field strength of a Kalb–Ramond field. It obeys the Bianchi identity $dH=0$ and contributes quadratically to the leading string background equations.

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.