Weyl Covariance on Curved Backgrounds

Coupling a CFT to a nondynamical metric turns conformal transformations into local geometric operations. A diffeomorphism moves points and tensor indices; a Weyl rescaling changes the local unit of length. Classically, an improved CFT responds covariantly. Quantum mechanically, the renormalized generating functional can acquire a local Weyl anomaly, so one must distinguish covariant separated-point observables from contact and curvature terms. The stress-tensor and curved-background conventions used here follow Osborn and Petkou 1994, §§2–3.

Required background. Currents and the Stress Tensor fixes the improved stress tensor. Smooth Manifolds and Tensors supplies metric and curvature notation. Helpful background. The Plane–Cylinder Map gives the basic conformal-frame example.

Background fields and Weyl weights

Work first in Euclidean signature; Lorentzian statements follow by a declared continuation compatible with the site’s $(+---)$ convention. Define the renormalized connected generating functional $W=-\log Z$ by

$$\delta W=\frac12\int d^dx\,\sqrt g\, \langle T^{\mu\nu}\rangle\,\delta g_{\mu\nu} +\int d^dx\,\sqrt g\, \langle\mathcal O_I\rangle\,\delta J^I.$$

This equation fixes every later sign. Under an infinitesimal Weyl rescaling,

$$\delta_\sigma g_{\mu\nu}=2\sigma(x)g_{\mu\nu}.$$

A scalar primary of dimension $\Delta$ transforms at separated points as

$$\mathcal O(x)\longmapsto e^{-\Delta\sigma(x)}\mathcal O(x)$$

when $g_{\mu\nu}\mapsto e^{2\sigma}g_{\mu\nu}$. A source coupled through $\int\sqrt g\,J\mathcal O$ therefore has Weyl weight $\Delta-d$. Tensor primaries also acquire the frame factors required by their indices.

The phrase “at separated points” matters. Renormalized composite operators can mix with curvature operators of the same dimension, and correlators can contain local terms supported when insertions coincide. A Weyl transformation can therefore add contact terms even when its noncoincident part transforms homogeneously. Curved-space scalar renormalization gives an explicit construction of this curvature mixing Brown and Collins 1980.

Improvement in a concrete theory

For a massless scalar in $d>2$, the curved-space action

$$S=\frac12\int d^dx\sqrt g\left[ g^{\mu\nu}\partial_\mu\phi\partial_\nu\phi +\xi_d R\phi^2\right], \qquad \xi_d=\frac{d-2}{4(d-1)},$$

is classically Weyl invariant when

$$\phi\mapsto e^{-(d-2)\sigma/2}\phi.$$

The curvature coupling is the curved-background completion of the flat-space improved stress tensor. Dropping it makes the minimally coupled scalar look scale covariant in flat space while failing local Weyl covariance on a curved background. With a boundary, additional boundary terms and boundary conditions are required; the bulk expression alone is not a complete variational problem.

More generally, an improvement

$$T_{\mu\nu}\longmapsto T_{\mu\nu} +(\nabla_\mu\nabla_\nu-g_{\mu\nu}\nabla^2)L$$

changes the trace by a total derivative. It is available only when the theory contains an appropriate scalar operator $L$ and when global or boundary conditions permit it. Improvement is not a universal license to set every trace term to zero.

Conformal maps as two geometric steps

Suppose a coordinate transformation pulls a metric back to

$$f^*g'=e^{2\omega(x)}g.$$

Transporting a CFT observable from $(M,g)$ to $(M',g')$ consists of:

1. the diffeomorphism $f$, which acts on coordinates and tensor indices;
2. a Weyl rescaling by $-\omega$, which contributes the primary weights and possibly an anomaly to $W$.

For scalar primaries away from coincident points,

$$\left\langle\prod_i\mathcal O_i(f(x_i))\right\rangle_{g'} =\prod_i e^{-\Delta_i\omega(x_i)} \left\langle\prod_i\mathcal O_i(x_i)\right\rangle_g,$$

provided the operator basis has been chosen so that no curvature mixing contributes. The plane–cylinder relation and stereographic projection to a sphere are applications of this rule. Vacuum energies or logarithmic radius dependence arise from the generating functional and cannot be reconstructed by multiplying local primary factors alone.

What can fail quantum mechanically

After regularization and renormalization,

$$\delta_\sigma W =\int d^dx\sqrt g\,\sigma(x)\,\mathcal A[g,J]$$

at a fixed point, up to source-derivative terms and boundaries. The anomaly density $\mathcal A$ is local, but its nontrivial coefficients are intrinsic CFT data. Adding a finite local counterterm shifts only the cohomologically trivial part.

The separation between cohomologically nontrivial anomaly coefficients and finite-counterterm shifts is reviewed in Duff 1994, §§2–4. A complete curved-background claim records:

Input	Question to fix	Failure if omitted
Signature and continuation	Euclidean calculation or Lorentzian state?	Wrong signs or boundary values
Stress-tensor definition	Which improvement and normalization?	Spurious trace or mismatched $C_T$
Curvature completion	Which $R\mathcal O$ terms are present?	Broken local Weyl covariance
Operator basis	Which curvature/contact mixing is subtracted?	Scheme-dependent quantity called universal
Regulator and counterterms	Which local terms are allowed?	Anomaly coefficient shifted illegally
Boundary conditions	Is $M$ closed, or what boundary data are used?	Missing surface variation and anomaly terms

Common pitfalls

Using a flat-space primary law at coincident points. Renormalized products contain contact terms and curvature mixing. Apply the homogeneous law only to separated insertions unless the local terms have been specified.

Calling every conformal map a diffeomorphism. The pulled-back metric generally differs by a Weyl factor. Both steps are needed, and the second can change the generating functional anomalously.

Assuming improvement is automatic. It depends on the operator spectrum and on boundary conditions. A flat-space total derivative can carry physical boundary information.

Exercises

Verify the classical Weyl invariance of the conformally coupled scalar action, ignoring boundary terms.

Solution

Use $\delta_\sigma\sqrt g=d\sigma\sqrt g$, $\delta_\sigma g^{\mu\nu}=-2\sigma g^{\mu\nu}$, $\delta_\sigma\phi=-(d-2)\sigma\phi/2$, and $\delta_\sigma R=-2\sigma R-2(d-1)\nabla^2\sigma$. Terms proportional to $\nabla\sigma\cdot\nabla(\phi^2)$ cancel after integration by parts precisely for $\xi_d=(d-2)/[4(d-1)]$.

References

• Brown, L. S., and Collins, J. C. “Dimensional Renormalization of Scalar Field Theory in Curved Space-Time.” Annals of Physics 130 (1980): 215–248. DOI.
• Duff, M. J. “Twenty Years of the Weyl Anomaly.” Classical and Quantum Gravity 11 (1994): 1387–1404. arXiv. DOI.
• Osborn, H., and Petkou, A. C. “Implications of Conformal Invariance in Field Theories for General Dimensions.” Annals of Physics 231 (1994): 311–362. arXiv. DOI.