Sphere Partition Functions and Universal CFT Data

The sphere partition function packages a CFT’s response to curvature in one number, but that number is not universally meaningful in every dimension. In even dimensions the logarithmic radius dependence is controlled by the Euler anomaly while the finite part is counterterm dependent. In odd dimensions the real finite part can be universal under the usual locality and symmetry assumptions. Zero modes, gauge volumes, and source contacts must be treated before either statement is applied.

Required background. Anomaly Coefficients and Central Charges fixes type-A normalization. Weyl Covariance on Curved Backgrounds fixes the generating functional. Helpful background. Defect Entropy and Monotonicity compares sphere-like observables localized on a defect.

Radius response on a round sphere

Define

$$F_{S^d}(R)=-\log Z[S^d_R]=W[S^d_R].$$

A constant change of radius is a Weyl rescaling, so in the convention of this chapter

$$\frac{dF_{S^d}}{d\log R} =\int_{S^d}d^dx\sqrt g\, \langle T^\mu{}_{\mu}\rangle.$$

For a two-dimensional CFT with $\langle T^\mu{}_{\mu}\rangle=cR/(24\pi)$, this gives $dF_{S^2}/d\log R=c/3$. For a four-dimensional CFT with

$$\langle T^\mu{}_{\mu}\rangle =\frac{cW^2-aE_4+b\nabla^2R}{(4\pi)^2},$$

the round sphere has $W^2=0$, $\nabla^2R=0$, and $\int E_4=64\pi^2$, hence

$$\frac{dF_{S^4}}{d\log R}=-4a.$$

These two signs follow from the displayed anomaly basis; changing the definition of $T_{\mu\nu}$ or $W$ changes them coherently. The coefficient of $\log(R/\epsilon)$ is universal, while a finite $R$-independent term can be shifted by local curvature counterterms.

Odd and even dimensions

The useful distinction is:

Dimension	Universal sphere datum on a closed sphere	Main qualification
Even $d$	coefficient of the logarithm, proportional to type-A data	finite part is scheme dependent
Odd $d$	real finite part $F=-\log\lvert Z\rvert$	parity-odd phases and topological counterterms are separate
Noninteger $d$ in an expansion	a chosen analytic continuation such as $\widetilde F$	continuation and subtraction convention must be stated

In three dimensions, the use of the real sphere free energy as an endpoint quantity and its perturbative tests are developed in Klebanov, Pufu, and Safdi 2011. This does not remove the zero-mode, phase, or endpoint qualifications stated here.

A frequently useful interpolation is

$$\widetilde F(d)=-\sin\!\left(\frac{\pi d}{2}\right)F_{S^d},$$

which connects odd-dimensional finite terms to even-dimensional anomaly residues in dimensional regularization Giombi and Klebanov 2015, §§1–2. Its normalization depends on how the even-dimensional pole is subtracted; it is a computational bridge, not a new dimension-independent theorem.

Source derivatives and integrated correlators

Deform the sphere action by constant sources $\lambda^I$:

$$S\longmapsto S+\lambda^I\int_{S^d}\sqrt g\,\mathcal O_I.$$

Then

$$\partial_I F =\int_{S^d}\sqrt g\,\langle\mathcal O_I\rangle,$$

and

$$\partial_I\partial_JF =-\int_{S^d}\sqrt g_x\int_{S^d}\sqrt g_y\, \langle\mathcal O_I(x)\mathcal O_J(y)\rangle_{\!c} +\text{local counterterms}.$$

The sign follows from adding $+\lambda^I\int\mathcal O_I$ to the action. Coincident-point singularities make the double integral regulator dependent; contact counterterms can alter the Hessian. On a conformal manifold, a protected combination can define the Zamolodchikov metric only after redundant directions and source coordinates are fixed.

Zero modes and normalization

A functional determinant with a zero eigenvalue is not defined by simply including that eigenvalue in $\log\det$. One must separate the zero mode, integrate or divide by its physical volume, and state the measure normalization. Examples include:

• the constant mode of a noncompact massless scalar;
• gauge transformations and ghost zero modes;
• Goldstone modes in a spontaneously broken description;
• collective coordinates around a saddle.

Different choices can add $\log R$ or volume factors that mimic an anomaly. The cure is a mode-by-mode definition plus a comparison to the flat-space or heat-kernel normalization, not an ad hoc deletion after the result is known.

The anomaly-to-sphere branch of the Weyl response diagram has this structured equivalent:

Stage	Retain	Do not confuse with
Regulated determinant or path integral	regulator, measure, zero-mode prescription	renormalized $F$
Local subtraction	divergent and finite curvature counterterms	universal coefficient
Even-dimensional output	logarithmic coefficient	arbitrary finite constant
Odd-dimensional output	real finite part in a symmetry-preserving scheme	parity-odd phase
Source differentiation	integrated connected correlator plus contacts	separated-point correlator alone

In the diagram, inspect the branch from background-source variation through the trace identity and local renormalization to sphere data; the flow conclusion remains a separate, hypothesis-dependent comparison.

Sphere data become universal only after local counterterms and zero modes are separated. In even dimensions the logarithmic branch is retained; in odd dimensions the appropriate real finite branch is retained. The diagram is schematic.

Reproduce the plane–cylinder anomaly and sphere/deformation checks with frozen inputs and declared tolerances.

Evidence scope and reproducibility

The theorem-level statements and computational conventions on this page were checked against the cited literature through 9 August 2026. A sphere calculation intended as evidence should publish the spectrum or integrand, regulator, counterterm basis, zero-mode measure, numerical precision, radius convention, and a scale-derivative check. The raw number $-\log Z$ without those data is not comparable across methods.

Common pitfalls

Calling the even-dimensional finite part universal. A finite local curvature counterterm shifts it. The logarithmic coefficient is the invariant datum.

Deleting zero modes silently. Their measure and physical quotient can contribute scale dependence. State the treatment before evaluating the determinant.

Equating a source Hessian with an ordinary integrated two-point function. Coincident singularities generate local terms. The metric or susceptibility is defined only after a subtraction and coordinate convention.

Exercises

Derive $dF_{S^4}/d\log R=-4a$ in the convention above.

Solution

The round sphere is conformally flat and has constant scalar curvature, so only $-aE_4/(4\pi)^2$ contributes. Since $\int_{S^4}\sqrt gE_4=32\pi^2\chi(S^4)=64\pi^2$, the integral is $-a(64\pi^2)/(16\pi^2)=-4a$.

References

• Giombi, S., and Klebanov, I. R. “Interpolating between $a$ and $F$.” Journal of High Energy Physics 2015, 117 (2015). arXiv. DOI.
• Klebanov, I. R., Pufu, S. S., and Safdi, B. R. “F-Theorem without Supersymmetry.” Journal of High Energy Physics 2011, 038 (2011). arXiv. DOI.

Further reading

• Pufu, S. S. “The F-Theorem and F-Maximization.” Journal of Physics A 50, 443008 (2017). arXiv. DOI.