Universal observables and scheme dependence

Two calculations can describe the same physics and still assign different finite values to a partition function. To decide whether a discrepancy matters, specify the observable and ask which changes can arise from allowed local terms in the sources.

Use sources and currents and fixed points and scaling. The main calculations require ordinary Euclidean QFT in three or four dimensions, with no supersymmetry prerequisite. We will separately state the additional assumptions for one four-dimensional supersymmetric result.

First compare a loop with a sphere free energy

Let $b$ denote all fixed background sources, including the metric. With dynamical fields $\varphi$,

$$Z[b]=\int\mathcal D\varphi\,e^{-S[\varphi;b]}, \qquad \mathcal F[b]=-\log Z[b]. \tag{T18.1}$$

$\mathcal F$ is dimensionless. It is often called the sphere free energy when the background is a sphere; it is not automatically a thermodynamic energy.

For a closed curve $C$, a Hermitian gauge connection $a$, and a representation $\mathcal R$ of dimension $d_{\mathcal R}$, define the ordinary Wilson insertion

$$\mathcal L_{\mathcal R}(C) =\frac1{d_{\mathcal R}}\operatorname{Tr}_{\mathcal R} \mathcal P\exp\!\left(i\oint_C a\right), \qquad \langle\mathcal L_{\mathcal R}\rangle =\frac{\int\mathcal D\varphi\,\mathcal L_{\mathcal R}e^{-S}}{Z}. \tag{T18.2}$$

The symbol $\mathcal P$ orders matrices along the curve. There are two normalizations here: the representation trace makes the trivial holonomy equal one, and division by $Z$ normalizes the functional integral. Since $[a_\mu]=1$ and $[dx^\mu]=-1$, the exponent is dimensionless.

A supersymmetric Wilson operator can add scalar couplings along the curve. For example, Drukker–Gross, (1.4), uses the fundamental trace divided by $N$ and a specified scalar coupling in four-dimensional $\mathcal N=4$ Yang–Mills. That operator is different from an ordinary Wilson loop; its matrix-integral evaluation is not needed here.

Now change the renormalization prescription by adding a finite source-only local counterterm $I_{\rm ct}[b]$. This is an integral of a local expression in the backgrounds, allowed by the symmetries and global definitions being retained. It contains no integrated matter or gauge variable. Directly,

$$Z'=e^{-I_{\rm ct}}Z, \qquad \mathcal F'=\mathcal F+I_{\rm ct}, \qquad \langle\mathcal L_{\mathcal R}\rangle' =\langle\mathcal L_{\mathcal R}\rangle. \tag{T18.3}$$

The last equality follows because the same factor multiplies numerator and denominator. It holds for a fixed definition of the insertion on the same background.

Here is a concrete four-dimensional shift. On a smooth closed manifold, the Euler curvature density is $E_4=R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma}-4R_{\mu\nu}R^{\mu\nu}+R^2$. Its integral divided by $32\pi^2$ is the Euler characteristic. Therefore

$$I_{\rm ct}=\frac{\alpha}{32\pi^2}\int d^4x\sqrt g\,E_4 \tag{T18.4}$$

is an allowed finite diffeomorphism-invariant metric counterterm, with dimensionless real $\alpha$. For the round $S^4$ of radius $L$, constant sectional curvature gives

$$R_{\mu\nu\rho\sigma}^2=\frac{24}{L^4},\qquad R_{\mu\nu}^2=\frac{36}{L^4},\qquad R^2=\frac{144}{L^4},\qquad \operatorname{Vol}(S^4)=\frac{8\pi^2L^4}{3}.$$

Consequently $E_4=24/L^4$ and

$$\int_{S^4}\sqrt g\,E_4=64\pi^2, \qquad \Delta\mathcal F_{S^4}=2\alpha, \qquad \Delta\langle\mathcal L_{\mathcal R}\rangle=0. \tag{T18.5}$$

All curvature squares are unchanged if the opposite overall Riemann-sign convention is used. No three-dimensional curvature convention is being imported.

The finite constant in $\mathcal F_{S^4}$ is thus ambiguous even though the theory is unchanged at separated points. This counterterm does not change the coefficient of a logarithm in $L$: its value is independent of $L$.

The normalized loop is immune to this bulk counterterm, not to every possible redefinition. Renormalizing the line insertion can require counterterms supported on the curve. A perimeter subtraction, a finite normalization, and—in Chern–Simons theories—a framing prescription may matter. Framing specifies a displaced copy of the curve used to regulate coincident points. Fix the representation, scalar couplings, contour, and defect prescription before comparing loop values. Requiring a weak-coupling limit of one does not forbid every coupling-dependent finite redefinition.

Why dimension changes the sphere conclusion

On a smooth closed round sphere, a local metric term with $2n$ derivatives scales as $L^{d-2n}$. Curvature counts as two derivatives. At a conformal point, a finite counterterm that introduces no new mass scale must have $2n=d$.

In four dimensions, curvature-squared terms meet this condition; (T18.4) is an explicit example. In three dimensions, no parity-even polynomial metric scalar with an even derivative count has dimension three. Terms proportional to the volume and to $\int\sqrt g\,R$ can remove power divergences, but their coefficients have dimensions three and one. Keeping arbitrary finite coefficients would introduce a scale rather than preserve the conformal prescription.

Thus the real finite part of the round-$S^3$ free energy at a conformal point is meaningful under these assumptions: smooth closed geometry, no other dimensionful sources, local counterterms regular near flat space, and a conformal subtraction prescription. Nonanalytic expressions such as $R^{3/2}$ are not permitted as counterterms regular around a flat background. An arbitrary normalization assigned separately to each manifold is not a local counterterm.

This statement does not remove all three-dimensional phases. Quantized parity-odd background Chern–Simons counterterms can change the phase of $Z$. Nor does it apply unchanged after turning on masses or extra backgrounds: for a scalar mass source $m$ of dimension one, terms such as $\int\sqrt g\,m^3$ and $\int\sqrt g\,mR$ are dimensionally available, subject to the actual symmetries. Supersymmetric source multiplets impose further restrictions and can produce their own allowed terms.

A local term changes a contact response

Return to the source convention $\mathcal F=-\log Z$. A local counterterm changes a source derivative by $\delta\mathcal F'=\delta\mathcal F+\delta I_{\rm ct}$. Differentiating twice can therefore change coincident-point responses without changing separated-point correlations. This is the mechanism illustrated by Closset et al., (1.1)–(1.2).

Consider a compact background $U(1)$ connection $a$ in three dimensions, normalized so that the charge lattice is integral and a gauge parameter has period $2\pi$. Work on oriented spin manifolds. The allowed counterterm is

$$I_n[a]=\frac{in}{4\pi}\int a\wedge da, \qquad n\in\mathbb Z. \tag{T18.6}$$

For a nontrivial bundle, the displayed three-form is only a local expression. Its global definition uses the curvature on a spin four-dimensional extension. Changing the extension gives $i\pi n\int c_1^2$, where $c_1=f/(2\pi)$ is the integral flux class. On a closed spin four-manifold, $\int c_1^2$ is even, so $e^{-I_n}$ is independent of the extension. This is why the coefficient cannot be an arbitrary real number. CT, (1.5)–(1.7), explains this global restriction; without the spin assumption the purely bosonic counterterm lattice is different.

In flat coordinates with $\epsilon^{123}=1$, varying the local expression gives

$$\begin{aligned} \frac{\delta I_n}{\delta a_\mu(x)} &=\frac{in}{2\pi}\epsilon^{\mu\rho\nu}\partial_\rho a_\nu(x),\\ \frac{\delta^2 I_n}{\delta a_\mu(x)\delta a_\nu(y)} &=-\frac{in}{2\pi}\epsilon^{\mu\nu\rho} \partial_{x^\rho}\delta^{(3)}(x-y). \end{aligned} \tag{T18.7}$$

The factor two comes from varying either copy of $a$ and integrating by parts. The minus sign in the second line exchanges the last two epsilon indices. It is a response of $-\log Z$; one must also keep the explicit source-contact terms when converting it to a current correlator, as in the prerequisite.

Define $\kappa$ to be the coefficient of $i\int a\wedge da/(4\pi)$ in the generating functional. Then (T18.6) changes $\kappa$ to $\kappa+n$. Its fractional class $\kappa\pmod1$ cannot be removed by these counterterms, even though the integer part is scheme dependent. The charge normalization and spin/global assumptions are essential. The contact term has support at $x=y$, but some of its information is still physical. CT, (1.3)–(1.7) and (2.2), makes this distinction.

For an explicit nonzero background, use a flat three-torus with coordinate periods $L$, and set $a=A[\cos(qz)\,dx+\sin(qz)\,dy]$, where $q=2\pi\ell/L$ and nonzero integer $\ell$ makes the source periodic. Here $[A]=[q]=1$. Exterior differentiation gives

$$a\wedge da=-A^2q\,dx\wedge dy\wedge dz, \qquad I_n=-\frac{inA^2qL^3}{4\pi}. \tag{T18.8}$$

The free energy shifts by this imaginary quantity, while $Z$ acquires the opposite phase in its exponent. It is nonzero even though the bundle in this example is trivial. Quantization is required because the same prescription must also make sense on nontrivial bundles. The counterterm remains purely a source term; it does not create a dynamical photon.

A mixed derivative that survives a restricted ambiguity

There is a useful specialized four-dimensional result. For an $\mathcal N=2$ superconformal theory—eight real Poincaré supercharges in its Lorentzian parent—on its canonical supersymmetric round $S^4$ background, let $\tau^I$ be local complex coordinates for exactly marginal couplings. The prescription retains an eight-supercharge $OSp(2|4)$ rigid subalgebra; the conformal theory can also have additional superconformal symmetries. With this supersymmetry-preserving prescription, the sphere partition function is related to a function $K$:

$$Z_{S^4}=e^{K/12}, \qquad K\longmapsto K+f(\tau)+\bar f(\bar\tau). \tag{T18.9}$$

$f$ is any locally holomorphic function. This is a Kähler ambiguity: $K$ is a potential for a metric on coupling space, and the metric is its mixed second derivative. No complex geometry is needed to see why the extra terms drop out:

$$G_{I\bar J} =\partial_I\partial_{\bar J}K =12\partial_I\partial_{\bar J}\log Z_{S^4} =-12\partial_I\partial_{\bar J}\mathcal F_{S^4}. \tag{T18.10}$$

Here $\partial_{\bar J}f=0$ and $\partial_I\bar f=0$. The sign follows from $\mathcal F=-\log Z$.

The relation and its allowed counterterm are sourced results, not consequences of dimensional analysis alone. Gomis–Ishtiaque v2, (1.1)–(1.2), (2.1)–(2.3), and (4.8)–(4.13), constructs the supersymmetric counterterm and evaluates its curvature part to $64\pi^2f(\tau)$ on $S^4$. The source normalizes marginal deformations as $\delta S=\pi^{-2}\int d^4x\,(\delta\tau^I O_I+\delta\bar\tau^{\bar I}\bar O_{\bar I})$; then $\langle O_I(x)\bar O_{\bar J}(0)\rangle=G_{I\bar J}/|x|^8$ at separated points. The theorem requires those supersymmetric counterterm restrictions. If one allowed an arbitrary nonholomorphic coefficient multiplying (T18.4), its mixed derivative would generally not vanish.

What is invariant therefore depends on the allowed prescriptions. A bulk-normalized loop, a real three-sphere constant, a four-dimensional logarithmic coefficient, a fractional contact class, and a mixed coupling derivative answer different questions.

Exercises

1. Extract the finite or logarithmic information

Let $D=L\partial_L$ at fixed cutoff $\varepsilon$ and fixed dimensionless couplings. Suppose the real regulated sphere free energies have the forms

$$\begin{aligned} \mathcal F_3&=c_3(L/\varepsilon)^3+c_1L/\varepsilon+F_3+o(1),\\ \mathcal F_4&=c_4(L/\varepsilon)^4+c_2(L/\varepsilon)^2 +A_{\log}\log(L/\varepsilon)+B_4+o(1). \end{aligned}$$

Find polynomials in $D$ that extract $F_3$ and $A_{\log}$ as $\varepsilon/L\to0$. Explain why the second procedure cannot extract an unambiguous $B_4$. Assume the remainder and the needed derivatives tend to zero.

Hint 1

$D$ has eigenvalue $j$ on $L^j$. A factor $D-j$ removes that power.

Hint 2

For the four-dimensional expression, remove the powers first and then differentiate once to remove the constant. Normalize the surviving coefficient.

Solution

The required operations are

$$\lim_{\varepsilon/L\to0}\frac{(D-3)(D-1)}3\mathcal F_3=F_3, \qquad \lim_{\varepsilon/L\to0}\frac{D(D-4)(D-2)}8\mathcal F_4=A_{\log}.$$

On a constant, $(D-3)(D-1)$ gives three. For the logarithm, $(D-4)(D-2)\log(L/\varepsilon)=8\log(L/\varepsilon)-6$; one further $D$ gives eight. The first two powers in each expression vanish identically.

The Euler counterterm changes $B_4$ by $2\alpha$, so no extraction from that unnormalized partition function makes its absolute finite constant unique without choosing a scheme. It leaves $A_{\log}$ fixed. The derivative assumption on the remainder matters: a small but rapidly oscillating regulator artifact need not remain small after differentiating. These operations are justified within the stated smooth asymptotic expansion, not for arbitrary numerical cutoff data.

2. Compare contact data in two prescriptions

Two descriptions have the same normalized separated-point current correlations and the same compact unit-charge lattice on spin manifolds. Their contact coefficients are $\kappa_A=1/3$ and $\kappa_B=7/3$. Can an allowed local counterterm match them? Compute the corresponding phase change on the torus background (T18.8). What changes if the second coefficient is instead $1/2$?

Hint 1

An allowed change is an integer, not an arbitrary real subtraction.

Hint 2

To change description B into A, use $n=\kappa_A-\kappa_B$. Remember that $Z$ transforms with $e^{-I_n}$.

Solution

For $7/3$, choose $n=-2$. The fractional classes already agree. Equation (T18.8) gives $I_{-2}=iA^2qL^3/(2\pi)$, so the partition function of description B is multiplied by $\exp[-iA^2qL^3/(2\pi)]$. A finite phase discrepancy on this sourced background can thus be a scheme discrepancy, despite equality at separated points.

If $\kappa_B=1/2$, the required shift is $-1/6$, which is forbidden by the declared counterterm lattice. Their fractional contact data differ. Equality of separated-point correlators alone would then be insufficient for the claimed comparison. This diagnostic assumes the same charge/source identification and global data; rescaling a current without tracking its compact charge lattice is not a remedy. Nor is the contact test alone a proof that two complete theories are dual.

3. Remove a holomorphic ambiguity locally

As an illustrative local function, take $Z_0(\tau,\bar\tau)=(\operatorname{Im}\tau)^{-p}$ with $p>0$ on the upper half-plane. This example is not an asserted partition function of a specified theory. Change it by $Z_1=Z_0\exp[(f+\bar f)/12]$, with $f=c\tau^2$ and constant complex $c$. Calculate the mixed response (T18.10) for both functions. Is $\mathcal F(\tau_1)-\mathcal F(\tau_2)$ equally invariant?

Hint 1

Write $\tau=x+iy$, $y>0$, and use $\partial_\tau=(\partial_x-i\partial_y)/2$ and $\partial_{\bar\tau}=(\partial_x+i\partial_y)/2$.

Hint 2

The mixed derivative is one quarter of the two-dimensional Laplacian. The real part of a holomorphic function is harmonic.

Solution

Since $\log Z_0=-p\log y$,

$$G_{\tau\bar\tau} =12\partial_\tau\partial_{\bar\tau}(-p\log y) =\frac{3p}{y^2}.$$

Writing $c=c_R+ic_I$ gives $f+\bar f=2c_R(x^2-y^2)-4c_Ixy$. Its Laplacian is zero, so $Z_1$ gives the same positive mixed response. Meanwhile $\mathcal F_1=\mathcal F_0-(f+\bar f)/12$. The difference between two distinct coupling points generally shifts by the corresponding difference of this harmonic function. A finite difference of free energies is therefore not automatically protected by the same argument as the mixed derivative.

Under a holomorphic change of coupling coordinate, the displayed component $G_{\tau\bar\tau}$ also transforms by the Jacobian factors of a metric. Comparing components without transforming the coupling coordinates would be another normalization error.

Before comparing two answers, identify the same sources and insertion, specify the permitted bulk and defect counterterms, and extract the information those terms cannot change.