Supersymmetric Partition Functions, Counterterms, and Universal Data

A curved-space partition function is first a generating functional for background fields, not automatically a scheme-independent number. Two supersymmetric regulators can differ by finite local functionals of those backgrounds. The physical task is therefore to classify the allowed supersymmetric counterterms and then form derivatives, differences, anomaly coefficients, or fractional contact terms that they cannot change.

Required background. Use the localization locus and determinant construction and the distinction between renormalization schemes and finite parts.

Helpful background. Anomaly polynomials and inflow identify logarithmic and contact terms that cannot be removed while preserving all desired symmetries.

Partition functions as background generating functionals

Let $\mathcal B$ denote a complete rigid supersymmetric background, including the metric and background gauge fields, scalars, and auxiliary fields. Define

$$W[\mathcal B]=-\log Z[\mathcal B].$$

Functional derivatives of $W$ generate integrated current correlators and contact terms. If two regulators preserve the same local symmetries and produce the same separated-point theory, locality implies

$$W_2[\mathcal B]-W_1[\mathcal B] =S_{\mathrm{ct}}[\mathcal B],$$

where $S_{\mathrm{ct}}$ is a finite local counterterm. Supersymmetry restricts it to the top component of a background supermultiplet, but does not usually force it to vanish Closset et al. 2012, §§4–5.

The classification proceeds in a fixed spacetime dimension:

1. specify the off-shell background supergravity and flavor multiplets;
2. list local invariant densities of the required mass dimension;
3. impose gauge invariance, supersymmetry, global quantization, and reality;
4. evaluate those densities on the chosen background;
5. quotient the proposed observable by the resulting finite shifts.

It is not enough to list ordinary bosonic curvature scalars. Their supersymmetric completions contain background auxiliary fields and can evaluate differently on two supersymmetric backgrounds with the same metric.

A three-dimensional contact-term benchmark

For a three-dimensional $N=2$ theory with Abelian background flavor multiplets, supersymmetric Chern–Simons counterterms can shift the round-sphere answer by a phase. With dimensionless real masses $\widehat m_a=r m_a$, the allowed shift has the schematic form

$$\Delta W =i\pi k_{ab}\widehat m_a\widehat m_b +2\pi i k_{aR}\widehat m_a +i\pi k_{RR},$$

supplemented by gravitational terms. Large-gauge invariance quantizes the permitted changes of the coefficients. Consequently, the integer part of a contact coefficient can be scheme dependent while its fractional part is a physical observable Closset et al. 2013, §§2–3.

On a reflection-positive round-sphere background these Chern–Simons shifts are pure phases. It is therefore useful to distinguish

$$F=-\log Z \qquad\text{from}\qquad \mathcal F=-\log|Z|.$$

The phase of $Z$ contains meaningful fractional contact data but depends on integer counterterm choices. The real quantity $\mathcal F$ removes those particular phase shifts, although other dimensions or backgrounds can admit different real local ambiguities.

At a three-dimensional conformal fixed point, derivatives with respect to a real mass provide a further check. In a convention with canonically normalized flavor current $j_a^\mu$,

$$\left. \frac{\partial^2\mathcal F}{\partial\widehat m_a\partial\widehat m_b} \right|_{\widehat m=0} =\frac{\pi^2}{2}\tau_{ab},$$

where $\tau_{ab}$ is the positive coefficient of the separated current two-point function. A sign change can arise if the differentiation variable is taken to be an imaginary supersymmetric mass instead; the source convention must accompany the formula Closset et al. 2012, §§3–4.

Four-dimensional sphere data and Kähler ambiguity

For a four-dimensional $N=2$ SCFT on $S^4$, the dependence on exactly marginal couplings $\tau^i$ determines the Kähler potential $K$ on the conformal manifold. In a common normalization,

$$Z_{S^4}\propto e^{K(\tau,\bar\tau)/12}.$$

A Kähler transformation

$$K\longmapsto K+f(\tau)+\overline{f(\tau)}$$

multiplies $Z_{S^4}$ by a holomorphic times antiholomorphic factor. This is precisely a supersymmetric local-counterterm ambiguity. The Zamolodchikov metric

$$g_{i\bar j}=\partial_i\partial_{\bar j}K$$

is invariant and is extracted by mixed derivatives of $\log Z_{S^4}$ Gerchkovitz, Gomis, and Komargodski 2014, §§2–4.

The logarithmic dependence on the sphere radius is separately controlled by the Weyl anomaly. Its coefficient is universal once the anomaly normalization is fixed, whereas a finite additive constant in $W$ need not be.

How to construct a universal quantity

Operation	Why it can be universal	Required check
Mixed derivative in independent sources	Annihilates holomorphic or low-degree local shifts	Classify every allowed source polynomial first
Difference between matched backgrounds	Cancels identical local terms	Match all background multiplets, not only the metric
Logarithmic scale coefficient	Fixed by an anomaly	Exclude zero-mode and boundary logarithms
Fractional contact coefficient	Integer shifts are counterterms	Fix charge lattice and large-gauge normalization
Absolute value of a unitary $S^3$ partition function	Removes pure Chern–Simons phases	Check that no allowed real counterterm remains

A ratio $Z_1/Z_2$ is not automatically universal. It is safe only if the same local counterterm evaluates equally in numerator and denominator or if its mismatch has been subtracted by a declared convention.

Failure modes

Ignoring the background multiplet. Equal metrics do not imply equal supersymmetric backgrounds; auxiliary fields can make a counterterm contribute differently.

Dropping every phase. A phase may contain the fractional part of a physical Chern–Simons contact term. Remove only the quantized scheme freedom.

Differentiating before classifying. A first or second derivative may still be shifted by a local polynomial. The necessary derivative order is dimension and background dependent.

Calling normalization a counterterm. Dividing by a decoupled sector, a gauge-group volume, or a supersymmetric Casimir factor changes the defined object. State that operation separately.

The strongest meaningful claim is always attached to a particular invariant combination, not to the bare symbol $Z$.

Exercises

Suppose $F(m)$ in three dimensions can shift by $i\pi k m^2$ with integer $k$. Which parts of $F''(0)$ are invariant?

Solution

The shift is $F''(0)\mapsto F''(0)+2\pi i k$. Its real part is invariant, and the imaginary part divided by $2\pi$ is defined modulo an integer. Thus the separated two-point coefficient extracted from the real part and the fractional contact term are both meaningful, but the full complex second derivative is not an absolute number.

References

• Closset, C., T. T. Dumitrescu, G. Festuccia, and Z. Komargodski. “Supersymmetric Field Theories on Three-Manifolds.” Journal of High Energy Physics 2013, no. 5 (2013): 017. DOI; Open PDF.
• Closset, C., T. T. Dumitrescu, G. Festuccia, Z. Komargodski, and N. Seiberg. “Comments on Chern–Simons Contact Terms in Three Dimensions.” Journal of High Energy Physics 2012, no. 9 (2012): 091. DOI; Open PDF.
• Closset, C., T. T. Dumitrescu, G. Festuccia, Z. Komargodski, and N. Seiberg. “Contact Terms, Unitarity, and F-Maximization in Three-Dimensional Superconformal Theories.” Journal of High Energy Physics 2012, no. 10 (2012): 053. DOI; Open PDF.
• Gerchkovitz, E., J. Gomis, and Z. Komargodski. “Sphere Partition Functions and the Zamolodchikov Metric.” Journal of High Energy Physics 2014, no. 11 (2014): 001. DOI; Open PDF.