Observable normalizations

Before comparing two numbers, identify what was held fixed: the operator, its source, the gauge group, the integration measure and the background. A factor of two in a current coefficient can be a notation change. Replacing an ordinary trace by a normalized trace inside the action can change the coupling. This reference makes those distinctions explicit; the linked lessons supply the complete derivations.

For Euclidean sources collectively denoted by $s$, use

$$Z[s]=\int\mathcal D\phi\,e^{-S[\phi;s]},\qquad \mathcal W[s]=-\log Z[s],\qquad F[s]=\operatorname{Re}\mathcal W[s]=-\log|Z[s]|. \tag{RF10.1}$$

The complex logarithm uses a continuous local branch where $Z\ne0$. A normalized expectation value includes division by the same $Z$. A sphere partition function is not, merely by being called a partition function, a thermal trace or an integer count.

Five different counting and generating objects

Here $\Gamma=(-1)^{\mathsf F}$ denotes fermion parity. It is distinct from a flavor charge and from the free energy $F$.

Object	Definition and essential data	What it measures
Witten index in quantum mechanics	$I(\beta)=\operatorname{Tr}_{\mathcal H}\Gamma e^{-\beta H}$, with $H=\{Q,Q^\dagger\}/2$, a specified Hilbert space and operator domains	A signed count of zero-energy states when the heat traces and pairing argument are valid
Three-dimensional superconformal index	A graded trace on the physical Hilbert space on $S^2$, using a chosen conformal supercharge and charges commuting with it	A weighted signed count of its cohomology, including states of nonzero conformal dimension
Hilbert series	$H_{\mathcal R}(t)=\sum_{n\ge0}(\dim\mathcal R_n)t^n$ for a specified nonnegatively graded ring with finite-dimensional graded pieces	Dimensions of graded ring components; no fermion-parity sign is inserted in this definition
Sphere partition function	The path integral on a specified supersymmetric sphere background, with a fixed contour, source prescription and counterterms	A generating function for integrated responses and normalized insertions
Topologically twisted index	A supersymmetric trace with the spatial R-symmetry flux and circle spin/R-holonomy specified; flavor fluxes and fugacities are additional data	A signed count in the twisted background, which differs from the radial superconformal background

For the site’s Witten index, the upper component is bosonic. Its positive-frequency supersymmetric oscillator has $I=+1$. Tong assigns the opposite parity to the upper component, so that example has the opposite sign in his convention. Also, $I=0$ does not imply that there are no supersymmetric states: one even and one odd zero-energy state cancel. Noncompact continuous spectra require additional care; subtracting two divergent traces is not the definition above. Tong, section 1.2.2

The three-dimensional $\mathcal N=2$ radial index uses the dimensionless conformal generator $D$, the physical superconformal charge $R$ and a sphere rotation $J_3$. Choose $q$ with charges $(D,R,J_3)=(1/2,1,-1/2)$. Then

$$H_q=\{q,q^\dagger\}=D-R-J_3\ge0,\qquad \mathcal I(x,z_a)=\operatorname{Tr}_{\mathcal H(S^2)} \Gamma e^{-\beta H_q}x^{D+J_3}\prod_a z_a^{F_a}. \tag{RF10.2}$$

The flavor charges $F_a$ commute with $q$ and $q^\dagger$; $D$ alone does not. The trace includes the gauge projection and required dynamical magnetic sectors. Background flavor fluxes, if present, are fixed separately. The regulator has $\beta>0$ here; the parameter in IDX3 (2.1) is $\beta_{\rm source}=-\beta$. Convergence of the fugacity expansion is part of the prescription.

By contrast, the simplest twisted $S^2\times S^1$ trace uses R-flux $-1$, periodic circle spin structure, and the corresponding trivial circle R-holonomy. With no rotational fugacity its cohomology trace has the insertion $\Gamma\prod_a y_a^{F_a}$. Changing the spin/R-holonomy choice changes that insertion; changing the R-charge lattice changes which fields are globally defined. The same manifold name $S^2\times S^1$ therefore does not identify this trace with (RF10.2). CK19 (6.1)–(6.5)

A Hilbert series is a third operation. For example, with $\deg X=1$, $\mathcal R=\mathbb C[X]/(X^2)$ has $H_{\mathcal R}=1+t$ because $1,X$ are independent and $X$ is nilpotent. Its reduced ring is $\mathbb C$, with series $1$, although both have the same single vacuum point. Neither series determines the Witten index without a Hilbert space, parity and supercharge. See chiral rings and vacuum points.

Current coefficients: an actual factor-of-two translation

At separated points in a three-dimensional conformal theory, define $I_{\mu\nu}(x)=\delta_{\mu\nu}-2x_\mu x_\nu/|x|^2$ and write $\langle J_\mu(x)J_\nu(0)\rangle=C_J I_{\mu\nu}(x)/|x|^4$. Two sources attach a coefficient called $\tau$ to different prefactors:

$$\frac{\tau_P}{16\pi^2} (\delta_{\mu\nu}\partial^2-\partial_\mu\partial_\nu)\frac1{|x|^2} = \frac{\tau_{CT}}{32\pi^2} (\delta_{\mu\nu}\partial^2-\partial_\mu\partial_\nu)\frac1{|x|^2}. \tag{RF10.3}$$

The left convention is P (8.3); the right is CT (2.2), with its separate contact term omitted here. For the same normalized current, equality requires $\tau_{CT}=2\tau_P$. Direct differentiation gives

$$(\delta_{\mu\nu}\partial^2-\partial_\mu\partial_\nu)|x|^{-2} =4I_{\mu\nu}(x)|x|^{-4},\qquad C_J=\frac{\tau_P}{4\pi^2}=\frac{\tau_{CT}}{8\pi^2}. \tag{RF10.4}$$

For the physical R-current of a free three-dimensional $\mathcal N=2$ chiral multiplet, whose scalar has R-charge $1/2$, $\tau_P=1/4$, so $\tau_{CT}=1/2$ and $C_J=1/(16\pi^2)$. This is a translation of notation, not a rescaling of $J$.

The current-correlator lesson fixes the accompanying stress-tensor convention. In that convention $C_T=3\tau_P/(2\pi^2)$ for the superconformal R-current. The specified squashing family gives

$$\left.\frac{d^2F}{db^2}\right|_{b=1}=\frac{\pi^2}{2}\tau_P. \tag{RF10.5}$$

This uses the full P background, the physical R-current and the stated supersymmetric conformal subtraction. It is not a formula for an arbitrary metric-only deformation or a trial R-current. The free chiral check is $F''(1)=\pi^2/8$. A change of the parameter $b$ also changes the numerical Hessian by the chain rule.

Three benchmark conventions

The normalized four-dimensional circular loop

For the fundamental half-BPS circle in $\mathcal N=4$ super Yang–Mills, set $\lambda=g^2N$ and $M=2\pi\ell a$, where $\ell$ is the sphere radius and $a$ the constant Hermitian localization scalar. The finite-$N$ matrix expectation is

$$W_U(\lambda,N)= \frac{\int dM\,e^{-2N\operatorname{Tr}M^2/\lambda} \,N^{-1}\operatorname{Tr}e^M} {\int dM\,e^{-2N\operatorname{Tr}M^2/\lambda}},\qquad W_{SU}=e^{-\lambda/(8N^2)}W_U. \tag{RF10.6}$$

$\operatorname{Tr}$ is the ordinary fundamental trace. The first integral is over all Hermitian matrices; the $SU(N)$ answer instead integrates traceless matrices, at the same $g$ and $\lambda$. This includes the scalar coupling of the half-BPS loop, not just gauge holonomy. It is an exact finite-$N$ formula for this observable, not a planar approximation. The finite-$N$ lesson connects the original proposal to localization and derives the center factor. The trace distinction appears explicitly in DG (5.1).

First-paper Seiberg–Witten periods

For four-dimensional pure $SU(2)$ $\mathcal N=2$ theory, this site uses the first paper’s electric and magnetic special coordinates $a,a_D$:

$$Z_\gamma=n_ea+n_ma_D,\qquad M_\gamma\ge\sqrt2\,|Z_\gamma|,\qquad \tau_{\rm eff}=\frac{da_D}{da}. \tag{RF10.7}$$

The W boson has $n_e=1$ and the smooth basic monopole has $n_m=1$. A fundamental Wilson probe has electric label $1/2$; it is not dynamical matter in the pure theory. The coordinates have mass dimension one. The bound is saturated by an existing BPS state; it does not assert the existence or stability of every charge pair. These are exact two-derivative low-energy data, and in general $a_D\ne\tau_{\rm eff}a$. The charge and period dictionary explains the conversion to the algebraic central charge $\mathcal Z=\sqrt2 Z_\gamma$. SW1 (4.3)–(4.6)

ABJM canonical normalization

Use compact $U(N)_k\times U(N)_{-k}$ with no additional quotient, positive integer $k$, and the massless round-sphere prescription of the small-rank calculation: $H=-i/\ell$, $A=V=C=0$, with no flavor masses or Fayet–Iliopoulos sources. The Cartan measure includes $1/(N!)^2$, and the dimensionless variables $s,t$ have phase $e^{i\pi k(\sum s_i^2-\sum t_i^2)}$. The exact normalization obeys

$$Z(0,k)=1,\qquad Z(1,k)=\frac1{4k},\qquad F(N,k)=-\log Z(N,k). \tag{RF10.8}$$

In the exact Fermi-gas formulation, $Z(1,k)=\operatorname{Tr}\rho$; the positive operator $\rho$ is not normalized to unit trace. Its Fredholm determinant generates the canonical coefficients $Z(N,k)$. That auxiliary particle-number generating function is not a supersymmetric index. Large-$N$ phase-volume and Airy approximations retain this measure convention but are not substitutes for the exact finite-$N$ coefficients. FG (3.3), (3.7), (3.9), (3.24)

What source and scheme changes do

For any one source parameter $s$, differentiating (RF10.1) gives

$$\mathcal W''(s)=\langle S''(s)\rangle- \langle S'(s)S'(s)\rangle_c. \tag{RF10.9}$$

The explicit second derivative includes quadratic source couplings, often called seagulls. Thus an integrated response is not obtained by integrating only the separated two-point function. The source-response lesson also fixes the supersymmetric path through auxiliary sources when a mass is varied.

A source-only local term $I_{\rm ct}[s]$ multiplies $Z$ by $e^{-I_{\rm ct}}$ and shifts $\mathcal W$ by $I_{\rm ct}$. It cancels between numerator and denominator of a fixed normalized insertion. A separate finite counterterm on the line can still redefine a Wilson-loop observable. Specify that defect prescription too.

For a compact, unit-charge flavor source on a spin three-manifold, an allowed integer Chern–Simons counterterm shifts the contact coefficient $\kappa$ by an integer; this is not an arbitrary continuous change of the separated coefficient $C_J$. The charge lattice and spin assumption matter. CT (1.3)–(1.7)

In four-dimensional $\mathcal N=2$ superconformal theories a different ambiguity occurs. For the renormalized supersymmetric sphere partition function, locally on the space of exactly marginal couplings $\tau^I$,

$$Z_{S^4}=e^{K/12},\qquad K\longmapsto K+f(\tau)+\overline f(\bar\tau),\qquad G_{I\bar J}=12\partial_I\partial_{\bar J}\log Z_{S^4}. \tag{RF10.10}$$

Here $K$ is a Kähler potential on coupling space, not on the Seiberg–Witten vacuum space. The metric $G$ is unchanged by this local holomorphic ambiguity. Its operator normalization uses the source term $\pi^{-2}\int\delta\tau^I O_I+\text{conjugate}$ and $\langle O_I(x)\overline O_{\bar J}(0)\rangle=G_{I\bar J}/|x|^8$. See GI (1.1)–(2.3) and universal observables.

Three quick convention tests

1. Replace the trace in the Gaussian action

Define $\operatorname{tr}=\operatorname{Tr}/N$. Someone replaces $\operatorname{Tr}$ by $\operatorname{tr}$ in the action of (RF10.6), retaining $2N/\lambda$, and uses the correctly normalized insertion $\operatorname{tr}e^M$. Has division by $Z$ repaired the change? Find the coupling actually being evaluated.

Solution

The new action is $(2/\lambda)\operatorname{Tr}M^2$, whereas the intended action is $(2N/\lambda)\operatorname{Tr}M^2$. It therefore evaluates $W_U(\lambda_{\rm eff},N)$ with $\lambda_{\rm eff}=N\lambda$. A normalized ratio cancels a constant measure factor, not a change in the distribution. To keep the same theory, use $(2N^2/\lambda)\operatorname{tr}M^2$. As a check, the Gaussian contraction gives $W_U=1+\lambda/8+O(\lambda^2)$; the mistaken action gives $1+N\lambda/8+O(\lambda^2)$.

2. Double the current

Replace the physical R-current by $J'=2J_R$ and choose its source $B'$ so that $B'J'=BJ_R$. What happens to $C_J$ and to the coefficient called $\tau_P$? Can the new coefficient be inserted unchanged into the stress-tensor relation?

Solution

$B'=B/2$, while $\langle J'J'\rangle=4\langle J_RJ_R\rangle$. Hence $C'_J=4C_J$ and, if the prefactor in (RF10.3) is retained, $\tau'_P=4\tau_P$. This is different from the factor-two source-notation translation in (RF10.3). The physical stress tensor has not changed. The relation $C_T=3\tau_P/(2\pi^2)$ assumes R-charge normalization in which the supercharges have unit magnitude; $J'$ does not obey that normalization. For compact sources, the connection periods and charge lattice must also be translated, not silently held fixed while dividing $B$ by two.

3. Continue a mass away from the real axis

On the round sphere with $H=-i/\ell$, $A=V=C=0$, take a constant flavor scalar $m$, auxiliary field $D_B=im/\ell$, vanishing flavor connection, and $x=\ell m$. An integer level-$n$ supersymmetric flavor counterterm gives $Z_n(x)=e^{i\pi n x^2}Z_0(x)$. Determine its effect on $F=-\log|Z|$ for real $x$, and then for $x=u+iv$ with $u,v$ real. Does equality of absolute values at real mass establish equality of complex partition functions?

Solution

The bosonic term in P (6.28) is $-nD_Bm/(2\pi)$. Multiplication by the sphere volume $2\pi^2\ell^3$ gives $I_{\rm ct}=-i\pi n x^2$, confirming the phase in the question. For real $x$ its modulus is one, so $F_n=F_0$ although $\mathcal W_n=\mathcal W_0-i\pi n x^2$ locally. For complex $x$, $|e^{i\pi n x^2}|=e^{-2\pi n uv}$ and $F_n-F_0=2\pi n uv$. Real-mass absolute values therefore miss the phase and do not determine the analytically continued absolute values. The statement compares two declared integer-counterterm prescriptions; it does not license arbitrary changes to the global flavor-source normalization.