Source derivatives and contact terms

A source derivative measures how a partition function changes under a specified probe. That response can contain information from separated points and a contribution supported only when the probe insertions coincide. We will calculate both and see why a contact term can retain physical information.

Use sources and currents, the first gauge-theory matrix integral, and supersymmetric Chern–Simons and FI terms as preparation. The SQED–XYZ section additionally uses the mass and FI dictionary. The initial source calculation needs no supersymmetry; its sphere application has the same three-dimensional $\mathcal N=2$ parent with four real supercharges as those lessons.

Differentiate the full sourced action

Let $B_\mu$ be a fixed Abelian flavor connection, and let $\varphi$ denote all dynamical fields. In flat Euclidean space define

$$Z[B]=\int\mathcal D\varphi\,e^{-S[\varphi;B]},\qquad \mathcal W[B]=-\log Z[B],\qquad j^\mu[B](x)=\frac{\delta S}{\delta B_\mu(x)}.$$

The metric and renormalization prescription are held fixed. A regulator must make the derivatives meaningful; source dependence of the regulated measure is included in $S$. With our plus-source convention, the first response is $\delta\mathcal W/\delta B_\mu=\langle j^\mu\rangle$. Differentiating the insertion and the normalized weight separately gives

$$\mathcal K^{\mu\nu}(x,y) \equiv\frac{\delta^2\mathcal W}{\delta B_\mu(x)\delta B_\nu(y)} =\left\langle\frac{\delta^2S}{\delta B_\mu(x)\delta B_\nu(y)}\right\rangle -\langle j^\mu(x)j^\nu(y)\rangle_c. \tag{O06.1}$$

The connected subtraction comes from the derivative of $1/Z$. The first term includes explicit quadratic source couplings. For example, a charged scalar’s $q^2B^2|\phi|^2$ contributes $2q^2\delta^{\mu\nu}\langle|\phi|^2\rangle\delta^{(3)}(x-y)$. It is needed even before we consider optional background counterterms.

It is useful to name the completed current correlation distribution $\mathcal C^{\mu\nu}=-\mathcal K^{\mu\nu}$. At separated points it is the connected current correlator. At coincidence its definition includes the subtraction of the explicit second derivative and the specified local renormalization terms. Thus $\mathcal C$ and $\mathcal K$ have opposite signs, while an uncompleted product $\langle jj\rangle_c$ need not equal either at coincidence. CT, (1.1)–(1.4) discusses this local-source dependence. Its later flavor coupling uses $-j_{\rm CT}\cdot B$; at the same source our current has the opposite sign, which leaves a two-current product unchanged.

Calculate a local Chern–Simons response

Choose orientation $\epsilon^{123}=+1$. A local term in the background effective functional has the form

$$\mathcal W_{\rm loc}[B] =\frac{i\kappa}{4\pi}\int d^3x\, \epsilon^{\mu\nu\rho}B_\mu\partial_\nu B_\rho. \tag{O06.2}$$

For this local differentiation, work with compactly supported variations in one trivialization, or globally defined potentials on a closed manifold. Global restrictions on its coefficient will be imposed below. The source has mass dimension one, so $\kappa$ is dimensionless.

Both occurrences of $B$ must be varied. Integration by parts makes their contributions equal:

$$\delta\mathcal W_{\rm loc} =\frac{i\kappa}{2\pi}\int d^3x\, \delta B_\mu\epsilon^{\mu\nu\rho}\partial_\nu B_\rho.$$

One more derivative gives the promised contact term:

$$\begin{aligned} \mathcal K^{\mu\nu}_{\rm loc}(x,y) &=-\frac{i\kappa}{2\pi}\epsilon^{\mu\nu\rho} \partial_{x^\rho}\delta^{(3)}(x-y),\\ \mathcal C^{\mu\nu}_{\rm loc}(x,y) &=+\frac{i\kappa}{2\pi}\epsilon^{\mu\nu\rho} \partial_{x^\rho}\delta^{(3)}(x-y). \end{aligned} \tag{O06.3}$$

The first minus sign comes from $\epsilon^{\mu\rho\nu}=-\epsilon^{\mu\nu\rho}$, not from dropping a term. The second line agrees with CT, (1.3). Its dimension is four, matching two dimension-two currents. It is transverse because $\epsilon^{\mu\nu\rho}\partial_\mu\partial_\rho=0$, and it obeys the required exchange symmetry $(\mu,x)\leftrightarrow(\nu,y)$.

A probe that detects the contact term

Take a flat three-torus with each coordinate of period $2\pi L$, volume $V_3=(2\pi L)^3$, and an integer $n\ne0$. The smooth, globally defined source

$$B_1=u\cos(nz/L),\qquad B_2=v\sin(nz/L),\qquad B_3=0$$

has $[u]=[v]=1$. Its curl is $(-nv\cos(nz/L)/L,-nu\sin(nz/L)/L,0)$, so

$$\mathcal W_{\rm loc}(u,v) =-\frac{i\kappa nV_3}{4\pi L}uv,\qquad \partial_u\partial_v\mathcal W_{\rm loc} =-\frac{i\kappa nV_3}{4\pi L}. \tag{O06.4}$$

Smearing the first line of (O06.3) with these two profiles gives the same answer: differentiating the sine produces $n\cos(nz/L)/L$, and the integral of $\cos^2$ is $V_3/2$. A contact response can therefore be measured by extended probes whose supports overlap. It vanishes for two smooth probes with disjoint supports. The torus profile is an ordinary source test, not a claim that these particular sources preserve supersymmetry.

In momentum space a derivative of a delta function is polynomial in momentum. With Fourier convention $\widehat f(p)=\int d^3x\,e^{ip\cdot x}f(x)$, the second line of (O06.3) becomes $\kappa\epsilon^{\mu\nu\rho}p_\rho/(2\pi)$. A generic momentum-dependent parity-odd form factor can also contribute at separated points. We are isolating its constant shift, which gives the one-derivative Chern–Simons contact term; higher-derivative local terms require their own dimensions and subtraction prescription. CT, (2.1)–(2.3) makes the local versus separated-point distinction away from a fixed point.

What a scheme change can remove

Fix the flavor group to be compact $U(1)$, with unit-normalized integral charges and flux $\int dB/(2\pi)\in\mathbb Z$. Work on spin manifolds, as appropriate for the fermionic theories here. An allowed background Chern–Simons counterterm changes $\kappa$ by an integer $N$.

The global definition is essential. Extending the connection over a spin four-manifold represents the counterterm as $iN(4\pi)^{-1}\int dB\wedge dB$. Two extensions differ by $i\pi N\int c_1^2$, which is a multiple of $2\pi i$ because the integral square is even on a closed spin four-manifold. This is the compact spin normalization derived in the Chern–Simons lesson and in CT, (1.5)–(1.7). A single-patch manipulation of $B\,dB$ does not define the functional on nontrivial bundles.

Thus the representative $\kappa$ is scheme dependent, but $\kappa$ modulo integers is not. An arbitrary real subtraction is not permitted in this fixed compact-source problem. A fractional effective coefficient can be carried by the complete quantum theory, including nonlocal or topological information; it is not an independently available fractional local counterterm. Exercise 3 makes this distinction concrete.

These statements compare counterterms with the source group, charge lattice and other renormalization conditions held fixed. Changing the normalization of the flavor generator changes the coefficient and its quantization rule. Mixed R-background terms have additional bundle and charge information, so the integer rule for this unit flavor connection must not be copied to every R-source convention.

Continue the source supersymmetrically on the sphere

Now return to the round sphere of radius $\ell$ with $H=-i/\ell$, $A=V=C=0$ and $R=-6/\ell^2$. The full three-dimensional parent is $\mathcal N=2$; the Euclidean background preserves both rigid spinor families. An external flavor vector multiplet contains a connection $B_\mu$, scalar $m$, two gauginos and auxiliary source $D_B$. Its constant four-supercharge configuration is

$$B_\mu=0,\qquad m\in\mathbb R,\qquad D_B=-mH=\frac{im}{\ell}, \qquad\lambda_B=\widetilde\lambda_B=0. \tag{O06.5}$$

This is P v3, (6.26). The auxiliary is prescribed, not integrated out. Consequently a mass derivative changes both $m$ and $D_B$. In terms of the dimensionless source $x=\ell m$,

$$\frac{d}{dx} =\frac1\ell\frac{\partial}{\partial m} +\frac{i}{\ell^2}\frac{\partial}{\partial D_B}. \tag{O06.6}$$

The partial derivatives on the right mean variation of the constant source values before restricting to (O06.5). Squaring this operator includes the mixed derivative and a minus sign on the $D_BD_B$ term. The sphere mass response therefore packages several operator insertions and their local terms; it is not merely a spatial integral of $\langle j_\mu j_\nu\rangle$.

The supersymmetric completion of the level-$N$ background counterterm is

$$S_N=\frac{N}{4\pi}\int\sqrt g\,d^3x\, \left(i\epsilon^{\mu\nu\rho}B_\mu\partial_\nu B_\rho -2mD_B+2i\lambda_B\widetilde\lambda_B\right).$$

This follows from P, (6.28) with the whole vector multiplet frozen; CT, (4.5) gives the corresponding background contact functional. Since $\operatorname{Vol}(S^3)=2\pi^2\ell^3$, evaluation on (O06.5) gives

$$S_N=-i\pi N x^2,\qquad Z_N(x)=e^{i\pi N x^2}Z_0(x),\qquad \mathcal W_N''(x)=\mathcal W_0''(x)-2\pi iN. \tag{O06.7}$$

The volume, factor of two and imaginary auxiliary all matter. The real-source modulus $|Z|$ is unchanged by this counterterm, but the imaginary second response changes. In the superconformal prescription, with the physical current and other sources fixed, CT, (7.6) extracts the flavor contact coefficient as $\kappa_{ff}=-(2\pi)^{-1}\partial_x^2\operatorname{Im}\mathcal W|_{x=0}$. Equation (O06.7) verifies its integer shift directly. Deriving the separated-point current coefficient from the real response requires the supersymmetric Ward identities in addition to this local calculation.

Match the topological source in SQED and XYZ

This section uses the deformation dictionary. Keep compact one-flavor $U(1)$ SQED with dynamical level $k=0$ and $W=0$. Its infrared XYZ description uses $X=M$, $Y=V_+$, $Z=V_-$ and $W=XYZ$. With axial mass $\mu$ and topological-source mass $\nu$, the XYZ real masses are $2\mu,-\mu+\nu,-\mu-\nu$ respectively. The topological charges are therefore $(0,+1,-1)$ in the fixed unit normalization.

On the SQED side the level-one mixed coupling is $i(2\pi)^{-1}\int B\wedge da$ together with its supersymmetric partners. Freezing the topological source gives $\xi=-\zeta=-\nu/(2\pi)$ and the sphere factor $e^{2\pi i\eta s}$, where $\eta=\ell\nu$ and $s=\ell\sigma$. These normalizations are part of the comparison, not adjustable factors after differentiation.

Take $\mu=0$ and the same trial matter R-charge $r=1/2$ used for the explicit integral. The XYZ trial charges are $(1,1/2,1/2)$, whose sum is two. This is a convenient supersymmetric observable; these trial charges are not asserted to be the interacting superconformal dimensions. In the matched topological contact prescription,

$$Z_0(\eta)=\int_{\mathbb R}\frac{e^{2\pi i\eta s}\,ds}{2\cosh\pi s} =\frac1{2\cosh\pi\eta}. \tag{O06.8}$$

The XYZ determinants on this slice give the same expression: the $R=1$, zero-mass $X$ factor is one, and the $Y,Z$ pair gives $1/(2\cosh\pi\eta)$. This evaluates a particular observable using the existing determinant and Fourier calculation. It does not reconstruct the full duality from the result.

Add an integer topological background counterterm $N$ on one side only. Then

$$\mathcal W_N(\eta)=\log(2\cosh\pi\eta)-i\pi N\eta^2, \qquad \mathcal W_N''(0)=\pi^2-2\pi iN. \tag{O06.9}$$

The separated-point content is unchanged, yet the second responses no longer match as complex numbers. Restoring the same counterterm prescription restores equality. The unit BF term couples a dynamical field to a source; the extra $N$ term contains only the source and does not change SQED’s dynamical level $k=0$.

This comparison tracks the topological-topological contact term on the stated mass slice. It does not declare all axial, R or gravitational contact data zero. Equality of moduli alone would discard the integer representative and could also miss physical fractional contact information. Conversely, a discrepancy that cannot be supplied by an allowed counterterm is not repaired by calling it a phase.

Exercises

1. Polarize a two-source counterterm

For two unit-normalized compact flavor sources on spin manifolds, let the symmetric integer matrix be $N_{ab}=\left(\begin{smallmatrix}n&p\\p&q\end{smallmatrix}\right)$. Write the bosonic supersymmetric counterterm, evaluate it on constant sphere sources $x_a=\ell m_a$, and find the mixed second derivative. Why is the coefficient of $x_1x_2$ twice the off-diagonal matrix entry?

Hint 1

Sum over both ordered pairs $(1,2)$ and $(2,1)$ in the matrix expression. Their integrated connection terms agree by integration by parts.

Hint 2

Use $D_a=im_a/\ell$ before integrating over the sphere. Keep the matrix convention separate from a convention that sums only over $a<b$.

Solution

With vanishing background gauginos, the density is

$$\mathcal L_N=\frac1{4\pi}\sum_{a,b=1}^2N_{ab} \left(i\epsilon^{\mu\nu\rho}B^a_\mu\partial_\nu B^b_\rho -2m_aD_b\right).$$

On a closed manifold the mixed connection piece is $ip(2\pi)^{-1}\int B^1\wedge dB^2$. Both ordered terms are required. The sphere action and response shifts are

$$S_N=-i\pi(nx_1^2+2px_1x_2+qx_2^2),\qquad \Delta\partial_{x_a}\partial_{x_b}\mathcal W=-2\pi iN_{ab}.$$

In particular the mixed entry is $-2\pi ip$, not $-4\pi ip$. The factor two already appeared in the polynomial before differentiation. The mixed global extension is $ip(2\pi)^{-1}\int dB^1\wedge dB^2$; its ambiguity is $2\pi ip$ times an integer, agreeing with the stated normalization. All $x_a$ and the action are dimensionless.

2. Remove a quadratic ambiguity without discarding it silently

For the SQED slice (O06.9), calculate the third derivative at a nonzero real $\eta$. Which part of the counterterm ambiguity does this operation remove? Does the value of $\mathcal W''$ modulo its allowed imaginary shifts become meaningless as a result?

Hint 1

Differentiate the full complex expression, including its quadratic phase. A higher derivative does not have to annihilate the nonlocal mass dependence.

Hint 2

Use $d\tanh x/dx=\operatorname{sech}^2x$. Distinguish choosing a response insensitive to a counterterm from proving that all lower responses lack information.

Solution

The derivatives are

$$\mathcal W_N''(\eta)=\pi^2\operatorname{sech}^2(\pi\eta)-2\pi iN, \qquad \mathcal W_N'''(\eta) =-2\pi^3\operatorname{sech}^2(\pi\eta)\tanh(\pi\eta).$$

The third derivative is independent of $N$ and nonzero for real $\eta\ne0$. At zero it vanishes by the reflection symmetry of the slice, so evaluating only there would hide this remaining mass dependence.

This operation removes the specified quadratic flavor-CS polynomial; it is not a license to add arbitrary source-dependent counterterms. The second response still has a well-defined equivalence class under $-2\pi iN$. In a conformal extraction that class includes the fractional contact coefficient. A quantity can be useful precisely because a discrete ambiguity is known, even when it is not an absolute real number.

3. Obtain a fractional contact coefficient from a compact theory

Consider a separate compact topological model on a closed oriented spin three-manifold, with dynamical connection $a$ and external connection $B$, both of unit flux normalization:

$$S=\frac{i}{4\pi}\int \left(k\,a\wedge da+2p\,B\wedge da+q\,B\wedge dB\right), \qquad k\ne0,\quad k,p,q\in\mathbb Z.$$

Calculate the local background response after eliminating $a$ in a trivial patch. For $k=3,p=1,q=0$, find its fractional class. On a closed two-cycle with fluxes $n_a,n_B\in\mathbb Z$, what does the gauge equation require? Why does the answer not authorize a stand-alone fractional counterterm for arbitrary $B$?

Hint 1

Complete the local Chern–Simons square using $a+(p/k)B$, or use the equation for the field strength. Such a shift need not preserve the compact flux lattice globally.

Hint 2

Integrate the field-strength equation over the two-cycle. Do not replace integer fluxes by continuously adjustable real numbers.

Solution

The variation gives $k\,da+p\,dB=0$. Locally, up to an exact form and boundary terms,

$$k\,a\,da+2p\,B\,da+q\,B\,dB =k\left(a+\frac pkB\right)d\left(a+\frac pkB\right) +\left(q-\frac{p^2}{k}\right)B\,dB.$$

The effective contact coefficient is therefore $\kappa=q-p^2/k$. For the specified integers it is $-1/3$, with fractional class $2/3$. Changing the allowed background counterterm shifts $q$ by an integer and leaves that class unchanged.

Globally the equation requires $k n_a+p n_B=0$. With $k=3,p=1$, the free integer flux of $B$ must be divisible by three. More generally this is a necessary condition on each free integral two-cycle; torsion sectors and holonomies require the full compact functional, not this differential-form calculation. The compact path integral implements the corresponding constraints. It cannot be replaced for every background bundle by a fractional local polynomial alone. This is the mechanism discussed in CT, (3.5)–(3.8).

A sphere check can be made with its supersymmetric completion. The normalized Gaussian Cartan ratio is $Z(\eta)/Z(0)=\exp[i\pi(q-p^2/k)\eta^2]$, obtained from the exponent $i\pi k s^2+2\pi ip\eta s+i\pi q\eta^2$ with a convergent Fresnel prescription. The same coefficient appears in the local contact response. This sphere calculation has trivial bundles; it verifies the phase coefficient, not the omitted global flux constraints.

A source response is ready for physical interpretation once its coupling, differentiation path and allowed local counterterms have been specified. Continue with superconformal currents and two-point functions to normalize the separated-point correlators that remain alongside these contact terms.

The geometric-dependence dossier uses these response conditions to distinguish one exact tangent from a finite partition-function comparison.

The duality benchmark applies these source conventions to a complete complex-valued SQED–XYZ comparison and a controlled large-mass limit.

Paper guides

Connect this lesson to the original papers and their notation.

• Paper guide: Section 8, physical observables