Reading three-dimensional localization with matter

The paper replaces a field integral by an ordinary integral and a loop by a character of a matrix. To reproduce an observable, however, one must carry the gauge quotient, determinant constants, trace normalization and complex phase through that replacement.

Read Anton Kapustin, Brian Willett and Itamar Yaakov, Exact Results for Wilson Loops in Superconformal Chern-Simons Theories with Matter, arXiv:0909.4559v4, 17 December 2010, JHEP 03 (2010) 089. PDF pages are one greater than printed pages. Use v4: footnote 1 on printed p. 2 records the restoration of the determinant’s factors of two; footnote 4 on printed p. 12 notes corrections in the quadratic gauge-sector action. These are verified version notes, not a claim to have inspected a separate journal erratum.

Prepare with localization in a field theory, the free chiral determinant, and the first gauge integral. The ABJM section below names its additional preparation. The paper’s conformal matter calculation uses the canonical scalar R-charge $1/2$. An arbitrary trial R-charge or unpaired chiral determinant requires the more general prescription in those lessons.

Translate the symbols before the determinants

Paper symbol or operation	Meaning to keep fixed
$a$ in (3.68)	A constant Cartan scalar, not the connection; on radius $\ell$, our variable is $s=\ell\sigma$
“Tr” in (2.1)	An invariant form containing $k/(4\pi)$; for one $U(N)$ node it is $k\operatorname{Tr}_{\rm fund}/(4\pi)$
$\operatorname{Tr}'$ in (3.1)	A positive form for the deformation; it need not equal the signed form in the physical action
$\det_{\rm Ad}$ in (3.68)	A product over nonzero roots; Cartan zero modes remain integration variables
$\mathcal R\oplus\mathcal R^*$ in (1.1)	A conjugate pair of chirals, not one chiral counted twice
$\operatorname{Tr}_{\mathcal S}/\dim\mathcal S$ in (2.5)	A normalized loop in a specified representation; this trace contains no level

KWY use unit radius and their own off-shell continuation, including $D=-\sigma$ at the saddle. The site’s continued auxiliary is $D=i\sigma/\ell$. Compare the localized phases and operators after the stated dictionary; do not infer a universal field-by-field continuation from these two saddle values.

For the calculations here choose the site’s positive single-node phase $e^{i\pi k\sum s_i^2}$. In KWY (3.7), (3.32) and (3.68) the corresponding displayed phase is negative. At matched real eigenvalues use $k_{\rm KWY}=-k$. We keep the supersymmetric regulated level; no extra bosonic level shift is appended. A normalized expectation can be insensitive to an overall constant in $Z$, but not to the phase of its integrand.

Read the reduction in this order

The selected complex odd operation in (3.2) is nilpotent; this is a particular localization choice, not the claim that every rigid supersymmetry squares to zero. The gauge-fixing extension in (3.10) belongs to that same calculation.

Checked source landmark	Reading question	Canonical explanation
(3.1)–(3.5), pp. 7–9	Which positive deformation and contour select the locus?	Deformation identity
(3.10)–(3.20), pp. 10–11	Which zero modes and gauge volumes remain?	The remaining scalar
(3.21)–(3.30), pp. 12–13	Why does an extra Vandermonde double count?	Completed gauge measure
(3.39)–(3.58), pp. 15–18	Which multiplicities survive cancellation?	Paired factors
(3.60)–(3.67), pp. 18–19	What fixes the constant and phase?	Finite determinant prescription
(3.68)–(3.69), p. 20	How do the measure and insertion combine?	Normalized integral

The gauge-measure link uses the two-node example for its detailed derivation; the elementary single-node reconstruction immediately below needs no ABJM dynamics. Appendix A verifies the exact deformations used in section 3. Appendix B evaluates the pure-gauge Gaussian sums. These are different calculations: an exact deformation is not established merely by recognizing the determinant it ought to produce.

The scalar operator (3.39) is the operator of the deformation, with a first-order term along a sphere vector. It need not equal the undeformed conformal scalar operator term by term. Similarly, (3.61) isolates a linear divergence that can require an FI subtraction. Opposite weights cancel it in the paired examples; that cancellation is not a prescription for discarding every phase in an unpaired theory.

First reconstruction: one gauge node and one insertion

Take pure compact $U(2)$ Chern–Simons theory in its $\mathcal N=2$ completion, with no matter and integer $k>2$. This is a changed theory used to check the paper’s pure-gauge result, not ABJM with one node erased. The defining gauge trace is ordinary fundamental trace.

The two eigenvalues $s_1,s_2$ can be permuted, giving Weyl factor $1/2!$. On an off-diagonal matrix unit, $[\operatorname{diag}(s),E_{ij}]=(s_i-s_j)E_{ij}$. These nonzero adjoint eigenvalues, called roots, are $s_1-s_2$ and its negative. Their completed positive-root factor is $[2\sinh\pi(s_1-s_2)]^2$. The polynomial Vandermonde has already combined with the nonzero-mode determinant in (3.20)–(3.30). It must not be multiplied in again.

Use a positive Gaussian regulator and then its Abel limit:

$$\begin{aligned} Z_\epsilon&=\frac12\int_{\mathbb R^2}ds_1\,ds_2\, e^{-(\epsilon-i\pi k)(s_1^2+s_2^2)} [2\sinh\pi(s_1-s_2)]^2,\\ w(s)&=\frac{e^{2\pi s_1}+e^{2\pi s_2}}2, \qquad \langle W\rangle=\lim_{\epsilon\downarrow0} \frac{\int d\mu_\epsilon\,w}{Z_\epsilon}. \end{aligned} \tag{PC14.1}$$

Here $d\mu_\epsilon$ is the full measure in the first line. Its root convention fixes the overall phase of this auxiliary $Z_\epsilon$; replacing the positive-root square by the source’s ordered all-root product changes an overall sign for one $U(2)$ node, which cancels in the displayed ratios.

Why is the insertion $w$? The supersymmetric line connection is $i a_\mu\dot x^\mu+\sigma|\dot x|$. On the locus $a_\mu=0$, its integral around a great circle is $2\pi\ell\sigma=2\pi s$. The trace in the two-dimensional fundamental is divided by two. Path ordering is trivial on the locus, not in the original non-Abelian operator. These are KWY (2.5), (3.8) and (3.32).

Put $c=(s_1+s_2)/2$, $d=(s_1-s_2)/2$. The absolute Jacobian is two and cancels the Weyl factor. Define $\mathfrak a=\epsilon-i\pi k$ and $Q_\epsilon=\exp(\pi^2/\mathfrak a)$. For either remaining Gaussian coordinate $z$,

$$G_\epsilon=\int_{\mathbb R}e^{-2\mathfrak a z^2}dz =\sqrt{\frac{\pi}{2\mathfrak a}}, \qquad \frac{\int e^{-2\mathfrak a z^2+bz}dz}{G_\epsilon} =e^{b^2/(8\mathfrak a)}. \tag{PC14.2}$$

The square-root branch is continuous from positive real $\mathfrak a$. Now the root factor is $2\cosh(4\pi d)-2$ and $w=e^{2\pi c}\cosh(2\pi d)$. Consequently

$$Z_\epsilon=\frac{\pi}{\mathfrak a}(Q_\epsilon^2-1), \qquad \langle w\rangle_\epsilon =\frac{Q_\epsilon(Q_\epsilon^2+1)}2. \tag{PC14.3}$$

For example, the relative-coordinate numerator uses $(2\cosh4\pi d-2)\cosh2\pi d =\cosh6\pi d-\cosh2\pi d$; this fixes the otherwise easy-to-miss factor two. Taking $\epsilon\downarrow0$ gives

$$q=e^{i\pi/k},\qquad \boxed{\langle W\rangle=q^2\cos(\pi/k).} \tag{PC14.4}$$

This is the $N=2$, level-reversed version of (3.33). All finite-regulator integrations above converge absolutely. Their finite exponential sums justify the limit for this example; no general matter contour theorem follows.

Restore the step from one loop to a link

The paragraph after KWY (3.33), printed p. 14/PDF p. 15, considers distinct loops in the same Hopf fibration. This is the family of circles generated by common phase rotation $(z_1,z_2)\mapsto e^{i\tau}(z_1,z_2)$ on the unit sphere in $\mathbb C^2$. They share a preserved operation. A fixed product of their closed insertions is again closed, so the same regulated Ward identity localizes it. The result contains the product of characters in one integral, not the product of separate integrals.

This explains the representation statement in that paragraph: $\operatorname{Tr}_{\mathcal S\otimes\mathcal T}(U\otimes U) =\operatorname{Tr}_{\mathcal S}U\,\operatorname{Tr}_{\mathcal T}U$. The normalization is $1/(\dim\mathcal S\dim\mathcal T)$. It says nothing about statistical independence.

The ribbon used to separate coincident points fixes the framing. We retain the site’s outward-boundary orientation on $S^3\subset\mathbb C^2$, for which equally oriented Hopf fibers link $+1$, and the single-node Gaussian calibration $e^{i\pi/k}$. KWY describe their phase and Hopf splitting as framing $-1$. Carry their level, orientation and phase together when comparing; a bare framing label does not translate a complex number. The pure-gauge examples below do not determine an all-order framing conversion in a matter theory.

The ABJM application

For this section, complete the round-sphere action, two-node matrix measure, rank-one partition function, and bosonic Wilson loop.

Keep compact $U(N)_k\times U(N)_{-k}$ with equal finite $N$, integer $k>0$, no further center quotient, masses or FI terms, and four chirals of R-charge $1/2$. The gauge rank is $N$ per node. The generic flat theory has twelve real Poincaré supercharges; levels one and two have sixteen. Its $\mathcal N=2$ presentation makes four manifest. The radius-$\ell$ background is $H=-i/\ell$, $A=V=C=0$, with the inherited curvature convention $R=-6/\ell^2$. The round calculation uses the established rigid subalgebra and a selected complex localizing operation; it does not count the full bulk symmetry by counting its labels.

Source landmark	What to reconstruct	Canonical location
(2.13) and section 2.2, pp. 6–7	Curved matter and the superconformal input	Curvature completion
(3.5), (3.37), pp. 9, 15	Vector zero modes and vanishing matter	The locus
(4.2)–(4.4), pp. 20–21	Two conjugate pairs per color pair	Bifundamental weights
(3.69), (4.1), pp. 20–21	Node and normalized representation	Loop insertion
(4.5)–(4.9), pp. 21–22	A perturbative expansion with its phase	Finite-rank contour check

The fundamental weights at the two nodes give masses $s_i-t_j$ and $-s_i+t_j$. One conjugate pair contributes $[2\cosh\pi(s_i-t_j)]^{-1}$; there are two such pairs, not four. Each node supplies its independent $1/N!$. Reconstructing the general (3.68), rather than discarding constants in the specialized (4.5), gives

$$\begin{aligned} Z(N,k)&=\lim_{\epsilon\downarrow0}\frac1{(N!)^2} \int_{\mathbb R^{2N}}\prod_i ds_i\,dt_i\, e^{\,i\pi k\sum_i(s_i^2-t_i^2)-\epsilon\sum_i(s_i^2+t_i^2)} \\[-2pt] &\quad\times \frac{\prod_{i<j}[2\sinh\pi(s_i-s_j)]^2 [2\sinh\pi(t_i-t_j)]^2} {\prod_{i,j}[2\cosh\pi(s_i-t_j)]^2}, \qquad w_L=\frac1N\sum_i e^{2\pi s_i}. \end{aligned} \tag{PC14.5}$$

The average uses this same regulated measure with $w_L$ inserted and is divided by $Z_\epsilon$ before taking the limit. For variables $\mu_i=2\pi s_i$, $\nu_i=2\pi t_i$, the measures are $d\mu_i/(2\pi)$ and $d\nu_i/(2\pi)$, the phase is $ik\sum(\mu_i^2-\nu_i^2)/(4\pi)$, and the insertion is $N^{-1}\sum e^{\mu_i}$. Neither center is discarded.

There are three source-reading cautions. First, (4.4) suppresses the Weyl division explicit in (3.68), and its following insertion sentence suppresses the $1/N$ explicit in (3.69). Second, its displayed first-node phase is negative; translate by level reversal or node exchange. Third, (4.5) reuses eigenvalue symbols after rescaling and sets $\alpha=2\pi iN/k$, which algebraically gives the positive first-node Gaussian phase. Its series (4.7) has that same positive linear phase. Keep these displayed formulas distinct rather than treating the phase change as a harmless real rescaling.

At rank one the canonical regulated calculation gives $Z(1,k)=1/(4k)$. It is a check of the complete measure, not a result established by ignoring the common-center integral. An inserted exponential is unbounded: the proved first-node loop answer $e^{i\pi/k}\sec^2(\pi/k)$ has the pole-free domain $k>2$ in that prescription. The source’s localization reduction alone does not prove every regulator limit or justify exchanging all integrations. Its section 4 gives a perturbative check; later exact finite-rank and large-rank evaluations are separate work.

Exit calculations

1. Lose constants and find the observable that notices

A reader copies (PC14.5) but deletes every two in $2\sinh$ and $2\cosh$, and also deletes $(N!)^{-2}$. Find the ratio of the mistaken partition function to the stated one at fixed finite regulator. Evaluate it for $N=2$. Would a normalized Wilson average using the same mistake in numerator and denominator detect this error? What if its $1/N$ were also deleted?

Hint 1

Each node has $N(N-1)/2$ positive roots, and every displayed hyperbolic factor is squared. There are $N^2$ color pairs in the denominator.

Hint 2

Separate a constant multiplying the measure from a constant multiplying only the observable.

Solution

The two vector products carry $2^{2N(N-1)}$, and the denominator carries $2^{2N^2}$. Thus the true hyperbolic ratio is $2^{-2N}$ times the ratio with all twos removed. Omitting the Weyl division gives in total

$$Z_{\rm wrong,\epsilon}=4^N(N!)^2Z_\epsilon.$$

At $N=2$ this factor is $64$. When the limit exists and $Z\ne0$, the free energy changes by $-N\log4-2\log(N!)$. A normalized average cancels the common measure factor, so matching that average cannot certify the absolute partition-function normalization.

Deleting $1/N$ from the insertion instead multiplies the average by $N$, and makes the trivial-connection value $N$ rather than one. This last mistake survives the ratio. None of these constants is a large-rank approximation or a change of framing.

2. Separate self-framing from mutual linking

In the positive-phase pure $U(1)_k$ example, put charge-$m$ and charge-$n$ loops on two distinct, equally oriented Hopf fibers. Here $m,n$ are integers and $k\ne0$. Evaluate their product average, and divide it by the two separate one-loop averages. What happens for $m=1,n=-1$? Use the same Gaussian regulator for every integral.

Hint 1

The product insertion is $e^{2\pi(m+n)s}$ in one integral. There is no root determinant for $U(1)$.

Hint 2

Complete the square with coefficient $\epsilon-i\pi k$, before taking its limit. Then separate $(m+n)^2$ into three terms.

Solution

The normalized finite-regulator integral is $\exp[\pi^2(m+n)^2/(\epsilon-i\pi k)]$. Therefore

$$\langle W_mW_n\rangle=e^{i\pi(m+n)^2/k}, \qquad \frac{\langle W_mW_n\rangle} {\langle W_m\rangle\langle W_n\rangle} =e^{2\pi i mn/k}. \tag{PC14.6}$$

The separate squares are the two self-framing phases; the cross term detects their mutual linking in the specified convention. For opposite unit charges the product average is one, but the ratio is $e^{-2\pi i/k}$. The two separate averages are each $e^{i\pi/k}$. The cancellation of the combined insertion does not make those two averages independent. These pure-gauge Gaussian identities do not specify a general matter-theory framing conversion.

3. A non-Abelian linked pair can vanish

Return to pure $U(2)$ in (PC14.1), $k>2$. Put a normalized fundamental loop on each of two distinct Hopf fibers. Calculate their joint average. Compare it with the square of (PC14.4), and specialize to $k=4$. Explain why a zero answer need not mean a missing supersymmetry or an ill-defined partition function.

Hint 1

The insertion is $w^2=e^{4\pi c}\cosh^2(2\pi d)$. Keep the same root factor and partition function as before.

Hint 2

Use $(2\cosh4\pi d-2)\cosh^2(2\pi d) =\tfrac12(\cosh8\pi d-1)$ and the finite Gaussian identity (PC14.2).

Solution

At finite regulator the center contributes $Q_\epsilon^2$. The relative numerator is $(Q_\epsilon^8-1)/2$, whereas its denominator is $2(Q_\epsilon^2-1)$. Thus

$$\begin{aligned} \langle W_1W_2\rangle &=\frac{q^2}{4}(1+q^2+q^4+q^6),\\ \frac{\langle W_1W_2\rangle}{\langle W\rangle^2} &=\frac{1+q^4}{1+q^2} =e^{i\pi/k}\frac{\cos(2\pi/k)}{\cos(\pi/k)} . \end{aligned} \tag{PC14.7}$$

The denominator is nonzero for the stated $k>2$. At $k=4$, the joint average vanishes, while $\langle W\rangle=i/\sqrt2$ and $\langle W\rangle^2=-1/2$. Also $Z_\epsilon$ tends to $\pi(-i\pi k)^{-1}(q^2-1)\ne0$, so normalization is well defined. The complex measure allows cancellation; it is not a probability measure for a positive random variable.

As an independent representation check, the unnormalized identity is $\chi_{\rm fund}^2=\chi_{\operatorname{Sym}^2}+\chi_{\det}$. The determinant representation has one weight $u_1u_2$, and the symmetric square has weights $u_1^2,u_1u_2,u_2^2$, where $u_i=e^{2\pi s_i}$. The same Gaussian calculation gives $\langle\chi_{\det}\rangle=q^2$ and $\langle\chi_{\operatorname{Sym}^2}\rangle=q^4+q^6+q^8$. Dividing their sum by $2^2$ reproduces the first line. A linked product is not a twice-wound single trace, nor a product of separately averaged traces. Sharing a supercharge justifies its localization; it does not remove its dynamics.

Return to (3.68)–(3.69) after these calculations. The remaining integral, its absolute normalization, its insertion and its prescription are four separate pieces of the claimed observable.