ABJM one-brane quotient

Two points can have the same fourth, fifth or even hundredth powers coordinate by coordinate and still represent different vacua. The missing information is the relative phase between coordinates. In this laboratory, that observation becomes a test of which gauge group a program has actually divided out.

Prepare with one brane and its vacuum quotient. That lesson derives the scalar equations, compact dualization and local metric. Here we retain those inputs in a runnable calculation, compare actual gauge orbits, and build a diagnostic that rejects a tempting but incorrect quotient.

Keep the compact data in the calculation

We use flat three-dimensional $U(1)_k\times U(1)_{-k}$ ABJM, with positive integer $k$, zero masses and zero FI parameters. Each gauge factor has rank one. Both circles admit unit-charge Wilson probes; there is no extra quotient of their common center. The generic full theory has twelve real Poincaré supercharges, with four manifest in its $\mathcal N=2$ description. The rank-one theories at $k=1,2$ have sixteen; neither that enhancement nor a large-rank gravity limit is established by this program.

The four scalar components are $Y=(A_1,A_2,B_1^*,B_2^*)$, all of relative gauge charge $+1$. At rank one the scalar potential vanishes, while the two vector scalars are fixed to the same value. There is no additional moment-map-zero constraint. This is the starting point of AB, section 2.3.

The global information must survive the change of variables. Write

$$b=a-\widehat a,\qquad c=\widehat a, \qquad \begin{pmatrix}a\\\widehat a\end{pmatrix} =\begin{pmatrix}1&1\\0&1\end{pmatrix} \begin{pmatrix}b\\c\end{pmatrix}. \tag{LAB09.1}$$

This matrix and its inverse have integer entries and determinant one. If the original fluxes divided by $2\pi$ are $(n_1,n_2)\in\mathbb Z^2$, the new ones are $(n_1-n_2,n_2)$. The program reconstructs the original pair exactly. In contrast, the paper’s difference and sum have correlated parity: $n_1+n_2\equiv n_1-n_2\pmod2$. Treating them as independent integers changes the allowed bundles.

In the inherited Euclidean convention, with weight $e^{-S_E}$, the compact spin Chern–Simons action has local form

$$S_{\rm top}=\frac{ik}{4\pi}\int(b\wedge db+2b\wedge dc), \qquad K=\begin{pmatrix}k&k\\k&0\end{pmatrix}. \tag{LAB09.2}$$

The code checks this level matrix by the integer basis change. The $b\wedge db$ term is retained. A local form expression does not replace the spin-CS patch definition on nontrivial bundles.

Introduce $f_c=dc$ as an independent two-form and the multiplier $-i\int\tau\,df_c/(2\pi)$. A unit flux $2\pi n$ makes a constant shift $\Delta\tau$ change its exponential by $e^{in\Delta\tau}$. Since every integer $n$ is allowed, the conjugate angle has period $2\pi$. Integration by parts and variation of $f_c$ give

$$kb+d\tau=0,\qquad \tau\longmapsto\tau-k\alpha, \qquad Y\longmapsto e^{i\alpha}Y. \tag{LAB09.3}$$

Consequently the transformations preserving $\tau=0$ have $\alpha=2\pi j/k$, modulo $2\pi$. The source’s dualization around (2.9) gives the same identification after its connection signs and correlated fluxes are translated as in the prerequisite. A local equation saying only that $b$ is flat would not fix these allowed holonomies.

Local invariant lifts are $X=e^{i\tau/k}Y$, with metric $\sum_I|dX^I|^2$. Changing the lift of $\tau$ rotates the whole tuple by a common phase. Choose a fixed positive scalar amplitude unit $v$, of mass dimension $1/2$, and set $x=X/v$ for the computation. The dimensionless identification is

$$x\sim\omega^j x,\qquad \omega=e^{2\pi i/k},\qquad j=0,\ldots,k-1. \tag{LAB09.4}$$

Distances printed below are in units of $v$. The unit $v$ is a choice of coordinates, not a mass or FI deformation.

An exact representation of phases

A floating-point root of unity does not satisfy $\omega^k=1$ exactly. A test such as “the distance is smaller than a tolerance” can therefore confuse numerical rounding with gauge equivalence.

Instead, the program represents $\omega$ by a symbol w and reduces polynomial expressions modulo its cyclotomic polynomial $\Phi_k(w)$. This is the minimal polynomial over the rationals whose roots are the primitive $k$th roots of unity. For example, $\Phi_3(w)=w^2+w+1$. A polynomial remainder of zero is an exact identity, not a tolerance decision. The coordinates used here have rational real and imaginary parts, so SymPy can perform this arithmetic exactly. When $k$ is divisible by four, it also uses $i=\omega^{k/4}$; the complex coefficient and the phase are then related, not independent symbols.

The compact character average provides a second useful test:

$$P_k(q)=\frac1k\sum_{j=0}^{k-1}\omega^{jq} =\begin{cases}1,&q\in k\mathbb Z,\\0,&q\notin k\mathbb Z.\end{cases} \tag{LAB09.5}$$

Here $q$ is an integer phase charge. When $\omega^q\ne1$, multiply the finite geometric series by $1-\omega^q$ to obtain $1-\omega^{kq}=0$; when $\omega^q=1$, all terms are one. This is also the selection rule produced by the compact flux sum. The released code evaluates the polynomial sum before comparing it with the divisibility rule. It neither truncates an infinite flux distribution nor claims to compute the complete gauge path integral.

Worked example: separate powers lose a relative phase

Take $k=3$ and compare

$$x=(1,2,1+i,0),\qquad z=(1,2\omega,1+i,0). \tag{LAB09.6}$$

Every separate cube agrees: $z_I^3=x_I^3$. Nevertheless, a common phase mapping $x$ to $z$ would have to preserve the nonzero first coordinate, hence be one. It then fails to map the second coordinate. The two vacua are distinct.

The mixed invariant $U(x)=x_1^{k-1}x_2$ detects the difference: $U(x)=2$, whereas $U(z)=2\omega$. Its total degree is $k$, so it is unchanged by the common phase. These are holomorphic functions of the target coordinates; because two of the original $Y$ fields were conjugated, this is not a claim about chirality in the chosen $\mathcal N=2$ superfield variables.

The local flat metric supplies a separate geometric diagnostic. The distance on this finite isometric quotient is

$$d([x],[z])^2=\min_{0\le j<k}\sum_{I=1}^4|x_I-\omega^j z_I|^2. \tag{LAB09.7}$$

A shortest covering-space line segment projects to a curve with the same length; taking the shortest endpoint image gives the quotient distance. For (LAB09.6), the three squared distances, in the order $j=0,1,2$, are $12,21,9$. Thus $d=3$, not zero.

The deliberately incorrect algorithm minimizes the phase separately in each coordinate. It returns zero, because it has replaced the common $\mathbb Z_k$ by $\mathbb Z_k^4$. The difference is the gauge group being used, not a numerical precision issue.

Reproduce the output

Download the complete lab (ZIP) with the program, pinned dependencies, reference output and run instructions.

Download the Python program, pinned requirements, and expected output into one directory. The tested environment is Python 3.13.7, SymPy 1.13.3 and mpmath 1.3.0. After installation, the program needs no network access, plotting package or site checkout.

python -m pip install -r requirements.txt
python one_brane_quotient.py
python one_brane_quotient.py --level 5
python one_brane_quotient.py --wrong-independent-phases

Mathematical object	Program representation	Interpretation
Integral connection basis	transform	Preserves the original flux lattice and transforms the level matrix
Primitive root $\omega$	w, cyclotomic, reduce	Exact polynomial arithmetic for phases
Covering tuples $x,z$	X, Z	Dimensionless coordinate values, not different branes
Common orbit	orbit, equivalent	One phase acts on all four coordinates
Quotient metric	distances, minimum	Numerical evaluation after exact equivalence is settled

The baseline passes 25 grouped exact assertions. It finds three points in a nonzero orbit, a trivial stabilizer there, and stabilizer order three at the origin. In the tested charge window, (LAB09.5) retains $q=-6,-3,0,3,6$. It reports equal separate cubes but inequivalent tuples, with mixed invariants 2 and 2*w and distance 3.0.

At $k=5$, the exact test still finds inequivalent tuples. The smallest squared distance is $6[1-\cos(2\pi/5)]=4.1458980337503$, and the distance is $2.0361478418205$. The changed input changes both the allowed discrete phases and which powers are invariant. It does not change the four-complex-dimensional local target dimension.

The final command is explicitly a deliberate-failure diagnostic. It reports wrong_independent_phases_detected: true while retaining the correct physical result. The code does not silently switch the model. It rejects nonpositive levels; the deliberate-failure mode also excludes $k=1$, where there is no nontrivial phase to distinguish.

Check the measure independently

The orbit calculation can be checked without comparing another list of points. Consider the convergent, dimensionless target-space Gaussian

$$I_k=\int_{\mathbb C^4/\mathbb Z_k}dV\,e^{-|x|^2} =\frac{\pi^4}{k},\qquad \langle|x_1^{k-1}x_2|^2\rangle=(k-1)!. \tag{LAB09.8}$$

This is a geometric integral, not the ABJM sphere partition function. Away from a measure-zero set, a fundamental region is $0\le\arg x_1<2\pi/k$, with the other coordinates unrestricted. Its volume measure is therefore $1/k$ times the covering measure. Four independent phase divisions would incorrectly give $1/k^4$.

The program uses direct radial quadrature. For a nonnegative integer $p$, $\int_{\mathbb C}d^2x\,|x|^{2p}e^{-|x|^2}=2\pi\int_0^\infty r^{2p+1}e^{-r^2}dr=\pi p!$. Apply this to the four coordinates and the one restricted angle to obtain (LAB09.8). The baseline outputs $I_3=32.469697011334$ and the normalized mixed moment $2$; at $k=5$ the moment is $24$.

A different coordinate check uses $W=x_1^k$ and $R_j=x_j/x_1$, $j=2,3,4$, on $x_1\ne0$. One local root gives $x_1=W^{1/k}$ and $x_j=W^{1/k}R_j$. The real volume measure is

$$dV=\frac1{k^2}|W|^{8/k-2}\,d^2W\prod_{j=2}^4d^2R_j. \tag{LAB09.9}$$

The complex Jacobian is $W^{4/k-1}/k$; its modulus squared gives this coefficient. Integrating the three ratio coordinates first again gives (LAB09.8). This tests the singular-looking coordinate measure, rather than assuming the orbit count fixes every Jacobian.

Exact phase decisions do not depend on mpmath precision. Distances and Gaussian quadratures use 50 decimal digits and print fourteen significant digits; --dps 70 repeats them at higher precision. The Gaussian checks require scaled discrepancies below $10^{-35}$ at the default setting. That threshold is a numerical check, not a certified integration-error bound. Finite character sums have no truncation error. The calculation does not test quantum corrections, monopole-operator multiplets, many-brane vacua or spin-CS phases on arbitrary spacetimes.

Exercises

1. Recover the missing relative phases

For $k>1$, explain why the four functions $x_I^k$ do not separate common-phase orbits. On $x_1\ne0$, show that $W=x_1^k$ together with $U_j=x_1^{k-1}x_j$, $j=2,3,4$, does separate them. How many complex local coordinates remain?

Hint 1

Separate powers allow a different $k$th root for each coordinate. Compare that freedom with the single common phase.

Hint 2

Divide $U_j$ by $W$, choose one root of $W$, and reconstruct all coordinates with that same root.

Solution

The points in (LAB09.6), with the same definition for general $k>1$, have identical separate powers but different mixed invariant $U_2$. Thus the separate powers identify too many points.

On the stated patch, $R_j=U_j/W=x_j/x_1$. Choose $s$ with $s^k=W$. Then $x_1=s$ and $x_j=sR_j$. Every other root is $\omega^a s$ and rotates all four reconstructed coordinates together, precisely the allowed equivalence. No extra relative-root choice remains.

There are four independent complex patch coordinates: $W\ne0$ and three ratios. As a differential check, the Jacobian of $(W,U_2,U_3,U_4)$ with respect to $(x_1,x_2,x_3,x_4)$ is $k x_1^{4(k-1)}$, which is nonzero on this patch. A discrete quotient does not remove a continuous target dimension.

2. Shorten the dual period incorrectly

A program declares $\tau\sim\tau+\pi$ while retaining all original integer $c$ fluxes. Test this declaration on unit flux. What residual phase group would it incorrectly infer from $\tau=0$? Why does agreement in an even-flux-only test not repair the calculation?

Hint 1

The shift changes the multiplier weight by $e^{in\pi}$. Test $n=1$ before testing a larger set.

Hint 2

With the proposed period, a residual transformation would satisfy $k\alpha\in\pi\mathbb Z$. Compare its number of solutions modulo $2\pi$ with the original theory.

Solution

Unit flux changes the weight by $-1$, so the proposed identification does not preserve the original compact theory. It would allow $\alpha=\pi j/k$, giving $2k$ phases instead of $k$, and would incorrectly replace the common quotient by $\mathbb Z_{2k}$.

Even flux alone gives $e^{i(2n)\pi}=1$, but the original group permits odd flux too, for example $n_1=n_2=1$, which has $b$ flux zero and $c$ flux one. Dropping these sectors is changed global input. A check restricted to even flux cannot justify that change or define the full modified gauge theory. The original unit-flux test fixes the minimal conjugate period to $2\pi$.

3. Two shortest paths without a fixed point

Let $u$ be a nonzero four-component vector with norm $r$ and compare $x=u$ with $z=e^{i\pi/k}u$, for $k>1$. Find the quotient distance and the two minimizing images. Does the nonuniqueness mean either endpoint is an orbifold fixed point? What changes as the relative angle passes through $\pi/k$?

Hint 1

Each squared image distance is $2r^2[1-\cos(\pi/k+2\pi j/k)]$.

Hint 2

The angles nearest zero are $+\pi/k$ and $-\pi/k$. A fixed point instead requires a nonidentity group element to leave the whole vector unchanged.

Solution

The minimizing images have $j=0$ and $j=k-1$, giving

$$d=2r\sin\frac{\pi}{2k}.$$

Their covering-space line segments project to two equal-length paths. Neither endpoint is fixed: $\omega^j u=u$ with $u\ne0$ implies $\omega^j=1$. The nonuniqueness is a change of the nearest image, not a stabilizer at either endpoint.

For a relative angle just below $\pi/k$, $j=0$ is nearest; just above, $j=k-1$ is nearest. The distance is continuous at the change, although its angle derivative changes branch. A numerical algorithm that returns one minimizing image may jump there without the physical distance jumping. At the origin all $k$ elements stabilize the point, which is a different geometric fact.

The lab has tested the compact data, the action of the residual group, and two ways of reading the quotient geometry. The one-brane result is already meaningful at fixed small $k$; it needs no large-rank approximation. Keep those conclusions separate from a full partition function or an enhancement proof.