One brane and the C4/Zk moduli space

At one brane the scalar potential vanishes for every value of the four complex scalars. Why is the vacuum space nevertheless a quotient, and why is that quotient discrete? The answer comes from the compact gauge fields, including information invisible in their local equations.

Use the R-symmetry organization of the fields and connections, flux, and holonomy. We study flat three-dimensional $U(1)_k\times U(1)_{-k}$ with no mass or FI deformation. Both factors are compact circles with unit-charge Wilson probes and integer fluxes. There is no additional quotient of their common center. The main convention is $k>0$, integer; reversing its sign gives the same vacuum quotient with order $|k|$. The generic full theory has twelve real Poincaré supercharges, four manifest in the chosen $\mathcal N=2$ description. Each gauge factor has rank one. The extra symmetries at levels one and two require their own analysis.

First solve the scalar equations

At rank one, $A_1,A_2,B_1,B_2$ are commuting complex numbers. Consequently $W=(4\pi/k)(A_1B_1A_2B_2-A_1B_2A_2B_1)$ and all its derivatives vanish. Write

$$\mu=|A_1|^2+|A_2|^2-|B_1|^2-|B_2|^2.$$

The two vector-auxiliary equations give $\sigma=\widehat\sigma=2\pi\mu/k$. Their matter couplings involve $\sigma-\widehat\sigma$, so the vector contribution to the potential vanishes too. In particular, there is no equation $\mu=0$. Nor is $\sigma$ an extra independent vacuum coordinate: its value is fixed by the matter fields. These conclusions follow from the ordered auxiliary equations before making the matrices commute; see AB, section 2.3 and Appendix A.1.

As in the preceding lesson, use the scalar components $Y^I=(A_1,A_2,B_1^*,B_2^*)$. All have charge $(+1,-1)$. With real connections $a,\widehat a$,

$$D Y^I=(d-i a+i\widehat a)Y^I,\qquad Y^I\longmapsto e^{i(\alpha_1-\alpha_2)}Y^I,\qquad a\longmapsto a+d\alpha_1,\quad \widehat a\longmapsto\widehat a+d\alpha_2. \tag{AB07.1}$$

Each $\alpha_i$ is an angle modulo $2\pi$. The kinetic normalization is $\sum_I|D_\mu Y^I|^2$, with $[Y]=1/2$ and $[a_\mu]=1$. The scalar potential supplies eight real zero-energy coordinates before the remaining gauge analysis.

Choose a basis that preserves the flux lattice

Define

$$b=a-\widehat a,\qquad c=\widehat a,\qquad a=b+c. \tag{AB07.2}$$

This is an invertible integer change of basis. On a closed oriented two-surface,

$$\frac1{2\pi}\int db=n_1-n_2,\qquad \frac1{2\pi}\int dc=n_2,\qquad n_1,n_2\in\mathbb Z. \tag{AB07.3}$$

Thus $b$ and $c$ are independent unit-flux compact connections. Matter couples only to $b$. Setting $\alpha=\alpha_1-\alpha_2$ gives $b\mapsto b+d\alpha$ and $Y\mapsto e^{i\alpha}Y$.

For the dualization it is convenient to display the Euclidean topological action; the physical scalar metric will remain real and positive. On a closed spacetime, or with boundary conditions that remove the displayed total derivative,

$$\begin{aligned} S_{\rm top} &=\frac{ik}{4\pi}\int(a\wedge da-\widehat a\wedge d\widehat a)\\ &=\frac{ik}{4\pi}\int\left(b\wedge db+2b\wedge dc\right). \end{aligned} \tag{AB07.4}$$

Indeed, the difference of the two densities is $d(b\wedge c)$. The $b\wedge db$ term has not been discarded. The term $b\wedge dc$ is called a BF coupling: one connection multiplies another connection’s curvature.

The compact action includes the usual spin Chern–Simons patch data when a bundle is nontrivial. The same integer change of basis sends its level matrix $\operatorname{diag}(k,-k)$ to $\left(\begin{smallmatrix}k&k\\k&0\end{smallmatrix}\right)$, so it respects that global definition as well as the local form expression.

The original paper instead uses the difference and sum of its two connections. The sum is not an independent unit-flux partner of the difference: its flux integer $n_+=n_1+n_2$ obeys $n_+\equiv n_1-n_2\pmod2$. After the difference has zero flux, the sum has flux in $4\pi\mathbb Z$. Our integer basis keeps this correlation automatic. In the paper’s gauge-sign convention, $a=-A_{(1)}$ and $\widehat a=-A_{(2)}$; this leaves the quadratic CS action unchanged. AB (2.7)–(2.9)

Dualize the neutral connection

Replace $dc$ locally by an independent real two-form $f_c$ and impose its Bianchi identity with a scalar $\tau$:

$$S_{\rm first} =\frac{ik}{4\pi}\int b\wedge db +\frac{ik}{2\pi}\int b\wedge f_c -\frac{i}{2\pi}\int\tau\,df_c. \tag{AB07.5}$$

The form convention is $f_c=\tfrac12 f_{\mu\nu}dx^\mu\wedge dx^\nu$. Flux quantization must accompany this local expression. A unit flux is $2\pi$, so its conjugate angle has period

$$\tau\sim\tau+2\pi. \tag{AB07.6}$$

One way to see the normalization is to let a small surrounding surface carry flux $2\pi n$. The last term then contributes $-in\tau$; its exponential is unchanged under a $2\pi$ shift for every integer $n$. In a global first-order path integral, the corresponding integer flux and winding sums are retained. A noncompact real multiplier would lose this information. Here $\tau$ is dimensionless.

Integrating the last term by parts makes the coefficient of $f_c$ equal to $i(kb+d\tau)/(2\pi)$. Its equation is therefore

$$\boxed{kb+d\tau=0.} \tag{AB07.7}$$

This is a first-order constraint, not a minimization of an imaginary Euclidean action. It implies $db=0$ locally. Invariance of the constraint and kinetic term requires

$$\tau\longmapsto\tau-k\alpha. \tag{AB07.8}$$

We can set $\tau=0$ by a local gauge choice. Since $\tau$ is an angle, the gauge transformations that preserve this choice satisfy $k\alpha\in2\pi\mathbb Z$. Their effective action on matter is

$$Y^I\sim e^{2\pi i/k}Y^I. \tag{AB07.9}$$

There are $k$ such transformations modulo the original $2\pi$ gauge period. This derives the residual group $\mathbb Z_k$.

The same quantization is visible without introducing $\tau$. On $S^1\times S^2$, take flat $b$ with holonomy $\vartheta=\oint_{S^1}b$ and a $c$ flux $2\pi n$ through $S^2$. The BF weight is $e^{-ikn\vartheta}$. Summing over all $n\in\mathbb Z$ restricts $\vartheta$ to $2\pi j/k$ modulo $2\pi$. The local equation “$b$ is flat” alone would have allowed every holonomy.

Read the metric, not only the identification

On a patch choose a lift of the angle $\tau$ and introduce

$$X^I=e^{i\tau/k}Y^I. \tag{AB07.10}$$

These are invariant under the continuous gauge transformation (AB07.8). Using $b=-d\tau/k$ gives

$$D Y^I=e^{-i\tau/k}dX^I,\qquad ds^2=\sum_{I=1}^4 dX^I\,d\overline X^I. \tag{AB07.11}$$

Changing the lift $\tau\to\tau+2\pi$ changes all four $X^I$ by $e^{2\pi i/k}$. Thus the local flat metric descends to

$$\boxed{\mathcal M_1=\mathbb C^4/\mathbb Z_k,\qquad (X^1,X^2,X^3,X^4)\sim e^{2\pi i/k}(X^1,X^2,X^3,X^4).} \tag{AB07.12}$$

For $k>1$ the only fixed point of a nonidentity group element is the origin. Away from it the quotient is locally flat; at the origin it is an orbifold singularity. For $k=1$ the quotient is simply $\mathbb C^4$. A negative nonzero level changes the direction of the generator, giving the same quotient by $\mathbb Z_{|k|}$.

This is the Abelian sigma-model result of AB, section 2.3. The dualization also explains why dividing the four scalars by a continuous phase would lose a physical direction. The variables $(Y,\tau)$ have nine real components, and the continuous gauge redundancy removes one; a discrete identification remains. No Maxwell term or independent Coulomb scalar was added.

The self-CS term in (AB07.4) has no local density after $b=-d\tau/k$. On a spacetime with nontrivial topology, flat connections can still carry global topological data. We are deriving the vacuum target and its local metric, not replacing the full compact gauge path integral on every spacetime by an unrestricted scalar integral.

Worked example: the phase circle survives

Take $Y^1=r e^{i\varphi}$ and $Y^{2,3,4}=0$, with $r>0$. The gauge-invariant local phase is $\chi=\varphi+\tau/k$. Substituting into the kinetic term gives

$$ds^2=dr^2+r^2d\chi^2,\qquad \chi\sim\chi+\frac{2\pi}{k}. \tag{AB07.13}$$

At fixed $r$, the circumference is $2\pi r/k$, while the distance from the origin is $r$. This slice is a flat cone of opening angle $2\pi/k$, not just a radial half-line. For $k=3$, the three covering points $r$, $r e^{2\pi i/3}$ and $r e^{4\pi i/3}$ describe the same vacuum, but a small change of $\chi$ is a genuine displacement.

Gauge-invariant coordinates need not be single-valued lifts $X^I$. A homogeneous polynomial of degree $d$ in $X$ survives the discrete identification precisely when $d$ is divisible by $k$. For positive $k$, degree-$k$ examples are

$$\mathcal U_{I_1\cdots I_k} =X^{I_1}\cdots X^{I_k} =e^{i\tau}Y^{I_1}\cdots Y^{I_k}. \tag{AB07.14}$$

The two phase transformations cancel, and the expression is periodic in $\tau$. All invariant polynomials are generated by these degree-$k$ monomials, with algebraic relations among them. The factors $e^{i\tau}$ belong to the dual description; translating them into local flux-insertion operators is a further task.

“Holomorphic” here refers to the four complex target coordinates $X$. Since two of the $Y$ components came from $B^*$, this is not an assertion that all these functions are chiral operators in the originally chosen $\mathcal N=2$ superspace.

Exercises

1. Check the tempting half-sum

Define $c_{\rm av}=(a+\widehat a)/2$ while keeping $b=a-\widehat a$. Show that the local topological density can be written as $ik\,b\wedge dc_{\rm av}/(2\pi)$. Why can $b,c_{\rm av}$ not both be declared independent unit-flux compact connections? Test the allowed original flux $(n_1,n_2)=(1,0)$, then the sector with zero difference flux.

Hint 1

The local change is $c_{\rm av}=c+b/2$. A change of local variables need not be an invertible integer change of the compact flux lattice.

Hint 2

Compute both fluxes before imposing $db=0$. Keep the allowed original Wilson probes fixed.

Solution

Since $dc_{\rm av}=dc+db/2$, the density in (AB07.4) is exactly $ik\,b\wedge dc_{\rm av}/(2\pi)$. But the fluxes are

$$\frac1{2\pi}\int db=n_1-n_2,\qquad \frac1{2\pi}\int dc_{\rm av}=\frac{n_1+n_2}{2}.$$

The original allowed pair $(1,0)$ gives $(1,1/2)$ in these coordinates. Moreover, integral difference flux and half-sum flux have correlated parity. Treating both as independent integers would remove allowed bundles and change the theory.

In the zero-difference-flux sector, $n_1=n_2=n$, so $c_{\rm av}$ has flux $2\pi n$. It can then reproduce the same dual angle and residual quotient. This restricted conclusion does not retroactively make the half-sum an independent unit-flux field in every original sector.

2. Put an invariant coordinate on the cone

On the one-complex-dimensional slice, let $z=(X^1)^k$. Derive the metric in $z$ and show that its apparent singular coefficient encodes the circumference in (AB07.13). State which part of the quotient is smooth.

Hint 1

Away from $z=0$, choose one local branch $X^1=z^{1/k}$ and differentiate. All branches give the same metric.

Hint 2

Write $z=r^k e^{i\theta}$, with $\theta$ of period $2\pi$.

Solution

The pullback of $|dX^1|^2$ is

$$ds^2=\frac1{k^2}|z|^{2/k-2}\,dz\,d\bar z =dr^2+\frac{r^2}{k^2}d\theta^2.$$

Thus $\chi=\theta/k$ has period $2\pi/k$, and a circle of radius $r$ has circumference $2\pi r/k$. For example $k=2$ gives $ds^2=dz\,d\bar z/(4|z|)$ and opening angle $\pi$. The coefficient is singular at the invariant coordinate’s origin because the quotient has a fixed point there. It is locally flat and smooth for $r>0$.

For the full four-complex-dimensional quotient, any nonzero vector also has trivial stabilizer under this common-phase action. Hence only the origin is singular for $k>1$; $k=1$ is smooth everywhere. These geometric facts alone do not count the full supersymmetry generators.

3. Invariants are coordinates with relations

For $k=2$, use $\mathcal U_{IJ}=X^IX^J$, $I,J=1,\ldots,4$. Count the distinct entries, derive one relation, and explain why ten entries do not imply ten complex vacuum dimensions. Reconstruct a generic vacuum on the patch $\mathcal U_{11}\ne0$, and check the continuous gauge invariance of the expression in dual variables.

Hint 1

The matrix $\mathcal U$ is symmetric and has rank at most one. All its two-by-two minors vanish.

Hint 2

Choose a square root of $\mathcal U_{11}$ and use the first row to determine the other $X^I$. Check the other square-root choice.

Solution

There are $4(4+1)/2=10$ symmetric entries, but for example $\mathcal U_{11}\mathcal U_{22}=\mathcal U_{12}^2$. More generally $\mathcal U_{IJ}\mathcal U_{KL}=\mathcal U_{IK}\mathcal U_{JL}$, so the entries are constrained.

On the stated patch choose $X^1=\sqrt{\mathcal U_{11}}$ and $X^j=\mathcal U_{1j}/X^1$. The rank-one relations recover every remaining entry. The alternative square root sends all four $X^I$ to their negatives, exactly the $\mathbb Z_2$ identification. The patch has four independent complex coordinates, for example $\mathcal U_{11}$ and the three ratios $\mathcal U_{1j}/\mathcal U_{11}$.

In the dual variables $\mathcal U_{IJ}=e^{i\tau}Y^IY^J$. Since $\tau\mapsto\tau-2\alpha$, the exponential contributes $e^{-2i\alpha}$ and the scalar product contributes $e^{2i\alpha}$. The expression is both continuously gauge invariant and single-valued under $\tau\mapsto\tau+2\pi$.

The vacuum coordinates, flat local metric and discrete identification have now followed from one compact Abelian action. Continue with many branes and global gauge-group data: diagonalization and permutations introduce further questions, and the allowed global gauge group and flux lattice must still be carried through the calculation.

The original ABJM paper guide locates this calculation in the source argument and gives a changed example to check while reading.

Paper guides

Connect this lesson to the original papers and their notation.

• Reading the original ABJM paper