Seifert fibrations and explicit examples

Before this lesson: Two supercharges and their Killing vector; Circle bundles and the Hopf fibration.

A nowhere-zero Killing vector gives a local flow, but its global orbits contain additional information. They may all close with one period, some may close earlier, or a generic orbit may never close. Which of these possibilities is a circle fibration, and what must happen to the spinors?

Start from two supercharges and their Killing vector and circle bundles and the Hopf fibration. We will use the Hopf quotient for a compact example, then resolve an exceptional fiber in ordinary smooth coordinates. The parent theory is three-dimensional $\mathcal N=2$ with four real flat-space supercharges. We construct an opposite-charge pair of independent Euclidean supersymmetry labels, without claiming a maximal solution count.

Allow a circle action to have finite stabilizers

A circle action is a family of transformations $T_t$ with $t\sim t+2\pi$ and $T_{t+t'}=T_tT_{t'}$ . At a point $p$ , its stabilizer consists of the parameters that return that point to itself. A free action has only the identity in every stabilizer; its quotient is an ordinary principal circle bundle. A locally free action permits finite stabilizers but no continuously fixed direction.

An oriented Seifert fibration is locally modeled by circles in

(D^2\times S^1)/\mathbb Z_q,\qquad (w,t)\sim\left(e^{2\pi i/q}w,\ t+\frac{2\pi b}{q}\right), \qquad \gcd(q,b)=1. \tag{M08.1}

Here $w$ is a complex disk coordinate, $t$ a period- $2\pi$ fiber angle, and $q,b$ are integers with $q\geq1$ . This equation declares our local convention for $b$ . The cyclic action is free on the total space: a power that fixes the circle coordinate has $q\mid nb$ , hence $q\mid n$ . The quotient three-manifold is therefore smooth.

At $w\ne0$ , the residual translation in $t$ closes after $2\pi$ . At $w=0$ , its smallest positive period is $2\pi/q$ . This central orbit is an exceptional fiber of multiplicity $q$ ; $q=1$ is regular. The base is locally the disk modulo $w\mapsto e^{2\pi i/q}w$ . It has an orbifold point: a quotient chart with a specified finite rotation group. That is a property of the base, not a conical singularity of the total three-manifold.

The local phase in (M08.1) is the unit-circle version of Closset–Kim (3.89). Their full Seifert data also specify the base genus, a bundle degree and the pairs attached to all exceptional fibers. Local data alone do not specify how the solid tori are glued into a closed manifold.

Descend the Hopf geometry to a lens space

For an integer $p\geq2$ , define our lens-space convention explicitly by

L(p,1)=S^3/\langle g\rangle,\qquad g(z_1,z_2)=\left(e^{2\pi i/p}z_1,e^{2\pi i/p}z_2\right). \tag{M08.2}

The sphere has $|z_1|^2+|z_2|^2=1$ and radius $\ell$ . The action is free, so the metric descends smoothly and the covering has $p$ sheets. This example has regular Hopf fibers; being a lens space does not require exceptional fibers in every choice of fibration.

Recall the normalized Hopf connection and northern fiber coordinate:

\Theta=d\chi_N+\frac{1-\cos\theta}{2}d\varphi, \qquad F=d\Theta=\frac12\sin\theta\,d\theta\wedge d\varphi .

On the quotient $\chi$ has period $2\pi/p$ . Set $\widehat\chi=p\chi$ and $\widehat\Theta=p\Theta$ . The overlap is $\widehat\chi_S=\widehat\chi_N+p\varphi$ , and

g=\frac{\ell^2}{p^2}\widehat\Theta^2+ \frac{\ell^2}{4}\left(d\theta^2+\sin^2\theta\,d\varphi^2\right), \qquad \frac1{2\pi}\int_{S^2}d\widehat\Theta=p . \tag{M08.3}

Thus the fiber length is $2\pi\ell/p$ and the volume is $2\pi^2\ell^3/p$ . The sign of the bundle number uses the base and fiber orientations of the prerequisite. The integer comes from the genuine quotient and its new period, not from arbitrarily rescaling a connection on the original bundle.

Descend the background and the two labels

The metric is only part of the construction. Use the sphere’s global frame $e^a=\ell\eta^a$ from the geometry prerequisite, with $de^a=\epsilon_{abc}e^b\wedge e^c/\ell$ and $\epsilon_{123}=1$ . Its spin connection is $\Gamma_a=i\gamma_a/(2\ell)$ in our gamma-matrix convention. Choose the covering-space fields

A=V=C=0,\qquad H=-\frac{i}{\ell},\qquad \zeta=u_+=\begin{pmatrix}1\\0\end{pmatrix},\qquad \widetilde\zeta=u_-=\begin{pmatrix}0\\1\end{pmatrix}. \tag{M08.4}

The R charges are respectively $+1$ and $-1$ . For either column, the full equation reduces to $(\nabla_a+H\gamma_a/2)u_\pm=0$ . This checks the local background in the source convention of P (5.9) and Appendix A. Constant dimensional normalizations of the labels have been suppressed.

Let $\lambda=2\pi/p$ . The common phase transformation of the ambient coordinates gives

g^*e^1=e^1,\qquad g^*(e^2+ie^3)=e^{2i\lambda}(e^2+ie^3). \tag{M08.5}

Consequently one spin lift of the deck transformation is $S_g=e^{-i\lambda\gamma_1}$ . It obeys $S_g^p=\mathbf1$ . It acts on $u_+$ by $e^{-i\lambda}$ and on $u_-$ by $e^{i\lambda}$ . The constant columns therefore do not automatically descend.

Specify a flat R line by the deck character $\chi_R=e^{i\lambda}$ . This means that a charge-one spinor is required to obey $\zeta(gx)=\chi_R S_g\zeta(x)$ , and an opposite-charge spinor uses $\chi_R^{-1}$ . Both constant columns now satisfy their conditions:

\chi_R S_gu_+=u_+,\qquad \chi_R^{-1}S_gu_-=u_- . \tag{M08.6}

Locally the R connection remains $A=0$ . Globally it has nontrivial transition data. For the loop lifted from $x$ to $gx$ , parallel transport in this line has holonomy $\chi_R^{-1}$ : a constant component upstairs must be converted back with the inverse gluing map. The distinction between a gluing character and a holonomy fixes this sign. The opposite phases cancel in the neutral mixed vector, which is the unit Hopf direction $K=E_1$ .

This constructs the spin structure and flat R line together. The compact R group is normalized so the labels have charges $\pm1$ ; matter representations must also admit the selected R bundle. Arbitrary real powers of a nontrivial line bundle cannot simply be assumed. This is the global qualification discussed in Closset–Kim, section 3.5.5.

The remaining auxiliary data are global as field strengths: $V=0$ is represented by $C=0$ , while

dB=-\frac1\ell\,\mathrm{vol}_g,\qquad \int_{L(p,1)}dB=-\frac{2\pi^2\ell^2}{p}. \tag{M08.7}

Hence $B$ cannot be one smooth global two-form. On a chart where $\chi_i$ has a real lift, a local potential is $B_i=-\ell^2\chi_i F/2$ , because $\mathrm{vol}_g=\ell^3\Theta\wedge F/2$ and $a_i\wedge F=0$ . Use ordinary local gauge potentials on a sufficiently fine atlas and the smooth $H$ in the rigid equations. No compact two-form gauge period or flux quantization has been stipulated here; choosing one would add a separate global restriction. The construction verifies the displayed pair on the quotient, not the classification of all solutions on all lens spaces.

Resolve an exceptional fiber and its spinors

Take $b=1$ in (M08.1), but use the universal fiber coordinate $t\in\mathbb R$ and impose the single identification

(w,t)\sim\left(e^{2\pi i/q}w,t+\frac{2\pi}{q}\right). \tag{M08.8}

The $q$ th power gives $t\sim t+2\pi$ . With $w=x+iy$ , the invariant flat metric $g=L^2dt^2+dx^2+dy^2$ has $L>0$ with length units; $t$ is dimensionless. Its smooth quotient coordinates are

X=e^{-it}w,\qquad \sigma=qt\pmod{2\pi},\qquad g=\left|dX+\frac{iX}{q}d\sigma\right|^2+ \frac{L^2}{q^2}d\sigma^2 . \tag{M08.9}

There is no singular coefficient at $X=0$ . The fiber vector is $\partial_t=q\partial_\sigma-iX\partial_X+i\overline X\partial_{\overline X}$ . At the core its orbit has period $2\pi/q$ ; away from the core the rotation of $X$ requires the full period $2\pi$ .

Now specify a complete local background on this solid torus:

A=-\frac12dt,\qquad V=H=0,\qquad B=C=0,\qquad \zeta=e^{-it/2}u_+,\qquad \widetilde\zeta=e^{it/2}u_- . \tag{M08.10}

In the covering Cartesian frame $(Ldt,dx,dy)$ , $\Gamma=0$ . Thus $(d-iA)\zeta=0$ and $(d+iA)\widetilde\zeta=0$ component by component. The source has physical frame component $A_1=-1/(2L)$ . The deck rotation has spin lift $S=e^{-i\pi\gamma_1/q}$ . The change of each exponential in (M08.10) is exactly the corresponding eigenvalue of $S$ , so both labels descend without an additional R transition. The globally defined connection is $A=-d\sigma/(2q)$ .

After $q$ deck steps, $S^q=-\mathbf1$ and both columns change sign. This specifies the antiperiodic spin structure on the period- $2\pi$ covering circle; it is consistent, not a failure of single-valuedness. The mixed vector is $K=L^{-1}\partial_t$ . We have constructed a smooth supersymmetric neighborhood of an exceptional fiber. Gluing it into a prescribed closed manifold still requires compatible overlap data outside that neighborhood.

A chosen flow need not define the fibration

The real-Killing-vector discussion in P, section 5.1 explicitly distinguishes closed and nonclosed orbits. For example, on $S^3$ the weighted phase flow $(z_1,z_2)\mapsto(e^{iat}z_1,e^{ibt}z_2)$ is an isometry of the round metric. If $a/b$ is irrational and both coordinates are nonzero, no positive time returns both phases. The underlying sphere still admits its ordinary Hopf fibration.

This geometric observation does not say that the particular background (M08.4) produces every such weighted vector as its mixed bilinear. Changing the chosen foliation or preserved labels requires rechecking the sources and equations. Regular fibers, exceptional fibers, and irrational Killing flows answer different global questions.

Exercises

1. Two exceptional orbits on the same sphere

On the unit sphere consider $(z_1,z_2)\mapsto(e^{2it}z_1,e^{3it}z_2)$ . Find the generic orbit period and the two exceptional periods. Why is the total space smooth? What changes if the two weights are $1,\sqrt2$ ? Does either calculation establish the spinors of a supersymmetric background?

Hint 1

When both coordinates are nonzero, both phases must return to one. At either axis only one phase remains.

Hint 2

The action is taking place on the original smooth sphere. A stabilizer of the orbit action is not a singular point of that sphere.

Solution

The generic return time satisfies $2t=2\pi m$ and $3t=2\pi n$ . Their smallest common positive solution is $t=2\pi$ . At $z_2=0$ the period is $\pi$ , giving multiplicity two; at $z_1=0$ it is $2\pi/3$ , giving multiplicity three. The generator is nowhere zero because the two nonzero weights cannot annihilate both coordinates on the unit sphere. Thus this is a locally free circle action with two exceptional fibers. The quotient base has the corresponding orbifold charts, while the total space remains $S^3$ .

For weights $1,\sqrt2$ , a generic return would require $t=2\pi m$ and $\sqrt2m=n$ for integers $m,n$ , which has no positive solution. The flow on each fixed-magnitude two-torus is irrational; the two axis orbits still close. It is not a circle action with a common period. The Hopf circle action on the same manifold remains available. Neither calculation specifies $A,V,H$ or spin/R gluing, so neither alone constructs a preserved pair.

2. Change the spin lift on the lens quotient

Keep the local fields (M08.4). For the lift $S_g$ , replace the R character by $\chi_m=e^{2\pi im/p}$ , $m\in\mathbb Z$ . Which values preserve the displayed pair? When is $-S_g$ also an allowed lift of the order- $p$ deck generator, and how must the R character change? Why would measuring only $dA$ miss the issue?

Hint 1

Apply $\chi_mS_g$ to $u_+$ and $\chi_m^{-1}S_g$ to $u_-$ separately.

Hint 2

A spin lift must respect the relation $g^p=1$ . For even $p$ , multiplication of the character by $-1$ is itself an order- $p$ character.

Solution

The two conditions are $e^{2\pi i(m-1)/p}=1$ and its inverse. Hence $m=1\pmod p$ . The alternative lift obeys $(-S_g)^p=(-1)^p\mathbf1$ , so it is allowed precisely for even $p$ . In that case choose $\chi'_R=-e^{2\pi i/p}$ , or $m'=1+p/2\pmod p$ . Then both combined spin/R maps act trivially on the corresponding columns, just as before. For odd $p$ that proposed lift violates the deck-group relation and is not an alternative spin structure.

All these local R potentials have $dA=0$ . Curvature cannot distinguish their finite holonomies. For example, with $p=5$ and the original lift, omitting the character gives nontrivial spin phases and loses the displayed pair, despite leaving every local differential equation unchanged. The calculation tests global descent; it does not count every possible solution in a different background.

3. Measure the core without creating a cone

For the exceptional neighborhood, restrict to $|w|\leq R_0$ . Compute its volume, the generic fiber length, and the core length. Then check the combined spin/R parallel transport around the core on both columns of (M08.10).

Hint 1

Use a fundamental interval of length $2\pi/q$ in $t$ for the volume. A generic fiber is not confined to one fixed representative $w$ in that interval.

Hint 2

Along the core, the covering Cartesian spin connection vanishes. Closing the loop uses $S^{-1}$ , while the charge-one R phase is $e^{i\int A}=e^{-i\pi/q}$ .

Solution

The covering volume form is $L\,dt\,dx\,dy$ , so the quotient volume is

\mathrm{Vol}=\frac{2\pi}{q}L\,\pi R_0^2 =\frac{2\pi^2LR_0^2}{q}.

The generic orbit has length $2\pi L$ , while the core has length $2\pi L/q$ . Equivalently, the orbifold base disk has area $\pi R_0^2/q$ ; multiplying by the generic fiber length gives the same volume. The shorter core is a measure-zero orbit, not a loss of smoothness of the three-metric (M08.9).

The spin parallel-transport map around the core is $S^{-1}=e^{i\pi\gamma_1/q}$ . On $u_+$ it contributes $e^{i\pi/q}$ , cancelled by the R phase $e^{-i\pi/q}$ . On $u_-$ it contributes $e^{-i\pi/q}$ , cancelled by the charge-minus-one phase $e^{i\pi/q}$ . Thus both full connections have a fixed column around the shorter orbit. After the generic period, the spin and R phases are each $-1$ and again cancel. This independently checks the equivariance condition rather than relying only on a local zero residual.

Use the Section 5 paper guide to locate this calculation in the original argument and compare its local and global assumptions.

The three-dimensional A-twist constructs the globally patched sphere-times-circle example and follows its integer flux into a free-chiral partition function.