Compactified Coulomb branches and their geometry

A massive particle can travel around a compact spatial circle and return to its starting point. At distances large compared with the circle, this process looks local. Its effect on the low-energy scalar kinetic terms is exponentially small when the circumference greatly exceeds the particle’s Compton wavelength. Near a massless-particle locus, that approximation fails precisely where the uncorrected geometry becomes singular.

We will calculate the correction for one light charged hypermultiplet and reconstruct the resulting metric. Prepare with the circle compactification and dual-photon bridge, BPS masses and stability, and the reproduction-to-research method. The local model is established; varying its radius and checking its convergence provides the first research exercise.

Choose the light particle and its charge unit

Start with four-dimensional $\mathcal N=2$ pure compact $SU(2)$ theory on $\mathbb R^3\times S^1$. The circle has circumference $2\pi R$, with periodic fermions and no supersymmetry-breaking twist. All eight real supercharges survive: the low-energy three-dimensional theory has full $\mathcal N=4$ supersymmetry. A choice of a manifest four-charge subalgebra does not reduce that full symmetry. The gauge rank is one, independently of the supersymmetry label.

On the Abelian Coulomb branch, the complex vector scalar supplies two real scalars. The Wilson line around the circle supplies a third. Dualizing the remaining three-dimensional photon locally supplies a fourth periodic scalar. Thus there are four real Coulomb coordinates per gauge rank. Eight supercharges require their two-derivative sigma-model metric to be hyper-Kähler: it admits three compatible complex structures obeying the quaternion algebra, with closed associated two-forms.

Use the primitive charge units from the bridge. Its periods and angles are $a_e=a/\sqrt2$, $a_m=\sqrt2a_D$, $\theta_e$ and $\theta_m$, with both angles of period $2\pi$. The BPS mass is $|p a_e+q a_m|$, where $(p,q)=(2n_e,n_m)$ in the original Seiberg–Witten basis. Near the monopole point, exchange electric and magnetic roles:

$$\begin{gathered} a_* =a_m,\qquad a_{m,*}=-a_e,\qquad \tau_*=-\frac1{\tau_c},\\ \theta=\theta_m,\qquad \theta_d=-\theta_e, \qquad \tau_c=2\tau_{\rm eff}. \end{gathered} \tag{D17.1}$$

Constant shifts of the local angular origins are allowed. The light monopole is now a unit electrically charged hypermultiplet, of physical mass $\rho=|a_*|$. This is a change of primitive electromagnetic frame, not a change from $SU(2)$ to $SO(3)$ or a replacement of the W-boson’s charge by one.

The isolated local theory is an infrared-free $U(1)$ vector multiplet coupled to this hypermultiplet. Its ultraviolet matching scale $\Lambda_*$ is distinct from the strong-coupling scale of the full $SU(2)$ theory. We take $\Lambda_*$ positive by a local phase choice. Work below that matching scale. When using the model near the $SU(2)$ singularity, also require $R M_{\rm heavy}\gg1$ for omitted heavy particles. This need not imply $R\rho\gg1$ for the light particle.

Turn the scalar metric into a Green-function problem

There are no magnetic charged particles in the isolated local model, so shifting the dual photon remains an exact symmetry. The corrected metric has the Gibbons–Hawking form

$$\begin{gathered} \boldsymbol{x}=(\operatorname{Re}a_*,\operatorname{Im}a_*,z), \quad z=\frac{\theta}{2\pi R}, \quad \psi=\frac{\theta_d}{2\pi},\\ ds^2=V\,d\boldsymbol{x}^{2} +V^{-1}(d\psi+\mathcal A)^2, \qquad d\mathcal A=*_{3}dV. \end{gathered} \tag{D17.2}$$

Here $\psi$ has period one, $z$ has period $1/R$, and $V$ is a positive harmonic function away from particle thresholds. The star uses the orientation $dx^1\wedge dx^2\wedge dz$. The target-space one-form $\mathcal A$ is a connection for the dual-photon circle; it is not the spacetime gauge potential. We use kinetic density $\tfrac12G_{IJ}\partial\phi^I\partial\phi^J$, as in the bridge.

Why does the curvature equation matter? Write $\eta=d\psi+\mathcal A$. For each cyclic triple $(i,j,k)$ of $(1,2,3)$, the candidate two-form is $\Omega_i=\eta\wedge dx^i-Vdx^j\wedge dx^k$. Its derivative vanishes because $d\mathcal A\wedge dx^i=(\partial_iV)dx^1\wedge dx^2\wedge dz$. In the orthonormal frame $e^0=V^{-1/2}\eta$, $e^i=V^{1/2}dx^i$, these forms are $e^0\wedge e^i-e^j\wedge e^k$. Their matrices obey $J_iJ_j=-\delta_{ij}+\epsilon_{ijk}J_k$. This gives both the required algebra and closed forms, rather than merely counting scalars.

The zero-winding, or semiflat, approximation has

$$\begin{gathered} \tau_* =\frac{1}{2\pi i}\log\frac{a_*}{\Lambda_*},\qquad V_{\rm sf}=\frac{R}{2\pi}\log\frac{\Lambda_*}{\rho},\\ \mathcal A_{\rm sf}=-\frac{\varphi}{4\pi^2}d\theta, \qquad \varphi=\arg a_*. \end{gathered} \tag{D17.3}$$

Indeed $V_{\rm sf}=R\operatorname{Im}\tau_*$. Combining the $V_{\rm sf}dz^2$ term with the fiber term in (D17.2) reproduces $|d\theta_d-\tau_*d\theta|^2/(4\pi^2R\operatorname{Im}\tau_*)$. The missing terms break continuous translation of $\theta$ down to its required periodicity.

Sum the circle momenta and expose the winding corrections

For a unit charged field, the circle momentum is $(n+\theta/(2\pi))/R$. Its three-dimensional mass is therefore

$$M_n^2=\rho^2+\frac{(n+\theta/(2\pi))^2}{R^2}. \tag{D17.4}$$

The complete hypermultiplet loop contributes $1/(4\pi M_n)$ to $V$. This coefficient and the nonrenormalization input for the isolated model are the sourced physical ingredients in GMN (4.4). The elementary radial loop integral $\int d^3p\,(2\pi)^{-3}(p^2+M^2)^{-2}=1/(8\pi M)$ explains the inverse-mass behavior; deriving that scalar integral alone would not fix all the hypermultiplet factors.

Set $\kappa=R\rho$ and $v=\theta/(2\pi)$. Summing the tower gives

$$V=\frac{R}{4\pi}\sum_{n\in\mathbb Z} \left[\frac1{\sqrt{\kappa^2+(n+v)^2}}-c_n\right]. \tag{D17.5}$$

The subtraction constants $c_n$ specify a holonomy-independent ultraviolet prescription and fix $\Lambda_*$. They are not adjustable separately for each value of $\theta$. A particularly clean observable of the local metric is the contrast $V(\theta)-V(0)$ at fixed $R,\rho$; its subtraction cancels.

To reorganize the sum, use a proper-time parameter $t$:

$$\begin{aligned} \frac1{\sqrt A}&=\frac1{\sqrt\pi} \int_0^\infty t^{-1/2}e^{-At}\,dt,\\ \sum_{n\in\mathbb Z}e^{-t(n+v)^2} &=\sqrt{\frac\pi t}\sum_{w\in\mathbb Z} e^{-\pi^2w^2/t}e^{2\pi iwv}. \end{aligned} \tag{D17.6}$$

The second line is Poisson summation of the Gaussian. The $w=0$ integral is logarithmically divergent before the subtraction. Its derivative with respect to $\kappa$ is $-2/\kappa$, giving (D17.3) after matching the constant. For $w\ne0$, the integral is $2K_0(2\pi|w|\kappa)$, where the modified Bessel function can be defined by $K_0(x)=\int_0^\infty e^{-x\cosh t}\,dt$. Pairing positive and negative windings yields

$$\begin{aligned} V&=V_{\rm sf}+V_{\rm inst},\\ V_{\rm inst}&=\frac{R}{\pi}\sum_{w=1}^\infty \cos(w\theta)K_0(2\pi wR\rho). \end{aligned} \tag{D17.7}$$

This is the periodic Green function written as a winding expansion. Each pair contains a particle and its opposite orientation around the circle. The large-positive-$x$ behavior $K_\nu(x)\sim\sqrt{\pi/(2x)}e^{-x}$ gives the expected factor $e^{-2\pi R\rho}$ and its fluctuation prefactor. Replacing the whole Bessel function by the exponential would lose that prefactor.

The connection must change too. A convenient local choice is

$$\mathcal A=\mathcal A_{\rm sf} +\frac{R\rho}{\pi}\sum_{w=1}^\infty \sin(w\theta)K_1(2\pi wR\rho)\,d\varphi. \tag{D17.8}$$

For one winding, write $V_w=(R/\pi)\cos(w\theta)K_0(c\rho)$ and the coefficient of $d\varphi$ as $B_w=(R\rho/\pi)\sin(w\theta)K_1(c\rho)$, with $c=2\pi Rw$. The identities $K_0'=-K_1$ and $(\rho K_1(c\rho))'=-c\rho K_0(c\rho)$ give

$$\partial_\rho B_w=2\pi R\rho\,\partial_\theta V_w, \qquad -\partial_\theta B_w=\frac{\rho}{2\pi R}\partial_\rho V_w. \tag{D17.9}$$

These are precisely the two nonzero components of $d\mathcal A=*dV$ in cylindrical coordinates. The same Bessel equation gives $\partial_\rho^2V_w+\rho^{-1}\partial_\rho V_w+(2\pi R)^2\partial_\theta^2V_w=0$. Thus the winding sum supplies the metric and its geometric consistency check.

Check the apparent singularity with the actual fiber period

The semiflat logarithm is singular all along $a_*=0$. In the momentum sum, a threshold instead requires both $a_*=0$ and $\theta=0$ modulo $2\pi$. Near that point the $n=0$ term gives $V\sim h/r$, where $h=1/(4\pi)$ and $r=\sqrt{\rho^2+z^2}$. Other terms are regular locally.

Use spherical base coordinates $(r,\chi,\varphi)$ and a local connection $\mathcal A=h\cos\chi\,d\varphi$. With $\varrho=2\sqrt{hr}$ and $\widetilde\psi=\psi/h$, the leading metric becomes

$$ds^2=d\varrho^2 +\frac{\varrho^2}{4}\left[ d\chi^2+\sin^2\chi\,d\varphi^2 +(d\widetilde\psi+\cos\chi\,d\varphi)^2\right]. \tag{D17.10}$$

Because $\psi$ has period one, $\widetilde\psi$ has period $4\pi$: one quarter of the bracket is the unit three-sphere metric in Hopf coordinates. Equation (D17.10) is smooth Euclidean four-space in polar coordinates at leading order. The divergent coefficient $V$ was a bad description at a shrinking circle, not a curvature singularity. This conclusion uses unit charge and the stated global fiber period; a different charge lattice can change it.

There is also a global patching check. Taking $\varphi$ around $2\pi$ changes $\mathcal A_{\rm sf}$ by $-d\theta/(2\pi)$, so $d\psi+\mathcal A$ must be patched. In the source’s chosen charge-character sign convention, the full angular transition is $\theta_d\mapsto\theta_d+\theta-\pi$, GMN (4.12)–(4.16). Its differential cancels the connection shift. The constant $-\pi$ is invisible to that differential check, but matters for globally defined charged coordinates; a local metric calculation must not claim to have fixed it independently.

Vary the radius with an error budget

Hold the local theory, $\rho$, $\Lambda_*/\rho=10$ and the angular conventions fixed. Vary $R$, so only $\kappa=R\rho$ changes. The dimensionless correction $V_{\rm inst}/R$ is:

$R\rho$	$\theta=0$	$\theta=\pi/2$	$\theta=\pi$
$0.35$	$0.03095615985$	$-0.00226022757$	$-0.02635539780$
$0.8$	$0.00114661555$	$-0.00000535287$	$-0.00113590915$
$1.5$	$0.00001035587$	$-0.00000000059$	$-0.00001035469$

Add $V_{\rm sf}/R=\log(10)/(2\pi)\simeq0.3664677994$ to obtain the positive metric coefficient. At $\theta=\pi/2$ the first harmonic of $V$ vanishes. The first harmonic of the connection does not vanish, because it contains $\sin\theta$. Even for $V$, the leading surviving term is $-RK_0(4\pi R\rho)/\pi$. A cancellation in one coefficient is not disappearance of the metric correction.

The table can be reproduced with Python and mpmath:

import mpmath as mp
mp.mp.dps = 50
W = 24
for kappa in map(mp.mpf, ['.35', '.8', '1.5']):
    vals = [mp.fsum(mp.cos(w*theta)*mp.besselk(0, 2*mp.pi*w*kappa)
                    for w in range(1, W+1))/mp.pi
            for theta in [0, mp.pi/2, mp.pi]]
    print(kappa, [mp.nstr(value, 12) for value in vals])

To bound the omitted terms, $\cosh t\ge1+t^2/2$ in the defining integral implies $K_0(x)\le\sqrt{\pi/(2x)}e^{-x}$. Set $x=2\pi R\rho$. After retaining windings $1$ through $W$,

$$|\delta V|\le\frac R\pi \sqrt{\frac{\pi}{2x(W+1)}} \frac{e^{-x(W+1)}}{1-e^{-x}}. \tag{D17.11}$$

The holonomy contrast has at most twice this bound. For the coefficient $B$ of $d\varphi$ in $\mathcal A_{\rm inst}$, replace the prefactor $R$ by $R\rho$ and the denominator $2x(W+1)$ inside the square root by $2[x(W+1)-1]$, provided $x(W+1)>1$. This follows from $K_1(x)=\int_0^\infty e^{-x\cosh t}\cosh t\,dt$ and $\cosh t\le e^{t^2/2}$. These are truncation bounds, not floating-point roundoff certificates.

An independent computation of the holonomy contrast uses (D17.5) directly, pairing $n$ and $-n$ before summing. For example, at $R\rho=0.35$, $\theta=\pi$, it gives $[V(\pi)-V(0)]/R=-0.0573115576500225$. A raw cutoff $|n|\le24$ still has error about $3.31\times10^{-5}$; merely increasing arithmetic precision cannot remove it. The large-$n$ paired summand begins $2v^2/n^3+(2v^4-6\kappa^2v^2)/n^5$. Subtracting and summing a controlled tail makes the momentum and winding calculations agree. Exercise 2 explains how to bound the simpler unaccelerated version.

The isolated one-hypermultiplet result uses the source’s nonrenormalization statement. Embedding it near the $SU(2)$ monopole locus retains a local approximation: heavy sectors and matching corrections are omitted. In particular, reducing $R$ at fixed heavy masses eventually violates $RM_{\rm heavy}\gg1$. The full finite-radius $SU(2)$ metric, mutually nonlocal BPS particles and wall-crossing consistency are further problems; this computation does not solve them by summing only the light hypermultiplet.

Exercises

1. Detect a wrong connection sign

Keep one nonzero winding in (D17.7), but reverse the sign of its contribution to (D17.8). Does the resulting positive metric still pass the hyper-Kähler check above? Explain why positivity alone cannot repair it.

Hint 1

Use the $d\rho\wedge d\varphi$ component of the curvature equation.

Hint 2

Evaluate $d\Omega_3$ at a point where $\partial_zV_w\ne0$.

Solution

The correct connection satisfies $\partial_\rho B_w=\rho\partial_zV_w$. Reversing its sign while retaining $V_w$ instead gives $-\rho\partial_zV_w$. Therefore the winding contribution to $d\Omega_3$ is $-2\partial_zV_w\,dx^1\wedge dx^2\wedge dz$, which is nonzero at generic $\theta$. The same failure appears in the second equation of (D17.9). This is a failure of the stated compatible hyper-Kähler structure even if $V$ stays positive. Positivity ensures a Riemannian kinetic term, not supersymmetric geometric consistency. A simultaneous reversal of the orientation and the associated conventions would be a different consistent convention change; changing one term alone is not.

2. Give the momentum cutoff an honest uncertainty

Let $f(y)=(\kappa^2+y^2)^{-1/2}$ and choose $|v|\le1/2$. Bound the omitted paired sum $\sum_{n>N}[f(n+v)+f(n-v)-2f(n)]$ and translate it into an error bound on $[V(\theta)-V(0)]/R$. Evaluate the bound for $N=24$, $v=1/2$.

Hint 1

Write the central second difference as an integral of $f''$ with a triangular weight.

Hint 2

Use $|f''(y)|\le2/y^3$ for $y>0$, then bound a decreasing sum by an integral.

Solution

For $v\ge0$, the difference is $\int_{-v}^{v}(v-|t|)f''(n+t)\,dt$; it is even in $v$. Since $f''(y)=(2y^2-\kappa^2)/(\kappa^2+y^2)^{5/2}$, its absolute value is bounded by $2/y^3$. The triangular weight integrates to $v^2$, so the omitted absolute sum is at most $2v^2\sum_{n>N}(n-1/2)^{-3}$. Using an integral from $N$ to infinity gives the conservative bound $v^2/(N-1/2)^2$. Including (D17.5), the error in the contrast divided by $R$ is at most

$$\frac{v^2}{4\pi(N-1/2)^2}.$$

At $N=24$, $v=1/2$, this is $0.00003602420622$, exceeding the actual $0.00003312643741$ error at $\kappa=0.35$. The bound is independent of arithmetic precision and requires no fitted convergence rate. It does not claim that all paired summands have the same sign at arbitrary $\kappa$.

3. Identify the limit that the calculation controls

Suppose $\rho/M_{\rm heavy}=10^{-3}$. Compare $R\rho=1$ with $R\rho=10^{-4}$. In each case, determine which of the light-particle winding approximation and the omission of heavy particles is parametrically justified. Does the smooth unit-charge center prove the full compactified $SU(2)$ metric is globally smooth?

Hint 1

There are two dimensionless products, with $RM_{\rm heavy}=10^3R\rho$.

Hint 2

The single-hyper model has an exact local resummation; the embedding in the non-Abelian theory has additional assumptions.

Solution

For $R\rho=1$, the first winding suppression is $e^{-2\pi}\simeq0.00187$, and higher windings can be bounded using (D17.11). Also $RM_{\rm heavy}=1000$, so heavy wrapped-particle effects are strongly suppressed. Local matching data must still be specified. For $R\rho=10^{-4}$, a few light windings are not a controlled approximation; many terms are needed or the momentum representation should be used. At the same time $RM_{\rm heavy}=0.1$, so heavy sectors are not parametrically suppressed. Resumming the light particle exactly does not fix that second failure.

The center calculation establishes the local unit-charge model’s smooth completion with the stated fiber period. It neither classifies every singularity nor incorporates every BPS sector of the full $SU(2)$ theory. Changing global charge data, including additional mutually nonlocal states or moving outside the local patch requires further analysis.

Reading beyond the local reproduction

As checked on 20 October 2026, the starting calculation is the established single-particle metric in GMN section 4.1. Read its global-coordinate discussion along with the Bessel formulas; omitting it hides the role of the charge lattice. Then read sections 5.6–5.7 to distinguish the construction with mutually nonlocal BPS data from its physical identification and large-radius expansion.

Fredrickson and Zimet’s 2025 work, Hyper-Kähler manifolds from Riemann–Hilbert problems I: Ooguri–Vafa-like model geometries, develops local multi-particle model geometries under explicit hypotheses. It provides a later mathematical direction, not an assertion that the general global physical problem has disappeared. The useful next project is to choose one specified additional BPS sector, translate its charge and angular conventions, and test the resulting correction in an overlap regime. Success requires a sourced spectrum, geometric consistency and separate control of omitted sectors; agreement between two implementations of the same light-particle formula is only the first benchmark.