Fractional Events, Calorons, and Monopole Constituents

On $\mathbb R^3\times S^1$, a charge-one caloron with nontrivial circle holonomy can resolve into monopole constituents carrying fractional topological charge. The fractions are fixed by holonomy gaps, their actions add to the BPST action, and their magnetic charges cancel. None of these statements applies without the circle, its boundary conditions, the holonomy, and the global gauge data.

Required background. Gauge instantons, topological charge, and moduli supplies the unit-charge four-dimensional solution. Topological sectors, boundary data, and global form explains why the compactification and bundle data are part of the theory.

Helpful background. Global form, matter representations, and the faithful gauge group supplies the quotient and line-operator distinctions that can change the allowed sectors.

Circle holonomy and constituent charges

Let the compact coordinate obey $x_4\sim x_4+L$. The formulas below use the simply connected gauge group $SU(N)$, the ordinary periodic bundle, and gauge transformations that are periodic as $SU(N)$-valued maps. Other global forms or center-twisted bundles are discussed separately below. In this setting, diagonalize the asymptotic Wilson line:

$$\Omega_\infty = \mathcal P\exp\!\left(i\int_0^L dx_4\,\mathcal A_4\right) = \operatorname{diag} \left(e^{2\pi i\mu_1},\ldots,e^{2\pi i\mu_N}\right).$$

Choose representatives ordered cyclically,

$$\mu_1\le\mu_2\le\cdots\le\mu_N\le\mu_1+1, \qquad \mu_{N+1}\equiv\mu_1+1,$$

Because $\det\Omega_\infty=1$, one first has $\sum_i\mu_i\in\mathbb Z$. A unique cyclic relabeling and lift to the real line can then be chosen so that $\sum_i\mu_i=0$. Define the holonomy gaps

$$\nu_i=\mu_{i+1}-\mu_i, \qquad \nu_i\ge0, \qquad \sum_{i=1}^{N}\nu_i=1.$$

For generic holonomy, all $\nu_i>0$. A unit-charge KvBLL caloron decomposes into $N$ fundamental monopole constituents with

$$Q_i=\nu_i, \qquad S_i=\frac{8\pi^2}{g^2(\mu)}\nu_i, \qquad m_{W,i}\sim\frac{2\pi\nu_i}{L},$$

at leading semiclassical order, with one common weak renormalization scale $\mu$. Choosing $\mu$ near the relevant W-boson masses limits logarithms; if the gaps are hierarchical, threshold matching and the one-loop determinants must distribute the resulting scale dependence rather than assigning a different classical coupling to each term.

Therefore

$$\sum_{i=1}^{N}Q_i=1, \qquad \sum_{i=1}^{N}S_i=\frac{8\pi^2}{g^2}$$

at that common scale. This classical recombination identity would be spoiled by naively evaluating each displayed action at a different running scale.

The first $N-1$ constituents carry magnetic co-roots $\alpha_i^\vee$; the Kaluza–Klein constituent carries the affine co-root

$$\alpha_N^\vee=-\sum_{i=1}^{N-1}\alpha_i^\vee.$$

Their total magnetic charge is zero, as required for a caloron that recombines into a four-dimensional instanton. The general $SU(N)$ constituent relation is derived in Kraan and van Baal 1998, pp. 389–395. Independent $SU(2)$ constructions and the detailed moduli are given by Lee and Lu 1998, §§ II–IV, pp. 025011-1–025011-12 and Kraan and van Baal 1998, §§ 2–5, pp. 627–659.

Center-symmetric example

At center-symmetric holonomy, the eigenvalues are evenly spaced and

$$\nu_i=\frac1N.$$

Every constituent then has

$$Q_i=\frac1N, \qquad S_i=\frac{8\pi^2}{g^2N},$$

while the $N$-constituent composite has $Q=1$ and $S=8\pi^2/g^2$. For $SU(3)$, the magnetic charges may be labeled

$$\alpha_1^\vee,\qquad \alpha_2^\vee,\qquad \alpha_3^\vee=-\alpha_1^\vee-\alpha_2^\vee.$$

The sum vanishes even though no individual constituent is magnetically neutral.

If two neighboring holonomy eigenvalues coincide, the corresponding $\nu_i$ vanishes and that constituent becomes massless and delocalized. The well-separated constituent description then degenerates. Trivial holonomy recovers the older periodic caloron in which the internal monopole structure is not resolved; see Harrington and Shepard 1978, pp. 2122–2125.

Boundary data are part of the result

The fractional charges above do not enlarge the charge lattice of ordinary finite-action fields on uncompactified $\mathbb R^4$. They arise because $\mathcal A_4$ acts as an adjoint Higgs field in the three-dimensional description and because a charge-one periodic solution can distribute its action among constituents. Removing the circle or failing to fix the holonomy removes the assumptions behind $Q_i=\nu_i$.

The global form of the gauge group determines which large gauge transformations and bundles are allowed. For example, passing from $SU(N)$ to $PSU(N)$ or admitting a center-twisted bundle changes the identifications and can change the allowed topological sectors; one must rederive the charge lattice rather than reuse the display above unchanged. Matter boundary conditions around $S^1$ also matter.

For well-separated constituents, write a fermion boundary condition as $\psi(x_4+L)=e^{2\pi iz}\psi(x_4)$. Its zero mode localizes on constituent $i$ when $z$ lies strictly between $\mu_i$ and $\mu_{i+1}$. As $z$ crosses a holonomy eigenvalue, the localization moves; at $z=\mu_i$ the limiting mode delocalizes and is no longer normalizable. This is derived explicitly in Chernodub, Kraan, and van Baal 2000, pp. 556–558. Therefore one must not assign fermion zero modes to a constituent using only its fractional charge.

The instanton–bounce boundary and mode comparison records the circle and holonomy assumptions alongside the uncompactified saddles. The moduli-to-measure chain shows which stages must be redone when BPST size is replaced by constituent moduli.

For the downstream regime in which such monopole events and correlated bions can be used to study confinement, continue to the controlled compactification mechanism chain. That analysis additionally requires a small circle, abelianization, stabilized holonomy, and a hierarchy among the W-boson, monopole, and bion scales.

When constituent semiclassics is controlled

A constituent expansion requires more than $S_i\gg1$. At generic holonomy, the lightest off-diagonal gauge boson has

$$m_{W,\min} = \frac{2\pi}{L}\min_i\nu_i.$$

At center-symmetric holonomy this becomes $m_W=2\pi/(NL)$, so a uniform weak-coupling and abelianization condition is

$$NL\Lambda\ll1.$$

More generally one requires $m_{W,\min}\gg\Lambda$, holonomy in a region where the gauge group abelianizes to its maximal torus, and all relevant constituent actions large. The mechanism that stabilizes that holonomy must be stated. Thermal pure Yang–Mills theory and a spatial circle with periodic adjoint fermions generally generate different holonomy potentials, so results cannot be transferred by changing the interpretation of $L$ alone.

The constituent gas must also be dilute. Long-range Coulomb interactions and fermion exchange can correlate events into magnetically or topologically neutral composites. Such composites have actions and selection rules derived from their constituents, but their amplitudes require their own quasi-zero-mode integrals. A list of constituents is not yet an effective potential.

Common pitfalls

Calling $Q_i=1/N$ universal. Equal fractions occur at center-symmetric holonomy. At generic holonomy the charges are the unequal gaps $\nu_i$.

Dropping the affine constituent. The Kaluza–Klein monopole supplies the affine co-root. Without it, neither the total magnetic charge nor the topological charge recombines correctly.

Treating compactified constituents as isolated $\mathbb R^4$ fractional instantons. The circle holonomy and boundary conditions are essential parts of the saddle.

Exercises

1. An $SU(3)$ holonomy has gaps $(\nu_1,\nu_2,\nu_3)=(1/6,1/3,1/2)$. Compute the constituent charges and actions and verify recombination.

Solution

The charges are $1/6$, $1/3$, and $1/2$. Their actions are

$$\frac{8\pi^2}{g^2} \left(\frac16,\frac13,\frac12\right).$$

Both sets sum to their unit-caloron values because $1/6+1/3+1/2=1$. The magnetic charges are the two simple co-roots and the affine co-root; those sum to zero independently of the $\nu_i$.

2. Explain what fails when $\nu_1\to0$.

Solution

The first constituent has $Q_1,S_1\to0$, and its characteristic core size grows as the corresponding holonomy-induced mass scale vanishes. It is no longer a localized, exponentially suppressed event separated from the other constituents. The abelian constituent description therefore loses its scale separation even though the total caloron charge remains one.

References

• Chernodub, Mikhail N., Thomas C. Kraan, and Pierre van Baal. “Exact Fermion Zero-Mode for the New Calorons.” Nuclear Physics B—Proceedings Supplements 83–84 (2000): 556–558. DOI.
• Harrington, Barry J., and Harvey K. Shepard. “Periodic Euclidean Solutions and the Finite-Temperature Yang–Mills Gas.” Physical Review D 17 (1978): 2122–2125. DOI.
• Kraan, Thomas C., and Pierre van Baal. “Monopole Constituents inside $SU(N)$ Calorons.” Physics Letters B 435 (1998): 389–395. DOI.
• Kraan, Thomas C., and Pierre van Baal. “Periodic Instantons with Non-Trivial Holonomy.” Nuclear Physics B 533 (1998): 627–659. DOI.
• Lee, Kimyeong, and Changhai Lu. “SU(2) Calorons and Magnetic Monopoles.” Physical Review D 58 (1998): 025011. DOI.