Topological Sectors, Boundary Data, and Global Form

Finite action splits a configuration space into topological sectors only after the spacetime, boundary conditions, bundle class, gauge group including its global form, and admissible defects have been fixed. For $SU(2)$ Yang–Mills fields on Euclidean $\mathbb R^4$ that approach a pure gauge at infinity, compactification gives a map $S^3_\infty\to SU(2)$ and an integer charge. Changing the global form or allowing nontrivial bundles and boundaries can change the charge lattice, so finite action alone never licenses an integer label or a $2\pi$ theta period.

Required background. Theta terms, periodicity, and vacuum sectors supplies the primary definition of the topological coupling and its dependence on global data.

Helpful background. Homotopy, degree, and winding and characteristic classes supply the mathematical classification. Global form and matter representations explains why a Lie algebra does not by itself specify the theory.

Shared comparison. The sector–theta–branch map shows how the charge lattice established here controls the theta transform, period test, branch envelope, and volume limit.

Sector decomposition from finite-action boundary data

Work in oriented Euclidean four-space with positive metric and $\epsilon_{1234}=+1$ . To keep the site’s Hermitian-generator convention while placing the coupling outside the action, define the mathematical connection $\mathcal A=gA$ and curvature

\mathcal F=\mathrm d\mathcal A-i\mathcal A\wedge\mathcal A=gF, \qquad \operatorname{tr}(T^aT^b)=\frac12\delta^{ab}.

Then

S_E=\frac1{g^2}\int_{M_4}\operatorname{tr}(\mathcal F\wedge *\mathcal F), \qquad Q=\frac1{8\pi^2}\int_{M_4}\operatorname{tr}(\mathcal F\wedge\mathcal F).

On $\mathbb R^4$ , finite action requires $\mathcal F\to0$ sufficiently rapidly. Under the standard no-defect boundary condition, $\mathcal A$ therefore approaches a pure gauge on $S^3_\infty$ . For $G=SU(2)$ , the boundary map has

Q=\deg(U)\in\pi_3(SU(2))\simeq\mathbb Z.

This is a statement about the completed configuration space, not merely about a local expression for $\mathcal F$ . The BPST solution supplies an explicit $Q=1$ finite-action representative Belavin et al. 1975, pp. 85–87, while the normalized characteristic-class derivation is given in Mariño 2015, § 4.3, pp. 112–124.

The self-duality identity independently checks the normalization. Since $*^2=1$ on two-forms in Euclidean four dimensions,

0\leq\frac1{2g^2}\int\operatorname{tr} \bigl[(\mathcal F\mp *\mathcal F)\wedge *(\mathcal F\mp *\mathcal F)\bigr] =S_E\mp\frac{8\pi^2}{g^2}Q,

and hence

S_E\geq\frac{8\pi^2}{g^2}\lvert Q\rvert.

Self-dual or anti-self-dual fields saturate the bound. The sign of $Q$ follows the declared orientation; reversing orientation exchanges the two cases.

Restricted functional integrals

If continuous finite-action deformations cannot change $Q$ , the regulated Euclidean functional integral decomposes as

Z=\sum_{Q\in\Lambda_Q}Z_Q, \qquad Z_Q=\int_{\mathcal C_Q}\!\mathcal D\Phi\,e^{-S_E[\Phi]}.

Here $\mathcal C_Q$ is the configuration space with the declared charge and $\Lambda_Q$ is the actual charge lattice. The symbol $Z_Q$ does not imply that a semiclassical instanton approximation is valid; it is a restriction of the regulated integral. It also does not imply that local operators can measure $Q$ in real time. Sector labels, superselection sectors, and saddle families are related notions only under additional hypotheses.

The theta-weighted transform is

Z(\theta)=\sum_{Q\in\Lambda_Q}e^{i\theta Q}Z_Q.

If $\Lambda_Q=\mathbb Z$ and all integer sectors are included, $Z(\theta+2\pi)=Z(\theta)$ . This one-line proof also identifies the missing premise in many incorrect periodicity claims.

Global form can change the charge lattice

The Lie algebra $\mathfrak{su}(2)$ admits distinct global theories. For gauge group $SU(2)$ on the compactification above, the usual instanton number is integral. For $SO(3)=SU(2)/\mathbb Z_2$ , bundles with nonzero second Stiefel–Whitney class can have fractional instanton number in the same normalization. On a spin four-manifold, half-integral sectors imply that a $2\pi$ shift of the continuous theta parameter exchanges the two choices of discrete theta data; returning to the same fixed $SO(3)_\pm$ theory can require $4\pi$ . On a general oriented non-spin four-manifold, quarter-integral sectors can instead require an $8\pi$ shift before the same continuous-theta description returns, with the discrete theta datum tracked throughout. Aharony, Seiberg, and Tachikawa derive these spin-sensitive relations from the allowed line-operator and bundle data in Aharony, Seiberg, and Tachikawa 2013, §§ 1–2.

Boundaries, surface operators, twisted boundary conditions, and compactification can likewise admit fractional constituents. Fractional local objects need not imply a fractional total charge: boundary or neutrality constraints may force an allowed collection to have integral total $Q$ .

Charge and periodicity comparison

The following table keeps normalization and global data adjacent. Its periods apply only under the stated assumptions.

Sector labels, global data, theta weights, and periodicity under declared spacetime assumptions
Sector label	Boundary and global-form data	Charge normalization	Theta weight	Periodicity	Volume limit	Branch caveat
n ∈ ℤ for a particle on a circle	Paths obey φ(β) = φ(0) + 2πn	n = (2π)⁻¹ ∫ dτ φ̇	e^iθn	2π	Large Euclidean time projects onto the ground state	Energy levels relabel under θ → θ + 2π; n is path winding, not a gauge-vacuum label
Q ∈ ℤ for SU(2) Yang–Mills	No defects; pure gauge on S³ at infinity; fundamental trace	Q = (8π²)⁻¹ ∫ tr(𝓕 ∧ 𝓕)	e^iθQ	2π	Take the regulated continuum limit before interpreting the infinite-volume energy density	Periodicity of Z does not require each candidate energy branch to be periodic
Fractional Q sectors for SO(3)_±	Nontrivial w₂ bundles allowed; discrete theta choice fixed; spin structure stated	Fractional part determined by the Pontryagin square of w₂	Continuous and discrete theta factors both enter	On spin four-manifolds a 2π shift can exchange SO(3)₊ and SO(3)₋, and a fixed theory can require 4π; on general oriented non-spin four-manifolds quarter-instantons can require 8π	The bundle sum and spin/non-spin class are part of the theory definition at every volume	Local correlators on ℝ⁴ need not reveal the distinction; line and bundle data do
Q ∈ ℤ in a closed two-dimensional CP^N−1 model	Closed oriented spacetime and standard compact U(1) auxiliary bundle	Q = (2π)⁻¹ ∫ f	e^iθQ	2π	Large volume may produce competing branches	The large-N branch pattern is model-specific and nonuniform near crossings
Fractional constituents on a compactified gauge background	Holonomy, magnetic flux, and boundary conditions explicitly fixed	Constituent charges sum according to the allowed bundle	Each constituent carries its own phase; allowed collections determine the total	Derived from the total charge lattice, not from one constituent	Decompactification and large-separation limits may not commute	A constituent is not automatically an independent sector of the uncompactified theory

This is also a nonvisual description of the global-data input to the sector–theta–branch figure used later in the chapter.

What finite action does not prove

Finite action is a necessary analytic condition in the $\mathbb R^4$ example, but it does not by itself establish:

that the charge is integral rather than fractional;
that every allowed topological class has a smooth classical representative;
that a representative is a stable saddle or dominates an observable;
that summing a dilute gas of representatives is controlled; or
that the sectors are superselected in the Lorentzian Hilbert space.

Those conclusions require, respectively, global bundle data, existence analysis, fluctuation and contour analysis, a separation/measure estimate, and a dynamical operator statement.

Exercises

Assume $Z(\theta)=\sum_{Q\in\mathbb Z}e^{i\theta Q}Z_Q$ . Derive the inverse transform and state the assumption that makes it valid.

Solution

Orthogonality of the characters of $\mathbb Z$ gives

Z_Q=\frac1{2\pi}\int_0^{2\pi}\!\mathrm d\theta\, e^{-i\theta Q}Z(\theta).

The derivation assumes an integral charge lattice with a complete $2\pi$ character domain and a sum for which the Fourier interchange is legitimate. It does not apply unchanged if a fixed theory has half-integer charges or if a $2\pi$ shift permutes distinct theories.

References

Aharony, Ofer, Nathan Seiberg, and Yuji Tachikawa. “Reading between the Lines of Four-Dimensional Gauge Theories.” Journal of High Energy Physics 2013, no. 8 (2013): 115. arXiv. DOI.
Belavin, A. A., A. M. Polyakov, A. S. Schwartz, and Yu. S. Tyupkin. “Pseudoparticle Solutions of the Yang–Mills Equations.” Physics Letters B 59, no. 1 (1975): 85–87. DOI.
Mariño, Marcos. Instantons and Large N: An Introduction to Non-Perturbative Methods in Quantum Field Theory. Cambridge University Press, 2015, § 4.3, pp. 112–124. DOI.