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 $\epsilon_{1234}=+1$. Keep the site’s Hermitian generators and define coupling-rescaled fields

$$A_g\equiv gA, \qquad F_g\equiv\mathrm dA_g-iA_g\wedge A_g=gF, \qquad \operatorname{tr}(T^aT^b)=\frac12\delta^{ab}.$$

The subscript is important: $A_g$ is not the site’s anti-Hermitian connection $-igA$. Rescaling the Hermitian field simply moves every explicit coupling into the overall action coefficient without changing the bundle or charge normalization.

Then

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

Equivalently, with $\widetilde F_{g,\mu\nu}^a=\tfrac12\epsilon_{\mu\nu\rho\sigma}F_{g,\rho\sigma}^a$,

$$Q=\frac1{32\pi^2}\int\mathrm d^4x\, F_{g,\mu\nu}^a\widetilde F_{g,\mu\nu}^a.$$

This component form is a normalization check: changing the trace convention changes the prefactor, but it must not change an integer charge computed for the same bundle.

On $\mathbb R^4$, finite action requires $F_g\to0$ sufficiently rapidly. Under the standard no-defect boundary condition, choose the asymptotic pure-gauge convention

$$A_g\big|_{S^3_\infty}=iU\,\mathrm dU^{-1}, \qquad U:S^3_\infty\longrightarrow SU(2).$$

Gauge transformations that approach the identity at infinity preserve this boundary representative. Transformations with nonzero winding change its integer label and must be treated according to the declared Hilbert space and global form; they are not silently included among the small redundancies.

For the Hermitian convention above, introduce the Chern–Simons three-form

$$\omega_3(A_g) =\operatorname{tr}\!\left( A_g\wedge\mathrm dA_g- \frac{2i}{3}A_g\wedge A_g\wedge A_g \right), \qquad \mathrm d\omega_3=\operatorname{tr}(F_g\wedge F_g).$$

Stokes’ theorem then turns the four-dimensional integral into a winding integral. With the boundary orientation induced from $\mathbb R^4$, define

$$\begin{aligned} Q &=\frac1{8\pi^2}\int_{\mathbb R^4} \operatorname{tr}(F_g\wedge F_g) =\frac1{8\pi^2}\int_{S^3_\infty}\omega_3(A_g)\\ &=-\frac1{24\pi^2}\int_{S^3_\infty} \operatorname{tr}\!\left[(U^{-1}\mathrm dU)^3\right] \equiv\deg(U) \in\pi_3(SU(2))\simeq\mathbb Z. \end{aligned}$$

The last equality fixes the sign convention for degree rather than assuming it. As a direct check, the self-dual BPST representative of size $\rho$ has

$$\frac1{32\pi^2}F^a_{g,\mu\nu} \widetilde F^a_{g,\mu\nu} =\frac{6\rho^4}{\pi^2(x^2+\rho^2)^4}, \qquad \int_{\mathbb R^4}\mathrm d^4x\, \frac{6\rho^4}{\pi^2(x^2+\rho^2)^4}=1.$$

Thus the chosen orientation assigns $Q=+1$ to the self-dual solution. This is a statement about the completed configuration space, not merely about a local expression for $F_g$. The BPST representative and normalization are given in Belavin et al. 1975, pp. 85–87, while the characteristic-class derivation is developed 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,

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

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 compactified-$\mathbb R^4$ example above, the usual instanton number is integral. We now leave that example and put the theory on a general closed oriented four-manifold $X$. This change is essential: $H^2(S^4,\mathbb Z_2)=0$, so the nonzero-$w_2$ bundles discussed next do not exist on $S^4$.

For $SO(3)=SU(2)/\mathbb Z_2$, a bundle with $w_2\in H^2(X,\mathbb Z_2)$ can have fractional instanton number in the same normalization. One useful general-manifold convention labels the discrete theta coupling by $p\in\mathbb Z_4$ and weights a bundle by

$$\exp\!\left(i\theta Q+ \frac{2\pi i p}{4}\int_X\mathcal P(w_2)\right), \qquad (\theta,p)\sim(\theta+2\pi,p+1).$$

On a spin $X$, the Pontryagin square is even, so $p$ and $p+2$ are indistinguishable: a $2\pi$ shift exchanges the two spin-theory choices conventionally denoted $SO(3)_+$ and $SO(3)_-$, and a fixed theory has period $4\pi$. On a general oriented non-spin $X$, quarter-integral sectors distinguish all four values of $p$, so returning to the same fixed theory can require $8\pi$. These periods are statements about the pair of continuous and discrete data, not about the Lie algebra alone. Aharony, Seiberg, and Tachikawa derive the line-operator exchange and the spin/non-spin bundle relations in Aharony, Seiberg, and Tachikawa 2013, §§ 1.2 and 6.1.

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	Local response convention	Symmetry or anomaly check
n ∈ ℤ for a particle on a circle	Paths obey φ(β) = φ(0) + 2πn	n = (2π)−1 ∫ dτ φ̇	eiθn	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	Derivatives of log Z give connected winding cumulants; no contact renormalization is needed in this quantum-mechanical model	Time reversal at θ = 0 or π constrains the spectrum, but the circle model carries no four-dimensional center anomaly
Q ∈ ℤ for SU(2) Yang–Mills	No defects; pure gauge on S³ at infinity; fundamental trace	Q = (8π²)−1 ∫ tr(Fg ∧ Fg)	eiθQ	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	χ = lim V4−1〈Q²〉c after a declared regulator and contact prescription	At θ = π, test CP together with the electric one-form symmetry in its background field; periodicity alone is not the anomaly
Fractional Q sectors for SO(3)	Closed oriented X; nontrivial w₂ bundles allowed; p ∈ ℤ₄ fixed; spin or non-spin structure stated	Fractional part determined by the Pontryagin square of w₂	eiθQ exp[2πip ∫ P(w₂)/4], with (θ,p) ∼ (θ + 2π,p + 1)	On spin X, p ∼ p + 2 and a fixed SO(3)± theory has 4π period; on general non-spin X, p ∈ ℤ₄ is distinguished and the fixed-theory period is 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	Define χ using the same (θ,p) family and bundle sum; a derivative that crosses to p + 1 compares different theories	Recompute CP and one-form-symmetry counterterms for the SO(3) line-operator spectrum; do not import the SU(2) result
Q ∈ ℤ in a closed two-dimensional CPN−1 model	Closed oriented spacetime and standard compact U(1) auxiliary bundle	Q = (2π)−1 ∫ f	eiθQ	2π	Large volume may produce competing branches	The large-N branch pattern is model-specific and nonuniform near crossings	χ is the connected flux variance per volume, with the chosen continuum prescription	CP and global-symmetry constraints depend on N and on the model's background-field completion
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	Differentiate only after the allowed constituent collection and total-charge normalization are fixed	Compactification may preserve or alter the relevant symmetry; match anomalies before transporting an infrared conclusion

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.

Expand the nonnegative square built from $F_g\mp *F_g$ and derive the bound $S_E\geq8\pi^2|Q|/g^2$. Which orientation-sensitive statement remains after the absolute value is taken?

Solution

Using $*^2=1$ on Euclidean two-forms and the symmetry of the inner product,

$$\frac1{2g^2}\int\operatorname{tr} \bigl[(F_g\mp *F_g)\wedge*(F_g\mp *F_g)\bigr] =S_E\mp\frac{8\pi^2}{g^2}Q\geq0.$$

Choosing the sign appropriate to $Q$ gives the absolute-value bound. Orientation still decides which fields are called self-dual and which sign of $Q$ they carry; reversing orientation exchanges instantons and anti-instantons even though the bound itself is unchanged.

On a spin four-manifold, suppose a fixed theory includes charges in $\tfrac12\mathbb Z$. Find the smallest shift that acts term by term as the identity on $e^{i\theta Q}$, and explain why a $2\pi$ shift can still be meaningful.

Solution

For every $Q=n/2$, the condition $e^{i\Delta\theta Q}=1$ requires $\Delta\theta=4\pi$ modulo multiples. A $2\pi$ shift multiplies half-integral sectors by $-1$ rather than returning each weight to itself. It can nevertheless map the theory to a distinct choice of discrete theta data, such as exchanging $SO(3)_+$ and $SO(3)_-$; the shift is then a map between theories, not a period of one fixed theory.

Continue from sectors to observables

Use Theta Parameters, Theta States, and Sector Sums to perform the character transform once the charge lattice is known. For a regulated measurement of $Q$, its index relation, and the continuum extrapolation, continue to Topology and Lattice Index Diagnostics.

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.