Characteristic Classes and Chern–Weil Theory

An invariant polynomial turns curvature into a scalar differential form that does not depend on a choice of local frame. The Bianchi identity makes that form closed, and a transgression formula shows that changing the connection changes it only by an exact form. Its de Rham class therefore depends on the bundle rather than on the chosen connection. This is the mechanism of Chern–Weil theory.

The distinction between form and class is essential. A normalized Chern form can represent the real image of an integral characteristic class, but an arbitrary invariant polynomial need not have integral periods, and real differential forms cannot detect torsion. This page develops that reusable mathematical statement and checks it on a monopole line bundle and a bounded four-dimensional gauge-theory normalization. Theta weighting, periodicity, instanton dynamics, anomalies, and vacuum sectors are left to their physical treatments.

Required background. Bundle Connections, Curvature, Gauge Transformations, and Bianchi Identities supplies principal and associated bundles, local connection forms, adjoint-valued curvature, and the Bianchi identity; de Rham Cohomology, Periods, Duality, and Intersection supplies the closed-versus-exact distinction, pullback on cohomology, integral periods, and the loss of torsion over the reals.

Chern–Weil setting and gauge-convention translation

Let $M$ be a smooth paracompact manifold and let $\pi:P_G\to M$ be a smooth principal bundle for a finite-dimensional Lie group $G$ with Lie algebra $\mathfrak g$. Compactness, a metric, and an orientation are not needed for the Chern–Weil theorem. They enter only when a particular characteristic number or geometric specialization requires them.

For local matrix calculations, choose a representation and write a connection and its curvature as

$$\nabla=\mathrm d+\mathbf A, \qquad \mathbf F = \mathrm d\mathbf A+\mathbf A\wedge\mathbf A.$$

The abstract theorem does not require this representation to be unitary. In the compact/unitary specialization used for the Chern and QFT formulas below, $\mathbf A$ and $\mathbf F$ are anti-Hermitian. Bold symbols will keep that mathematical convention distinct from the site’s Hermitian-generator convention. Products of matrix-valued forms mean wedge product of the form parts and matrix multiplication of the coefficients. The complete unitary translation used below is

$$\begin{aligned} \mathbf A&=-i g_{\mathrm{YM}}A, & \mathbf F&=-i g_{\mathrm{YM}}F, \\ D&=\mathrm d-i g_{\mathrm{YM}}A, & F&=\mathrm dA-i g_{\mathrm{YM}}A\wedge A. \end{aligned}$$

The Hermitian generators satisfy

$$[T^a,T^b]=if^{abc}T^c, \qquad \operatorname{tr}_{\rho}(T^aT^b) =T(\rho)\delta^{ab},$$

with $T(F)=1/2$ in the fundamental representation of $SU(N)$. The trace representation and coupling placement are part of every normalization, not silent notation.

A characteristic class is a natural cohomology assignment

Fix a coefficient ring $R$ and a bundle type, such as principal $G$-bundles or rank-$r$ complex vector bundles. A characteristic class is a rule assigning to each bundle $E\to M$ of that type a cohomology class

$$c(E)\in H^\bullet(M;R)$$

that is invariant under bundle isomorphism and natural under pullback. Thus, for every smooth map $f:N\to M$,

$$c(f^*E)=f^*c(E).$$

For a compact Lie structure group, Freed 2002, § 1.4, journal p. 299, PDF gives the equivalent classifying-space formulation: a universal cohomology class pulls back along the bundle’s classifying map.

The word “characteristic” refers to the bundle, not to an auxiliary connection used to calculate the class. Such classes can obstruct a bundle from being trivial and can distinguish many bundles, but they are not, in general, complete bundle classifiers. Equal characteristic classes do not by themselves imply that two bundles are isomorphic.

The definition allows coefficient systems that differential forms cannot model, such as $\mathbb Z$ and $\mathbb Z_2$. Chern–Weil theory first lands in real de Rham cohomology. Identifying one of its forms with the real image of an integral class is an additional normalization theorem.

Invariant polynomials remove patch dependence

A degree-$k$ invariant polynomial will mean a symmetric $k$-linear map

$$\Phi:\mathfrak g^{\times k}\longrightarrow\mathbb R$$

such that

$$\Phi( \operatorname{Ad}_gX_1, \ldots, \operatorname{Ad}_gX_k) = \Phi(X_1,\ldots,X_k)$$

for every $g\in G$. Equivalently, $\Phi\in(\operatorname{Sym}^k\mathfrak g^*)^G$. When $G$ is disconnected, invariance under the full adjoint action of $G$ is required; infinitesimal invariance under the identity component alone is not enough.

Extend $\Phi$ to $\mathfrak g$-valued forms by wedging their form parts. We use the shorthand

$$\Phi(\mathbf F^k) := \Phi(\mathbf F,\ldots,\mathbf F) \in\Omega^{2k}(M),$$

with no implicit factor of $k!$. Any factorial, trace, or $2\pi$ factor belongs to the chosen polynomial.

Chern–Weil theorem. For every $\Phi\in(\operatorname{Sym}^k\mathfrak g^*)^G$, the form $\Phi(\mathbf F^k)$ is globally defined and closed. Its de Rham class is independent of the connection and natural under pullback. The next three steps establish each clause rather than treating the theorem as a black box.

On an overlap $U_i\cap U_j$, local curvature matrices obey

$$\mathbf F_j = g_{ij}^{-1}\mathbf F_i g_{ij}.$$

Adjoint invariance then gives

$$\Phi(\mathbf F_j^k)=\Phi(\mathbf F_i^k).$$

The local scalar forms therefore glue to one global $2k$-form on $M$. A non-invariant polynomial fails at this first step: its local expressions need not agree and need not be gauge invariant. Nakahara 2003, § 11.1, pp. 419–425 develops the invariant-polynomial construction and its extension to Lie-algebra-valued forms.

The Bianchi identity makes the global form closed

For a $p$-form $X$ valued in the matrix Lie algebra, set

$$D_{\mathbf A}X = \mathrm dX +\mathbf A\wedge X -(-1)^pX\wedge\mathbf A.$$

Infinitesimal adjoint invariance says that the commutator contributions cancel after all slots of $\Phi$ are summed. Consequently,

$$\mathrm d\,\Phi(\mathbf F^k) = k\,\Phi( D_{\mathbf A}\mathbf F, \mathbf F^{k-1}).$$

The Bianchi identity $D_{\mathbf A}\mathbf F=0$ now gives

$$\boxed{ \mathrm d\,\Phi(\mathbf F^k)=0 }.$$

Ordinary $\mathrm d$ appears on the scalar-valued result only after invariance has removed the connection commutators. Replacing $D_{\mathbf A}\mathbf F$ by $\mathrm d\mathbf F$ before that cancellation would be wrong in a non-Abelian bundle. This proof is local and algebraic; it uses neither a metric nor a field equation.

Transgression removes the connection from the class

Take two connections $\mathbf A_0$ and $\mathbf A_1$ on the same principal bundle. Their difference

$$\mathbf a:=\mathbf A_1-\mathbf A_0$$

is a global $\operatorname{ad}P_G$-valued one-form, even though neither local potential is generally a global matrix-valued one-form. The affine path

$$\mathbf A_t=\mathbf A_0+t\mathbf a, \qquad 0\leq t\leq1,$$

has curvature

$$\mathbf F_t = \mathbf F_0 +tD_0\mathbf a +t^2\mathbf a\wedge\mathbf a.$$

Differentiating gives the covariant identity

$$\dot{\mathbf F}_t=D_t\mathbf a.$$

Because $\Phi$ is symmetric and $\mathbf F_t$ has even degree,

$$\begin{aligned} \frac{\mathrm d}{\mathrm dt} \Phi(\mathbf F_t^k) &= k\,\Phi( D_t\mathbf a, \mathbf F_t^{k-1}) \\ &= k\,\mathrm d\, \Phi( \mathbf a, \mathbf F_t^{k-1}). \end{aligned}$$

The second line again uses invariance, now together with $D_t\mathbf F_t=0$. Integrating from $0$ to $1$ produces the transgression formula

$$\Phi(\mathbf F_1^k)-\Phi(\mathbf F_0^k) = \mathrm d\, T_\Phi(\mathbf A_1,\mathbf A_0),$$

where

$$T_\Phi(\mathbf A_1,\mathbf A_0) = k\int_0^1 \Phi( \mathbf a, \mathbf F_t^{k-1})\, \mathrm dt.$$

Thus two characteristic forms can differ, but their de Rham classes agree:

$$[\Phi(\mathbf F_1^k)]_{\mathrm{dR}} = [\Phi(\mathbf F_0^k)]_{\mathrm{dR}}.$$

This is connection independence at the correct level. Nakahara 2003, Theorem 11.1, pp. 422–425 proves closedness and exact connection variation in this normalization-free form. Freed 2002, § 1.2, journal pp. 297–298, PDF independently derives closedness and relative transgression from an invariant polynomial and two global connections.

There are two useful qualifications.

First, a comparison with a “zero connection” is global only when that endpoint really is a connection on the same bundle—for example, after choosing a global trivialization. On a nontrivial bundle, the familiar Chern–Simons primitive is generally local or defined relative to another genuine global connection.

Second, the cohomology statement remains valid when $M$ has boundary, but an integral over a chain $C$ with boundary satisfies

$$\int_C \bigl[ \Phi(\mathbf F_1^k)-\Phi(\mathbf F_0^k) \bigr] = \int_{\partial C} T_\Phi(\mathbf A_1,\mathbf A_0).$$

The boundary contribution vanishes automatically only when $C$ is a cycle or when additional boundary conditions make it vanish.

Gauge invariance, connection independence, and integrality

Three statements that are often compressed into “topological” have different inputs.

1. Gauge invariance of the form. Curvature transforms by conjugation, so adjoint invariance makes $\Phi(\mathbf F^k)$ exactly unchanged.
2. Connection independence of the class. Transgression makes the difference between two representatives exact.
3. Integrality of a characteristic number. This requires a specifically normalized universal class and a matching integral cycle; it does not follow from closedness or transgression.

Naturality follows directly as well. A pullback connection has curvature $f^*\mathbf F$, so

$$\Phi((f^*\mathbf F)^k) = f^*\Phi(\mathbf F^k).$$

Passing to cohomology gives the pullback law required in the definition of a characteristic class.

Chern classes and the Chern character

Let $E\to M$ be a rank-$r$ Hermitian complex vector bundle with a unitary connection. In a unitary frame, its curvature $\mathbf F$ is anti-Hermitian. Define the Hermitian matrix-valued two-form

$$X:=\frac{i\mathbf F}{2\pi}.$$

The total Chern form is the formal determinant

$$\begin{aligned} c(E,\nabla) &:=\det(\mathbf1+X) \\ &=1+c_1(\nabla)+c_2(\nabla)+\cdots+c_r(\nabla). \end{aligned}$$

The determinant is expanded in the graded-commutative algebra of differential forms. Its first two positive-degree terms are

$$\begin{aligned} c_1(\nabla) &=\operatorname{tr}X =\frac{i}{2\pi}\operatorname{tr}\mathbf F, \\ c_2(\nabla) &=\frac12 \left[ (\operatorname{tr}X)\wedge(\operatorname{tr}X) -\operatorname{tr}(X\wedge X) \right] \\ &=\frac1{8\pi^2} \left[ \operatorname{tr}(\mathbf F\wedge\mathbf F) -\operatorname{tr}\mathbf F\wedge \operatorname{tr}\mathbf F \right]. \end{aligned}$$

The classes $[c_j(\nabla)]_{\mathrm{dR}}$ are the real images of the integral Chern classes $c_j(E)\in H^{2j}(M;\mathbb Z)$. This integrality is a theorem about the displayed normalization, not a consequence of the Chern–Weil argument alone. For compact $G$, the normalized invariant polynomials whose universal classes lie in the integral lattice have integer periods on integral cycles; Freed 2002, § 1.4, journal p. 299, PDF states this integral-lattice criterion. Frankel 2012, § 22.5, pp. 608–616 gives an independent obstruction-theoretic account for Chern forms.

The Chern character packages different invariant polynomials:

$$\begin{aligned} \operatorname{ch}(E,\nabla) &:=\operatorname{tr}e^X, \\ \operatorname{ch}_j(\nabla) &:=\frac1{j!}\operatorname{tr}(X^j). \end{aligned}$$

The first components are

$$\begin{aligned} \operatorname{ch}_0&=r, \\ \operatorname{ch}_1&=c_1, \\ \operatorname{ch}_2 &=\frac12\operatorname{tr}(X\wedge X) \\ &=-\frac1{8\pi^2} \operatorname{tr}(\mathbf F\wedge\mathbf F) \\ &=\frac12(c_1^2-2c_2). \end{aligned}$$

Unlike the individual Chern classes, Chern-character components are generally rational combinations of integral classes. The Chern character is included here because it is the characteristic expression that enters the next chapter step toward index theory; no index theorem is assumed or proved on this page. Nakahara 2003, §§ 11.2–11.3, pp. 426–435 derives the determinant expansion, naturality, and the relation between $\operatorname{ch}_j$ and $c_j$.

Translation to the site’s Hermitian generators

The convention map

$$\mathbf F=-i g_{\mathrm{YM}}F$$

implies

$$X = \frac{i\mathbf F}{2\pi} = \frac{g_{\mathrm{YM}}F}{2\pi}.$$

Therefore, in a declared unitary representation $\rho$,

$$\begin{aligned} c_1(\nabla) &= \frac{g_{\mathrm{YM}}}{2\pi} \operatorname{tr}_{\rho}F, \\ c_2(\nabla) &= \frac{g_{\mathrm{YM}}^2}{8\pi^2} \left[ \operatorname{tr}_{\rho}F\wedge \operatorname{tr}_{\rho}F - \operatorname{tr}_{\rho}(F\wedge F) \right], \\ \operatorname{ch}_2(\nabla) &= \frac{g_{\mathrm{YM}}^2}{8\pi^2} \operatorname{tr}_{\rho}(F\wedge F). \end{aligned}$$

For $SU(N)$, the fundamental trace of $F$ vanishes, and hence

$$c_1=0, \qquad c_2=-\operatorname{ch}_2.$$

The last sign is not optional: it follows from translating the anti-Hermitian curvature before expanding the determinant. Changing the trace representation also changes the quadratic normalization because

$$\operatorname{tr}_{\rho}(F\wedge F) = T(\rho)F^a\wedge F^a.$$

Thus a formula with an unspecified trace does not yet define a normalized characteristic number.

Other curvature classes and what real forms miss

Chern–Weil theory is not limited to complex Chern classes. For a real vector bundle $V$ with a fiber metric and metric connection, let $\boldsymbol\Omega$ be the curvature in an orthonormal frame. The Pontryagin classes may be normalized through complexification,

$$p_j(V)=(-1)^j c_{2j}(V\otimes\mathbb C),$$

and the first Pontryagin form is

$$p_1(\nabla) = -\frac1{8\pi^2} \operatorname{tr}_{\mathbb R} (\boldsymbol\Omega\wedge\boldsymbol\Omega).$$

No orientation of $V$ is needed for $p_j$. If $V$ is oriented and has even rank $2m$, the normalized Pfaffian

$$e(\nabla) = \operatorname{Pf}\!\left( \frac{\boldsymbol\Omega}{2\pi} \right)$$

represents the real image of its Euler class. Reversing the bundle orientation reverses the Euler class. These statements require the displayed real, metric-connection curvature convention; a source using the opposite curvature sign must be translated before its Pfaffian or trace formula is used. Nakahara 2003, §§ 11.4.1–11.4.2, pp. 436–441 derives these Pontryagin and Euler normalizations from skew-symmetric curvature matrices.

Ordinary real Chern–Weil forms do not produce Stiefel–Whitney classes in $H^\bullet(M;\mathbb Z_2)$, and they cannot distinguish integral classes that differ only by torsion. In particular, a flat connection makes every positive-degree Chern–Weil form vanish, yet it can still have nontrivial holonomy, and its bundle can retain torsion characteristic data. Vanishing real curvature classes therefore does not imply global triviality. Nakahara 2003, opening of § 11.6, p. 448 notes that Stiefel–Whitney classes are not curvature expressions, while Freed 2002, § 1.1, journal p. 296, PDF separates curvature from flat holonomy and the full integral class.

The two-patch monopole is a first Chern number

Return to the coupling-absorbed real $U(1)$ connection on $S^2$ from the prerequisite pages. On northern and southern patches,

$$\begin{aligned} a_N&=\frac n2(1-\cos\theta)\,\mathrm d\varphi, \\ a_S&=-\frac n2(1+\cos\theta)\,\mathrm d\varphi, \end{aligned}$$

with transition function $g_{NS}=e^{in\varphi}$ and

$$a_N-a_S=n\,\mathrm d\varphi.$$

Their common real curvature is

$$f = \frac n2\sin\theta\, \mathrm d\theta\wedge\mathrm d\varphi.$$

The corresponding anti-Hermitian forms are $\mathbf A_i=-ia_i$ and $\mathbf F=-if$. The first Chern form is therefore

$$c_1(\nabla) = \frac{i\mathbf F}{2\pi} = \frac f{2\pi}.$$

With $S^2$ oriented by $\sin\theta\,\mathrm d\theta\wedge\mathrm d\varphi$,

$$\left[ \frac f{2\pi} \right]_{\mathrm{dR}} = c_1(L_n)\otimes1 \in H^2(S^2;\mathbb R), \qquad \int_{S^2}\frac f{2\pi}=n.$$

This identifies the transition-function winding, quantized flux, and first Chern number as three descriptions of the same integer in this example.

Now change the connection on the same line bundle. The patchwise differences glue to one global real one-form $\alpha$, so

$$a_i'=a_i+\alpha, \qquad f'=f+\mathrm d\alpha.$$

Hence

$$c_1(\nabla')-c_1(\nabla) = \mathrm d\!\left(\frac{\alpha}{2\pi}\right),$$

which is the degree-one transgression formula. The curvature representative changes, while its class and integral over $S^2$ do not. Nakahara 2003, Example 11.2, pp. 432–433 checks the same monopole Chern character in an anti-Hermitian convention.

Four-dimensional instanton-number normalization

Let $X$ be a closed oriented smooth four-manifold and let $E\to X$ be the fundamental bundle associated to a principal $SU(N)$ bundle. To match later gauge-theory language, one may equip $X$ with a Euclidean metric, but neither that metric nor a Hodge star enters the following characteristic number. With

$$\operatorname{tr}_F(T^aT^b)=\frac12\delta^{ab},$$

define the site-compatible topological charge

$$\begin{aligned} Q &:= \int_X\operatorname{ch}_2(\nabla) \\ &= \frac{g_{\mathrm{YM}}^2}{8\pi^2} \int_X \operatorname{tr}_F(F\wedge F). \end{aligned}$$

Since $c_1(E)=0$ and $\operatorname{ch}_2=-c_2$,

$$Q = -\langle c_2(E),[X]\rangle \in\mathbb Z.$$

This equation is the promised instanton-number prerequisite. The positive-sign physics expression $Q$ is the negative of the second Chern number in the determinant convention used above; reversing the orientation of $X$ reverses both. Mariño 2015, § 4.3, pp. 112 and 116–122 derives the normalized $\operatorname{Tr}(F\wedge F)$ topological charge and its winding-number interpretation with a different coupling placement; the displayed map $\mathbf F=-ig_{\mathrm{YM}}F$ gives the translation to the site convention.

No instanton solution, self-duality equation, action bound, or tunneling claim follows from this characteristic number alone. The gauge group’s global form, boundaries, admissible sectors, and charge normalization can also change the physical periodicity statement. Those questions belong to Theta Terms, Periodicity, and Vacuum Sectors.

Common pitfalls

Treating the representative as connection-independent. A characteristic form can vary with the connection. Transgression proves that its variation is exact, so the cohomology class—not generally the form—is unchanged.

Inferring integrality from closedness. Multiplying an invariant polynomial by an arbitrary real number preserves gluing, closedness, and transgression. Only special normalizations correspond to integral universal classes.

Suppressing the trace representation. Quadratic traces differ by Dynkin indices. A missing trace label can turn an integer normalization into a rescaled number.

Confusing $c_2$ with $\operatorname{ch}_2$. For an $SU(N)$ bundle in the conventions above, $c_2=-\operatorname{ch}_2$. Importing an instanton formula without translating anti-Hermitian versus Hermitian curvature reverses this sign.

Using a global zero potential on a nontrivial bundle. Local potentials can be set to convenient forms in trivializations, but a nontrivial bundle need not admit one global matrix potential. Relative transgression between two actual connections remains global.

Concluding that flat means trivial. Flatness erases positive-degree real Chern–Weil forms. It does not erase holonomy or torsion information.

Ignoring the integration chain. A degree-$2k$ characteristic form pairs with a $2k$-cycle. On a chain with boundary, connection variation can leave a transgression term on the boundary.

Exercises

1. Trace invariance and gluing

For a matrix representation $\rho$, show that

$$\Phi(X,Y)=\operatorname{tr}_{\rho}(XY)$$

is adjoint invariant. Use this to explain why $\operatorname{tr}_{\rho}(\mathbf F\wedge\mathbf F)$ is a global closed four-form.

Solution

Cyclicity of the trace gives

$$\begin{aligned} \operatorname{tr}_{\rho} (gXg^{-1}gYg^{-1}) &= \operatorname{tr}_{\rho}(gXYg^{-1}) \\ &= \operatorname{tr}_{\rho}(XY). \end{aligned}$$

The local curvature expressions therefore agree on overlaps. The Bianchi identity then gives

$$\mathrm d\, \operatorname{tr}_{\rho} (\mathbf F\wedge\mathbf F) = 2\operatorname{tr}_{\rho} (D_{\mathbf A}\mathbf F\wedge\mathbf F) =0.$$

2. Quadratic transgression

Let $\mathbf a=\mathbf A_1-\mathbf A_0$ and $\mathbf A_t=\mathbf A_0+t\mathbf a$. Derive

$$\operatorname{tr}(\mathbf F_1\wedge\mathbf F_1) - \operatorname{tr}(\mathbf F_0\wedge\mathbf F_0) = \mathrm d\left[ 2\int_0^1 \operatorname{tr}(\mathbf a\wedge\mathbf F_t) \,\mathrm dt \right].$$

Solution

Since $\dot{\mathbf F}_t=D_t\mathbf a$,

$$\begin{aligned} \frac{\mathrm d}{\mathrm dt} \operatorname{tr}(\mathbf F_t\wedge\mathbf F_t) &= 2\operatorname{tr} (D_t\mathbf a\wedge\mathbf F_t) \\ &= 2\,\mathrm d\, \operatorname{tr}(\mathbf a\wedge\mathbf F_t). \end{aligned}$$

The second line uses trace invariance and $D_t\mathbf F_t=0$. Integrating in $t$ gives the claimed identity.

3. Chern–Weil does not imply integrality

Suppose a line bundle over $S^2$ has $\int_{S^2}i\mathbf F/(2\pi)=1$. For a real number $\lambda$, does $\lambda i\mathbf F/(2\pi)$ still define a connection-independent Chern–Weil class? Must its period be integral?

Solution

Scalar multiplication preserves adjoint invariance, closedness, and the transgression argument. The new period is

$$\int_{S^2} \lambda\frac{i\mathbf F}{2\pi} =\lambda.$$

Taking $\lambda=\sqrt2$ gives a nonintegral period. The form defines a valid real Chern–Weil class, but it is not the normalized first Chern form. This separates the Chern–Weil theorem from the integrality theorem.

4. Translate the four-dimensional charge

For an $SU(N)$ bundle, start from $\mathbf F=-ig_{\mathrm{YM}}F$ and derive $c_2$, $\operatorname{ch}_2$, and their integer evaluations on a closed oriented four-manifold. State one physical conclusion that does not follow.

Solution

Tracelessness gives $c_1=0$. Therefore

$$\begin{aligned} c_2 &= \frac1{8\pi^2} \operatorname{tr}_F (\mathbf F\wedge\mathbf F) \\ &= -\frac{g_{\mathrm{YM}}^2}{8\pi^2} \operatorname{tr}_F(F\wedge F), \\ \operatorname{ch}_2 &=-c_2 = \frac{g_{\mathrm{YM}}^2}{8\pi^2} \operatorname{tr}_F(F\wedge F). \end{aligned}$$

Thus

$$\int_X c_2\in\mathbb Z, \qquad \int_X\operatorname{ch}_2 =-\int_X c_2\in\mathbb Z.$$

These equalities do not construct a classical instanton, determine an action, or establish theta periodicity. Each of those statements needs additional geometric and physical input.

Synthesis and continuations

Chern–Weil theory answers the principal question in three steps: adjoint invariance makes $\Phi(\mathbf F^k)$ global, Bianchi makes it closed, and transgression makes its de Rham class independent of the connection. Normalized determinant and trace polynomials then recover the real representatives of Chern, Pontryagin, and Euler classes, while the Chern character supplies the rational combination used in index formulas. The monopole example identifies transition winding, flux, and $c_1$; the four-dimensional example fixes the sign and trace normalization needed before using an instanton charge.

Continue according to the question:

• Fredholm and Dirac Index Theorems and Zero-Mode Counting combines characteristic forms with analytic operator data;
• Theta Terms, Periodicity, and Vacuum Sectors develops the physical weighting, global-form qualifications, boundaries, and vacuum interpretation of the normalized four-dimensional charge;
• de Rham Cohomology, Periods, Duality, and Intersection supplies the real period and torsion distinctions used throughout this page.

References

• Theodore Frankel, The Geometry of Physics: An Introduction, third edition, Cambridge University Press, 2012, §§ 22.1c and 22.5, pp. 587–590 and 608–616. These sections provide a determinant expansion, connection-independence argument, and an interpretation of integral Chern periods.
• Daniel S. Freed, “Classical Chern–Simons Theory, Part 2”, Houston Journal of Mathematics 28 (2002), 293–310, especially § 1, journal pp. 296–299. The cited discussion supplies the curvature-versus-integral-class distinction, invariant-polynomial closedness, relative transgression, flat holonomy, and the integral lattice of normalized Chern–Weil polynomials.
• Marcos Mariño, Instantons and Large N: An Introduction to Non-Perturbative Methods in Quantum Field Theory, Cambridge University Press, 2015, § 4.3, pp. 112 and 116–122. The cited section supports the normalized $\operatorname{Tr}(F\wedge F)$ charge and its winding-number interpretation. Its coupling placement is translated to the site’s $D=\mathrm d-i g_{\mathrm{YM}}A$ convention above.
• Mikio Nakahara, Geometry, Topology and Physics, second edition, Institute of Physics Publishing, 2003, Chapter 11, especially §§ 11.1–11.4, pp. 419–441, and § 11.6, p. 448. These sections support invariant-polynomial closedness, Chern forms, the monopole example, Pontryagin and Euler forms, and the curvature limitation for Stiefel–Whitney classes.