2026-10-01 · chapter Yang–Mills and the mass gapPart 1 Mathematical preparationparallel transport, curvature, holonomy

Vector bundles and connections — a rule for comparing separate spaces at each point along a path

Suppose a field $\psi$ takes values in $\mathbb C^N$ at each point ($N = 3$ for the colour of quarks). In gauge theory we allow the basis of $\mathbb C^N$ at each point (which direction to call "the first colour") to be chosen afresh point by point (3-01). Then $\psi(x+h) - \psi(x)$ is a difference of components written in separate bases, and its value changes when the bases are rechosen. Tangent vectors of the sphere are the same: the tangent planes at the north pole and at the equator are different things. So we prepare a rule $P$ for carrying one space to the other along a path, and subtract after carrying ($e_i$ is a coordinate direction).

The carrying rule is called a connection, and the spaces at each point bundled together a vector bundle. Example. In electromagnetism, $N = 1$, and the carrying rule was the phase factor $e^{\frac{iq}{\hbar}\int\mathbf A\cdot d\mathbf l}$ (2-03). This chapter extends that to matrices, and treats the "gluing of two sheets" of 1-06 §08 in the same language.

A vector bundle $E\to M$ attaches to each point x of a manifold M (1-05) an N-dimensional space $E_x$ (the fibre), so that over a small open set $U_\alpha$ it can be identified with $U_\alpha\times\mathbb C^N$. N is called the rank, and one with $N = 1$ a line bundle. It can be defined in the same way with fibre $\mathbb R^N$, and this is called a real vector bundle. A basis giving the identification is a local frame, and on an overlap the components in the two frames are related by a matrix.

$g_{\alpha\beta}$ is called the transition function, and the formula on the right, where three overlap, the cocycle condition; when the transition functions take values in a group G (U(1), SU(N) and so on), G is called the structure group. Given transition functions satisfying (2), a bundle can be made by gluing Known. A smooth choice of an element of $E_x$ at each point is a section, and the field $\psi$ is a section.

Trivial bundles and twisted bundles

A bundle that can be identified with $M\times\mathbb C^N$ ($M\times\mathbb R^N$ if real) over the whole space is called trivial. The following three are not trivial Known: the Möbius strip (a real line bundle with fibre $\mathbb R$; covering the circle with two arcs, the transition function is +1 on one part of the overlap and −1 on the other), the tangent bundle $TS^2$ (a real bundle of rank 2), and the monopole bundle of §08 ($n\ne 0$). A trivial bundle would have a section that is nowhere 0. On the Möbius strip, the value of a section written in the frame changes sign in one turn, so by the intermediate value theorem it is 0 somewhere. On $TS^2$ the hairy ball theorem (1-05) forbids it.

Making the fibre complex untwists it with the same transition functions. $-1 = e^{i\pi}$ can be moved continuously inside U(1) to 1 through $e^{it}$ (t from π to 0), and moving transition functions continuously keeps the bundle isomorphic Known, so the complex line bundle with transition functions +1 and −1 on the overlap is trivial. On the circle, whether a twist remains is decided by whether the transition function can be moved continuously to 1 inside the group where it takes values. Inside $\{\pm1\}$ it cannot; inside U(1) it can. In §08 the winding number in U(1) carries this impossibility of moving. Bundles over $\mathbb R^4$ are trivial Known, but on $S^4$, with infinity regarded as one point, and on periodic lattices, non-trivial bundles appear (3-05).

Once a frame is fixed, the first-order term of the $P$ of (1) becomes a matrix-valued 1-form $A = A_i\,dx^i$. $A_i(x)$ is an element of the Lie algebra $\mathfrak g$ of the structure group (1-03), and for U(N), SU(N) it is an anti-Hermitian matrix.

This is what we get by putting $P_{x+he_i\to x} = 1 + hA_i + O(h^2)$ in (1). For $D'\psi' = g\,D\psi$ to hold even when the frame is changed to $\psi' = g\psi$ ($g(x)\in G$), since $(d + A')(g\psi) = dg\,\psi + g\,d\psi + A'g\psi$ equals $g(d\psi + A\psi)$,

is needed. This is the gauge transformation. The $A_\alpha$ and $A_\beta$ of an overlap are also related by (4) with $g = g_{\alpha\beta}$: gluing and gauge transformation are the same formula.

Example (U(1)). Putting $A = -\frac{iq}{\hbar}\,\mathbf A\cdot d\mathbf x$, $g = e^{iq\chi/\hbar}$, (3) is the covariant derivative of 2-03 and (4) is $\mathbf A\mapsto\mathbf A + \nabla\chi$. Physics books write $A = -i\,A^a_\mu T_a\,dx^\mu$ with Hermitian $T_a$ (1-04), and call the real $A^a_\mu$ the gauge field (for where the coupling constant is put, see 3-02). The difference of two connections changes as $g(A_1 - A_2)g^{-1}$ (the $dg$ term cancels), and is a quantity that can be added and subtracted. The set of all connections is then an affine space — a set with no fixed origin, in which the difference of two points is an element of a vector space. Choosing one reference connection $A_0$, every connection can be written as $A_0$ + (a $\mathfrak g$-valued 1-form) (3-04).

Along a path $\gamma : [0, 1]\to M$, carry v so that the covariant derivative is 0.

The solution is unique, and $P_\gamma$ is called parallel transport. If A is anti-Hermitian, $\frac{d}{dt}|v|^2 = -v^*(A + A^*)v = 0$, so $P_\gamma$ is unitary. The $A(\dot\gamma)$ at different times do not commute, so it cannot be written $\exp(-\int A)$; instead the solutions over short intervals are multiplied in order of time.

$A_k = A(\dot\gamma(t_k))$ is the path-ordered product, with later times placed on the left. Joining paths gives $P_{\gamma_2\gamma_1} = P_{\gamma_2}P_{\gamma_1}$, and under gauge transformations it only undergoes the change of frame at the two ends.

Example: once round the sphere

Carrying a tangent vector so that it does not bend as seen from inside the sphere is the Levi-Civita parallel transport; on the unit sphere it is $\dot v = -(v\cdot\dot\gamma)\,\gamma$ (allowing only change in the normal direction). A curve that keeps its own velocity vector $\dot\gamma$ under this parallel transport is called a geodesic; on the sphere these are great circles. Going east round the circle of latitude at colatitude $\theta_0$, the angle relative to the south-pointing and east-pointing frame $(e_\theta, e_\varphi)$ changes by $\alpha(\varphi) = \alpha_0 - \varphi\cos\theta_0$, and in one turn

The right-hand side is the area of the spherical cap enclosed by the circle of latitude. Circles of latitude other than the equator are not great circles (not geodesics), so the vector turns relative to the frame. (8) is a special case of the Gauss–Bonnet theorem, "on a surface of curvature 1 the angle turned in one circuit equals the area enclosed" Known. For a geodesic triangle, whose three sides are geodesics, the angle turned also equals the area; the bottom row of Table 1 is an example.

Detour: Foucault's pendulum

Whether parallel transport depends on the path can be measured by the mismatch after going round a small loop. In the difference of covariant derivatives taken in swapped order, the derivatives of $\psi$ cancel, and only matrix multiplication remains.

Gathered into forms, it is $F = dA + A\wedge A$. $dA$ is the exterior derivative component by component, and the wedge product of matrix-valued forms multiplies the component matrices and takes the wedge product of $dx$.

The second equality is the average with the formula with the names i and j swapped ($dx^j\wedge dx^i = -dx^i\wedge dx^j$). For a scalar 1-form α, $\alpha\wedge\alpha = 0$ (1-06), but matrices do not commute, so it is not 0. The coefficient of $F = \sum_{i\lt j}F_{ij}\,dx^i\wedge dx^j$ is the $F_{ij}$ of (9), exactly the formula of the Jaffe–Witten problem description (1-03). F is called the curvature, and in physics the field strength.

Checking with a small loop

Going once anticlockwise round a square of side ε in the x and y directions from the point p,

Putting (4) into (9) gives $F' = gFg^{-1}$. For U(1), F does not change under gauge transformations and is (apart from the factor $-iq/\hbar$) the electric and magnetic fields themselves (1-06 §09), but in the non-abelian case F turns with the frame, and what does not change are combinations such as $\operatorname{tr}(F_{ij}F_{kl})$ (the action of 3-03).

ComputedTable 2 Integrating (5) with RK4 for a smooth $\mathfrak{su}(2)$-valued connection and comparing with (10) (operator norm, $|F_{xy}| = 0.369$).

Halving ε halves the error too ($O(\varepsilon)$). With $dA$, where the commutator is dropped, the error approaches $|[A_x, A_y]| = 0.298$ and does not vanish. For a random g, (7) differs by $1.5\times10^{-12}$, and $F' = gFg^{-1}$ by $3\times10^{-7}$ (the error of differencing) Computed.

A connection that curves even when constant

In the non-abelian case F is not 0 even when A is constant. Take the group to be SO(3) (the adjoint representation of SU(2), 1-04) and the fibre $\mathbb R^3$ (the real vector bundle of §02). Here $L_x, L_y, L_z$ are the real antisymmetric matrices $(L_a)_{bc} = -\varepsilon_{abc}$, the Hermitian $L_a$ of 1-04 §02 multiplied by $-i$. They satisfy $L_a v = e_a\times v$, $[L_x, L_y] = L_z$, and $e^{tL_z}$ is the rotation by angle t about the z axis (the same as the $e^{-itL_3}$ of 1-04). Putting $A_x = aL_x$, $A_y = aL_y$, we have $dA = 0$ and yet $F_{xy} = a^2L_z$. A constant A for U(1) can be removed by $g = e^{A_xx + A_yy}$ (pure gauge), but here $F' = gFg^{-1}\ne 0$, so no g can remove it.

θ/s² approaches 1 as s → 0, and the axis is almost −z, agreeing with the $1 - s^2L_z$ of (10) (a positive rotation about −z). This is the geometric face of "the gauge field interacting with itself" (3-02).

The parallel transport along a path γ that leaves the point $x_0$ and returns is called the holonomy $\mathrm{Hol}_\gamma$. By (7), since both ends are the same point, it changes by conjugation under gauge transformations, and its trace does not change.

The gauge-invariant $W(\gamma)$ is called the Wilson loop.

• It remains even when flat. If $F = 0$, the holonomy does not change under continuous deformation of the loop Known. In a space with a hole, the holonomy of a loop that cannot be shrunk can be other than 1; this is the Aharonov–Bohm effect (2-03).
• Curvature is the infinitesimal version of holonomy. If M is connected, the Lie algebra of the set (subgroup) of holonomies of loops at $x_0$ is spanned by the curvatures $F_{ij}(y)$ at all points y carried to $x_0$ along every path joining y and $x_0$ (Ambrose–Singer 1953) Known.
• The lattice. Placing parallel transports $U_\ell\in G$ on the edges and using the plaquette product $U_p$ in place of curvature is Wilson's lattice (5-01). From (10), with spacing a, $U_p\approx\exp(-a^2F_{\mu\nu})$, hence $\operatorname{Re}\operatorname{tr}(1 - U_p)\approx -\tfrac{a^4}{2}\operatorname{tr}F_{\mu\nu}^2 \ge 0$ (μ, ν fixed and not summed; $F_{\mu\nu}^2 = -F_{\mu\nu}^*F_{\mu\nu}\le 0$ because $F_{\mu\nu}$ is anti-Hermitian) Known.

A gauge transformation is a smooth map $g : M\to G$, which moves a section to $g\psi$, the connection by (4), and the curvature to $gFg^{-1}$. The whole $\mathcal G$ is an infinite-dimensional group. As a metaphor, it is a rechoosing of the orientation of the ruler at each point; to be exact, it can be read either as a change of frame (passive) or as an automorphism of the bundle — a map of the bundle that maps each fibre linearly and isomorphically and does not move the points of the base — (active), and in both readings the formula is (4). What has meaning in physics are quantities that do not change under $\mathcal G$: Wilson loops, the $\operatorname{tr}(F\wedge *F)$ of the action, the topological quantity $\operatorname{tr}(F\wedge F)$ (3-05).

Curvature satisfies an identity independent of the gauge. $A\wedge F$ and $F\wedge A$ are also fixed by the same rule as in §05 (multiply the component matrices and take the wedge product of $dx$), and the order of the matrices matters.

This is the Bianchi identity. Putting in $F = dA + A\wedge A$, it vanishes by $d(A\wedge A) = dA\wedge A - A\wedge dA$, $d^2 = 0$ and the associativity of the wedge product alone. For U(1) it is $dF = 0$ (the two Maxwell equations without sources). For a $\mathfrak{su}(2)$-valued connection in three dimensions, the left-hand side is within the differencing error $3.5\times10^{-7}$; dropping the commutator gives 0.553 Computed.

The $\mathcal A/\mathcal G$ obtained by dividing the set of all connections $\mathcal A$ by $\mathcal G$ is the space of configurations that physics can distinguish, and choosing a representative continuously from each orbit (a global gauge fixing) is in general impossible (Singer, 3-04, 4-08). In the translation table of Wu–Yang (1975), gauge potential = connection, phase factor = parallel transport and field strength = curvature are listed, and only the entry for sources is "?".

In 1-06 §08, the potential of the point magnetic charge $\hat{\mathbf r}/r^2$ could be written only with two sheets, north and south, and the difference on the overlap was $2\,d\varphi$. Multiplying the magnetic charge by $g_m$ to get the magnetic field $g_m\hat{\mathbf r}/r^2$ (total flux $\Phi = 4\pi g_m$), the difference is $2g_m\,d\varphi$, and multiplying by $-iq/\hbar$ to make a connection as in the example of §03 gives

On the other hand, (4) in the commutative case is $A_N - A_S = -g_{NS}^{-1}dg_{NS}$, so the transition function on the equator is $g_{NS} = e^{in\varphi}$, $n = 2qg_m/\hbar$. In the language of bundles, it is the U(1) bundle obtained by gluing the north and south sheets along the equator by this $g_{NS}$. $g_{NS}$ must return to itself after going once round the equator, so n is an integer.

Seen through phase factors

Seen through phase factors, the meaning of n takes a measurable form. The phase factor of a closed path C is the parallel transport (the holonomy of C, §06) obtained by putting $A = -\frac{iq}{\hbar}\,\mathbf A\cdot d\mathbf x$ in (6). For U(1) no path ordering is needed, and by Stokes' theorem (1-06)

($\Phi_C$ is the magnetic flux enclosed by C, and we take q > 0; for details see 2-03). $\Phi_0$ is called the flux quantum, and writing $f = \Phi/\Phi_0$, we have $n = 2qg_m/\hbar = q\Phi/(2\pi\hbar) = f$. From here on n and f are the same quantity; when we write f we also consider values that are not necessarily integers.

Computing the phase factor of the circle of latitude C at colatitude ϑ with $A_N$, usable on the northern cap, the enclosed flux is $\Phi(1-\cos\vartheta)/2$, giving $e^{i\pi f(1-\cos\vartheta)}$. Computing with $A_S$, usable on the southern cap, C goes round the boundary of the southern cap in the opposite direction, giving $e^{-i\pi f(1+\cos\vartheta)}$. The ratio is $e^{2\pi i f}$, and they agree only when f is an integer. This is the condition for the two descriptions to be a connection on one bundle, and is the reading, in the language of bundles, of Dirac's quantisation condition (1931).

Read from curvature

The integer can also be read from curvature. Using Stokes' theorem on the northern and southern caps, and (4) on the equator (commutative, so $A_N - A_S = -g^{-1}dg$),

The left-hand side is $\frac{i}{2\pi}\cdot\bigl(-\frac{iq}{\hbar}\bigr)\Phi = f$, and the right-hand side is the winding number of the transition function (1-06 §07); the n = f seen above appears here too. $c_1$ is called the first Chern number. Changing the connection continuously cannot make an integer jump, so it is fixed by the bundle alone Known.

In four dimensions

SU(2) bundles over $S^4$ are classified by an integer $c_2 = \frac{1}{8\pi^2}\int\operatorname{tr}(F\wedge F)$ (the sign depends on the book) Known. On $\mathbb R^4$ too, solutions of the Yang–Mills equations with finite action approach a pure gauge $-(dg)g^{-1}$ ($g : S^3\to SU(2)$) far away, and the winding number of g is the instanton number Known. This is not an unconditional fact. To be exact, it rests on the theorem that such solutions extend to connections on a bundle over $S^4$, obtained by adding one point at infinity (Uhlenbeck's removable singularity theorem); for general fields that are not solutions of the equations, the same is said by assuming suitable decay far away. For details see 3-05, 1-12 and the article on four dimensions.

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