Each worksheet prints cleanly on A4 or Letter. Answer keys are on a separate page.

The Lumenary · Research pathways

Gauge theory and the geometry of physics

Name: ______________________

Date: ______________________

You will learn to

Describe principal bundles, connections, gauge transformations and curvature, and prove that curvature transforms by conjugation and satisfies the Bianchi identity
Write Maxwell's equations as $dF = 0$ , $d{\star}F = J$ and derive the Yang–Mills equations $d_A{\star}F_A = 0$ as Euler–Lagrange equations
Define instantons, prove the energy bound $\mathrm{YM}(A) \ge 8\pi^2|k|$ with equality exactly for (anti-)self-dual connections, and explain the topological charge as a characteristic number
Explain the Aharonov–Bohm effect as holonomy of a flat connection, classifying flat $U(1)$ connections on the punctured plane
State Donaldson's theorem and the Yang–Mills existence and mass gap problem, separating theorems from open problems

1.
Under the $U(1)$ gauge transformation $a \mapsto a + d\lambda$ of the electromagnetic potential, which of these quantities are unchanged? (Choose all that apply.)
- A.The field strength $F = da$
- B. $\oint_\gamma a$ for a closed loop $\gamma$
- C.The value $a_x$ of the potential at a point $x$
- D. $\int_\gamma a$ along an open path from $x$ to $y$
2.
A $U(1)$ -bundle over $S^2$ is built from trivial bundles over the two hemispheres glued along the equator by the transition function $e^{3i\phi}$ , so that the potentials satisfy $a_N - a_S = 3\,d\phi$ . What is the first Chern number $\frac{1}{2\pi}\int_{S^2} F$ ?
3.
Writing the electromagnetic field as a $2$ -form $F$ on space-time, which Maxwell equations are equivalent to the Bianchi identity $dF = 0$ ?
- A. $\operatorname{div} B = 0$ and $\partial_t B + \operatorname{curl} E = 0$
- B. $\operatorname{div} E = \rho$ and $\operatorname{curl} B - \partial_t E = j$
- C.The continuity equation $\partial_t \rho + \operatorname{div} j = 0$
- D.All four Maxwell equations
4.
In the proof of the instanton bound $\mathrm{YM}(A) \ge 8\pi^2|k|$ , we use $\operatorname{tr}(F \wedge F) = \operatorname{tr}(F^+ \wedge F^+) + \operatorname{tr}(F^- \wedge F^-)$ . Which fact makes the cross terms vanish?
- A. $\Lambda^+$ and $\Lambda^-$ are orthogonal, and $\alpha \wedge \beta = \langle \alpha, \star\beta\rangle\,d\mathrm{vol}$ , so $\Lambda^+ \wedge \Lambda^- = 0$
- B.The Bianchi identity $d_A F = 0$
- C.Chern–Weil theory
- D.The Yang–Mills equations $d_A{\star}F = 0$
5.
An $SU(2)$ -bundle over the round $S^4$ has $c_2 = -2$ . What is the smallest possible Yang–Mills energy $\mathrm{YM}(A)$ of a connection on it, in units of $8\pi^2$ ?
6.
Electrons of charge $q$ pass on both sides of a thin shielded solenoid of flux $\Phi$ . For which values of $\Phi$ is the interference pattern identical to the pattern with $\Phi = 0$ ?
- A.Exactly when $q\Phi/\hbar \in 2\pi\Z$
- B.Only when $\Phi = 0$
- C.For every $\Phi$ , since the field outside the solenoid is zero
- D.Exactly when $q\Phi/\hbar \in \pi\Z$
7.
In the Lean file, gauge g A changes each link variable by U(x → y) ↦ g x * U * (g y)⁻¹. Which statement is the theorem holonomy_gauge for a path p starting at x (with endpoint x p its end)?
- A.holonomy (gauge g A) x p = g x * holonomy A x p * (g (endpoint x p))⁻¹
- B.holonomy (gauge g A) x p = g x * holonomy A x p * (g x)⁻¹ for every path
- C.holonomy (gauge g A) x p = (g (endpoint x p))⁻¹ * holonomy A x p * g x
- D.holonomy (gauge g A) x p = holonomy A x p
8.
By Donaldson's theorem, which of these unimodular forms cannot be the intersection form of a smooth closed simply connected oriented $4$ -manifold? (By Freedman, all of them are intersection forms of topological ones.)
- A. $-E_8 \oplus -E_8$
- B. $\langle -1 \rangle \oplus \langle -1 \rangle$
- C.The hyperbolic form $H = \left(\begin{smallmatrix} 0 & 1 \\ 1 & 0 \end{smallmatrix}\right)$
- D. $\langle 1 \rangle$
9.
Which of these are established theorems? (Choose all that apply.)
- A.Donaldson: a definite intersection form of a smooth closed simply connected 4-manifold is diagonalisable.
- B. $\R^4$ admits smooth structures not diffeomorphic to the standard one.
- C.Furuta: a smooth closed spin 4-manifold with indefinite form has $b_2 \ge \frac{10}{8}|\sigma| + 2$ .
- D.A smooth closed spin 4-manifold has $b_2 \ge \frac{11}{8}|\sigma|$ .
- E.Quantum Yang–Mills theory on ℝ⁴ exists and has a mass gap.
- F.Every smooth homotopy 4-sphere is diffeomorphic to S⁴.
10.
A $U(1)$ lattice gauge field on an $n \times n$ torus is given by link angles ax[x][y] (edge $(x,y) \to (x+1,y)$ ) and ay[x][y] (edge $(x,y) \to (x,y+1)$ ), indices mod $n$ . Write reduce(t) returning the representative of $t$ modulo $2\pi$ in $(-\pi, \pi]$ , plaquette(ax, ay, x, y) returning the reduced angle $a_x(x,y) + a_y(x+1,y) - a_x(x,y+1) - a_y(x,y)$ , and charge(ax, ay) returning $\frac{1}{2\pi}\sum$ of all plaquettes. The tests check that the charge is a (near-)integer and gauge invariant.
Computational — Practice

Answer key — Gauge theory and the geometry of physics

1.
$F = da$ is unchanged because $d^2 = 0$ , and $\oint_\gamma a$ is unchanged because $\oint_\gamma d\lambda = 0$ for a closed loop. The potential at a point changes by $d\lambda$ , and the integral along an open path by $\lambda(y) - \lambda(x)$ — exactly the endpoint factors $g(x)$ , $g(y)^{-1}$ of the Lean theorem holonomy_gauge.
2.
By Stokes on each hemisphere, $\int_{S^2} F = \oint_{\text{eq}} a_N - \oint_{\text{eq}} a_S = \oint 3\,d\phi = 6\pi$ , so $\frac{1}{2\pi}\int F = 3$ . It is the degree of the transition function $S^1 \to U(1)$ and does not depend on the connection: a Dirac monopole of charge $3$ .
3.
$dF = 0$ is the homogeneous pair: $\operatorname{div} B = 0$ (the spatial component) and Faraday's law $\partial_t B + \operatorname{curl} E = 0$ (the mixed components). They hold automatically for $F = da$ . The inhomogeneous pair $d{\star}F = J$ (Gauss and Ampère–Maxwell) involves the Hodge star, hence the metric, and the sources; applying $d$ gives charge conservation.
4.
Pointwise, $\alpha \wedge \beta = \langle \alpha, \star\beta \rangle\, d\mathrm{vol}$ for $2$ -forms in dimension $4$ . If $\alpha \in \Lambda^+$ and $\beta \in \Lambda^-$ then $\star\beta = -\beta$ and $\langle \alpha, \beta \rangle = 0$ because the $\pm1$ -eigenspaces of the self-adjoint operator $\star$ are orthogonal. So $F^+ \wedge F^- = 0$ . The Lean theorem abs_wedge_le is the resulting pointwise inequality.
5.
By Theorem 4, $\mathrm{YM}(A) \ge 8\pi^2|{-2}| = 16\pi^2$ , i.e. $2$ in units of $8\pi^2$ . Equality holds for self-dual connections (here $k \lt 0$ ), which exist on $S^4$ by the ADHM construction; so the minimum is $2$ .
6.
The two paths together form a loop winding once around the solenoid, so the relative phase is the holonomy $\exp(iq\Phi/\hbar)$ . The pattern is unchanged exactly when this is $1$ , i.e. $q\Phi/\hbar \in 2\pi\Z$ , $\Phi \in (h/q)\Z$ . By Theorem 5 these are exactly the fluxes for which the flat connection outside is gauge equivalent to the trivial one.
7.
For a path x → y → z, the transformed transport is g(x) U₁ g(y)⁻¹ · g(y) U₂ g(z)⁻¹ = g(x) U₁U₂ g(z)⁻¹. By induction, for any path the result is g x * holonomy A x p * (g (endpoint x p))⁻¹. For closed paths this is conjugation, which the trace (Wilson loop) does not see.
8.
$-E_8 \oplus -E_8$ is negative definite and even, hence not isomorphic to $\langle -1 \rangle^{16}$ ; by Donaldson's theorem no smooth closed simply connected $4$ -manifold has it, although Freedman's topological manifold does. (For $-E_8$ alone, Rokhlin's theorem already forbids a smooth structure, since a smooth spin $4$ -manifold has signature divisible by $16$ .) The other forms belong to $\overline{\mathbb{CP}^2}\#\overline{\mathbb{CP}^2}$ , $S^2 \times S^2$ and $\mathbb{CP}^2$ .
9.
Theorems: Donaldson's theorem (1983), the existence of exotic $\R^4$ (Freedman–Donaldson, with uncountably many by Taubes), and Furuta's 10/8 theorem (2001). Open: the 11/8 conjecture, Yang–Mills existence and mass gap, and the smooth 4-dimensional Poincaré conjecture.

10.

import math


def reduce(t):
    r = math.fmod(t + math.pi, 2 * math.pi)
    if r <= 0:
        r += 2 * math.pi
    return r - math.pi


def plaquette(ax, ay, x, y):
    n = len(ax)
    return reduce(ax[x][y] + ay[(x + 1) % n][y]
                  - ax[x][(y + 1) % n] - ay[x][y])


def charge(ax, ay):
    n = len(ax)
    total = sum(plaquette(ax, ay, x, y) for x in range(n) for y in range(n))
    return total / (2 * math.pi)

Each link appears in exactly two plaquettes, once with each sign, so the unreduced angles sum to $0$ ; reducing each changes it by a multiple of $2\pi$ , so the reduced sum is $2\pi Q$ with $Q$ an integer. A gauge transformation adds $\lambda(x) - \lambda(y)$ to each link, which cancels around every plaquette.