2026-10-01 · chapter Yang–Mills and the mass gapPart 3 Yang–Mills theory (classical)the space of connections and the orbit space

Gauge theory as geometry — divide a straight space by orbits, and representatives can no longer be chosen continuously

In mechanics, the set of all positions of a particle was seen as one space (the configuration space) (2-01). For fields too, we regard one connection $A$ as one point and write the set of all of them as $\mathcal A$ (infinite-dimensional). Configurations related by gauge transformations represent the same physics (3-01), so the true configuration space is the collection of bundles of related configurations. We call one such bundle an orbit, and the set of all of them the orbit space $\mathcal A/\mathcal G$. The gauge transformation is (4) of 1-07 and its lattice version ((7) of 1-07).

The $U_\ell$ in (1) is the parallel transport carrying from $x$ to $y$ along the edge $\ell$ ((7) of 1-07). The $U_{x,\mu}$ of 5-01 is read in the direction carrying from $x+a\hat\mu$ to $x$, so in the notation of this chapter it corresponds to $U_\ell$ ($\ell:x+a\hat\mu\to x$) ($U_\ell=U_{x,\mu}$; on a reversed edge, the inverse).

Example (lattice). On a lattice with group elements on the edges (5-01), a configuration is a tuple $G^{E}$ of $E$ elements, one per edge, and a gauge transformation is a tuple $G^{V}$ of $V$ elements, one per vertex; both are finite-dimensional. There are two questions. What shape is the orbit space? Can one representative be chosen from each orbit continuously (§07)?

Conventions. As in 1-07, $G=SU(N)$, $\mathfrak g$ is the anti-Hermitian traceless matrices, and a connection is a $\mathfrak g$-valued 1-form $A=\sum_\mu A_\mu\,dx^\mu$ (the $\mathcal A=-igA$ of the table in 3-02 is written $A$ in this chapter, and $\mathcal A$ is used for the set of all connections). The base space $M$ is $\mathbb R^4$, $S^4$, $S^3$ or the circle. The difference of two connections is a $\mathfrak g$-valued 1-form, and adding a 1-form to a connection gives a connection again (1-07 §03). So $\mathcal A$ is a vector space with no fixed origin (an affine space).

The right-hand formula follows because the $-(dg)g^{-1}$ in (1) appears only once, with weight $(1-t)+t=1$. Moving $t$ from 1 to 0 shrinks the whole of $\mathcal A$ continuously to one point $A_0$. We call this contractible, and it is the starting point of §07. On the tangent directions $a,b$ we put an inner product.

For anti-Hermitian matrices $-\operatorname{tr}(a_\mu a_\mu)\ge0$, and the gauge transformation $a\mapsto gag^{-1}$ preserves this inner product. Example. The lattice $U(1)^E$ is a torus and has loops that cannot be shrunk, so it is not contractible (that the configuration space of a lattice is not contractible matters in §08).

The set of all gauge transformations $\mathcal G$ (smooth $g:M\to G$, with pointwise product) is an infinite-dimensional group. The tangent to an orbit comes from putting $g=e^{t\xi}$ ($\xi:M\to\mathfrak g$) into (1) and differentiating.

$D_A$ is the covariant derivative in the adjoint representation (3-02 §03). Integrating (3) by parts (the inner product of two $\xi$ is $-\int\operatorname{tr}(\xi\eta)\,d^dx$, and boundary terms are dropped) determines the adjoint $D_A^*$.

That a direction $a$ is orthogonal to the orbit is the same as $D_A^*a=0$, and at $A=0$ it is the Landau condition $\partial_\mu a_\mu=0$ Known. The elements that fix. Multiplying $g\cdot A=A$ by $g$ on the right gives $gA-dg=Ag$, that is,

which is the equation of parallel transport (1-07 §04). $g$ is determined by its value at one point $m$, and $g(m)$ must commute with the holonomy of $A$ (Singer 1978 §2) Known. So the based gauge group $\mathcal G_m=\{g:g(m)=1\}$ moves every connection (acts freely), and the only elements that fix an irreducible connection, one whose holonomies have no common invariant subspace, are the centre $\mathbb Z_N$ (1-04). A reducible connection such as $A=0$ is fixed by a larger group (all constant $G$); its orbit becomes thinner, and it is a corner of the orbit space.

On a lattice of $L$ edges forming a ring, the only gauge-invariant quantity is the conjugacy class of the holonomy round the ring, $U=U_{L-1}\cdots U_0$ (1-07 §06). For $U(1)$ it is the angle $\Phi$ ($U=e^{i\Phi}$), and the orbit space is a circle. For $SU(2)$ it is fixed by the eigenvalues $e^{\pm i\theta}$ ($0\le\theta\le\pi$), and the orbit space is the segment $[0,\pi]$; the ends $U=\pm1$ are corners where the elements that fix jump up to the whole of $SU(2)$. Seen as $SU(2)\cong S^3$ (unit quaternions, 1-12), the conjugacy class with angle $\theta$ is a sphere of radius $\sin\theta$, which collapses to a point at the ends. The measure of weight 1 on a group that does not change under multiplication is called Haar measure. For $SU(2)$ it is the uniform distribution on $S^3$ (1-08 §03). Pushed down to the orbit space, it gets the weight of the area of the orbit (numerator) divided by the volume of $S^3$ (denominator) (the Weyl integration formula) Known.

ComputedTable 1 The distribution of the angle of the holonomy $\theta$ for a ring whose 4 edges are each chosen at random with the Haar measure of $SU(2)$ (2 million samples; sampling error at most 0.0004).

It agrees with (7), and the same holds for a single edge (a product of Haar-distributed elements is again Haar-distributed). Orbits near the corners are small, and the probability of landing there is small too. This weight plays the same role as the Faddeev–Popov determinant multiplied in when the gauge is fixed (the determinant of $M$ in (9) of §05). More precisely, when conjugation in a finite-dimensional group is fixed to diagonal form, the Faddeev–Popov determinant becomes a constant times $\sin^2\theta$ (the Weyl factor). The determinant in infinite-dimensional gauge fixing is a formal analogue of it (4-08 §02).

Detour: the orbit space and the gap

Fixing the gauge means choosing a map $s:\mathcal A/\mathcal G\to\mathcal A$ that picks one representative from each orbit (a slice). A rule often used is to choose the point on the orbit closest to the origin. From (4)(5),

and this is 0 for every $\xi$ when $\partial_\mu A_\mu=0$, the Landau gauge. Singer calls the surface $\mathcal S_A=\{A+\tau:D_A^*\tau=0\}$ through $A$ orthogonal to the orbits the generalised Coulomb gauge, and states that near $A$ the surface meets each orbit exactly once (a local slice) Known. Far away the surface becomes tangent to orbits. The tangent $D_{A+\tau}\xi$ to the orbit at $A+\tau$ lies in the surface when the following operator has a non-zero solution.

There the Faddeev–Popov determinant (the determinant of $M(\tau)$) is 0. For $A=0$ it is $M(\tau)=-\partial_\mu D_{\tau,\mu}$. The range where $M$ is positive is called the Gribov region, its boundary the horizon, and each point where an orbit meets the surface a Gribov copy. Singer's Theorem 8 shows that in every direction of the surface, far enough out, there is a point where it is tangent to an orbit. Gribov (1978) found, for $SU(2)$ in $\mathbb R^4$, that the Coulomb gauge meets the orbit of the origin once more far away (as presented by Singer) Known. How to count them in quantum theory is in 4-08 §05 and §06.

Example (Figure 1). A ring of $L$ edges of $U(1)$. With $g_0=1$ at the base point and the rest $g_x=e^{i\alpha_x}$, the angle of an edge transformed by (1) is $\psi_\ell=\theta_\ell+\alpha_{\ell+1}-\alpha_\ell$ (indices go round with $L$). We measure the distance from the origin with the same function as Mehta et al. (2014).

For small $\psi$, $E\approx\tfrac12\sum_\ell\psi_\ell^2$, the lattice version of (8). At stationary points all the $\sin\psi_\ell$ are equal; there are the uniform points $\psi_\ell=\varphi$ and points where only some edges have $\pi-\varphi$. The uniform points are the following $L$.

The matrix of second derivatives there (the lattice Faddeev–Popov matrix) is $\cos\varphi_k$ times the Laplacian of the ring, so for $\cos\varphi_k>0$ it is a minimum (a copy inside the Gribov region), and for negative values a maximum. Non-uniform points are not minima ($L=3$–$9$) Computed.

Only when $\Phi$ is between 90° and 270° are $\cos\varphi_0>0$ and $\cos\varphi_2>0$ both possible, and there are two copies inside the Gribov region. When $\Phi$ crosses 90° going up, the point $k=2$ and three saddle points merge into one, and the second derivative becomes 0. Beyond this, $k=2$ becomes a minimum. This is the horizon. At $\Phi=180°$ the $E$ of the two minima are equal, and the boundary of the region that collects only the closest points (the fundamental modular region) is here. That the signed sum of critical points is 0 (Neuberger's 0/0) is in 6-11 §05.

Taking $\Phi$ once round in Figure 1, the uniform representative of (11) moves continuously from $\varphi=\Phi/L$ to $(\Phi+2\pi)/L$ and arrives at the representative $k=1$. The orbit has returned to where it was, but the chosen representative is another copy. For $L=3$ in Figure 1, it crosses the horizon at $\Phi=270°$ on the way, and where it arrives is a maximum (a copy outside the Gribov region). Arriving at another copy while staying inside the region happens for $L\ge5$ (Table 3). The two copies are related by the gauge transformation that turns the point $x$ by the angle $\alpha_x=2\pi x/L$ (the transformation of winding number 1 on the continuous circle, transferred to the lattice) (on the last edge too, $\alpha_0-\alpha_{L-1}\equiv2\pi/L$).

Keep choosing representatives continuously, and after one round they disagree: this is the smallest form of the mechanism that prevents a continuous global gauge fixing. Whether the path hits the horizon ($\cos\varphi=0$, that is, $\Phi=\pi L/2$) on the way depends on $L$.

ComputedTable 3 The result of following the minimum $\Phi=0$ once round along $\Phi$, and the number of copies inside the Gribov region (minimum–maximum over 4000 values of $\Phi$).

For $L\ge5$ the representative stays inside the Gribov region during the round and arrives at the copy $k=1$; for $L\le4$ it hits the horizon on the way or at the end. In neither case does it return to the original representative Computed.

Example (the continuous circle). For a $U(1)$ connection $A=i\,a\,dx$ on a circle of length $\ell$, the Landau condition makes $a$ constant, and the gauge transformation $e^{2\pi inx/\ell}$ of winding number $n$ takes $a$ to $a-2\pi n/\ell$, so there are infinitely many copies. Only $a\ell\bmod2\pi$ is gauge-invariant; choosing a representative continuously means choosing a real number continuously from an angle (taking a continuous logarithm), which is impossible because after one round it is off by $2\pi$ Known.

Singer (1978) takes as base space $S^4$ ($\mathbb R^4$ with one point at infinity added; this corresponds to the condition that gauge transformations approach a constant value far away) or $S^3$ (space at one time), and for compact connected semisimple groups (groups whose Lie algebra is a direct sum of simple ones; in the paper, for simplicity, $SU(N)$, $N>1$) states in the abstract “It is shown that no continuous choice of exactly one connection on each orbit can be made.”, concluding “No gauge fixing is possible.” Known. Precisely, it says there is no continuous global gauge fixing; local slices and discontinuous choices are not forbidden.

Homotopy groups. $\pi_k(X)$ is the set of classes of based continuous maps from the $k$-dimensional sphere into $X$, grouped by continuous deformation into one another. $\pi_0$ is the connected components, $\pi_1$ the classes of loops, and $\pi_3(SU(2))$ is the degree of 1-12, with $\pi_k(X\times Y)=\pi_k(X)\times\pi_k(Y)$. The proof has four steps (Singer 1978 §2).

1. $\mathcal A$ is contractible (§02). So is $\mathcal A^*$, with the set of reducible connections $C$ removed: for every $\pi_j(\mathcal A^*)=0$ (Theorem 2). The removed set $C$ leaves, at every point, infinitely many directions orthogonal to it, so the image of a sphere can be pushed off $C$.
2. $\hat{\mathcal G}=\mathcal G/\mathbb Z_N$ (the centre $\mathbb Z_N$ fixes every connection, so it is divided out) acts freely on $\mathcal A^*$, and $\mathcal A^*\to\mathcal N^*=\mathcal A^*/\hat{\mathcal G}$ is a principal bundle with fibre $\hat{\mathcal G}$ (Theorem 1). A principal bundle is a bundle in which over each point of the base (here $\mathcal N^*$) sits a copy of the group $\hat{\mathcal G}$ (the fibre; here one orbit), and $\hat{\mathcal G}$ acts freely and transitively within each copy (every element other than 1 moves every point, and any point can be moved to any point). If there is a section, the whole is the product of the base and the group.
3. If there were a continuous gauge, it would be a section of the principal bundle, the bundle would be the product $\mathcal A^*\cong\mathcal N^*\times\hat{\mathcal G}$, and from $\pi_j(\mathcal A^*)=0$ and the product formula, $\pi_j(\hat{\mathcal G})=0$ would follow for every $j$.
4. But some $\pi_j$ of $\hat{\mathcal G}$ is not 0 (Theorem 3). Contradiction (Corollary 4).

The tool of the fourth step is (13) (Theorem 5). Singer notes that $\mathcal G_m$ acts freely on the whole of $\mathcal A$, and by (13) some homotopy group of it is not 0, so $\mathcal A\to\mathcal A/\mathcal G_m$ has no section either.

KnownTable 4 $\pi_j(\mathcal G_m)$ ($M=S^r$) read from (13). The non-abelian columns follow Singer's text (which cites Toda's tables).

Note: the $U(1)$ column is read from the homotopy groups $\pi_j(\mathcal G_m)=\pi_{j+r}(U(1))$ of the space of based maps (the $\pi_k$ of $U(1)$ is 0 for $k\ge2$); Singer's Theorem 5 is a statement for $SU(N)$.

For $S^3$, the contradiction comes just from the fact that gauge transformations of different winding numbers cannot be connected (the circle of §06 is the abelian version, by $\pi_0(\mathcal G_m)=\pi_1(U(1))=\mathbb Z$). The higher homotopy groups of $U(1)$ are 0, so for $S^3$ and $S^4$ the fourth step does not work, and Singer too writes that in the abelian case “An integral submanifold provides the gauge for quantum electrodynamics.” The obstruction comes from the topology made by non-commutativity and the condition at infinity Known.

The Faddeev–Popov procedure (4-08 §03) is correct on a local slice but cannot be used globally as it is. Singer named the route of covering the orbit space by regions where local slices work and adding them up with a partition of unity (assigning to each piece of the cover a weight such that the weights add up to 1 everywhere), and wrote “This partition of unity argument would be useful if the covering used could be made explicit.” Later work has avoided the obstruction by three routes.

• The lattice. Wilson's lattice integrates with Haar measure, so it is finite without fixing the gauge (4-02, 5-01) Known. The gauge that sets all edges of a tree joining all vertices (a maximal tree) to 1 is a global continuous slice for the based gauge group Known. Set $g=1$ at the root vertex, and moving along the tree to neighbouring vertices, fix $g$ in turn so that each edge becomes 1 (if $g_x$ is fixed on the edge $\ell:x\to y$, then $g_y=g_xU_\ell^{-1}$). $g$ is determined uniquely by products and inverses of the edge elements, and depends continuously on $U$. Since the based gauge group acts freely and transitively on the configurations of the tree edges, each orbit has exactly one representative, and it can be chosen continuously. $SU(N)^E$ is not contractible ($\pi_3(SU(N))=\mathbb Z$; for $SU(2)$, the degree of 1-12), so this does not contradict Singer's theorem. Instead the symmetry of the lattice is broken. Fixing a covariant gauge BRST-symmetrically (so that a formal symmetry remains even after fixing) produces 0/0 (6-11, 4-08 §08; for BRST, 4-08 §04).
• Local gauges. On Bałaban's renormalisation group on the three-dimensional periodic lattice, the problem description writes “One defines these gauges in phase cells: The choices vary locally in space-time, as well as on different length scales.” (Jaffe–Witten §6.5) Known. It is a method of changing slices by place and by scale (6-06 §05, 6-05 §05). Bałaban's four-dimensional work is said to use the same idea (6-07 §07; the specific choice in the four-dimensional papers has not been checked in the text).
• The orbit space itself. Stochastic quantisation does not fix the gauge and takes the orbit space as the state space. A review writes “The space O is a non-linear space if G is non-Abelian, which makes non-trivial even the construction of the state space” (Chevyrev 2022), and Chandra–Chevyrev–Hairer–Shen, on the two-dimensional torus, took as state space the space of orbits of connections as distributions (1-11) and constructed a stochastic dynamics on it (abstract) Known (6-09).

6-11 §05 treats this as one of the obstacles other than divergence (obstacle 4), and separates it into the part settled on the lattice and the 0/0 that remains when weak-coupling analysis needs gauge fixing.

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Next chapter to read: 3-05 Instantons and topology (the integer that separates $S^4$ bundles, and the orbits of least action). The quantum theory of gauge fixing is 4-08.

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