2026-10-01 · chapter Yang–Mills and the mass gapPart 1 Mathematical preparationManifolds and tangent spaces

Manifolds and tangent spaces — gluing charts together, and differentiating without depending on coordinates

Spaces that are not flat appear in Yang–Mills theory. The gauge group SU(2) is, as a space, the sphere $S^3$ (1-02 §05). The stage of the problem statement is $\mathbb{R}^4$, but the official description also touches on approximation by the four-dimensional torus $T^4$ and on general four-dimensional manifolds Known.

Such spaces sometimes cannot be covered by a single set of coordinates. As a metaphor: however a world map is drawn, it tears at a pole or at the date line. To be exact, there is no one-to-one correspondence, continuous in both directions, between the sphere $S^2$ and an open set of $\mathbb{R}^2$ Known. The sphere is closed and bounded, and so is its continuous image, but a non-empty open set of $\mathbb{R}^2$ is not.

So we glue several charts together smoothly, and call geometric only those quantities that do not depend on the choice of chart. It is the same stance as saying that only quantities independent of the gauge are observed (2-03 §03).

A pair $(U,\varphi)$ of a subset $U$ of a set $M$ and a $\varphi$ that maps $U$ one-to-one onto an open set of $\mathbb{R}^n$ is called a chart (coordinate neighbourhood), and $\varphi(q)=(x^1,\dots,x^n)$ are the coordinates of $q$. A collection of charts covering $M$ that satisfies the following two conditions is called an atlas Known. For any two charts $(U_\alpha,\varphi_\alpha)$, $(U_\beta,\varphi_\beta)$, the images of the overlap $\varphi_\alpha(U_\alpha\cap U_\beta)$ and $\varphi_\beta(U_\alpha\cap U_\beta)$ are open sets of $\mathbb{R}^n$ (because derivatives cannot be defined except on open sets). And the change of reading on the overlap (the transition map) is $C^\infty$.

Example: the circle $S^1$. The angle $\theta$ does not make a single chart (on $0\le\theta\lt 2\pi$ the image is not open, and on $0\lt\theta\lt 2\pi$ one point is missing). So we use stereographic projection. Let $u_N$ be the position where the line joining the north pole $N=(0,1)$ and the point $p=(\cos\theta,\sin\theta)$ meets the horizontal axis, and $u_S$ the position obtained in the same way from the south pole $S=(0,-1)$.

$u_N$ covers everything except $N$, $u_S$ covers everything except $S$, and $u_S=1/u_N$ on the overlap is smooth, so (1) is satisfied.

The sphere $S^2$ too can be covered by the projections $u=(x_1,x_2)/(1-x_3)$, $u'=(x_1,x_2)/(1+x_3)$ from the two poles onto the equatorial plane, and the gluing is $u'=u/|u|^2$ Known (difference at most 2×10⁻¹³ at about 2000 random points Computed).

Definition A (smooth) manifold of dimension $n$ is a pair of a set $M$ and an atlas covering $M$. Every chart that can be added without breaking (1) is added (a maximal atlas) Known. Textbooks also impose topological conditions (the Hausdorff property and so on), but in the examples of this book they are satisfied automatically.

Examples: open sets of $\mathbb{R}^n$ (such as $\mathrm{GL}(n,\mathbb{R})\subset\mathbb{R}^{n^2}$) Known, the sphere $S^n$, products (the torus $T^2=S^1\times S^1$, $T^4$), groups of matrices (§04). Non-examples: the crossing point of a figure 8. Removing the crossing point from a small neighbourhood of it leaves four pieces, so it cannot have the same shape as an interval, which splits into two when one point is removed. The vertex of the double cone $x^2+y^2=z^2$ is the same.

This definition does not use an outside space. Things that differ only as atlases (for example, the atlas on $\mathbb{R}$ made from the chart $x\mapsto x^3$) are carried to the usual one by a diffeomorphism (§05), so we regard them as the same structure. Even counted this way, genuinely different smooth structures can sit on the same space. A smooth structure that is homeomorphic to $\mathbb{R}^4$ (there is a one-to-one correspondence continuous in both directions) but not diffeomorphic to it is called an exotic $\mathbb{R}^4$, and Taubes is credited with first showing that there are uncountably many of them (of the cardinality of the continuum, that is, as many as there are real numbers) Known (1-12 §07, 3-06, the article on four dimensions).

Many manifolds appear as solution sets of equations. For a smooth $F:\mathbb{R}^N\to\mathbb{R}^m$ and a value $c$, the following holds (the regular value theorem) Known.

What lies inside is the implicit function theorem: solve $m$ of the coordinates as functions of the remaining $N-m$, and use those $N-m$ as a chart. For example, $F(A)=A^{\mathsf T}A$ maps $\mathbb{R}^9$ to the symmetric matrices (6 entries), and $F^{-1}(I)=\mathrm{O}(3)$ has dimension $9-6=3$. $\det$ is continuous and takes only $\pm1$, so $\mathrm{O}(3)$ splits into the part with $\det A=1$ (the rotation group $\mathrm{SO}(3)$) and the part with $\det A=-1$, both of dimension 3. This agrees with a rotation being fixed by 2 numbers for the direction of the axis and 1 for the angle (1-02 §04).

Table 1 ComputedThe rank of $dF$ was counted at 200 random points, and for the groups the kernel at the identity (the tangent space of §06) was also found. The rank is the number of square roots of eigenvalues of $dF\,dF^{\mathsf T}$ (singular values) that cannot be regarded as 0 at a relative threshold of 10⁻⁶. The points of the orthogonal group were taken from the part with $\det=1$.

SL(2, ℝ), SL(3, ℝ) (kernel: trace 0), O(4) (the part with $\det=1$ is SO(4)) and U(3) also gave dimensions 3, 8, 6, 9. The cone drops rank at the vertex. A closed group of matrices is always a manifold (a Lie group) Known, and this backs up the discussion of 1-03 §02.

A function $f:M\to\mathbb{R}$ is called smooth when $f\circ\varphi^{-1}$ is $C^\infty$ in every chart. On an overlap (4) holds, so if it is smooth in one chart it is smooth in the other (the reason for requiring smooth gluing). Maps $F:M\to N$ are the same, and a bijection whose inverse is also smooth is called a diffeomorphism.

Example: height on the circle. Assign to the point $p=(\cos\theta,\sin\theta)$ of $S^1$ the height $f(p)=\sin\theta$. From (2), $u_N^2=\frac{1+\sin\theta}{1-\sin\theta}$ and $u_S^2=\frac{1-\sin\theta}{1+\sin\theta}$; solving these for $\sin\theta$, the $f$ read in the two charts of §02 is

The denominators are both at least 1, so both formulas are $C^\infty$, and $f$ is smooth. Substituting $u_S=1/u_N$ on the overlap carries one formula to the other, a concrete instance of (4) Computed.

The values of partial derivatives differ from chart to chart. At the point $\theta=0$ ($u_N=u_S=1$) we have $df/du_N=1$, $df/du_S=-1$: the scales of the two charts run in opposite directions. Even so, the rate of change $\frac{d}{dt}f(\gamma(t))|_{t=0}$ along a curve $\gamma$ is written without coordinates. If $\gamma(t)=(\cos t,\sin t)$, the coordinate speeds $v_N=du_N/dt$, $v_S=du_S/dt$ at $t=0$ are $1$ and $-1$ (the quantities of the table of Figure 1). By the chain rule, in chart N it is $1\times1$ and in chart S $(-1)\times(-1)$, both $1=\cos 0$ Computed. This is the seed of differentiation independent of coordinates.

In a curved space an arrow stretching from point to point sticks out of the space, so we redefine an arrow as a direction attached to one point. Three views give the same thing Known. (a) Velocity of a curve: regard curves whose velocities read in a chart are equal as the same (independent of the chart by (1)). (b) Directional derivative: $v(f)=\frac{d}{dt}f(\gamma(t))|_{t=0}$. Linear, and satisfies the product rule. (c) Components: using the chain rule in a chart $x$,

and one can write $v=\sum_iv^i\,\partial_i$. The whole set of these directions is the tangent space $T_pM$ at $p$ (a vector space of dimension $n$), and $\partial_1,\dots,\partial_n$ is called the coordinate basis.

When cut out by equations. For a curve inside $M=F^{-1}(c)$ we have $F(\gamma(t))=c$, so $dF_p(\gamma'(0))=0$, and counting dimensions gives $T_pM=\ker dF_p$ Known. At the identity of SO(3), differentiating $A^{\mathsf T}A=I$ gives $X^{\mathsf T}+X=0$: the tangent space is the set of all antisymmetric matrices, and agrees with the Lie algebra of 1-03 §02 Known (the right-hand column of Table 1) Computed.

The tangent vector $v$ is fixed before any chart, and the components $v^i$ are its shadow cast onto a chart. The components in another chart $x'$ are obtained by the chain rule Known.

The components change by the Jacobian matrix, the basis by its inverse, and $\sum_iv^i\partial_i$ is invariant. Example: in $(x,y)=(1,1)$ ($r=\sqrt2$, $\theta=\pi/4$), $\partial_x=\tfrac{1}{\sqrt2}\,\partial_r-\tfrac12\,\partial_\theta$. For the circle of Figure 1, $v_N=1/(1-\sin\theta)$ and $v_S=-1/(1+\sin\theta)$, and (6) is $v_S=-v_N/u_N^2$. The agreement of the two right-hand columns of the table of Figure 1 is the check of this Computed ($v_N$ is large near the north pole because the scale of chart N stretches).

The differential of a map. A smooth $F:M\to N$ carries tangent vectors by carrying curves.

$dF_p:T_pM\to T_{F(p)}N$ is linear, and read in charts it is the Jacobian matrix. (6) is the case of the identity map read in two charts (a linear example is the Lorentz transformation, 2-04). Treating $df$ as "something that eats a vector and returns a number" is done in 1-06.

A smooth assignment $X=\sum_iX^i(x)\,\partial_i$ of a tangent vector to each point is called a vector field. A curve following the arrows (an integral curve) is a solution of an ordinary differential equation, and is locally unique Known. The point reached by advancing $p$ by time $t$ is written $\varphi_t(p)$ and called the flow.

The flow of a linear field $X(x)=Ax$ is $\varphi_t(x)=e^{tA}x$, that is, the matrix exponential of 1-01. Examples in the plane: the rotation $R=-y\,\partial_x+x\,\partial_y$ is rotation by the angle $t$, the dilation $E=x\,\partial_x+y\,\partial_y$ is scaling by $e^t$, and $x\,\partial_y$ is $(x,y)\mapsto(x,y+tx)$.

Detour: the hairy ball theorem

Applying two fields to a function one after the other produces second derivatives, but swapping the order and subtracting makes the second-order terms vanish (partial derivatives can be taken in either order), and a new vector field is left. This is the bracket Known.

For coordinate basis fields, $[\partial_i,\partial_j]=0$. Conversely, if any two of $n$ fields forming a basis commute, then locally there is a chart whose coordinate basis they are (a theorem on commuting frames, a form of the Frobenius theorem) Known.

Geometric meaning. Going $t$ by the flow of $X$, $t$ by $Y$, $-t$ by $X$ and $-t$ by $Y$, the path does not close, and the displacement appears with size $t^2$ in the direction of $[X,Y]$ Known. For example, if $X=\partial_x$ and $Y=x\,\partial_y$, then $(x,y)\to(x+t,y)\to(x+t,y+tx+t^2)\to(x,y+tx+t^2)\to(x,y+t^2)$, and the displacement is exactly $t^2\partial_y=t^2[X,Y]$.

A closes, being a coordinate basis. C also commutes: $R=\partial_\theta$ and $E=\partial/\partial(\log r)$ are the coordinate basis of the chart $(\log r,\theta)$. D is the same group SE(2) as the parallel parking of 1-03 §04, seen as motions of points, and the displacement is exactly $(t(1-\cos t),\,t\sin t)$. E is an example of a linear field $X_A(x)=Ax$, and from (9) $[X_A,X_B]=X_{BA-AB}$: the sign is opposite to that of the matrix commutator.

"The amount by which a small quadrilateral fails to close" appears three times: in the multiplication of a group (1-03 §04), in flows (this section), and in curvature (1-07: the mismatch of parallel transport around a small loop) Known.

• Spacetime: $\mathbb{R}^4$ can be covered by a single chart. The official description writes the gauge field of electromagnetism (a connection for the group $\mathrm{U}(1)$) as "locally a 1-form on spacetime", and treats the non-abelian case by replacing this with the group $G$ (Jaffe–Witten §1) Known. The meaning of 1-forms and connections is given in 1-06 and 1-07.
• The four-dimensional sphere: on Euclidean $\mathbb{R}^4$ (with the time coordinate entering distances with the same sign as space; 2-04, 4-02), solutions of the Yang–Mills equations (3-03) with finite action $\int|F|^2$ ($F$ is the field strength, 3-02) are said to extend smoothly, after a change of gauge, to $S^4$, obtained by adding the point at infinity (Uhlenbeck 1982, 3-05) Known.
• Gauge groups and lattices: the tangent space at the identity is the Lie algebra (1-04). The space of lattice configurations is a finite product of copies of the group, and is a compact manifold (here meaning bounded and closed in the space of lined-up matrix entries; 5-01).
• The space of gauge orbits: it is infinite-dimensional, and it has been argued that a continuous gauge fixing cannot be chosen globally (Singer 1978, 3-04) Known.

Choosing a chart and choosing a gauge, transition maps and transition functions, charts of the sphere and the obstruction to gauge fixing are much alike. But this is a metaphor. To be exact, the gauge side is about a structure that places a group at each point (a principal bundle, defined in 1-07) and connections on it, and this chapter is its groundwork.

What this chapter can say

Sources and reproduction

Next chapter: 1-06 Differential forms and the exterior derivative — on to things that eat tangent vectors and return numbers (1-forms), and Stokes' theorem.

← 1-04 SU(2) and SU(3)Contents1-06 Differential forms and the exterior derivative →