2026-10-01 · chapter Yang–Mills and the mass gapPart 3 Yang–Mills theory (classical)Donaldson and exotic R⁴

Detour: Donaldson and exotic R⁴ — the space of solutions of classical Yang–Mills tells smoothness in four dimensions apart

Being the same in the topological sense (homeomorphic) means there is a one-to-one correspondence that is continuous in both directions. For smooth manifolds (1-05) we ask for a correspondence that is smooth in both directions (a diffeomorphism).

When two are homeomorphic but not diffeomorphic, one is called an "exotic" structure on the other. Structures that differ only in how the atlas is written (the $x\mapsto x^3$ of 1-05) are related by a diffeomorphism, so they do not count. The first example is said to be the seven-dimensional sphere (Milnor 1956) Known.

Example. The smooth structure on $\mathbb R^n$ is unique for $n\ne4$. The introduction of Gompf (1983) writes “it is a standard fact that for n ≠ 4, exotic Rⁿ's cannot exist”, and continues that applying the high-dimensional theory of smoothing, $\mathbb R^4$ should have one structure (“there should only be one structure on R4, as it has no cohomology”) Known. In fact there are uncountably many (§07, the article on four dimensions). Why is the subject of this chapter.

The closed surfaces inside a closed (compact, without boundary) oriented four-dimensional manifold $X$ fall into classes that do not change under continuous motion, and classes can be added. This collection is $H_2(X;\mathbb Z)$, and for simply connected $X$ it is $\mathbb Z^r$ ($r=b_2$; the Betti number of 1-06 §08). Precisely, the elements of $H_2$ are homology classes, and here we use only the intuition (in four dimensions every class is said to be representable by an embedded surface). Two surfaces in general position meet at finitely many points (1-12 §07), and the sum of $\pm1$, fixed by orientations, does not change when the surfaces are moved.

$\det Q=\pm1$ (unimodular) is due to Poincaré duality Known. The numbers read from the form are the rank $r$, the signature $\sigma=b^+-b^-$ (the number of positive eigenvalues minus the number of negative ones), and the parity (even if every $Q(a,a)$ is even). If $b^-=0$, it is positive definite.

Example. In $\mathbb{CP}^2$ two lines $\mathbb{CP}^1$ meet at one point, so $(1)$; $\overline{\mathbb{CP}^2}$ with the orientation reversed gives $(-1)$; and $S^2\times S^2$ gives $H=\bigl(\begin{smallmatrix}0&1\\1&0\end{smallmatrix}\bigr)$. The connected sum (remove a ball and glue) becomes a direct sum, and $\#r\,\mathbb{CP}^2$ is the identity matrix $\langle1\rangle^r$ (the standard form). The form of the Kummer surface (K3) is $E_8\oplus E_8\oplus H^{\oplus3}$ ($E_8$ is the negative-definite sign; Gompf writes this $H^{\oplus3}$ as $H$ of rank 6) Known.

The smallest example of a positive-definite even form is $E_8$. The following Cartan matrix has determinant 1 and is even and positive definite Computed. Gompf states that the only even definite unimodular form of rank 8 is $E_8$ (“the unique symmetric form over Z which has rank 8, is even, and is negative definite”).

Seen topologically only, four dimensions resembles higher dimensions. Freedman (1982) showed that every symmetric form of determinant $\pm1$ can be realised as the form of a closed simply connected topological manifold, and that the homeomorphism class is fixed by the form and a $\mathbb Z/2$ quantity (the Kirby–Siebenmann obstruction; 0 if smooth) (as presented in Donaldson 1983 §I.1) Known. There is also a topological manifold $|E_8|$ whose form is $E_8$.

One restriction from smoothness was already known. As Donaldson presents it, “Rohlin's theorem asserts that any even form coming from a smooth simply connected 4-manifold has signature divisible by 16” Known.

Example. $E_8$ ($\sigma=8$) is excluded by (4), and $|E_8|$ has no smooth structure. $E_8\oplus E_8$ ($\sigma=16$) passes (4), and (4) says nothing about odd $E_8\oplus\langle1\rangle$. What settled these is the theorem of §04.

Detour: why smoothness does not matter in five or more dimensions

Theorem 1 of Donaldson (1983) is “Let X be a compact smooth simply connected oriented 4-manifold with the property that the associated form Q is positive definite. Then that form is equivalent, over the integers, to the standard diagonal form” Known. That is, a change of integer basis turns it into the identity matrix. Reversing the orientation, the same holds for negative definite forms, and the simply connected assumption is said to have been removed later (Donaldson 1987). The proof uses one property of numbers, a lemma on the number $n(Q)$ of $a$ in $Q(a,a)=1$, counting $\pm a$ as one pair (Lemma 2 of the paper) Known.

The proof is by induction on the rank. If $Q(a,a)=1$, every $b$ splits with integer coefficients as $b=m\,a+c$ ($m=Q(b,a)$, $c\perp a$), and $Q(b,b)=m^2+Q(c,c)$. Since the form is positive definite, $Q(b,b)=1$ is $b=\pm a$ or "$m=0$ and $Q(c,c)=1$", so $n(Q)=n(Q|_{a^\perp})+1$, and the rank also drops by 1. When the $a$ run out, the form is standard if the remaining rank is 0, and otherwise $n(Q)\lt r$.

Example. Table 1 gives $n(Q)$ counted by enumeration Computed. $D_n^+$ is the union of the set $D_n$ of all integer vectors with even sum of components and that set shifted by $(\tfrac12,\dots,\tfrac12)$ (an integer form of determinant 1 when $n$ is a multiple of 4). Even forms that pass (4) are excluded by the theorem if $n(Q)\lt r$.

Conventions. We use the same geometric notation as 3-05 (anti-Hermitian $A$, $c_2=\frac1{8\pi^2}\int\operatorname{tr}F\wedge F$). Donaldson (1983) also uses the same sign, with an SU(2) bundle of $c_2(P)=-1$ over a positive-definite $X$, and self-dual connections Known. Self-dual means $F^-=0$, where $F^-$ is the anti-self-dual part of $F$ (1-12 §03; the same condition as the $F=*F$ of 3-03). Many textbooks reverse the orientation and write in terms of negative definite forms and anti-self-duality ($c_2=k\gt0$) (the detour of 3-05, 1-12 §07). Changing the orientation, it is the same story. The collection of solutions with those related by gauge transformations identified is the moduli space, the part of the orbit space of 3-04 made of solutions.

$k$ is the $Q$ of 3-05. $b_1$ is the number of one-dimensional holes (the Betti number of 1-06 §08; 0 if simply connected). The dimension comes from the index of the linearised equation. The index is an estimate of the dimension of the solution space of a linear equation; for a finite system of linear equations it corresponds to "the number of unknowns − the number of equations" (precisely, the dimension of the kernel minus the dimension of the cokernel; here we use only the name). Donaldson (1983) §II writes this index as $8|c_2(P)|-\tfrac32(\chi(X)-\tau(X))$ Known. $\tau$ is the signature $\sigma$ and $\chi=b_0-b_1+b_2-b_3+b_4$ is the Euler number (a relative of the $V-E+F$ of 1-06); in closed four dimensions $b_0=b_4=1$, $b_3=b_1$, $b_2=b^++b^-$, so $\tfrac32(\chi-\tau)=3(1-b_1+b^-)$, that is, (6). In the anti-self-dual convention $b^-$ is replaced by $b^+$.

For $S^4$ ($b_1=b^-=0$), $8k-3$. Removing from the $8k$ parameters of 3-05 §04 the 3 constant gauge rotations (identified because they are related by gauge transformations) gives $8k-3$, which for $k=1$ is 5: centre 4 + size 1. For the $X$ of the theorem ($b_1=b^-=0$, $k=1$) it is also 5.

Example. For $k=1$ on $S^4$, Donaldson writes $\mathcal M(S^4)=SO(5,1)^+/SO(5)$ as the open five-dimensional ball $B^5$. $SO(5,1)^+$ is the group of conformal transformations (angle-preserving transformations) of $S^4$. Donaldson explains that a solution is fixed by a "centre" and a "size", and states for the closed ball $\bar B^5=\mathcal M(S^4)\cup S^4$ that “a point x of S4 represents the limit point of a sequence of connections whose curvature becomes concentrated in diminishing balls about x” Known. The limit of "shrinking to a point" in 3-05 §05 is the edge. Figure 1 reads the centre $a$ (on the $x_1$ axis) and the size $\lambda$ as a point $w=a+i\lambda$ of the upper half-plane, and is the slice mapped by the linear fractional transformation $z=(w-i)/(w+i)$ that maps the upper half-plane to the unit disc (coordinates for the figure).

The hyperbolic distance in Figure 1

The hyperbolic distance in the figure's readout is the shortest length joining the points $(a,\lambda)$ and $(0,1)$ when the length of curves in the upper half-plane $\lambda\gt0$ is measured as $\int\sqrt{da^2+d\lambda^2}/\lambda$. This metric $(da^2+d\lambda^2)/\lambda^2$ (a slice of the metric of upper half-space) is invariant under the group of conformal transformations $SO(5,1)^+$, but it is not the $L^2$ metric that enters naturally on the space of solutions (the one measuring the size of a change of connection $\delta A$ by $\int|\delta A|^2$). The hyperbolic distance of $a=0$ is $|\ln\lambda|$ ($\ln(1/\lambda)$ if $\lambda\le1$) Computed. In this metric the edge is infinitely far away. However small the size $\lambda$ is made, it remains a solution, and the family does not break off before the edge because in four dimensions the action does not depend on the scale (1-12 §04).

ComputedThe numbers in this figure were computed on this machine. The density $48\lambda^4/(|x-a|^2+\lambda^2)^4$ ((9) of 1-12) and $z$ are computed on the spot. Static values: shrinking as $\lambda=1,\ 0.1,\ 0.01$ at $a=0$ gives $1-|z|=1,\ 0.182,\ 0.0198$, hyperbolic distance $0,\ 2.30259,\ 4.60517$, fraction of the action inside radius 0.1 $0.0003,\ 0.5000,\ 0.9997$ (the closed formula and numerical integration agree).

The $\mathcal M$ of (6) consists of three parts (Donaldson 1983 §I.2, §II, §III) Known.

(i) Irreducible points. Connections whose holonomies (1-07) have no common invariant subspace; a small perturbation of the equation makes them an oriented smooth five-dimensional manifold.

(ii) Reducible points. The case where the bundle splits as $E=L\oplus L^{-1}$ and the connection preserves the splitting; from $c_2(E)=-c_1(L)^2$, $Q(\alpha,\alpha)=1$ ($\alpha=c_1(L)$). Since $b^-=0$, there is a self-dual connection for each $\pm\alpha$; since $b_1=0$, it is unique up to gauge equivalence, and their number is $n(Q)$. The group of gauge transformations that leaves the connection fixed (the group that fixes) is as large as U(1), so these are the "corners" of 3-04 §04. Precisely, at points where the group that fixes is large, the orbit space stops being a smooth manifold and the neighbourhood takes the shape of a cone. The neighbourhood is $\mathbb C^3/S^1$: dividing the unit sphere $S^5$ of $\mathbb C^3$ by the circle $S^1$ gives $\mathbb{CP}^2$, and adding the radial direction gives a cone over $\mathbb{CP}^2$.

(iii) The end. Solutions of small size look like BPST solutions concentrated at one point of $X$. Taubes (1982) constructed them under $b^-=0$. Donaldson §III uses the convergence results of Uhlenbeck (1982) to show that sequences that escape are only those that concentrate. Theorem 11 states that a neighbourhood of the end is diffeomorphic to $X\times(0,\lambda_0)$. The edge of Figure 1 is here a copy of $X$.

Cutting out the neighbourhoods of the cone vertices and gluing $X$ onto the end gives an oriented compact five-dimensional manifold $W$ whose boundary is $X$ and $n(Q)$ copies of $\pm\mathbb{CP}^2$ (the relation of sharing a boundary is called cobordism). The signature is “an invariant of oriented cobordism”. That it is a cobordism invariant means that the signature of the whole boundary of an oriented compact five-dimensional manifold (here $X$ and the sum of $n(Q)$ copies of $\pm\mathbb{CP}^2$, with orientations) is 0; from this follows $\sigma(X)=\sum\varepsilon_i\,\sigma(\mathbb{CP}^2)$ ($\varepsilon_i=\pm1$ is how the orientations enter), and the following can be said.

The reasons for the four relations are as follows. The first equality: since the form is positive definite, $b^-=0$, so $\sigma=b^+=r$. The second equality: cobordism invariance (the paragraph above). The first inequality: each $\varepsilon_i\le1$, so the sum is at most the number of terms $n(Q)$. The second inequality: (5). Both ends are $r$, so everything in between is equality, $n(Q)=r$, and the equality case of (5) gives the standard form. Example. For $E_8\oplus E_8$ there are no cones and the boundary is only $X$, so $\sigma=16$ cannot be 0, a contradiction. Figure 2 is the balance for each form. Most of the paper is the analysis of (i)–(iii).

ComputedThe numbers in this figure were computed on this machine. The rank and $n(Q)$ are the enumerated values of Table 1 ($\langle1\rangle^4$ is $n(Q)=4$). Without JavaScript, putting $r$ and $n(Q)$ from Table 1 into (7) gives the same verdict.

The introduction of Taubes (1987) writes “Donaldson's theorem ... has the surprising corollary that there exists an exotic smooth structure on R4. This corollary was deduced by M. Freedman” Known. The line has three steps (as presented by Taubes). (a) $|E_8\oplus E_8|$ cannot be smoothed (§04). (b) With one point removed it can be smoothed (Freedman), and the neighbourhood of the removed point is diffeomorphic, outside a compact set, to the infinity of some smooth $R$ homeomorphic to $\mathbb R^4$. (c) If $R$ were standard, one could cut along a large smooth $S^3$ and cap it with a ball, and $|E_8\oplus E_8|$ would become smooth. So the following can be said.

Gompf (1983) made a $R^4_{DF}$ of the same kind from K3 and $\#3\,S^2\times S^2$, and writes “there is a compact set in R4DF ... which cannot be enclosed by any smooth S3. This behavior is impossible in the standard R4.” Known.

Example. Gompf §2 uses $\mathbb{CP}^2\#9\overline{\mathbb{CP}^2}$ (form $\langle1\rangle\oplus9\langle-1\rangle$, basis $e_0,\dots,e_9$) instead of the Kummer surface to make another exotic $\mathbb R^4$. The key is $x=3e_0+e_1+\dots+e_8$. $x\cdot x=9-8=1$, and the orthogonal complement $E_8\oplus\langle-1\rangle$ (negative definite and not diagonalisable) contradicts the theorem, so cutting off by surgery (cutting out part of a manifold and gluing it back differently) the part of form $\langle1\rangle$ spanned by $x$ is possible topologically but cannot be done smoothly. Table 2 is the check Computed.

How many are there? Taubes showed, by gauge theory on manifolds with periodic ends, that the $R_r$ obtained as images of balls of radius $r$ under the homeomorphism of (b) are not diffeomorphic to one another for distinct $r$ larger than some $r_0$. Theorem 1.1 is “There exists an uncountable family of diffeomorphism classes of oriented 4-manifolds which are homeomorphic to R4.” Known. $r$ runs over the interval $r\gt r_0$, and distinct $r$ give distinct classes, so there are at least as many as there are real numbers (the classes of manifolds have at most the cardinality of the reals, so it is said to be exactly the cardinality of the continuum). On the other hand, whether $S^4$ has an exotic structure is said to be unknown.

Although the equation is the same as in 3-05, Donaldson uses the shape of the collection of solutions as a tool, and uses neither probability nor time evolution. What the problem statement (0-03) asks about is the probability measure weighted by the same action (4-02) and the spectrum of its Hamiltonian.

On the quantum measure, smooth classical solutions are expected not to be typical configurations (by analogy with the Cameron–Martin space of 1-09) Physics. The "four" that matters in the problem is the four at which the dimension of the coupling constant vanishes (0-04). Example. §2 of the problem statement says “Classical properties of gauge theory are within the reach of established mathematical methods”, and after describing the role in the study of four-dimensional manifolds, continues “one does not yet have a mathematically complete example of a quantum gauge theory in four-dimensional space-time” Known.

There are two bridges at the level of physics. The abstract of Witten (1994) writes, on Donaldson's invariants, “instead of defining four-manifold invariants by counting SU(2) instantons, one can define equivalent four-manifold invariants by counting solutions of a non-linear equation with an abelian gauge group” (background in 7-11) Physics. §5 of the problem statement says that the clustering that follows from the mass gap “may make it possible to apply mathematical results established on R4 to any 4-manifold”, and concludes that the gap “may also be important in mathematical applications of four-dimensional quantum gauge theories to geometry” Physics.

What this chapter can say

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Next chapter to read: 3-07 Yang–Mills inside the Standard Model. For the physics of Seiberg–Witten, 7-11; for the "four" that matters in the problem, 0-04.

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