2026-10-01 · chapter Yang–Mills and the mass gapPart 3 Yang–Mills theory (classical)instantons and topology

Instantons and topology — the integer in the lower bound on the action comes from the winding number at infinity

Topology deals with properties that do not change under continuous deformation. The winding number of a closed curve $z(t)$ that does not pass through the origin,

is an integer, so it cannot change when the curve is moved continuously; it changes only at the moment the curve crosses the origin. Example. $z=e^{it}+c$ ($c$ real) gives 1 for $|c|\lt1$ and 0 for $|c|\gt1$. Moving $c$ along the real axis, the curve passes through the origin at the moment it crosses $|c|=1$, and only there does the value jump. The first Chern number worked the same way (1-07 §08). This chapter derives, from the side of the field, where the integer $Q$ in (12) of 3-03 comes from.

Conventions. $A$ is an anti-Hermitian matrix-valued 1-form, $D=d+A$, $F=dA+A\wedge A$, and $g$ is absorbed into $A$ ($A=-ig\,A_{\text{phys}}$; 3-03 §01, the table of notation). In this notation the action, the second Chern number and $Q$ take the following form. $Q$ is the $Q$ of 3-03 and the $k$ of 1-12 (the sign is in the detour of §03).

In the physics notation (the $D_\mu=\partial_\mu-igA_\mu$ of 3-03, without absorbing $g$ into $A$), with $\tilde F_{\mu\nu}=\tfrac12\varepsilon_{\mu\nu\rho\sigma}F_{\rho\sigma}$ (the components of the $*F$ of 3-03),

which is exactly the $Q$ of (12) of 3-03.

For U(1), $F\wedge F=d(A\wedge dA)$, and the integral was fixed by the boundary alone (1-06). In the non-abelian case it holds too, once one correction is added Known.

Checking it needs only one rule. For a matrix-valued $p$-form $\alpha$ and $q$-form $\beta$, the trace of the product of the component matrices does not change when the order is swapped, and only the sign from exchanging the differential-form parts remains.

Let $A^k$ be the wedge product of $A$ taken $k$ times. Expand $d\operatorname{tr}A^3$ into three terms by the product rule $d(\alpha\wedge\beta)=d\alpha\wedge\beta+(-1)^p\alpha\wedge d\beta$ (1-06), and move $dA$ to the front in each term.

The sum is $d\operatorname{tr}A^3=3\operatorname{tr}(dA\wedge A^2)$. Together with $d\operatorname{tr}(A\wedge dA)=\operatorname{tr}(dA\wedge dA)$ ($d\,dA=0$), $d\,\mathrm{CS}=\operatorname{tr}(dA\wedge dA)+2\operatorname{tr}(dA\wedge A^2)$. Expanding $\operatorname{tr}F\wedge F=\operatorname{tr}\bigl((dA+A^2)\wedge(dA+A^2)\bigr)$ gives the same two terms ($\operatorname{tr}(A^2\wedge dA)=\operatorname{tr}(dA\wedge A^2)$) and $\operatorname{tr}A^4$, which is 0 since $\operatorname{tr}A^4=\operatorname{tr}(A\wedge A^3)=-\operatorname{tr}(A^3\wedge A)=-\operatorname{tr}A^4$. The name comes from Chern–Simons (1974).

There are two consequences. First, if $A$ is smooth on the ball $B_R$, Stokes' theorem (1-06) gives

Example. For the solution of §04 the right-hand side is $R=\rho,3\rho,10\rho$ at $-0.5,-0.972,-0.999708$, agreeing with the fraction inside in Table 1 of 1-12 (Table 2) Computed. Second, for $\delta A$ that is 0 far away, the Bianchi identity (3-03 §04) gives $\delta\operatorname{tr}F\wedge F=2\,d\operatorname{tr}(\delta A\wedge F)$, and the integral does not change. Adding $c_2$ to the action does not change the equations, and the θ term of §06 starts here.

If the action is finite, far away $A\to-(dg)g^{-1}$ ($g:S^3\to SU(2)$, pure gauge). For solutions, the basis is the abstract of Uhlenbeck (1982), “every Yang-Mills field over R4 with bounded functional (L2 norm) may be obtained from a field on S4” Known; for general fields we assume decay far away (1-07). For pure gauge $dA=-A^2$, so $\mathrm{CS}=\tfrac13\operatorname{tr}(g^{-1}dg)^3$, and the $R\to\infty$ of (4) is the following formula.

Checking with the identity map and with numbers

We use quaternions (1-12 §02; this chapter needs only the following rules). $x=x_1+x_2i+x_3j+x_4k$ is multiplied with $i^2=j^2=k^2=ijk=-1$, $\bar x=x_1-x_2i-x_3j-x_4k$, $|x|^2=x\bar x$. The set of quaternions of length 1 is $S^3$, which under multiplication is the same group as SU(2). The degree $\deg g$ is the integer obtained by counting the preimages of a generic target point, with $+1$ for points that preserve orientation and $-1$ for points that reverse it (1-12 §06).

The orientation of $S^3$ is the one that, as the boundary of the ball $B^4$, puts the outward normal first (at the point $x=1$, $dx_2\wedge dx_3\wedge dx_4$ is positive). For the identity map $g(x)=x$, at the point $x=1$, $g^{-1}dg=i\,dx_2+j\,dx_3+k\,dx_4$. The signed sum over permutations of $i,j,k$ is $-6$, so $\operatorname{tr}(g^{-1}dg)^3=-12\,dx_2\wedge dx_3\wedge dx_4$ (the quaternion $1$ is the $2\times2$ identity matrix, with trace 2). It is invariant under left multiplication, so it is the same everywhere, and multiplying by the area $2\pi^2$ and dividing by $24\pi^2$ gives $c_2=-1$. For general $g$ too, the integrand is $-1$ times the pullback of the normalised volume form, and the integral is $-1$ times the degree Known.

Fields of finite action split into classes (sectors) by $Q$, which cannot be connected to one another continuously. This is the statement that SU(2) bundles over $S^4$ are classified by $c_2$ (1-07).

A group divided by its centre — SO(3) bundles

Taking as group $SO(3)=SU(2)/\mathbb Z_2$, SU(2) divided by its centre $\{\pm1\}$ (1-04 §07), the Lie algebra is the same, so the local formulas do not change. The difference shows in how the bundle is glued. On $S^4$ (fields on $\mathbb R^4$ that approach pure gauge far away), the gluing is a map from $S^3$ to the group, and since $S^3$ is simply connected, a map to SO(3) lifts to a map to SU(2). So every SO(3) bundle over $S^4$ comes from an SU(2) bundle Known.

On a closed four-dimensional manifold $X$ with $H^2(X;\mathbb Z_2)\ne0$ (for example the four-dimensional torus $T^4$), there are SO(3) bundles that do not lift to SU(2) bundles. The obstruction is the second Stiefel–Whitney class $w_2\in H^2(X;\mathbb Z_2)$, and SO(3) bundles are said to be classified by $w_2$ and the first Pontryagin class $p_1$ (Dold–Whitney 1959) Known. Measuring the number by $-p_1/4$, so that it agrees with $c_2$ for bundles coming from SU(2), it is not necessarily an integer for $w_2\ne0$, and half-integers also appear for $T^4$. In physics this is said to be known as the fractional topological charge of 't Hooft's twisted boundary conditions (1979, 1981) Known. The integer $Q$ of this chapter is about SU(2) bundles (or $S^4$). Centre symmetry is in 8-03.

Detour: the Q of physics and the c₂ of mathematics have opposite signs

With the quaternions $x=x_1+x_2i+x_3j+x_4k$ of §03, reading $i,j,k$ as $-i\sigma_1,-i\sigma_2,-i\sigma_3$, the purely imaginary quaternions are elements of su(2). With $dx=\sum_\mu e_\mu dx_\mu$ ($e_\mu=1,i,j,k$), the solution of size $\rho$ has the following form Known.

The $i$ components of $dx\wedge d\bar x$ are $-2(dx_1\wedge dx_2+dx_3\wedge dx_4)$, and the components of $j,k$ are also self-dual (1-12 §03). So $F=*F$, and it is a solution (3-03 §07). Far away, with $g=x/|x|$, $A\to\operatorname{Im}(x\,d\bar x)/|x|^2=g\,d\bar g=-(dg)g^{-1}$; $g$ is the identity map of $S^3$ (degree 1), so $Q=1$, $c_2=-1$. The density is $48\rho^4/(|x|^2+\rho^2)^4$, the fraction of the action inside radius $R$ is $1-3/(1+u)^2+2/(1+u)^3$ ($u=R^2/\rho^2$; (9) of 1-12), and $S_E=8\pi^2/g^2$ is equality in the lower bound (the same property as (13) of 3-03). The abstract of BPST (1975) reads “We find regular solutions of the four dimensional euclidean Yang-Mills equations” Known. The mirror image $\operatorname{Im}(\bar x\,dx)/(\cdots)$ is anti-self-dual, with $Q=-1$.

The parameters are 8: centre 4, size 1, and constant gauge rotation 3; for $Q=k$ there are $8k$ (Tong's lecture notes: $4kN$ for SU(N)) Known. The construction that gives all self-dual fields of finite action by linear algebra is ADHM (1978).

A self-dual field has the least action in each $Q$, so for $Q=2$, wherever two lumps are placed the action is $16\pi^2/g^2$, and no force acts between the lumps. There is a concrete family. Here only, in the component notation of 3-03 ($A_\mu=A^a_\mu T^a$, $D_\mu=\partial_\mu-igA_\mu$), we set $g=1$. $\bar\eta$ is 't Hooft's symbol $\eta$ of 3-03 §08 with the sign of $\eta_{ai4}$ changed ($\bar\eta_{aij}=\varepsilon_{aij}$, $\bar\eta_{ai4}=-\bar\eta_{a4i}=-\delta_{ai}$).

$1/|x|^2$ is a four-dimensional harmonic function, and this family is self-dual and $Q=k$ regardless of $a_i,\rho_i$. The $A$ of (7) diverges at the poles $a_i$, but this singularity can be removed by a gauge transformation singular at that point. $k=1$ is thereby taken to the solution of §04 translated to $a_1$.

The density can be written with $\phi$ alone. The notation is $\square=\sum_\mu\partial_\mu^2$ (the Euclidean four-dimensional Laplacian; not the d'Alembertian of 2-04), $\psi_\mu=\partial_\mu\phi$, $H_{\mu\nu}=\partial_\mu\partial_\nu\phi$, $|H|^2=\sum_{\mu\nu}H_{\mu\nu}^2$, $\psi\cdot H\psi=\sum_{\mu\nu}\psi_\mu H_{\mu\nu}\psi_\nu$, $|\psi|^2=\sum_\mu\psi_\mu^2$.

With two poles, the $F$ from differences satisfies $F=*F$ and (8) with relative error $10^{-9}$ Computed. Corrigan–Fairlie (1977) treated solutions of this type as a relation between solutions of a scalar field theory of $\varphi^4$ type and solutions of the SU(2) gauge field. Jackiw–Nohl–Rebbi (1977) extended the known family of $5k$ parameters, representing the position and size of each lump, to solutions with $5k+4$ parameters (both from the abstracts) Known.

ComputedThe numbers in this figure were computed on this machine. (8) is computed on the spot ($\rho_i=1$, poles at $x_4=\pm d/2$).

The total action is $16\pi^2$ for every $d$. For small $d$ the density at the origin approaches $48/(d/2)^4$ ($7.661\times10^6$ against $7.68\times10^6$ at $d=0.1$), and the 0.896 inside radius $d$ is the fraction $1-\frac{3}{25}+\frac{2}{125}$ for the BPST solution of size $d/2$ ((9) of 1-12). One lump shrinks while keeping its action, and at $d=0$ it becomes the $Q=1$ solution of $\phi=1+2/|x|^2$, one lump's worth vanishing into a point. This is the concentration of 1-12 §05 happening within a family of solutions (3-06).

By §02, adding a constant multiple of $Q$ to the action does not change the classical equations. In the quantum weight it stays imaginary even in imaginary time (4-02; the sign depends on conventions).

Known(i) In perturbation theory (the expansion around $A=0$; 4-03) the integral of a total derivative is 0 at every order, and it cannot be seen. (ii) Since $Q$ is an integer, θ is an angle of period $2\pi$. (iii) In spacetime it is proportional to $\mathbf E^a\cdot\mathbf B^a$ (§01, 2-04); $\mathbf E$ changes direction under spatial reflection and $\mathbf B$ under time reversal, so $\theta\ne0,\pi$ breaks spatial reflection and time reversal. The abstract of Jackiw–Rebbi (1976) describes θ vacua (§07) as “a family of quantum mechanical vacua, parametrized by a CP-nonconserving angle” (CP is the transformation combining the exchange of particles and antiparticles with spatial reflection). At θ = 0 and π it is invariant under time reversal (Gaiotto–Kapustin–Komargodski–Seiberg 2017, hereafter GKKS, abstract: “time reversal invariant at θ = 0 and θ = π”).

A particle on a circle — the smallest model of θ

There is quantum mechanics with the same structure (GKKS Appendix D.1). To a particle on the angle $q\sim q+2\pi$ we add a term linear in the velocity.

The added term is $\frac{d}{dt}\frac{\theta q}{2\pi}$ and does not change the equations (2-01), but $\theta q/2\pi$ shifts by $\theta$ in one round, and the weight of periodic paths in imaginary time gets a factor $e^{i\theta\times\text{winding number}}$. This is the same form as (9). With periodic wave functions, the eigenvalues of the eigenfunctions $e^{inq}$ of $H=\frac12(-i\,d/dq-\theta/2\pi)^2$ are (10) (2-05). θ plays the role of a magnetic flux threading the circle (GKKS footnote 27; 2-03), and “for θ = π and only in this case the ground state is two-fold degenerate”. Adding $-K\cos q$ opens the crossing, while for $-K\cos2q$ the two-fold degeneracy remains exactly (same appendix: “still maintains some exact two-fold degeneracy”).

ComputedThe numbers in this figure were computed on this machine. V = 0 is (10); the others are a table diagonalised with $e^{inq}$ ($|n|\le40$) (13 points of $0\le\theta\le\pi$), extended by periodicity and symmetry.

From here on the level is that of physics Physics. Setting $A_4=0$, the classical vacua are three-dimensional pure gauges $-(dh)h^{-1}$ (with $h:\mathbb R^3\to SU(2)$ approaching 1 far away, regarded as $\mathbb R^3\cup\{\infty\}=S^3$; a different sphere from the $S^3$ at infinity of §03), numbered by the degree $n$, and the gauge transformation $T$ of degree 1 shifts $n$ by one. Applying (4) to a slab of imaginary time, $Q$ is the difference of the vacuum numbers at $\tau\to\pm\infty$, and the BPST solution is a path in imaginary time from $n$ to $n+1$. The weight is

Computed, and since every derivative at $g^2=0$ is 0, it appears at no order of perturbation theory (the double well of 2-05, 4-07, 4-12). In 1976 't Hooft computed “the tunneling process described by the Belavin-Polyakov-Schwarz-Tyupkin field configuration”, and Jackiw–Rebbi and Callan–Dashen–Gross described the family of vacua (abstracts) Known. The states that are invariant under small gauge transformations (4-01) and change only by a phase under $T$ are the θ vacua.

For the particle on a circle, the valleys $q=2\pi n$ of the line before it is wrapped onto the circle are $|n\rangle$, $q\mapsto q+2\pi$ is $T$, and (12) has the form of a Bloch wave (GKKS also read $q$ as the Chern–Simons quantity of Yang–Mills). When the valleys are deep the lowest level becomes a narrow band, fixed by the action $S_0=\int_0^{2\pi}\sqrt{2K(1-\cos q)}\,dq=8\sqrt K$ of the path from valley to valley. The band has the form $E_0(0)+\frac W2(1-\cos\theta)$, and $W$ approaches the asymptotic form of the Mathieu equation (a rewriting of DLMF 28.8.2, $\frac{16}{\sqrt\pi}K^{3/4}e^{-S_0}(1-\frac{7}{64\sqrt K})$) (Table 5).

In the dilute gas approximation of Yang–Mills (regarding instantons as scattered far apart), the vacuum energy density is said to be proportional to $-\cos\theta\,e^{-8\pi^2/g^2}$ Physics. It is an approximation that regards typical configurations as superpositions of smooth solutions, and it is mathematically unsettled (Dictionary C). More precisely, in four dimensions the coupling runs with the scale (4-06), so the integral over the instanton size $\rho$ diverges on the side of large $\rho$, and the approximation is said to be uncontrolled even at the level of physics Physics. The quantity that does not depend on the approximation is the topological susceptibility $\chi_t=\langle Q^2\rangle/V$ (the second derivative of $E(\theta)=-\frac1V\ln\sum_QZ_Qe^{i\theta Q}$ at $\theta=0$).

The space of fields on the lattice (5-01) is connected, and “the concept of a topological sector has therefore no a priori well-defined meaning” (Lüscher 2010) Known. Lattice fields with small action density have an integer $Q$, which is said to return to the Chern number in the classical continuum limit (that they have an integer is from the abstract of Lüscher 1982; 6-11 §09). For SU(3), the fraction of fields $V$ smoothed by the Wilson flow that satisfy $\max_p\operatorname{Re}\operatorname{tr}\{1-V(p)\}\lt0.067$ is 0%, 8% and 70% at $a=0.1,\ 0.07,\ 0.05$ fm, and the sector is “a dynamical property of the theory” (same, 2010) Physics. The construction of Magnen–Rivasseau–Sénéor (1993) for SU(2) with a fixed infrared cutoff (a cutoff on the long-distance side) is restricted to the trivial sector (6-07) Known. In HMC the effective dynamical critical exponent of $Q^2$ is about 5 (Schaefer–Sommer–Virotta 2011; 5-07) Physics.

Neither the θ term nor instantons appear in the problem statement of Jaffe–Witten (full-text search). GKKS argue, from the 't Hooft anomaly of time reversal and centre symmetry (8-03) (an obstruction to making a trivial vacuum while keeping both symmetries), that “at θ = π the vacuum cannot be a trivial non-degenerate gapped state”, and state that the vacuum at θ = 0, by contrast, is trivial, gapped and non-degenerate (both at the level of physics) Physics. "A unique vacuum with a gap" is a claim that depends on θ, and $-K\cos2q$ and $\theta=\pi$ in Figure 2 are its quantum-mechanical version.

What this chapter can say

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Next chapter to read: 3-06 Detour: Donaldson and exotic R⁴. The dictionary of topological charge is 8-04.

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