2026-10-01 · chapter Yang–Mills and the mass gapPart 9 Where things stand, and the tasksRicci flow and the renormalisation group

Candidates for new concepts — what plays the part of "Ricci flow"

The Clay problem statement (Jaffe–Witten 2000) §6, having said that a construction on $T^4$ can be expected from extensions of known methods, writes: "no present ideas point the direction to establish the existence of a mass gap that is uniform in the volume. Nor do present methods suggest how to obtain the existence of the infinite volume limit $T^4\to\mathbb R^4$". §6.5 is "New ideas are needed". The same §6, however, also treats the construction on $T^4$ as incomplete. What can be said goes only as far as "the setters asked for new ideas on the infrared side" (6-10 §01). Known

We write "uniform in the volume" as a formula. For the gap $\Delta_L$ of a box of side $L$,

are different claims. Example: for a free field of mass $m$ the gap is at least $m$ for every $L$, and the right-hand one also holds. If $\Delta_L=1/L$ held, only the left-hand one would hold.

Among the new concepts that worked on other great problems, two types can be seen: a flow with a monotone quantity (Perelman, the Poincaré conjecture), and the identification of two objects (Wiles, Fermat's Last Theorem) (this book's arrangement).

A Riemannian metric $g$ is an inner product on the tangent space at each point (1-05), with components written $g_{ij}$. The Ricci curvature $\mathrm{Ric}$ (components $R_{ij}$) is, for a direction $v$ of length 1, the sum of the bending (sectional curvatures) of $n-1$ mutually orthogonal planes containing $v$ ($n$ is the dimension of the manifold). Equivalently, it is $n-1$ times the average. Precisely, it is a contraction of the curvature tensor. We use only the following example.

On a sphere of radius $r$, the bending of every plane is $1/r^2$. Adding $n-1$ of them gives the coefficient $n-1$. The trace of $\mathrm{Ric}$ (the sum of the diagonal components in an orthonormal basis) is called the scalar curvature $R$. On this sphere $R=n(n-1)/r^2$.

Ricci flow is the equation that moves a metric by its Ricci curvature, and Perelman's paper starts from this equation, introduced by Hamilton (1982). Known

Positively curved directions shrink. Along the flow the scalar curvature obeys $\partial_tR=\Delta R+2|\mathrm{Ric}|^2$. $\Delta$ is the Laplacian on the manifold (not the $\Delta$ of the mass gap), which in flat coordinates is $\sum_i\partial^2/\partial x_i^2$ (a different symbol from the gap $\Delta_L$ of §01). $|\mathrm{Ric}|^2$ is the sum of squares of the components of $\mathrm{Ric}$ in an orthonormal basis.

If the right-hand side were $\Delta R$ alone it would be the heat equation, and just as heat evens out temperature, it would shave the peaks of $R$ and fill its valleys. Precisely, the non-negative term $2|\mathrm{Ric}|^2$ is added upward everywhere. For equations of heat-equation type, a solution cannot create a new minimum in the interior (the maximum principle). So the minimum of $R$ does not decrease along the flow. Perelman's introduction, presenting Hamilton's work, writes this equation and conclusion in this form. Known

Example: a sphere of initial radius $r(0)$ stays round and only its radius changes. From $g=r^2\hat g$ ($\hat g$ is the metric of the unit sphere) and (2),

and at time $t_*$ (the extinction time) it shrinks to a point. For the unit two-sphere $t_*=1/2$. For a general shape the curvature becomes infinite somewhere. Perelman writes "The most natural way of forming a singularity in finite time is by pinching an (almost) round cylindrical neck", and continues the flow by surgery that cuts the thin neck and puts small caps on it.

We split Perelman's proof into three components. The first, the flow, is (3) of §02. The second is a quantity that increases monotonically along the flow. We define the tools first. $M$ is a closed manifold (a compact manifold without boundary), $f$ an auxiliary function on $M$, and $dV$ the volume element determined by the metric ($\int_M dV$ is the volume of $M$). $\nabla f$ is the gradient of $f$, and $|\nabla f|^2$ the square of its length. $\nabla_i\nabla_j f$ is the second derivative of $f$, which in flat coordinates is $\partial_i\partial_j f$.

When the metric is moved by (3), $f$ is moved at the same time by $\partial_t f=-\Delta f+|\nabla f|^2-R$. This way of moving preserves the measure $e^{-f}dV$ up to a change of coordinates, and the total integral $\int_M e^{-f}dV$ does not change. Then

holds (Perelman §1.2). The right-hand side is the integral of a square, so it is not negative. Known

The flow behaves as if it climbs the slope of $\mathcal F$. Precisely, the flow $\partial_t g_{ij}=-2(R_{ij}+\nabla_i\nabla_j f)$ with the measure $e^{-f}dV$ held fixed is a gradient flow in the direction that increases $\mathcal F$, and a change of coordinates takes it to the pair of (3) and the equation for $f$ above (Perelman §1.1–1.2). On the choice of the measure held fixed, Perelman writes "the choice of $m$ is analogous to the choice of gauge". The $m$ of the original is this measure, not the mass of §06. Whatever the choice, the flow is the same up to a change of coordinates. Known

Further, seen up to changes of coordinates and of scale, the flow has no non-trivial periodic orbits (Perelman §2, §3). The trivial ones are called solitons (solutions that keep their shape under changes of coordinates and scale alone), and in this view they correspond to fixed points. Known Example: on a flat torus $R_{ij}=0$, and taking $f$ constant, the right-hand side of (5) is 0 and the flow stands still (a fixed point).

The third component is to classify in advance the shapes of the regions of large curvature, and to remove by surgery only what has been classified. What could play the same roles for Yang–Mills: for the flow, the renormalisation group and gradient flow (§05); for the monotone quantity, $c$ and $a$ (§06); for classification and surgery, large fields, instantons and topological sectors (§07). As another type there is identification (§08). Every one of these correspondences is an analogy.

Within one-loop perturbation theory, the place where the correspondence stops being an analogy is the two-dimensional non-linear σ model. The field $\phi$ is a map from the plane to a manifold $M$. In this section we write the metric of $M$ as $h$ (components $h_{ij}$).

The coupling constant is not a number but the metric $h$ itself. The abstract of Friedan (1980) writes "the renormalization group acting on the infinite-dimensional space of Riemannian metrics". At one loop, for the scale $\ell=\ln(\mu_0/\mu)$ (larger means seen more coarsely),

and setting $t=\ell/4\pi$ gives (3) itself. Physics "$\cdots$" stands for corrections at two loops and beyond. From two loops a term quadratic in the curvature is added (of the form $R_{iklm}R_j{}^{klm}$, where $R_{iklm}$ are the components of the curvature tensor), and the flow departs from Ricci flow. The abstract of Friedan (1985) says the coefficients of the renormalisation group were computed in the two-loop approximation. Physics

Example: taking $M$ to be the sphere $S^{N-1}$ of radius $r$ gives the $O(N)$ model, with $r^2=1/g^2$. With $n=N-1$ in (2),

The more coarsely one looks, the more the sphere shrinks and the stronger the coupling becomes. This is the "asymptotic freedom" of the abstract of Polyakov (1975). The scale $\ell_*$ at which it shrinks away is $4\pi$ times the extinction time $t_*$ of (4), and it determines the one-loop $\Lambda$ (4-07). For four-dimensional $SU(N)$ too, $1/g^2$ decreases along the straight line $1/g_0^2-2b_0\ell$.

Perelman touches on this relation in section 2 of his introduction, writing "this connection between the Ricci flow and the RG flow suggests that Ricci flow must be gradient-like; the present work confirms this expectation". That the flow is gradient-like (moves along the slope of some quantity) was expected from the intuition of the renormalisation group, and (5) confirmed it: that is how Perelman puts it. Known

ComputedThe straight lines of the figure are formula (8) and $b_0=11N/(48\pi^2)$ drawn on the spot. Table 1 was computed on this machine.

Table 1 The scale at which the one-loop $1/g^2$ reaches 0. $\ell_*=4\pi t_*$ was checked against formula (4). Computed

Before the straight line reaches 0 the coupling becomes large and the one-loop formula can no longer be used. As near a singularity of Ricci flow, what matters is what happens after the coupling becomes $O(1)$. The gap of the two-dimensional $O(N)$ model ($N\ge3$) is a conclusion at the level of physics. Physics Douglas's report (2004) goes no further than citing an example in which a gap was shown for a regularised version of the $O(N)$ model (Kopper 1999). Its later status as a theorem has not been checked in this chapter.

Detour: a flat circle does not shrink

In four dimensions there are two different "flows". The flow on the space of theories (the renormalisation group, 6-05) has no picture of a sphere, but the one-loop $1/g^2$ is a straight line of the same form (Table 1). Bałaban's ultraviolet stability on $T^4$ comes with a hypothesis, and the limit is only along subsequences (6-07). Where the method of Bauerschmidt–Brydges–Slade reached $d=4$ is the side where the coupling flows to 0 in the infrared (the $|\phi|^4$ model and the weakly self-avoiding walk), the exact opposite of the infrared of Yang–Mills. What is missing is "a controlled step after the coupling has reached $O(1)$" (this book's arrangement).

The flow on the space of fields (gradient flow) smooths a single configuration along the slope of the action. For an abelian field, following the gradient of the action $S=\frac14\int F_{\mu\nu}F_{\mu\nu}\,d^4x$ backwards gives

(for a non-abelian field, replace $\partial$ by the covariant derivative). Since it is a gradient flow, the action always decreases. Example: after a Fourier transform, the component perpendicular to the wave number $k$ decays as $e^{-k^2t}$, and fluctuations with $|k|\gg1/\sqrt t$ disappear. The component parallel to $k$ (the gauge direction) does not move. In Ricci flow the same thing happens through invariance under changes of coordinates, and DeTurck (1983) fixed it by adding a term. The same device appears in the stochastic quantisation of Yang–Mills (6-09 §07).

The abstract of Lüscher (2010) says that theory and numerics "suggest" that the flowed field is "a smooth renormalized field", and Carosso–Hasenfratz–Neil (2018) used it numerically as a continuous real-space renormalisation group. Waldron (2019) showed that, for smooth initial data on a closed four-dimensional manifold, Yang–Mills flow has no finite-time singularity. Known The relation between the two flows is at the level of suggestion and numerics.

We prepare two tools. The stress tensor (energy–momentum tensor) $T_{\mu\nu}$ is the field that describes the flow of energy and momentum. Its trace $\Theta=T^\mu{}_\mu$ (in two dimensions $T_{11}+T_{22}$) measures how much the action changes when the scale is changed slightly. For a theory that does not change under a change of scale (a fixed point), $\Theta=0$. The central charge $c$ is a number carried by the theory at a fixed point, and roughly counts the massless degrees of freedom (1 for one free scalar field). In this chapter we use only the name, and see its value through the sum rule (10) below.

In two dimensions there is the $c$-theorem of Zamolodchikov (1986), which constructs a function that decreases along the flow and equals the central charge at fixed points. Physics As a form we can compute, we look at the sum rule for correlations of $\Theta$, cut off at distance $r_0$.

Here $\Theta$ is the trace of the stress tensor made directly from the action (in the normalisation $\Theta=2\pi T^\mu{}_\mu$, the prefactor becomes $3/4\pi$). The content of the sum rule is that the value at $r_0\to0$ is the difference between the ultraviolet and infrared central charges. This sum rule is attributed to Cardy (1988). Physics

For a free scalar field of mass $m$, $\Theta=-m^2\phi^2$ and $\langle\Theta(x)\Theta(0)\rangle=2m^4G(x)^2$. $G(x)=K_0(m|x|)/2\pi$ is the propagator (1-11), and $K_0$ is a modified Bessel function (decaying like $e^{-m|x|}$ far away). Therefore

We checked $c(0)=1$ by numerical integration as $1-8\times10^{-12}$. Computed Since the integrand is never negative, $c(r_0)$ decreases. In general, $\langle\Theta(x)\Theta(0)\rangle\ge0$ for $x\ne0$ follows from translation and rotation invariance and reflection positivity (4-10). A property that is easily lost under regularisation (6-11 Obstacle 5) is where the monotonicity comes from. Known

ComputedThe curve is an interpolation of a table of 85 points from numerical integration of formula (11) on this machine (embedded with the trailing 0s omitted).

Table 2 $c(r_0)$ for one field. For two fields with mass ratio 100, at $m_1r_0=10^{-3},10^{-2},0.1,1$ it is 1.999, 1.708, 0.999, 0.708, staying on the step at 1 in between. Computed

The step shows that the flow passes near an intermediate fixed point (the scales at which only the light field looks massless); in an infrared with a gap, $c=0$.

When a four-dimensional fixed point is placed on a curved background, the Euler density (a quadratic expression in the curvature whose integral is proportional to the Euler number of the manifold) appears with a coefficient in the expectation value of $\Theta$. That coefficient is the Euler anomaly $a$ (not the lattice spacing $a$; in this chapter only the name). The abstract of Komargodski–Schwimmer (2011) derives a monotonically decreasing function joining the ultraviolet and infrared $a$, "establishing the $a$-theorem". Physics The $a$ of free fields is positive, and the contributions of a real scalar, a Weyl fermion and a gauge field are in the ratio $1:\tfrac{11}{2}:62$ (appendix A of the same paper). Known

The ultraviolet of pure Yang–Mills approaches free gluons, so $a>0$. If there is a gap in the infrared, no massless degrees of freedom remain, so $a=0$. Physics The two do not contradict each other, but $a_{\text{UV}}>a_{\text{IR}}$ does not force a gap. That there is a gap in the infrared is itself the unproved part of this book's subject. A separate "classification of infrared endpoints", corresponding to the classification of singularities, is needed (an analogy).

In four-dimensional Yang–Mills, the fact that the action does not change under changes of scale creates the same kind of difficulty as singularities in which curvature concentrates at a point. If $A_\lambda(x)=\lambda A(\lambda x)$, then $F_\lambda(x)=\lambda^2F(\lambda x)$, and

For $d=4$ this does not depend on $\lambda$. Example: an instanton of size $\rho$ has action $8\pi^2/g^2$ whatever $\rho$ is (3-05), and as $\rho\to0$ the action does not decrease even when the curvature concentrates at a point.

As a ruler for the size of the curvature we use $\|F\|_{L^p}=\bigl(\int|F|^p\,d^dx\bigr)^{1/p}$ ($p=2$ is the $L^2$ of 1-10, and squared it is the action). The compactness of Uhlenbeck (1982) (from a sequence of connections whose curvature is bounded in $L^p$, one can extract, after changes of gauge, a subsequence converging in a weak sense) works with $L^p$ estimates for $p>d/2$, and the $L^2$ of $d=4$ (the action itself) lies exactly on the borderline. On the borderline, curvature can concentrate at finitely many points and small instantons can pinch off like bubbles (bubbling; in this chapter only the name). Known

On the quantum side, what could play the part of surgery are the large-field regions treated separately in Bałaban's method (6-05 §04), small instantons, and topological sectors (6-11 Detour). None of them has reached the stage of "classify, then remove".

The Wiles type shows that two objects are the same, and brings in from one of them a property not visible in the other ($R=T$; a summary at the level of common knowledge). Jaffe–Witten §6.6 writes that, since the gap is not visible classically, "a nonclassical change of variables or “duality transformation”" may be needed. Known

Examples where a gap was shown by this type start from an identity of Gaussian integrals. For a Gaussian field $\varphi$ with covariance $V$,

The right-hand side is the weight of particles with charges $q_j$ interacting through the potential $V$; the left-hand side is an expectation value of the field. Summing over the gas of charges, the left-hand side becomes a field theory containing $\cos\varphi$. This is called the sine-Gordon model (a field theory with a $\cos\varphi$ term in its action; in this chapter only the name). Brydges–Federbush (1980) used this transformation to prove exponential decay of correlations (Debye screening) in the Coulomb gas at low density or high temperature. Known It is an abelian model of statistical mechanics.

Fragments for Yang–Mills itself are the gauge–string duality for $SO(N)$ at strong coupling and large $N$ of Chatterjee (2019), the representation of Wilson loops as sums over surfaces of Cao–Park–Sheffield (7-04 §09; the picture is in §05), and "unbroken centre symmetry ⇒ confinement" of Chatterjee (2021) (an implication, not an equivalence). At the level of physics there are Seiberg–Witten (7-11) and holography (7-09). The problem asks for fixed $N$, so large $N$ alone is not enough.

The proposal to give a symbol to $1/0$ is not included, since it carries little information (6-12 §10).

First, how to read the table. The "Component" column says which of the three components of §03 the candidate corresponds to. They divide into flow (i–iii), identification (v, vi) and surgery (vii); (iv) corresponds to none. "How far" is the range reached as a theorem; "Where it stops" is the first place not reached.

Lining up the places where they stop, they gather roughly in two. One is that small numbers disappear in the infrared, where the coupling becomes $O(1)$. (i) has no step there; the continuum limit obtained in (iii) is of a design that ties the coupling to 0; and (iv) is a power series, so it does not reach $e^{-1/g^2}$. The other is that they work only in a different limit, $N\to\infty$ or strong coupling: (v) and (vi) are of this kind. (i) tries to go from the ultraviolet side to the infrared and stops; (vi) has a picture of the infrared but stops without being able to return to finite $N$. They stop facing in opposite directions (this book's arrangement).

The study on this machine treated in Part 10 (Assumption H, 10-01) is closest to type (i): it cuts out the physical premise "a single scale" as an inequality on the lattice (the judgement of closeness is this book's arrangement).

What this chapter can say

Sources and reproduction

Next chapter to read: 9-04 The list of open tasks (what would move if what were known).

← 9-02 Where the two ways of digging partContents9-04 The list of open tasks →