2026-10-15 · chapter Yang–Mills and the mass gapPart 12 The second road: from the direction of spacefractional instantons, the dilute gas, the lattice action · Kiichi, Shiori and Rin

Vortices and fractional instantons

The topological charge $Q$ of the continuum theory and the lower bound on the action of an instanton are as in 3-05: the Euclidean action satisfies

and equality is attained by solutions of the (anti-)self-duality equation $F=\pm{*F}$ Cited. On a periodic four-dimensional torus or on $\mathbb R^4$, $Q$ is an integer. On a bundle with 't Hooft's twist, however, the fractional part of $Q$ is fixed by the twist: for $SU(N)$ with $n_{12}=n_{34}=1$, $Q\equiv-1/N\pmod1$ (when this chapter writes $SU(N)$, the $N$ is the degree of the group, different from the lattice $N$ of this part) Cited (equation (2.24) of Tanizaki–Ünsal; the original source is 't Hooft 1981). For $SU(2)$, $Q\in\tfrac12+\mathbb Z$, and from (1)

(ibid., equation (2.25): the lower bound is $1/N$ of the ordinary instanton action $8\pi^2/g^2$) Cited.

A solution attaining this lower bound is a fractional instanton. Tanizaki–Ünsal state that on $\mathbb R^2\times T^2$ (with the twist on $T^2$) the action density of this solution is localised in the directions of $\mathbb R^2$ and takes a vortex-like shape, and that when it lies inside a Wilson loop in the plane it multiplies the loop by the central element $-1$ ($e^{2\pi i/N}$ for $SU(N)$), that is, it behaves as a centre vortex (7-02) Cited (around equation (2.26)). In the words of the previous chapter, this lump is the cheapest inhabitant of the sector $(1,1)$, carrying the surface $\Pi_q$ not as a thin surface but as a smooth lump.

A word on size. The action of the lump is fixed by (2) and does not depend on size, but on the twisted $T^2$ the lump is fixed to an extent of order the size $L$ of the $T^2$ (a footnote of the same paper cites earlier numerics in which, for $SU(N)$, the radius of the lump is proportional to the square root of the degree of the group) Cited. Since the size is fixed, no infrared divergence arises in the counting of the dilute gas. This contrasts with the integral over instanton sizes on a periodic $\mathbb R^4$, which diverges in the infrared (3-05).

The state of the literature Cited. Tanizaki–Ünsal write that almost no analytic forms of self-dual solutions with fractional topological charge are known (except for abelian constant solutions on tori of particular aspect ratios), and cite reports that numerical searches by lattice cooling have found configurations attaining the lower bound (2) within the discretisation errors of the lattice, and that the limit in which the directions of $\mathbb R^2$ are stretched to infinity is also well defined (ibid. §2.3.1). Analytic existence is not shown in the literature within the range this book checked.

A strand with an assumption Assumption (strand: "Proposition F" in the record of this book's work; evidence in [05] of Table 3). We assume the following property.

The strand has three steps. (i) Map a charge-1 $SU(2)$ doubly periodic instanton on $T^2_{2L\times L}\times\mathbb R^2$ by the Nahm transform to data on the dual side (the construction of doubly periodic instantons and the bijectivity of the transform are said to be in the literature; not yet checked). (ii) The composite $\tau$ of translation by a half period in the long direction and a central gauge transformation corresponds, by the assumption, to the composite of translation by a half period and a constant shift on the dual side, and if the singular points are placed at a quarter period, data invariant under $\tau$ can be chosen. (iii) The corresponding solution is invariant under $\tau$ up to gauge, and when brought down to the quotient $T^2_{L\times L}\times\mathbb R^2$, since $\tau$ contains the central twist, it becomes an anti-self-dual solution of charge $\tfrac12$ on the bundle with $n_{12}=1$, whose action equals the lower bound (2).

If this strand closes, the lower bound (2) is attained in the continuum theory. What is not closed is the assumption itself (the exact form of equivariance), the check of the literature in (i), and the confirmation that the topology of the bundle ($n_{12}$ and the flux at infinity in the plane) comes out correctly on passing to the quotient. As another strand there is a variational strand combining the Bogomolny bound, exclusion of bubbles by Uhlenbeck compactness, and exclusion of splitting by concentration compactness, but that the minimal value equals the lower bound must itself be shown separately, so it too is Assumption. This book uses neither strand as a premise of the chapters that follow.

We assume that the lumps are scattered in the plane far apart from one another (dilute), and count ignoring the interactions between lumps. This is the dilute-gas approximation, and the counting from here on transcribes §2.3.2 of Tanizaki–Ünsal to $SU(2)$ Cited (a copy of the formulae). The correctness of the approximation itself is a physical estimate Reading.

Note: the original source places the small $T^2$ in directions 3 and 4 and the plane in directions 1 and 2, so the indices are swapped relative to this book ($T^2$ in 1 and 2, the plane in 3 and 4). Below we write with this book's indices.

The weight of one lump. The weight of one lump (self-dual, $Q=+\tfrac12$) and of one antilump ($Q=-\tfrac12$) in the plane, with the $\theta$ angle set to 0, is

($N=2$ in equation (2.27) of the original source). The prefactor $K$ has mass dimension 2 and $K\sim1/L^2$; its exact value is fixed by an analysis of the fluctuation determinant, which the original source does not give Cited.

Rewriting with asymptotic freedom. Writing the coupling at scale $L$ in terms of $\Lambda$ with the one-loop running coupling (4-06), for $SU(2)$ we have $8\pi^2/g^2(L)=\tfrac{22}{3}\ln\frac1{\Lambda L}$ (at the one-loop level; for general $SU(N)$, $\tfrac{11N}{3}$), so

(the last step of equation (2.32) of the original source) Cited. The exponent $11/3$ does not depend on the degree of the group (for $SU(N)$ the action of the lump is $S_I/N$, and it cancels against the $N$ of the coefficient $11N/3$).

The Wilson loop. If one lump lies inside a loop $C$ of area $A$, the value of the loop is multiplied by $-1$ (a centre vortex). If lumps and antilumps are each scattered in the plane as independent Poisson points of density $K e^{-\pi^2\beta}$, the number $n$ inside follows a Poisson distribution with mean $\rho A$ ($\rho=2Ke^{-\pi^2\beta}$), and

This is a one-line computation, $\mathbb E[(-1)^n]=e^{-\lambda}\sum_k(-\lambda)^k/k!=e^{-2\lambda}$ for the Poisson distribution Proved here (a computation within the model). The original source obtains the same answer by putting in the constraint on the total charge (the difference between the numbers of lumps and antilumps is a multiple of $N$) through a Fourier sum: at $\theta=0$ the string tension of the fundamental representation is

(equation (2.37) of the original source; $N$ is the degree of the group, and $|R|$ is the number of boxes of the Young diagram of the representation modulo $N$), and for the fundamental representation of $SU(2)$, $\sin^2(\pi/2)=1$ Cited. Combining (5) and (4),

Reading (a physical estimate; $\sim$ is proportionality with the constant of the prefactor omitted).

Relation to the bias $\theta$. Applying the same counting to the parity of the number of lumps in the whole plane (area $A_{\text{plane}}=(Ma)^2$), the $\theta=\mathbb E[(-1)^{|Q|}]$ of Proposition 3 of 12-01 becomes

and one reads both that $-\ln\theta$ is proportional to the area of the plane (the type of an area law) and that its rate is fixed by the weight $e^{-\pi^2\beta}$ of the fractional instanton Reading (the dilute-gas estimate). The statement, on the lattice, of the part of this form saying "the rate is $e^{-\beta S}$ with $S$ close to $\pi^2$" is §06.

Numerics on the lattice also support the same picture. Bergner–González-Arroyo–Soler (2025) report, for the $SU(2)$ lattice on a four-dimensional torus with two directions far smaller than the others and twisted boundary conditions, that at small sizes of the $T^2$ the topological charge and the string tension arise from a gas of vortex-like fractional instantons, and that as the size increases the density increases and the spacing of the structures approaches a constant Cited (checked in the abstract; the numbers in the text not yet checked).

The exponent of (4) is the difference of two contributions Reading (a physical estimate).

• $11/3$: asymptotic freedom. The smaller the $T^2$, the weaker the coupling, and lumps become rare at the rate $(\Lambda L)^{11/3}$. The coefficient $11/3$ is exactly the one-loop coefficient ($11/3=-1/3+4$ of 4-06).
• $-2$: the two dimensions of the plane. The size of a lump is $L$, so the number of places where one can be put per unit area of the plane is $1/L^2$ ($K\sim1/L^2$). This corresponds to the entropy of the lump positions.

Since the difference $11/3-2=5/3$ is positive, if one holds $\Lambda$ fixed and widens the $T^2$ (increases $L$), the string tension in physical units increases. It is the direction in which confinement strengthens as one widens, and reaching order $\Lambda^2$ at $L\sim1/\Lambda$ is consistent with dimensional counting. Counting per square of the size of one $T^2$ (area $L^2$) gives $\sigma L^2\sim(\Lambda L)^{11/3}$, which can be restated as: the probability that a lump is in that square increases with $L$.

The abelian group $U(1)$ has no localised lumps carrying fractional flux. The flux can only spread uniformly over the $T^2$, and the behaviour in the direction of widening the $T^2$ is reversed. This difference is treated as "the only place where non-abelianness enters" in 13-03, and there too this counting is an estimate at the one-loop level. Within the scope of this chapter, (4)–(8) are all statements in the dilute region $\Lambda L\ll1$, and they do not extend into the band $L\sim1/\Lambda$.

Matching units. The action $S^z(U)=\sum_p(1-\tfrac12z_p\operatorname{tr}U_p)$ of 12-01 is the quantity before multiplying by $\beta$. In the classical continuum limit of 5-01, for $SU(2)$ and $\beta=4/g^2$, $\beta S^z$ approaches the continuum action $S_E$, so a lump of action $4\pi^2/g^2$ corresponds to $S^z\approx\pi^2$ (that is, $\beta S^z\approx\pi^2\beta$). All the numbers below are values of $S^z$ divided by $\pi^2$.

Cooling numerics Reading. Table 1 gives the result of searching for the minimal action $S_{\min}(N,M)$ of the sector $(1,1)$ by repeatedly lowering $S^z$ (cooling) through the operation of replacing each link by the direction of the sum of the remaining sides of the plaquettes containing it (the staple). The action of the configuration at which cooling stopped is an upper bound on the minimal action, since that configuration exists (within floating-point computation). The possibility that the true minimum is lower still is set aside only as a reading, on the grounds that different initial configurations arrived at the same value.

How to read it Reading: (i) every value is smaller than $\pi^2$ and approaches $\pi^2$ as $N$ grows (the lattice values approach the continuum value from below; a physical reading consistent with the discretisation correction of the Wilson action being negative). (ii) On elongated lattices ($M\gt N$), at $N=2$ the value hardly changes when the plane is widened, $M=4\to6$. The lump is localised in the directions of the plane and its action does not depend on the size of the plane; that is, the lattice supports the premise of the dilute gas (that a lump is a single localised object). (iii) That the values on symmetric lattices are larger than on elongated ones can be read as the lump being squeezed to order $N$ in the directions of the plane as well on a symmetric lattice. In a separate cooling that measured the distribution of the action on a slab of size $16$ in the directions of the plane ($4\times4\times16\times16$), $S^z/\pi^2\approx0.96$, about $91\%$ of the action lay within radius 2 of the centre of the lump in the directions of the plane, and about $99.6\%$ within radius 4 ($=N$) Reading.

A lump confined to a column, and certified upper bounds Computer-assisted. Paper 3 (one of the papers that came out of this book, a draft; A-07) divides the plane into square blocks of side $\ell$, puts flux into the column $T^2_N\times[0,\ell]^2$ above one block, and defines the minimal action when the outer frame of the column is fixed to the vacuum that absorbs the twist $U^0$ (12-01 §05) as

The values obtained by normalising the saved configurations exactly, checking that the outer frame agrees with the vacuum, and re-evaluating the action in 40-digit arithmetic are upper bounds on each $S_*$:

($N=4$ is a value at which cooling had not converged). Conditions: the group $SU(2)$, the Wilson action, an evaluation of the action of concrete configurations, and only in the direction of upper bounds. The label "Computer-assisted" attaches only to these upper bounds (actions of concrete configurations evaluated with guaranteed digits), not to the minimal values themselves. All are smaller than $\pi^2$, so "the minimal cost on the lattice is smaller than the continuum value $\pi^2$" can be said directly from the upper bounds for $N=2,3,4$.

Dependence on the width of the column Reading. A table from the same paper (cooling values):

Elementary properties of $S_*$ Proved here: (a) the minimum is attained (compactness). (b) $S_*(N,\ell)\le2N^2$ (place one thin surface in the column). (c) It does not increase in $\ell$ (embed the minimal configuration of a smaller column into a larger column, filling the rest with vacuum; the plaquettes on the border are vacuum against vacuum, with action 0). (d) Relation to the minimal action on a periodic lattice: $S_*(N,\ell)\ge S_{\min}(N,k\ell)$ ($k\ge1$; extend the configuration of the column to the periodic lattice, filling the outside with vacuum). Hence $\lim_\ell S_*(N,\ell)$ exists.

The values on periodic lattices (cooling) are $S_{\min}(2,M)=6.740117,\ 6.677345,\ 6.673724,\ 6.673486,\ 6.673485$ ($M=4,6,8,12,16$) and $S_{\min}(3,16)=9.072494$, and $S_*(N,\ell)$ approaches them from above as $\ell$ grows. The difference $S_*(N,\ell)-S_{\min}(N,\infty)$ is read as decreasing like $e^{-m_N\ell}/\ell$ in terms of the twist mass $m_N=2\operatorname{arsinh}\sin(\pi/2N)$ of (8) of 12-01 ($m_2\approx1.317$, $m_3\approx0.962$) Reading (the draft of Paper 3 sets the observed slopes $1.41$, $1.02$ beside the predicted values $1.45$, $1.06$; how the predicted values are obtained has not been checked in this book). The reading is that the tail of the lump is cut off by the mass of the adjoint fluctuations on the twisted $T^2$.

In summary, the minimal cost on the lattice is roughly $0.68\pi^2,\ 0.92\pi^2,\ 0.96\pi^2$ for $N=2,3,4$ (values on periodic lattices; $N=4$ is cooling on the elongated lattice $4\times8$), and approaches $\pi^2$ from below as $N$ grows. "$S_{\min}(N,\infty)\to\pi^2$ as $N\to\infty$" is numerical only, with no proof Reading.

With $N$ and the lattice spacing fixed, a bound of area-law type $-\ln\theta\ge c\,M^2$ follows elementarily at every $\beta$ (12-04). The rate $c$ there is fixed by the cost of the thin surface, $c\approx e^{-2\beta N^2}$ (the configuration of the upper bound $2N^2$ in Fact 3 of 12-01). $2N^2$ grows with $N$ and has nothing to do with fractional instantons.

The main theorem of Paper 3 (Theorem A) is a statement that replaces this rate by $S_*(N,\ell)$. We transcribe the form of the draft with its conditions Assumption (this book has not yet placed the proof in the text; the evidence column is §3–§6 of the draft of Paper 3. The proof uses chessboard estimates and a Morse–Bott lemma by citation, and only the existence of the rigidity constants is shown, with their values not made explicit).

Combined with the upper bounds of (10), the rate drops from $2N^2=8,18,32$ to about $\le0.68\pi^2,\ 0.92\pi^2,\ 0.98\pi^2$ ($N=2,3,4$). However, the terms of the prefactor $\beta^{-a}$ and $\varepsilon(\beta)$ are crude, so (11) beats the thin-surface bound only from some large, non-explicit $\beta$ onwards. (11) is not uniform in $N$, and says nothing about the continuum limit or the mass gap. The bound in the reverse direction (an upper bound from the dilute gas, uniform in $M$) is open. Where these stand is treated in 14-01 and 14-02.

Setting the estimate (8) beside (11), the position is that, of the part of the estimate saying "the rate is the action of the fractional instanton", only the direction of confinement (the bound from below), on a lattice with fixed $N$, has the form of a statement.

(a) The dilute gas. (5)–(8) are counting that ignores interactions, a physical estimate in the dilute region $\Lambda L\ll1$ at the one-loop level. The value of the prefactor $K$ is not known.

(b) Cooling values. The values of Tables 1 and 2 are upper bounds on the minimal action, not lower bounds. Of "approaches $\pi^2$ from below", "below $\pi^2$" can be said for $N=2,3,4$ from the upper bounds of (10), and "approaches" is a numerical reading.

(c) Continuum existence. The existence of a solution attaining the lower bound (2) on the twisted $T^2\times\mathbb R^2$ is a strand with an assumption in this book, and is not shown.

(d) The band. Every statement of this chapter is a statement with $\Lambda L\ll1$ or with $N$ fixed, and says nothing about widening the $T^2$ beyond $1/\Lambda$ (12-09).

(e) Paper 3. Theorem A is a statement of a draft, and this book's label is "Assumption". It is not raised until the proof of the draft has been checked line by line and placed in the text.

Dependency graph (the numbers of premises are smaller than the number of the conclusion):

[01] CIT BPS    S ≥ 8π²|Q|/g²                                     continuum
[02] CIT HALF   twist gives Q ∈ 1/2 + ℤ, bound π²β ⟸ [01]        SU(2), n₁₂ = n₃₄ = 1
[03] CIT LOC    lump localised in plane, centre vortex, size ~ L
[04] CIT NOAN   analytic existence not known
[05] ASM NAHM   strand of existence under equivariance            literature not checked
[06] CIT FUG    weight of one lump K e^(−π²β)
[07] CIT AF     K e^(−π²β) ~ Λ²(ΛL)^(5/3)                          one loop
[08] PRV POIS   Poisson model ⟨W⟩ = e^(−2ρA)
[09] CIT TR     T_R ~ Λ²(ΛL)^(5/3) sin²(π|R|/N)                    θ = 0
[10] RDG SIG    σ = 4K e^(−π²β) ~ Λ²(ΛL)^(5/3) ⟸ [06][07][08][09]  dilute, one loop
[11] RDG THETA  −ln θ ≈ 4K e^(−π²β)(Ma)² ⟸ [08][10]
[12] CIT BGAS   numerics of the lattice gas (abstract)
[13] RDG 53     reading of 5/3 = 11/3 − 2 ⟸ [10]
[14] RDG COOL   cooling values, fraction of localisation          upper bounds
[15] PRV SPROP  properties (a)–(d) of S_*
[16] CMP CERT   S_* ≤ 0.6762π², 0.9193π², 0.9707π²                 upper; N = 4 not converged
[17] RDG ELL    ℓ dependence, e^(−m_N ℓ)/ℓ
[18] RDG BELOW  approaches π² from below ⟸ [14][15][16][17]
[19] ASM THMA   Theorem A of Paper 3                               N fixed, draft

The way to check is to see by eye that every number to the right of "⟸" is smaller than the number on the left. There are no circles. The only nodes with proofs placed in the text are [08] and [15], both one-line arguments within a model or on a finite lattice. The "Assumption" of [05] and [19] enters as a premise of no node. The nodes on the conclusion side of this chapter, [10], [11], [13] and [18], are all "Reading", and there is neither a node of confinement nor a node of the mass gap.

What this chapter can say

Sources and reproduction

Words used in this chapter — fractional instanton, topological charge $\tfrac12$, the BPS bound, centre vortex, dilute gas, the prefactor $K$, string tension, $5/3=11/3-2$, cooling, a lump confined to a column $S_*(N,\ell)$, a certified upper bound, the twist mass $m_N$. Their meanings are in A-02 Glossary.

Next chapter to read: 12-03 Containers — conditioning on the frame makes things independent — divide the plane into blocks, and bound the bias by a product of quantities per column.

← 12-01 The twisted T² and 't Hooft fluxContents12-03 Containers — conditioning on the frame makes things independent →