2026-10-15 · chapter Yang–Mills and the mass gapPart 12 The second road: from the direction of spacetwist, sectors, the flux free energy · Kiichi, Shiori and Rin

The twisted T² and 't Hooft flux

Let the lattice be $\Lambda=(\mathbb Z/N)^2\times(\mathbb Z/M)^2$ ($N,M\ge2$). Directions 1 and 2 form the small $T^2$, and directions 3 and 4 the plane. On each link $(x,\mu)$ we place $U_{x,\mu}\in SU(2)$, and we write the value of a plaquette $p=(x;\mu,\nu)$ ($\mu\lt\nu$) as

(as in 5-01). Given a sign $z_p\in\{+1,-1\}$ for each plaquette, we set the action and the partition function to be

$dU_\ell$ is the normalised Haar measure. If $z\equiv+1$, (2) is exactly the Wilson action of 5-01, and each term lies in $[0,2]$ because $|\operatorname{tr}U|\le2$. A plaquette with $z_p=-1$ is "a plaquette with one central element $-1$ inserted", and there the action is 0 when $U_p=-1$.

The signs of the two sectors. We put the twist of the small $T^2$ in by placing $z_p=-1$ on exactly one plaquette of each $(1,2)$ plane (the $N\times N$ plane with $x_3,x_4$ held fixed), namely the $(1,2)$ plaquette at $(x_1,x_2)=(N-1,N-1)$. The position is the same for every $(x_3,x_4)$. We write this as $z^{(0)}$ (all others are $+1$). The sign $z^{(1)}$ with one unit of flux added in the directions of the plane is obtained from $z^{(0)}$ by taking, above a unit square $q=(q_3,q_4)$ of the plane, the surface

and placing $z_p=-1$ on its $N^2$ $(3,4)$ plaquettes. $\Pi_q$ is a "surface winding the $T^2$" standing above the point $q$ of the plane, and in the plane it behaves like a point (reason 2 of 12-00 §03).

Where the surface is placed, $q$, has no effect on the partition function. The next lemma shows this.

Lemma 1 (moving the surface) Proved here For a link $\ell$, write $\delta_\ell z$ for the field in which the signs of all plaquettes containing $\ell$ are flipped. Then $Z[\delta_\ell z]=Z[z]$. In particular, (a) the surface $\Pi_q$ can be moved to the surface $\Pi_{q'}$ of a neighbouring square, and the partition function of $z^{(1)}$ does not depend on $q$. (b) The partition function of the sign $z^{(0)}\prod_{q\in Q}(\text{surface }\Pi_q)$, with surfaces placed over a set $Q$, is determined only by the parity of $|Q|$.

Proof. The change of variables $U_\ell\mapsto-U_\ell$ preserves the Haar measure. Since $-1$ is a central element, in a plaquette that contains $\ell$ exactly once, $U_p$ changes to $-U_p$, and other plaquettes do not change (a plaquette of the lattice contains a given link at most once). Hence $S^{\delta_\ell z}(-U_\ell,\ldots)=S^z(U)$, and the integrals are equal. (a): apply the lemma in turn to all the links in direction 3 above the point $(a,b+1)$ of the plane (the $N^2$ links obtained by running over $x_1,x_2$). The only $(3,4)$ plaquettes containing these links are those on $\Pi_{(a,b)}$ and $\Pi_{(a,b+1)}$, each containing exactly one. A $(\mu,3)$ plaquette ($\mu\le2$) contains two links in direction 3, both above the same point of the plane, so its sign is flipped twice and returns to what it was. Other plaquettes contain no link in direction 3. So the signs on the two surfaces $\Pi_{(a,b)}$ and $\Pi_{(a,b+1)}$ are flipped, and the surface moves one step over. Direction 4 is the same. (b): moving surfaces so that two overlap makes them cancel, $(-1)^2=+1$, so this follows from (a) and the connectedness of the squares of the plane. ∎

Lemma 1 is a basic property of the lattice version of 't Hooft's twist: it says that it is not the field $z$ itself but only the parity of the number of $-1$ piercing each plane (the flux $n_{\mu\nu}\in\{0,1\}$) that affects the partition function Cited (the general form is 't Hooft 1979; the form with a lattice $\mathbb Z_2$ sign field is Mack–Petkova 1980). In this chapter we use only cases (a) and (b).

Definition 2 (sectors and the flux free energy) We write $Z(0)=Z[z^{(0)}]$, $Z(1)=Z[z^{(1)}]$, and call the flux pairs $(n_{12},n_{34})=(1,0)$ and $(1,1)$ sectors. The flux in the mixed planes ($(1,3)$ and so on) is taken to be 0. We then set

$\Delta F$ is the difference of free energy for putting in one unit of flux $n_{34}$ in the directions of the plane ('t Hooft's flux free energy), and $\theta$ is the "bias" between the two sectors. The right-hand equality of (4) comes from $(1-e^{-\Delta F})/(1+e^{-\Delta F})=\tanh(\Delta F/2)$. Since $Z(0),Z(1)\gt0$, $|\theta|\lt1$. $\theta\ge0$ (that is, $Z(1)\le Z(0)$) needs a separate argument through the transfer matrix, and is treated in 13-04.

The reason this quantity matters for confinement is that the Tomboulis–Yaffe inequality bounds the expectation of a Wilson loop in the plane from above by a ratio of twisted partition functions Cited. If $\theta$ decreases in the size $M$ of the plane as $-\ln\theta\ge cM^2$, the area law follows through that inequality. The exact form of the inequality, and that it is not enough for $\theta$ merely to move away from 1, are treated in 13-01.

Instead of keeping the two sectors as separate measures, we combine them into one measure in which the placement of the surfaces is also a random variable. Let $T$ be the set of all unit squares of the plane ($M^2$ of them), and for each subset $Q\subset T$ form the sign $z^Q=z^{(0)}\prod_{q\in Q}(\Pi_q\text{: sign }-1)$.

Definition (the twist-summed ensemble) Cited We call the probability measure on pairs $(U,Q)$ of a configuration and a set of surfaces

the twist-summed ensemble. The form that sums the $\mathbb Z_2$ signs of plaquettes as random variables is of the type of the representation of Mack–Petkova (1980), who sum over all closed sign fields (where the product of the six on each cube is $+1$). Here we use a small form that sums only the surfaces involved in the flux $n_{34}$ of the plane.

Proposition 3 (the bias as one expectation) Proved here $\displaystyle \theta=\mathbb E_{\tilde\mu}\bigl[(-1)^{|Q|}\bigr]$.

Proof. By Lemma 1 (b), $Z[z^Q]$ is $Z(0)$ if $|Q|$ is even and $Z(1)$ if it is odd. Of the $2^{M^2}$ subsets of $T$, exactly half have an even number of elements. Hence $\Xi=2^{M^2-1}(Z(0)+Z(1))$ and $\sum_Q(-1)^{|Q|}Z[z^Q]=2^{M^2-1}(Z(0)-Z(1))$, and the ratio is the $\theta$ of (4). ∎

In the representation of Proposition 3, the placement $Q$ of the surfaces is a record of "whether a vortex is there" above the points of the plane, and $(-1)^{|Q|}$ is the parity of the number of vortices in the whole plane. That the bias $\theta$ is small means that the parities are mixed almost fairly, that is, vortices well up freely in the plane. In $\mathbb R^4$, the vortex surface linked with a loop is an extended surface and cannot be written in this form of "a record above points" (12-00 §03). This representation is the starting point both for rewriting things as a product of biases per column by dividing the plane into blocks (the container identity, 12-03) and for the identities of the map that inserts one vortex (12-05).

The proofs of the three facts reduce to properties of configurations in which every plaquette takes a central value ($\pm1$). First we check that action 0 is the same thing.

Lemma 4 (configurations with action 0) Proved here A necessary and sufficient condition for $S^z(U)=0$ is that $U_p=z_p\cdot\mathbf 1$ for every plaquette.

Proof. An element of $SU(2)$ can be written $U=\cos\alpha\,\mathbf 1+i\sin\alpha\,(n\cdot\sigma)$ ($|n|=1$), and $\operatorname{tr}U=2\cos\alpha$ (1-04). Each term $1-\tfrac12z_p\operatorname{tr}U_p=1-z_p\cos\alpha_p$ of (2) is at least 0, and it is 0 only when $z_p\cos\alpha_p=1$, that is, when $\alpha_p=0$ ($U_p=\mathbf 1$) if $z_p=1$, and $\alpha_p=\pi$ ($U_p=-\mathbf 1$) if $z_p=-1$. ∎

Next, if all plaquettes take central values, the commutators of the holonomies around a plane are determined by the product of the signs of that plane. We write $P_\mu(x)$ for the product of links along the straight path that goes once around in direction $\mu$ from the base point $x$ (of length $N$ if $\mu=1,2$, of length $M$ if $\mu=3,4$).

Lemma 5 (the commutator is the product of the signs of the plane) Proved here If $U_p=z_p\cdot\mathbf 1$ for every plaquette, then for $\mu\ne\nu$ and a base point $x$

holds. The product on the right runs over all plaquettes of the $(\mu,\nu)$ plane (a two-dimensional torus) through $x$.

Proof. Cut the $(\mu,\nu)$ plane through $x$ open into a square $R$ with $x$ at a corner. The path that goes once anticlockwise around the boundary of $R$ from $x$ consists, in this order, of one round in direction $\mu$, one round in direction $\nu$, one round backwards in direction $\mu$ and one round backwards in direction $\nu$, and its holonomy is the left-hand side of (6). On the other hand, on a disc of the square lattice, the holonomy of the boundary equals a product, in some order, of the values of the plaquettes conjugated by parallel transport along paths to the base point (the lattice non-abelian Stokes theorem: an induction that peels plaquettes off one at a time and shrinks the boundary path). If all plaquette values are central elements $z_p\mathbf 1$, conjugation does not change the values and the order does not matter, so the product is $\prod_{p\subset R}z_p$. ∎

For $z^{(0)}$, every $(1,2)$ plane has exactly one $-1$, so $\varepsilon_{12}=-1$, and the other planes have $\varepsilon=+1$. For $z^{(1)}$, in addition, the $(3,4)$ planes have exactly one $-1$ (the one meeting the surface $\Pi_q$), so $\varepsilon_{34}=-1$. This is the lattice version of the relation, in 't Hooft's twisted boundary conditions of the continuum theory, that the consistency of the gluing in two directions is off by the central element $e^{\pi i n_{\mu\nu}}$ Cited ('t Hooft 1979).

Finally, we state how far the holonomies determine a configuration.

Lemma 6 (determined by holonomy) Proved here If two configurations $U,U'$ satisfying $U_p=z_p\mathbf 1$ (for every $p$) for the same $z$ have the same four holonomies at the base point 0 ($P_\mu=P'_\mu$, $\mu=1,\ldots,4$), then $U'$ is a gauge transform of $U$.

Proof. For each lattice point $x$ choose a fixed path $\gamma_x$ from 0 to $x$ (the path that goes in the positive sense in direction 1, direction 2, direction 3 and direction 4, in that order), and write $U(\gamma_x)$ for the product along it. For a link $\ell=(x\to y)$, consider the closed path $c_\ell=\gamma_x\,\ell\,\gamma_y^{-1}$. In the two-dimensional skeleton of the toroidal lattice, a closed path based at 0 can be deformed, by sweeping finitely many plaquettes, into a word in the four straight rounds (the lattice version of $\pi_1(T^4)=\mathbb Z^4$, a standard topological fact Cited). The plaquettes swept in the deformation take central values, so the holonomy of $c_\ell$ is the product of "the product of $z$ over the swept plaquettes" and "a word in the $P_\mu$", both of which are the same for $U$ and $U'$. Hence $U(\gamma_x)U_\ell U(\gamma_y)^{-1}=U'(\gamma_x)U'_\ell U'(\gamma_y)^{-1}$, that is, setting $g(x)=U'(\gamma_x)^{-1}U(\gamma_x)$ gives $U'_\ell=g(x)\,U_\ell\,g(y)^{-1}$. ∎

The configuration that absorbs the twist. Set $\Omega_1=i\sigma_1$, $\Omega_2=i\sigma_2$, and define the configuration $U^0$ as follows. Of the links in direction 1, only those at $x_1=N-1$ (those crossing the seam of the $T^2$) are $\Omega_1$; of the links in direction 2, only those at $x_2=N-1$ are $\Omega_2$; all other links (including those in directions 3 and 4) are $\mathbf 1$.

Lemma 7 Proved here $\Omega_1\Omega_2\Omega_1^{-1}\Omega_2^{-1}=-\mathbf 1$. Moreover, among the plaquettes of $U^0$, only the $(1,2)$ plaquette at $(x_1,x_2)=(N-1,N-1)$ is $-\mathbf 1$, and all others are $\mathbf 1$. Hence $S^{z^{(0)}}(U^0)=0$.

Proof. From $\sigma_1\sigma_2=i\sigma_3=-\sigma_2\sigma_1$ we get $\Omega_1\Omega_2=-\sigma_1\sigma_2=-i\sigma_3$ and $\Omega_1^{-1}\Omega_2^{-1}=(-i\sigma_1)(-i\sigma_2)=-i\sigma_3$, and the product is $(-i\sigma_3)^2=-\mathbf 1$. Looking at the plaquette (1), all $(1,2)$ plaquettes not containing the seam are $\mathbf 1$. At $x_1=N-1$, $x_2\ne N-1$ we get $\Omega_1\cdot\mathbf 1\cdot\Omega_1^{-1}\cdot\mathbf 1=\mathbf 1$; at $x_2=N-1$, $x_1\ne N-1$ we get $\mathbf 1\cdot\Omega_2\cdot\mathbf 1\cdot\Omega_2^{-1}=\mathbf 1$; at the corner $(N-1,N-1)$ we get $\Omega_1\Omega_2\Omega_1^{-1}\Omega_2^{-1}=-\mathbf 1$. The links in directions 1 and 2 do not depend on $x_3,x_4$, so all plaquettes involving directions 3 and 4 are $\mathbf 1$. By Lemma 4 the action is 0. ∎

Lemma 8 (an anticommuting pair) Proved here If $A,B\in SU(2)$ satisfy $ABA^{-1}B^{-1}=-\mathbf 1$, then for some $g\in SU(2)$ we have $gAg^{-1}=i\sigma_1$, $gBg^{-1}=i\sigma_2$. Moreover, the only elements of $SU(2)$ that commute with both $i\sigma_1$ and $i\sigma_2$ are $\pm\mathbf 1$.

Proof. Taking the trace of $ABA^{-1}=-B$ gives $\operatorname{tr}B=-\operatorname{tr}B$, hence $\operatorname{tr}B=0$ and $B=i\,b\cdot\sigma$ ($|b|=1$). Likewise $BAB^{-1}=-A$ gives $A=i\,a\cdot\sigma$. Since $(a\cdot\sigma)(b\cdot\sigma)=(a\cdot b)\mathbf 1+i(a\times b)\cdot\sigma$, $AB=-BA$ is equivalent to $a\cdot b=0$. A pair $(a,b)$ of orthogonal unit vectors can be moved to $(e_1,e_2)$ by a rotation $R\in SO(3)$ ($(a,b,a\times b)$ is a positively oriented orthonormal basis). $SU(2)\to SO(3)$ is surjective, and taking $g$ with $g(v\cdot\sigma)g^{-1}=(Rv)\cdot\sigma$ gives the first half of the claim. Second half: if $C=c_0\mathbf 1+i\,c\cdot\sigma$ commutes with $\sigma_1$ then $c\times e_1=0$, and if it commutes with $\sigma_2$ then $c\times e_2=0$; hence $c=0$ and $C=\pm\mathbf 1$. ∎

Fact 1 (uniqueness of the vacuum that absorbs the twist) Proved here In the sector $(1,0)$, the configuration with action 0 ($S^{z^{(0)}}(U)=0$) is $U^0$ alone, up to gauge transformations and central transformations in directions 3 and 4 (the operation of multiplying by $-1$ all the links in direction 3 of some direction-3 cut $x_3=c$; likewise for direction 4). On the small $T^2$ alone (the two-dimensional lattice $(\mathbb Z/N)^2$, twist $n_{12}=1$), it is unique up to gauge transformations only.

Proof. By Lemma 4, $U_p=z^{(0)}_p\mathbf 1$. By Lemma 5, the holonomies at the base point satisfy $P_1P_2P_1^{-1}P_2^{-1}=-\mathbf 1$, and the other pairs commute. Re-choosing the gauge at the base point by Lemma 8, $P_1=i\sigma_1$, $P_2=i\sigma_2$. $P_3,P_4$ commute with both, so they are $\pm\mathbf 1$. The holonomies of $U^0$ are $(i\sigma_1,i\sigma_2,\mathbf 1,\mathbf 1)$. If one multiplies the direction-3 links of the cut $x_3=c$ by $-1$, every plaquette containing them contains two (the two direction-3 sides of the $(\mu,3)$ and $(3,4)$ plaquettes lie at the same $x_3$), so the action does not change and only $P_3$ changes sign. Direction 4 is the same. So all four tuples $(i\sigma_1,i\sigma_2,\pm\mathbf 1,\pm\mathbf 1)$ are obtained from $U^0$ by central transformations, and by Lemma 6 configurations with action 0 whose holonomies agree are related by gauge transformations. In two dimensions there are no $P_3,P_4$, so gauge transformations alone suffice. ∎

As the proof of Lemma 8 shows, $(-i\sigma_1,i\sigma_2)$ is moved to $(i\sigma_1,i\sigma_2)$ by conjugation with $g=i\sigma_2$. Central transformations in the directions of the small $T^2$ are absorbed into gauge transformations. Central transformations in the directions of the plane are not absorbed, and there are four configurations with action 0 as gauge classes.

In the quadratic part of the fluctuations around the vacuum $U^0$, the covariant Laplacian of the adjoint field appears. On the small $T^2$ (the two-dimensional lattice $(\mathbb Z/N)^2$), for a field $\phi$ with values in $\mathfrak{su}(2)$ we set

Since $\operatorname{Ad}$ is an orthogonal transformation, $-\Delta_U$ is symmetric and non-negative with respect to the inner product of $\mathfrak{su}(2)$.

Fact 2 Proved here When $U=U^0$, for each component $\phi^1,\phi^2,\phi^3$ in the basis $i\sigma_1,i\sigma_2,i\sigma_3$ of $\mathfrak{su}(2)$, $-\Delta_{U^0}$ splits into scalar Laplacians under the following signed boundary conditions.

Every component is antiperiodic in at least one direction. Hence the smallest eigenvalue of $-\Delta_{U^0}$ is $4\sin^2(\pi/2N)\gt0$, and there are no zero modes.

Proof. $\operatorname{Ad}(i\sigma_1)$ preserves $i\sigma_1$ and changes the signs of $i\sigma_2,i\sigma_3$ ($\sigma_1\sigma_2\sigma_1=-\sigma_2$ and so on), and $\operatorname{Ad}(i\sigma_2)$ preserves $i\sigma_2$ and changes the signs of $i\sigma_1,i\sigma_3$. Only the links of $U^0$ that cross the seam are $\Omega_\mu$, so (7) becomes, component by component, a scalar Laplacian with a sign $\pm1$ attached only to the bonds crossing the seam. On a ring of length $N$ with a sign $s$ on one bond, the difference operator $2f(x)-f(x+1)-f(x-1)$ has eigenfunctions $f(x)=e^{ikx}$ for $k$ satisfying $e^{ikN}=s$, with eigenvalues $2-2\cos k=4\sin^2(k/2)$. If $s=+1$ then $k\in\frac{2\pi}{N}\mathbb Z$ (minimum 0); if $s=-1$ then $k\in\frac{2\pi}{N}(\mathbb Z+\tfrac12)$ (minimum $4\sin^2(\pi/2N)$, at $k=\pi/N$). In two dimensions it is the sum over the two directions, which gives the values in the table. ∎

For the vacuum $U\equiv\mathbf 1$ of a periodic $T^2$ without twist, every component is periodic in both directions, and constant $\phi$ give three dimensions' worth of zero modes. The role of the twist is to remove these zero modes and give the fluctuations the lower bound $4\sin^2(\pi/2N)$. When $N$ is large, $4\sin^2(\pi/2N)\approx\pi^2/N^2$, and restoring the unit of the lattice spacing $a$ gives $(\pi/L)^2$ ($L=Na$).

Reading the same gap as a mass, the lattice mass $m_N$ is given by $4\sinh^2(m_N/2)=4\sin^2(\pi/2N)$, that is,

(the relation between the decay rate of the lattice propagator and the gap, $2\cosh m-2=4\sinh^2(m/2)$) Proved here. This $m_N$ appears in 12-02 §05 as the rate of convergence of the action of a fractional instanton confined to a column.

Fact 3 Proved here In the sector $(1,1)$ there is no configuration with action 0. Hence the minimal action

is positive. Moreover, $S_{\min}(N,M)\le2N^2$.

Proof. If there were a configuration with action 0, then by Lemmas 4 and 5 the holonomies at the base point would satisfy $P_1P_2P_1^{-1}P_2^{-1}=-\mathbf 1$, $P_3P_4P_3^{-1}P_4^{-1}=-\mathbf 1$ ($\varepsilon_{34}=-1$), with the mixed pairs commuting. By Lemma 8, $P_3$ commutes with both $P_1,P_2$ and so is $\pm\mathbf 1$, and the same holds for $P_4$. Then $P_3P_4P_3^{-1}P_4^{-1}=\mathbf 1$, a contradiction. The configuration space $SU(2)^{\#\text{links}}$ is compact and the action is continuous, so the minimum is attained, and since it is not 0 it is positive. Upper bound: evaluating $U^0$ in the sector $(1,1)$, on the $N^2$ plaquettes of the surface $\Pi_q$ we have $U_p=\mathbf 1$, $z_p=-1$, giving $1-\tfrac12(-1)(2)=2$ per plaquette, and 0 elsewhere. ∎

The configuration of the upper bound is a "thin vortex surface", which pays action $2N^2$ on a surface one plaquette thick. Fact 3 says only that the minimal action is not 0; it does not say how small it is (whether there is a smooth lump far cheaper than the thin surface). That is the story of fractional instantons in the next chapter; lattice numerics read $S_{\min}$ as of order $\pi^2$, much smaller than $2N^2$ (12-02 §05).

In the continuum theory, the topological charge of a bundle with twist $n_{12}=n_{34}=1$ is restricted to half-integers, and the action is bounded below by $8\pi^2|Q|/g^2\ge4\pi^2/g^2$ Cited ('t Hooft 1981, equations (2.24) and (2.25) of Tanizaki–Ünsal). Fact 3 is the weakest form (positivity) of the lattice version of this lower bound. That $S_{\min}$ is close to $\pi^2$ on the lattice has not been shown as a lattice version of the continuum lower bound.

• Fact 1 says that the classical vacuum of the small $T^2$ is isolated. On a periodic $T^2$, the whole set of commuting tuples of holonomies (the family of flat connections) is vacuum, and fluctuations move freely over it, so semiclassical counting cannot begin. The twist shrinks it to one point (and its central images).
• Fact 2 says that fluctuations around the isolated vacuum have the lower bound $4\sin^2(\pi/2N)$. This is the classical side of the grounds for the estimate of 12-00 §03, "weak coupling up to the size of the $T^2$". The quantum side (control of the ultraviolet) remains an assumption.
• Fact 3 says that putting in a flux $n_{34}$ in the directions of the plane always costs action, that is, vortices cannot be placed "for free". What moves the bias $\theta$ away from 1 is the competition between the cost of vortices and the entropy of their positions, and Fact 3 is the weakest form of the lower bound on that cost.

All three are statements on a finite lattice with $N$ fixed. None of the statements of this chapter includes increasing $N$ (widening the $T^2$ as a physical length). That is the problem of the crossover band (12-09).

(a) The continuum limit. None of the three facts says anything about $a\to0$. The gap of Fact 2 is a lower bound on classical fluctuations, not a gap of the quantum effective action.

(b) Confinement. The definition of $\theta$ and Proposition 3 are where the tools of confinement sit, and they do not say how small $\theta$ is. Bounds with the lattice spacing fixed are in 12-04, and the semiclassical rate in 12-02.

(c) Other groups. The proofs rely on computations with the Pauli matrices of $SU(2)$ (Lemma 8). The case of $SU(N)$ is not treated in this chapter.

(d) The value of the minimal action. That $S_{\min}(N,M)$ is close to $\pi^2$, and that it approaches $\pi^2$ as $N$ grows, are numerical readings and are not shown in this chapter.

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

[01] CIT FLUX   only the flux matters; the twist relation         't Hooft, Mack–Petkova
[02] PRV MOVE   Lemma 1: moving the surface                       surface Π_q
[03] DEF DF     ΔF, θ = tanh(ΔF/2)                                n₁₂ = 1
[04] CIT TY     Tomboulis–Yaffe                                   13-01
[05] PRV SUM    Prop. 3: θ = E[(−1)^|Q|] ⟸ [02]
[06] PRV ZERO   Lemma 4: action 0 ⟺ U_p = z_p
[07] PRV COMM   Lemma 5: commutator = product of plane signs      central values
[08] CIT TOP    closed path deforms into a word of rounds
[09] PRV HOL    Lemma 6: determined by holonomy ⟸ [07][08]
[10] PRV TE     Lemma 7: U⁰ has action 0
[11] PRV PAIR   Lemma 8: anticommuting pair, commutant ±1         SU(2)
[12] PRV F1     Fact 1 ⟸ [06][07][09][10][11]
[13] PRV F2     Fact 2: gap 4sin²(π/2N) ⟸ [10]                    (ℤ/N)²
[14] PRV MASS   m_N = 2 arsinh sin(π/2N) ⟸ [13]
[15] PRV F3     Fact 3: 0 < S_min ≤ 2N² ⟸ [06][07][11]
[16] CIT BPS    continuum lower bound 4π²/g²                      continuum

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. This chapter has no assumption rows. The cited rows are [01], [04], [08] and [16], and the only citation used by the proofs of the three facts is [08] (through Lemma 6, for Fact 1 only). Facts 2 and 3 close without citations.

What this chapter can say

Sources and reproduction

Words used in this chapter — 't Hooft's twist, sector, flux $n_{\mu\nu}$, the surface $\Pi_q$, the flux free energy $\Delta F$, the bias $\theta$, the twist-summed ensemble, the configuration that absorbs the twist $U^0$, central transformation, the adjoint Laplacian, antiperiodic, the minimal action $S_{\min}$. Their meanings are in A-02 Glossary.

Next chapter to read: 12-02 Vortices and fractional instantons — what lies inside the minimal action of the $(1,1)$ sector, and the counting of the dilute gas.

← 12-00 Entrance — space, not timeContents12-02 Vortices and fractional instantons →