2026-10-09 · chapter Yang–Mills and the mass gapPart 11 Further considerations — beyond Assumption H, and other viewssymbols, where they are defined, units · Kiichi, Shiori and Rin

The whole book　Contents and reading orders

Table of notation for Part 11

What this page shows — A list of the symbols used in Part 11. For each symbol, its meaning, where it is defined, its unit, and whether it is the same quantity as the symbol with the same letter in Part 10 or a different one.

Prerequisite chapters — None. The notation of Part 10 is in 10-01 §01 and A-01 Table of notation.

Said first — Quantities are counted in lattice units (the lattice spacing is 1). Langevin time has the dimension of length squared (diffusion), and the value of $C_P$ measured in physical units is $a^2C_P$ . Numbers are values under the conditions of the chapter of definition, and their labels are in that chapter.

List of symbols

Table 1 Symbols of Part 11 ("Same" = the same quantity as the same letter in Part 10; "New" = a symbol added in Part 11)
Symbol	Meaning	Where defined	Unit (lattice) / physical reading	Correspondence with Part 10
$\ell$	A ruler of length given from outside (independent of the gap). In four dimensions $\xi_{\rm RG}$ ; for the 0+1 double well $1/\omega_b=1$	11-03 §02	Length / $a\ell$	New (in four dimensions the same quantity as $\xi_{\rm RG}$ of 10-01 §07)
$\xi$	Correlation length $1/m_{\rm lat}$ . In 0+1, $\hbar/\Delta$	(1) and (17) of 10-01	Length / $a\xi$	Same
$\xi_{\rm RG}$	The bookkeeping length (the length fixed by counting the steps that coarsen the lattice) $=1/a$	10-01 §07	Length / 1	Same
$\xi_{XY}$	The static length of the two-dimensional $U(1)$ strip, $-1/\ln(I_1(\beta)/I_0(\beta))$ (the inverse of the string tension)	11-01 §02, 11-02	Length	New
$R=c\ell$	Side of the local window or box	11-03 §03	Length	New
$c$	The ratio of the window length to the ruler	(2) of 11-03	Dimensionless	New
$\mathcal D$	The algebra of time-averaged functionals (weights with bounded support)	(1) of 11-03	—	New (fixes the collection over which the supremum in (7) of 10-01 is taken)
$\mathcal D_c$	The subfamily of $\mathcal D$ whose weights have support in the window $[0,c\ell]$	(2) of 11-03	—	New
$\mathcal F_1$	The family of time-integrated single-time functions	11-04	—	A name given to the family of test functions of Proposition B of 10-01
$C_P$	The Poincaré constant $\sup\operatorname{Var}/\mathcal E$ (path measure)	(7) of 10-01. In Part 11 the supremum is taken over $\mathcal D$	Length squared (Langevin time) / $a^2C_P$	Same ((β) of 10-01 Table 3)
$C_P^{\rm loc}(c)$	The local Poincaré constant (the supremum over $\mathcal D_c$ )	(2) of 11-03	Length squared	New
$C_P^{\rm loc,0}(R)$	The value of the massless reference in the same box with the same graduation, $=\Gamma(R)R^2$	(4)(5) of 11-03	Length squared	New
$K(c)$	The constant of the local H↑ in absolute form, $C_P^{\rm loc}(c)\le K(c)\ell^2$	(3) of 11-03	Dimensionless	The local version of $C$ in (14) of 10-01 ( $K$ in Lean)
$K_{\rm rel}$	The ratio $C_P^{\rm loc}/C_P^{\rm loc,0}$	(5) of 11-03	Dimensionless	New
$\Gamma$	The coefficient of the value for the reference box, $C_P^{\rm loc,0}(R)/R^2$ . $\Gamma_\infty=0.0740$ for the scalar, $\Gamma_Q=0.0277$ for the gauge quotient	11-03 §03, Table 1	Dimensionless	New
$\lambda_W$	The Dirichlet value of the box within the test space multiplied by $R^2$ ( $4\pi^2$ for the scalar, $72.0$ for the gauge quotient). For a window function $W$ , $\\|\nabla W\\|^2/\\|W\\|^2$	(7) of 11-01, 11-03 Table 1	Dimensionless	New
$\kappa$	The constant comparing the Langevin metric with the metric of the kinetic term ((A3)). A different quantity from the correction κ to the raw rate on the strip in 11-02 §05 (which is neither κ_BE nor this κ)	Proposition B of 10-01	Dimensionless	Same
$\kappa_{\rm BE}$	The lower bound on the Bakry–Émery curvature. A different quantity from the comparison constant κ of Proposition B	11-06 §01	Inverse length squared	New
$s$	The rate of rarity (the exponent with which the weight of defects falls as e^(−s/ħ)). The same quantity as ℓ of Part 10	11-06 §03	Dimensionless (in units of action)	The rarity exponent ℓ of 10-09
$K_*$	The threshold of the bridge $\kappa/(\lambda_W\Gamma)$	11-04	Dimensionless	New
$\ell_c$	The length at which conservation breaks on the two-dimensional $U(1)$ strip (the bend in the relaxation of the flux dipole)	11-01 §02, 11-02	Length	New
$\ell_*$	The length at which conservation breaks (the general name. The candidate definition is (2) of 11-02: the smallest R at which the ratio reaches half its saturation value)	11-02	Length	New
$R_*$	The crossing length $\sqrt{d\lambda_W/h}$ (the boundary between the regimes of transport and of the barrier)	(7) of 11-01	Length	New
$\lambda_Q$	The relaxation rate of the topological charge of the strip, $1/\tau_{\rm int}(Q)$	11-02	Inverse Langevin time	New
$\tau_{\rm int}$	Integrated autocorrelation time	(1) of 11-01, 10-02 §05	Langevin time (in the originals, numbers of sweeps or molecular-dynamics time)	Same
$m$	The gap as an inverse time, $m=\Delta/\hbar$ . In four dimensions $m_{\rm lat}$	(1) and (17) of 10-01	Inverse length	Same ( $m_{\rm lat}$ )
$m_{\rm phys}$	The mass in physical units $m\ell$ ( $m_{\rm lat}\xi_{\rm RG}$ in four dimensions)	(12)(16) of 10-01	Dimensionless (physical units)	Same
$\Delta$	The gap of the Hamiltonian $E_1-E_0$	Proposition B of 10-01	Energy	Restricted to one meaning (§02 below)
$\omega_b$	The angular frequency at the bottom of the double well $\sqrt{V''(\pm1)}=1$	(7) of 11-03, 11-04	Inverse time	New (a quantity of the model of 10-04)
$S_0$	The kink action $2/3$	10-04 §03	Action	Same
$\varphi(x)$	The window factor $1-(1-e^{-x})/x$	The bracket in (17) of 10-01, 11-04	Dimensionless	Same (different from the plaquette angle $\varphi_p$ )
$\varphi_p$	The plaquette angle of the two-dimensional strip	11-01 §02, 11-02	Angle	New
$Q,\ Q_s$	The topological charge, the smoothed topological charge. On the two-dimensional strip $Q=\sum_p[\varphi_p]/2\pi$	11-01 §02, 11-03 §04	Integer ( $Q_s$ is real)	The same as $Q$ in the note of 10-01 §05
$\chi_t,\ h,\ d,\ g$	The topological susceptibility, the density of breaking of conservation, the transport coefficient, the coefficient of the boundary layer	(4)(5) of 11-01	Lattice units ( $h,d,g$ depend on the graduation of $\mathcal E$ )	New
$C_P^{\rm sec}(n),\ C_{\rm trans}$	The Poincaré constant of a sector, the quantity between sectors	(8)(10) of 11-03	Length squared	New

Symbols with the same letter but different meanings

$\Delta$ . In 10-01 of Part 10, $\Delta$ was used in two meanings: $\Delta(\beta)=m_{\rm lat}/a$ of (1) (the mass in physical units) and $\Delta=E_1-E_0$ of Proposition B (the gap of the Hamiltonian). In Part 11, $\Delta$ is restricted to the meaning $E_1-E_0$ of Proposition B, and the mass in physical units is written $m_{\rm phys}$ .

$K$ . (14) of 10-01 writes the constant of H↑ as $C$ , and the Lean theorems write the same constant as $K$ . $K(c)$ in Part 11 is its local version, $K_{\rm rel}$ is the constant of the ratio form, and $K_*$ is the threshold of the bridge. The three are different quantities.

$\varphi$ . The window factor $\varphi(x)$ carries an argument, and the plaquette angle of the two-dimensional strip is $\varphi_p$ with a subscript.

$Q$ . The four-dimensional topological charge and the topological charge of the two-dimensional strip are written with the same letter. Both are "global integers not reflected in the gap", examples of $Q$ of the pattern of 11-01. The two-dimensional numbers do not imply anything for four dimensions.

Symbols carrying $\lambda$ . $\lambda_1$ (the Langevin gap $=1/C_P$ , 10-01), $\lambda_W$ (the Dirichlet value of the box) and $\lambda_Q$ (the relaxation rate of the topological charge of the strip) are different quantities.

Revised 2026-10-09: first version.
2026-10-09: added to Table 1 where the candidate definition of ℓ_* is, the two meanings of κ, κ_BE, and the rate of rarity s.