Conventions — Geometric QCD

Conventions

Geometry and operators

Modules I–IV use Euclidean ${\mathbb{R}}^4$, ${\delta }_{\mu \nu }$ and ${ϵ}_{1234}=1$. For two-forms, $(\ast B{)}_{\mu \nu }={ϵ}_{\mu \nu \rho \sigma }{B}_{\rho \sigma }/2$ and ${\ast }^2=1$. The spinor rotating-surface example in V.5 uses Minkowski signature $(+---)$ and states that choice explicitly. Euclidean self-duality is not carried unchanged into Lorentzian signature.

The anti-Hermitian connection is $\mathcal{A}=igA$ when $A$ is Hermitian. Write $D=\partial +\mathcal{A}$, $\mathcal{F}=[D,D]=igF$, and Euclidean $\{{\gamma }_{\mu },{\gamma }_{\nu }\}=2{\delta }_{\mu \nu }$. Define ${\sigma }_{\mu \nu }=[{\gamma }_{\mu },{\gamma }_{\nu }]/2$. Then $/D^2=D^2+{\sigma }_{\mu \nu }{\mathcal{F}}_{\mu \nu }/2$.

Parallel sections obey $DU=0$ and have $U=\mathcal{P}{e}^{-\int \mathcal{A}}$ along the stated orientation. The source’s ordered covariant-derivative product and the common Wilson convention $\mathcal{P}{e}^{+ig\oint A}$ use their explicitly stated transport orientation. I.3–I.5 keep the orientation and area insertion together. Reversing one sign without the other is invalid.

Loop quantities and color normalization

$w_C[A]=N_c^{-1}\mathrm{tr}{U}_C[A]$ is the unaveraged observable; $W[C]=⟨w_C[A]⟩$. The full two-loop correlator is $⟨w_{C_1}{w}_{C_2}⟩$; its connected part subtracts $W[C_1]W[C_2]$.

The worked color algebra uses $\mathrm{tr}{T}^a{T}^b={\delta }^{ab}/2$ and $\lambda =g^2{N}_c$. In I.5 the normalized color insertion has coefficient $\lambda /2$. Source equations written with coefficient $\lambda$ have their own loop-operator normalization, denoted ${\lambda }_{\mathrm{loop}}$ when comparing. The induced coupling must be matched, not inferred by identifying those symbols.

I.4 defines $A_{\mu \nu }$ as an oriented, point-split antisymmetric insertion and ${\Delta }_{\mu }$ as the difference of neighboring dot derivatives. The vector operator is ${\mathcal{L}}_{\nu }={\Delta }_{\mu }{A}_{\mu \nu }$; $L=\oint {\dot{C}}_{\nu }{\mathcal{L}}_{\nu }$. The surrounding holonomy remains inside every traced insertion.

Fourier transforms and harmonic extension

Finite examples use $\hat{f}(p)=\int e^{ipx}f(x)dx$, inverse measure $dp/(2\pi )$, and $\widehat{f^{\prime }}=-ip\hat{f}$. The coordinate Dirac kernel in V.3 uses $e^{iq\cdot \tau }$ and $d^{4}q/(2\pi {)}^4$.

For the disk, $z=u+iv$, $d^{2}z=dudv$, and ${\partial }_z=({\partial }_u-i{\partial }_v)/2$. The Hilbert transform has Fourier multiplier $-i\mathrm{sgn}n$. Thus $H\cos n\theta =\sin n\theta$, and $f=(C+iHC)/2$ solves the fixed boundary harmonic problem. The null condition is $f^{\prime }\cdot f^{\prime }=0$, a bilinear square, not $|f^{\prime }{|}^2=0$.

Dimensions and distinct symbols

Natural units are used. A length has mass dimension $-1$; ordinary and proposed areas have dimension $-2$; $\kappa$ and string tension $\sigma$ have dimension $2$. $\sqrt{\sigma }$ is a mass. The quoted fit scale is $\sqrt{\sigma }\simeq 0.417$ GeV.

The auxiliary surface-fermion mass in Module IV is not the endpoint quark mass in Module V. The RG scale ${\Lambda }_{\mathrm{QCD}}$ is not the spinor norm $\Lambda ={\lambda }^{†}\lambda$. The metric factor $e^{2\rho }$, geometric density denoted $e$ in the source, and boundary einbein must be distinguished.

In V.6–V.7, $\mathcal{E}=E/\sqrt{\sigma }$, $x=m/\sqrt{\sigma }$, $K$, $\beta$, and $J$ are dimensionless. The source’s stationary phase is written in these dimensionless variables throughout the worked derivation.