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Finite SU(2) Wilson plaquette measure

Definition

Assume ${A C}_{ω}$ . Use the matrix group $G_{2} = S U (2) = {U \in M_{2} (C) : U^{*} U = I_{2}, \det U = 1}$ with matrix multiplication and the embedded smooth-manifold structure on $M_{2} (C) ≅ R^{8}$ used for special unitary matrix groups. Its defining representation is the identity smooth homomorphism $ρ_{2} (U) = U$ on $W = C^{2}$ with positive-definite Hermitian form $h (v, w) = v^{*} w$ . In real coordinates every element has the form

U (x) = (\begin{array}{cc} x_{0} + i x_{3} & x_{2} + i x_{1} \\ - x_{2} + i x_{1} & x_{0} - i x_{3} \end{array}), x \in S^{3} : = {x \in R^{4} : \sum_{j = 0}^{3} x_{j}^{2} = 1} .

Let $σ_{3}$ be the induced round surface-area measure on $S^{3}$ , whose total area is $2 π^{2}$ , and let $μ_{2}$ be the pushforward of $σ_{3} / (2 π^{2})$ under $x \mapsto U (x)$ . In spherical coordinates $x_{0} = \cos θ$ , $(x_{1}, x_{2}, x_{3}) = \sin θ n$ , $0 \leq θ \leq π$ , $n \in S^{2}$ , its coordinate density is $d μ_{2} = (2 / π) \sin^{2} θ d θ d ω_{2} (n)$ , where $n = (\sin φ \cos ψ, \sin φ \sin ψ, \cos φ)$ and $d ω_{2} = (1 / (4 π)) \sin φ d φ d ψ$ for $0 \leq φ \leq π$ , $0 \leq ψ < 2 π$ . Define the real class function $τ (U) : = \frac{1}{2} {Tr}_{2} (U) = x_{0}$ .

For each oriented face $p$ , let $U_{p}$ be the product of its link matrices in boundary traversal order, using inverse matrices on reverse edges. For dimensionless $β \geq 0$ , define the pure-gauge Wilson plaquette action on any finite oriented lattice by

S_{β} (U) : = β \sum_{p} (1 - τ (U_{p})), β = \frac{4}{g_{0}^{2}}

when a dimensionless bare coupling $g_{0} > 0$ is used. The finite link measure is the product of $μ_{2}$ over positively oriented edges, and its normalized expectation is the finite product integral with density proportional to $e^{- S_{β}}$ . The one-plaquette normalizing factor and one-plaquette probability are

z_{β} : = \int_{G_{2}} e^{β τ (U)} d μ_{2} (U), d ν_{β} (U) : = z_{β}^{- 1} e^{β τ (U)} d μ_{2} (U) .

Since $τ (U) = x_{0} \in [- 1, 1]$ , every $β \geq 0$ gives $e^{- β} \leq z_{β} \leq e^{β}$ ; hence $z_{β}$ is finite and positive and $ν_{β}$ is a normalized probability measure.