Independent classical Euclidean SU(2) Yang–Mills model

Assume countable choice ${\mathrm{A}\mathrm{C}}_{\omega }$, meaning that every countable family of nonempty sets has a choice function, for the specified smooth gauge-data and principal-connection constructions. Let $(X,g_X)$ be a connected, compact, oriented, finite-dimensional Hausdorff second-countable smooth four-manifold without boundary, with a supplied Riemannian metric. The alternative flat-space branch is standard oriented Euclidean $M={\mathbb{R}}^4$ together with a fixed smooth trivialization outside a ball and a specified smooth extension $(\bar{P},\bar{A})$ over $X=S^4$ with its unit round metric, which is conformal to the flat metric in the stereographic chart; here $\bar{A}$ is a smooth principal connection one-form on $\bar{P}$. In the compact branch set $M=X$ and $(P,A)=(\bar{P},\bar{A})$. The extension and its transition map are part of the field data. Finite action alone is not asserted to imply such an extension. Write $g=g_X$ in the compact branch and $g=\delta$ in the flat branch.

Fix the global structure group $G=SU(2)=\{U\in M_2(\mathbb{C}):U^{\ast }U=I_2,\det U=1\}$, not its quotient, with Lie algebra $\mathfrak{s}\mathfrak{u}(2)=\{X\in M_2(\mathbb{C}):X^{\ast }=-X,{\mathrm{tr}}_{\mathbf{2}}X=0\}$ and commutator bracket. For $U\in SU(2)$ define ${\mathrm{Ad}}_{U}X:=UX{U}^{-1}$ for $X\in \mathfrak{s}\mathfrak{u}(2)$. Let $\bar{P}\to X$ be a smooth right principal $SU(2)$-bundle: the right action is smooth, free, and transitive on each fibre, and local trivializations are equivariant. Its bundle-sector code $[\bar{P}]$, the nonempty minimum-rank set of codes of smoothly isomorphic bundles in the supplied non-Abelian gauge-data convention, belongs to the model-specified set $\mathrm{\Sigma }$ of admitted bundle sectors. In the flat branch the physical bundle and connection are $P=\bar{P}{\mathrm{\mid }}_M$ and $A=\bar{A}{\mathrm{\mid }}_P$. An isomorphism here is a smooth equivariant diffeomorphism over ${\mathrm{id}}_X$, and $\mathrm{\Sigma }$ may be a singleton. For the defining representation, use the Pauli matrices ${\sigma }_1=\left(\begin{array}{cc}0& 1\\ 1& 0\end{array}\right)$, ${\sigma }_2=\left(\begin{array}{cc}0& -i\\ i& 0\end{array}\right)$, and ${\sigma }_3=\left(\begin{array}{cc}1& 0\\ 0& -1\end{array}\right)$, and put $T_a=-i{\sigma }_a\mathrm{/}2$. The earlier lemma Compactness and normalized invariant form for SU(2) states that $T_1,T_2,T_3$ form a real basis of $\mathfrak{s}\mathfrak{u}(2)$ and that $\kappa (X,Y)=-2{\mathrm{tr}}_{\mathbf{2}}(XY)$ is a positive-definite symmetric real bilinear form with $\kappa (T_a,T_b)={\delta }_{ab}$ and $\kappa ({\mathrm{Ad}}_{U}X,{\mathrm{Ad}}_{U}Y)=\kappa (X,Y)$ for all $U\in SU(2)$ and $X,Y\in \mathfrak{s}\mathfrak{u}(2)$. Writing $\bar{\omega }:=\bar{A}$, this principal connection one-form satisfies $R_g^{\ast }\bar{\omega }={\mathrm{Ad}}_{g^{-1}}\bar{\omega }$ and $\bar{\omega }({\xi }_{\bar{P}})=\xi$ for each $g\in SU(2)$ and $\xi \in \mathfrak{s}\mathfrak{u}(2)$, where $R_g(p)=pg$ and ${\xi }_{\bar{P}}(p)={\frac{d}{dt}\mid }_{0}p{\exp }_{SU(2)}(t\xi )$. In a local section $s_i$ of $\bar{P}$, write $A_i=s_i^{\ast }\bar{\omega }$ and $F_i=d{A}_i+\frac{1}{2}[A_i\wedge A_i]$; the alternating wedge–Lie bracket is normalized by $\frac{1}{2}[A_i\wedge A_i](Y,Z)=[A_i(Y),A_i(Z)]$. The Hodge star is determined by $\alpha \wedge {\ast }_{g_X}\beta =⟨\alpha ,\beta {⟩}_{g_X}d{\mathrm{vol}}_{g_X}$ on two-forms, acts on the form part of $\mathfrak{s}\mathfrak{u}(2)$-valued forms, and satisfies ${\ast }_{g_X}^2=1$ there.

The principal-curvature result gives $F_j={\mathrm{Ad}}_{u_{ij}^{-1}}{F}_i$ on an overlap where $s_j=s_i{u}_{ij}$. Hence the $F_i$ are local representatives of one global curvature $F_{\bar{A}}$; in the compact branch put $F_A=F_{\bar{A}}$ and in the flat branch put $F_A=F_{\bar{A}}{\mathrm{\mid }}_M$. For smooth $\mathfrak{s}\mathfrak{u}(2)$-valued $p$- and $q$-forms $\alpha ={\sum }_{a=1}^3{T}_a{\alpha }^a$ and $\beta ={\sum }_{b=1}^3{T}_b{\beta }^b$, with real-valued coefficient forms ${\alpha }^a$ and ${\beta }^b$, define their coefficientwise pairing by

$\kappa (\alpha \wedge \beta ):={\sum }_{a,b=1}^3\kappa (T_a,T_b){\alpha }^a\wedge {\beta }^b\mathrm{.}$

This is independent of the chosen basis. The $\mathrm{Ad}$-invariance of $\kappa$ and the stated curvature transition law make the local four-forms $\kappa (F_i\wedge {\ast }_g{F}_i)$ agree on overlaps; they therefore define a global four-form, denoted $\kappa (F_A\wedge {\ast }_g{F}_A)$ in the action. Adopt, independently of any Lorentzian theory, the source-free pure Euclidean $SU(2)$ model on exactly these supplied geometric data. No matter fields are present. Fix a dimensionless coupling $g_{\mathrm{Y}\mathrm{M}}>0$. In natural units $c=\mathrm{\hslash }=1$, $A_{\mu }$ has mass dimension one and $F_{\mu \nu }$ mass dimension two. Adopt the dimensionless action

$S_E[A]=\frac{1}{2g_{\mathrm{Y}\mathrm{M}}^2}{\int }_M\kappa (F_A\wedge {\ast }_g{F}_A)\mathrm{.}$

In the flat-space branch require finite action in addition to the specified smooth compactification extension. Smooth gauge transformations are equivariant bundle automorphisms covering the identity; the action is evaluated on the supplied admissible connection. Classical stationarity means vanishing first variation for smooth compactly supported adjoint-bundle-valued one-form variations preserving the supplied extension/boundary data. On a closed base every smooth variation has compact support.

This is a classical Euclidean model assumption. It neither derives a Euclidean theory by Wick rotation from a Lorentzian postulate nor asserts empirical confirmation, reflection positivity, a functional measure, gauge fixing, a convergent quantum integral, or a continuum quantum theory. The closed-manifold formulation is the explicitly adopted geometric extension of the cited flat Euclidean action; finite action alone is not claimed to prove removable singularities or compactification.

David Tong, Lecture 1 §§1.1–1.1.1, printed pp. 6–9

SU(2) matrix realization and normalized positive invariant form

Local curvature transition law