Euclidean Yang–Mills sectors and instanton interpretation

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 $(P,A)=(\bar{P},\bar{A})$. In the flat branch set $P=\bar{P}{\mathrm{\mid }}_M$ and $A=\bar{A}{\mathrm{\mid }}_P$. 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 $G=SU(2)$, not its quotient, and 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. An isomorphism here is a smooth equivariant diffeomorphism over ${\mathrm{id}}_X$, and $\mathrm{\Sigma }$ may be a singleton. Writing $\bar{\omega }:=\bar{A}$, this smooth $\mathfrak{s}\mathfrak{u}(2)$-valued principal connection one-form satisfies $R_g^{\ast }\bar{\omega }={\mathrm{Ad}}_{g^{-1}}\bar{\omega }$ and $\bar{\omega }({\xi }_{\bar{P}})=\xi$, where ${\xi }_{\bar{P}}(p)={\frac{d}{dt}\mid }_{0}p{\exp }_{SU(2)}(t\xi )$. 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 earlier lemma Principal curvature, Cartan structure equation, and Bianchi identity shows that the $F_i$ are local representatives of the global curvature $F_{\bar{A}}$; $F_A$ denotes its restriction to $M$, and in the compact branch $F_A=F_{\bar{A}}$.

Work under the independent physical assumption in the earlier postulate Independent classical Euclidean SU(2) Yang–Mills model, on precisely the geometric data specified above: it is source-free pure $SU(2)$, contains no matter fields, and adopts the action below independently of the Lorentzian model; no Wick-rotation inference is made. Use anti-Hermitian defining-representation generators with ${\mathrm{tr}}_{\mathbf{2}}(T_a{T}_b)=-\frac{1}{2}{\delta }_{ab}$, the positive invariant form $\kappa (X,Y)=-2{\mathrm{tr}}_{\mathbf{2}}(XY)$, and a fixed dimensionless coupling $g_{\mathrm{Y}\mathrm{M}}>0$. In natural units $c=\mathrm{\hslash }=1$, $A_{\mu }$ has mass dimension one, $F_{\mu \nu }$ has mass dimension two, and $S_E\mathrm{/}\mathrm{\hslash }$ is dimensionless. The conditional Euclidean action is

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

In the flat-space branch admissibility also requires $S_E[A]<\mathrm{\infty }$ on ${\mathbb{R}}^4$. For $U,V\in \mathfrak{s}\mathfrak{u}(2)$ define $B_{\mathrm{t}\mathrm{o}\mathrm{p}}(U,V):=-{\mathrm{tr}}_{\mathbf{2}}(UV)=\kappa (U,V)\mathrm{/}2$. The mathematical charge used here is the supplied trace-normalized charge of the compactified connection,

$\nu (\bar{P},\bar{A})=\frac{1}{8{\pi }^2}{\int }_X{B}_{\mathrm{t}\mathrm{o}\mathrm{p}}(F_{\bar{A}}\wedge F_{\bar{A}})=-\frac{1}{8{\pi }^2}{\int }_X{\mathrm{tr}}_{\mathbf{2}}(F_{\bar{A}}\wedge F_{\bar{A}})=-⟨c_2({\bar{E}}_{\mathbf{2}}),[X]⟩\in \mathbb{Z},$

where ${\bar{E}}_{\mathbf{2}}=\bar{P}{\times }_{\mathbf{2}}{\mathbb{C}}^2$ and the final equality uses the supplied instanton-number definition with its full-Axiom-of-Choice and Chern–Weil hypotheses. This sign makes a positive-charge self-dual configuration satisfy $F_{\bar{A}}={\ast }_{g_X}{F}_{\bar{A}}$ and a negative-charge anti-self-dual configuration satisfy $F_{\bar{A}}=-{\ast }_{g_X}{F}_{\bar{A}}$ on the compactified base. The charge label is not asserted to classify every connected component or every gauge orbit.

For each sector code $s\in \mathrm{\Sigma }$, the model supplies one representative smooth right principal bundle ${\bar{P}}_s\to X$ with $[{\bar{P}}_s]=s$, taking ${\bar{P}}_{[\bar{P}]}=\bar{P}$ for the selected bundle. In the flat-space branch use its restriction $P_s={\bar{P}}_s{\mathrm{\mid }}_M$ and its specified smooth compactification extension. Let ${\mathcal{A}}_{\nu }(\mathrm{\Sigma })$ denote the set of tagged pairs $(s,A)$, where $s\in \mathrm{\Sigma }$ and $A$ is an admissible smooth connection on that supplied representative whose compactified charge is $\nu$. Each connection space is a set of smooth one-forms on the specified total space, and the supplied set-indexed family makes this tagged union a set. The representatives are part of the model data; changing them transports connection data by bundle isomorphisms, and no universal choice of representatives is asserted. In the flat-space branch, admissible connections are required to have the specified smooth extension and finite Euclidean action.

A classical Euclidean saddle is a smooth admissible connection whose first variation of $S_E$ vanishes for every smooth compactly supported connection variation preserving the boundary data. An instanton (respectively anti-instanton) saddle is a nonzero-charge saddle satisfying the positive (respectively negative) self-duality equation. Let $\mathcal{A}(\mathrm{\Sigma }):={⨆}_{\nu \in \mathbb{Z}}{\mathcal{A}}_{\nu }(\mathrm{\Sigma })$ be the tagged disjoint union of the admissible connection spaces over the model-specified bundle sectors. A source observable is a complex-valued function $O:\mathcal{A}(\mathrm{\Sigma })\to \mathbb{C}$ that is constant on connection data related by a smooth diffeomorphism $\mathrm{\Phi }:P\to P$ satisfying $\pi \circ \mathrm{\Phi }=\pi$ and $\mathrm{\Phi }(pg)=\mathrm{\Phi }(p)g$ for all $p\in P$ and $g\in SU(2)$. No continuity, measurability, or other regularity of $O$ is assumed. Gauge transformations are these smooth principal-bundle automorphisms.

The notation

$Z_{\mathrm{f}\mathrm{o}\mathrm{r}\mathrm{m}\mathrm{a}\mathrm{l}}(\theta ;O)={\sum }_{\nu }{e}^{i\theta \nu }{\int }_{{\mathcal{A}}_{\nu }(\mathrm{\Sigma })}\mathcal{D}AO[A]e^{-S_E[A]}$

denotes only a formal continuum sector expression over connections, with gauge redundancy left untreated. Here $\theta$ is dimensionless and has period $2\pi$ when $\nu$ is integral. No continuity, measurability, or other regularity of the formal expression is asserted. The notation defines no functional measure, gauge fixing or orbit-volume prescription, determinant, convergent integral, Hilbert-space sector, or continuum quantum field theory, and selects no finite regulator or limit.

Tong, §§1.1–1.1.1, complete instanton lecture

Vandoren and van Nieuwenhuizen, §§1–2, complete instanton lecture notes

Countable-choice assumption for the supplied connection construction

Full choice assumption in the integral charge comparison

Specified global form, bundle sector, and representation data

Right-action principal connection form

Graded wedge–Lie bracket convention

Cartan curvature and Bianchi identity

Gauge transformations as base-fixing principal automorphisms

Current Lorentzian SU(2) model postulate; it has no Euclidean branch

Trace-normalized charge and self-duality definition on the compact base

Independent classical Euclidean SU(2) Yang–Mills model