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The trivial Euclidean vacuum lies in the zero-charge sector

Physical scope

Exact classical configuration of the conditional SU(2) Euclidean Yang–Mills formulation on compact oriented S4; no empirical or continuum-quantum claim.

Example

Assume full AC, meaning that every family of nonempty sets has a choice function; in particular, countable choice holds. Let $M = S^{4}$ be a connected compact oriented smooth four-manifold with a chosen orientation and any smooth Riemannian metric $g$ , let $S U (2) = {U \in M_{2} (C) : U^{*} U = I_{2}, \det U = 1}$ , and set $P = M \times S U (2)$ with projection $π (x, U) = x$ and right action $(x, U) \cdot V = (x, U V)$ . In the compact branch of the independent Euclidean model, assume that the bundle-sector code $[P]$ belongs to its model-specified set $Σ$ of admitted sectors, and supply $P$ itself as the representative of this sector. Here $[P]$ is the nonempty minimum-rank set of codes of smooth principal bundles isomorphic to $P$ over $M$ , in the convention of the earlier Euclidean sector definition. In its global product trivialization take the zero connection potential $A = 0$ , with curvature $F_{A} = d A + \frac{1}{2} [A \land A]$ . In the independent Euclidean $S U (2)$ model, use natural units $c = ℏ = 1$ , fix a dimensionless $g_{Y M} > 0$ and $κ (X, Y) = - 2 {tr}_{2} (X Y)$ with its coefficientwise extension to Lie-algebra-valued forms; the action is $S_{E} [A] = (2 g_{Y M}^{2})^{- 1} \int_{M} κ (F_{A} \land *_{g} F_{A})$ . The compactified charge is $ν (P, A) = - (8 π^{2})^{- 1} \int_{M} {tr}_{2} (F_{A} \land F_{A})$ . A classical saddle is a smooth admissible connection whose first variation of $S_{E}$ vanishes for every smooth compactly supported connection variation preserving the supplied boundary data. With these action, charge, and saddle definitions, this configuration has $F_{A} = 0$ , $ν (P, A) = 0$ , and $S_{E} [A] = 0$ . It is a classical saddle in the trivial sector, and it is not a nonzero-charge instanton saddle.

Example references

Tong, §1.1, the $k = 0$ vacuum

Euclidean action, charge, and saddle definitions

Independent Euclidean model assumption

Full Axiom of Choice used by the charge supplier

Facts & Assumptions

Given: Full AC; the Euclidean SU(2) model and conventions in the statement with $g_{Y M} > 0$ ; $M = S^{4}$ with its chosen orientation and Riemannian metric; the admitted trivial bundle $P$ supplied as its sector representative; and the zero connection.

[A1]

The underlying principal bundle is $P = M \times S U (2)$ with the product right action, its sector code satisfies $[P] \in Σ$ , and the supplied representative in that sector is $P$ itself. In the compact branch every smooth connection on this bundle is admissible. The zero potential $A = 0$ in the global product trivialization represents the smooth principal connection $ω_{(x, U)} (v, W) = U^{- 1} W$ for $v \in T_{x} M$ and $W \in T_{U} S U (2)$ .

[A2]

Full AC means that every family of nonempty sets has a choice function; it supplies the charge definition's full-choice hypothesis. The Axiom of Choice, Trace-normalized instanton number and self-dual sectors

[F1]

The local curvature is $F_{A} = d A + \frac{1}{2} [A \land A]$ . Euclidean Yang–Mills sectors and instanton interpretation

[F2]

The conditional Euclidean action is $S_{E} [A] = \frac{1}{2 g_{Y M}^{2}} \int_{M} κ (F_{A} \land * F_{A})$ with $κ = - 2 {tr}_{2}$ . Euclidean Yang–Mills sectors and instanton interpretation

[F3]

The SU(2) charge is $ν = - \frac{1}{8 π^{2}} \int_{M} {tr}_{2} (F_{A} \land F_{A})$ . Euclidean Yang–Mills sectors and instanton interpretation

[F4]

A classical saddle is a smooth admissible connection whose first variation of $S_{E}$ vanishes under every smooth compactly supported connection variation preserving the boundary data. An instanton saddle additionally has nonzero charge and self-dual curvature; there is no boundary in this compact branch. Euclidean Yang–Mills sectors and instanton interpretation

[F5]

The Euclidean action and its coupling are independent model inputs with $g_{Y M} > 0$ . Independent classical Euclidean SU(2) Yang–Mills model

Verification

1.1givenA1F1algebra

The form $ω$ in [A1] is smooth, reproduces the vertical generator $ξ$ on $(0, U ξ)$ , and satisfies $R_{V}^{*} ω = {Ad}_{V^{- 1}} ω$ , so it is a principal connection. Its pullback by $x \mapsto (x, I_{2})$ is $A = 0$ . Since $[P] \in Σ$ with representative $P$ , this connection is admissible. The exterior derivative and wedge–Lie bracket of $A = 0$ both vanish, so [F1] gives $F_{A} = 0$ .

2.1A2F2F3step 1.1algebraF5

The charge integrand in [F3] is zero, so $ν (P, A) = 0$ ; the action integrand in [F2] is also zero, so $S_{E} [A] = 0$ .

3.1F1F3F4step 1.1step 2.1algebra

Since $* 0 = 0$ , both $F_{A} = * F_{A}$ and $F_{A} = - * F_{A}$ hold, but [F4] calls an instanton saddle nonzero-charge. Therefore this zero-charge configuration is not a nonzero-charge instanton; its classical stationarity is verified next.

4.1A1F1F2F4step 1.1algebra∎

Let $a$ be any smooth $s u (2)$ -valued one-form on $M$ and vary $A_{t} = t a$ , with $t \in R$ . This potential defines the smooth principal form $ω_{t} (v, W) = {Ad}_{U^{- 1}} (t a_{x} (v)) + U^{- 1} W$ on the same admitted bundle. These exhaust all connection-variation directions in the global trivialization, and all are compactly supported because $M$ is compact. With $G_{a} = \frac{1}{2} [a \land a]$ , the curvature is $F_{A_{t}} = t d a + t^{2} G_{a}$ . Expanding the quadratic action in [F2] gives only terms of orders $t^{2}, t^{3}, t^{4}$ ; each coefficient is integrable because $M$ is compact and $a$ is smooth. Hence ${\frac{d}{d t} ∣}_{0} S_{E} [A_{t}] = 0$ for every admissible variation, which verifies the saddle condition in [F4]. Thus the admitted zero connection is a classical saddle in the trivial sector.

Depends on

Dependency tree · 2 levels

Sources

David Tong, TASI Lectures on Solitons, Lecture 1: Instantons