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The zero-charge regular datum and framed flat ASD sector

Example

Fix $N \geq 2$ and equip $V = C^{0} = 0$ and $W = C^{N}$ with their standard Hermitian products. A raw ADHM datum consists of $B_{i} \in End (V)$ , $I : W \to V$ , and $J : V \to W$ , with moment maps

$μ_{C} = [B_{1}, B_{2}] + I J, μ_{R} = [B_{1}, B_{1}^{†}] + [B_{2}, B_{2}^{†}] + I I^{†} - J^{†} J .$

Stability means the span of all vectors $B_{i_{1}} \dots B_{i_{m}} I (w)$ is $V$ , costability means that no nonzero common invariant subspace lies in $\ker J$ , and the regular zero-FI quotient is the set of data with both moments zero and both conditions, modulo $U (V)$ . It consists of the unique empty datum at $k = 0$ .

Give $S^{4} = H P^{1}$ its unit round metric and positive stereographic orientation. A framed ASD object is a smooth Hermitian rank- $N$ bundle $E \to S^{4}$ with a smooth unitary connection $A$ , a unitary framing at infinity, $F_{A}^{+} = (F_{A} + *_{g} F_{A}) / 2 = 0$ , and

$q_{N} (E, A) = - \frac{1}{8 π^{2}} \int_{S^{4}} {tr}_{N} (F_{A} \land F_{A}) = 0,$

modulo unitary bundle isomorphisms preserving the framing and connection. This sector consists of the trivial rank- $N$ bundle with its flat connection and fixed framing, and its group of framing-preserving parallel unitary automorphisms is trivial.

Example references

Conditional zero-FI regular ADHM and framed ASD correspondence

ADHM linear data, stability, costability and FI levels

Euclidean Yang–Mills self-dual action bound

$S^{n}$ is simply connected for every $n \geq 2$

Donaldson, §§2–3, printed pp. 3–5

Nakajima–Yoshioka, §3.1, printed p. 17

Facts & Assumptions

Given: $N \geq 2$ , $k = 0$ , the raw ADHM definitions, and a smooth framed unitary ASD connection on the round $S^{4}$ with $q_{N} = 0$ .

[F1]

For $V = C^{0}$ , each ADHM endomorphism or framing map is the unique zero-dimensional map; stability and costability hold vacuously, and both $U (0)$ and $G L (0)$ are trivial. ADHM linear data, stability, costability and FI levels

[F2]

On the real vector space $u (N)$ of skew-Hermitian matrices, $B (X, Y) = - {tr}_{N} (X Y)$ is a positive-definite real bilinear form. The charge formula in the Example is $q_{N} = (8 π^{2})^{- 1} \int B (F_{A} \land F_{A})$ . Under full AC the general action-bound supplier gives the ASD zero-charge implication; step 1.2 proves the needed identity directly here, so that supplier is not used without its axiom premise. Trace-normalized instanton number and self-dual sectors, Euclidean Yang–Mills self-dual action bound

[A1]

Only for the additional comparison in step 2.1, assume full AC and the exact ordinary quotient, GAGA, Ward and monad supplier hypotheses of the conditional correspondence theorem. The direct classification below does not invoke that general correspondence. Conditional zero-FI regular ADHM and framed ASD correspondence, The Axiom of Choice

[F3]

The sphere $S^{4}$ is simply connected. A flat connection has local parallel frames and path-independent parallel transport on coordinate balls. $S^{n}$ is simply connected for every $n \geq 2$ , A flat connection admits local parallel frames, Flat connections have locally path-independent parallel transport on a coordinate ball

Proof

Given: The hypotheses in the Example.

1.1F1algebra

When $k = 0$ , $V = 0$ . There is exactly one quadruple of maps, all with zero-dimensional domain or codomain. Both moment equations are the unique equality of endomorphisms of $0$ ; the span of the empty family is $0 = V$ , and no nonzero subspace of $V$ exists. Thus this datum is stable and costable, and the quotient by $U (0)$ is one point.

1.2givenF2F3construct

The Hodge identity and $*_{g} F_{A} = - F_{A}$ give $\int_{S^{4}} ∣ F_{A} ∣_{B}^{2} d v o l = \int B (F_{A} \land *_{g} F_{A}) = - 8 π^{2} q_{N} = 0$ , where the norm uses the round metric and the positive form [F2]. Its continuous nonnegative integrand must vanish everywhere: a positive value would give a positive integral on an open neighborhood. Hence $F_{A} = 0$ . For two piecewise smooth paths from $\infty$ to $x$ , their concatenation is a loop and is null-homotopic by [F3]. Pull back a cover by coordinate balls with local parallel frames to the compact homotopy square; a Lebesgue number gives a finite subdivision with each cell mapped into one such ball. Transport along a continuous edge is defined using that ball's parallel frame; on overlaps parallel-frame changes are locally constant, so the definition agrees along each connected edge. Each cell has identity boundary transport. Sweeping through successive cells cancels the transport on shared edges with its inverse and leaves identity transport around the external boundary. Thus transport from $\infty$ is path-independent. Transporting the fixed framed orthonormal basis gives smooth parallel sections, as seen in local parallel frames, and a framed unitary isomorphism with the trivial flat bundle. A parallel unitary automorphism preserving the framing is identity, since its value everywhere is transported from its identity value at $\infty$ . Conversely, the trivial framed flat connection is ASD and has $q_{N} = 0$ . Thus the sector has exactly one class and its based parallel stabilizer is trivial.

2.1A1F1F2step 1.1step 1.2∎

Pairing the unique empty datum with the trivial framed flat bundle gives the bijection of the two singleton sets established in steps 1.1 and 1.2. For the additional comparison, impose all hypotheses [A1], including full AC: the conditional correspondence then applies, and its bijection necessarily sends that datum to this class, with $q_{N} = 0$ . The direct classification and its no-stabilizer conclusion were proved before this comparison and require no irreducibility assumption.

The zero-charge regular datum and framed flat ASD sector

Example

Example references

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources