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Determinant flatness and charge normalization for framed U(N) ASD data

Statement

Assume full Axiom of Choice. Let $S^{4} = H P^{1}$ have the unit round metric and positive stereographic orientation $d x^{0} \land d x^{1} \land d x^{2} \land d x^{3}$ , where $z_{1} = x^{0} + i x^{1}$ , $z_{2} = x^{2} + i x^{3}$ . Let $E \to S^{4}$ be a smooth Hermitian rank- $N$ bundle, $N \geq 2$ , with a smooth unitary connection $A$ and a unitary framing $Φ : E_{\infty} \to C^{N}$ . Its curvature $F_{A}$ is an anti-Hermitian endomorphism-valued two-form; set $F_{A}^{+} = (F_{A} + *_{g} F_{A}) / 2$ , where $*_{g}$ is the Hodge star for the stated metric and orientation, and assume $F_{A}^{+} = 0$ . The determinant-line connection is the connection induced by $A$ on $\det E = Λ^{N} E$ . Its framing at $\infty$ determines a unit vector in $(\det E)_{\infty}$ . With

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

and determinant Chern-form convention $c (A) = \det (I - F_{A} / (2 π i))$ , one has

$q_{N} (E, A) = - ⟨ c_{2} (E), [S^{4}] ⟩ \in - Z_{\geq 0} .$

Thus $k = - q_{N} = ⟨ c_{2} (E), [S^{4}] ⟩$ is a nonnegative integer, with the sign corresponding to the anti-self-dual equation $F_{A}^{+} = 0$ .

The induced determinant-line connection is flat. Its framing at $\infty$ determines a unique global unit parallel section, so the framed unitary bundle reduces to a framed $S U (N)$ bundle.

Statement references

Zero-FI regular ADHM and framed twistor data

Trace-normalized instanton number and self-dual sectors

De rham cohomology of spheres

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

Characteristic forms represent topological characteristic classes over the reals

Facts & Assumptions

Given: Full AC, a smooth framed Hermitian rank- $N$ bundle on the specified compact oriented round $S^{4}$ , a smooth unitary connection, and $F_{A}^{+} = 0$ .

[A1]

Full AC is the choice-function principle; restricting its choice function to any countable family gives countable choice. The Axiom of Choice

[F1]

In local unitary frames $F_{A} = d A + A \land A$ is anti-Hermitian, the Hodge star is that of the specified orientation, and $q_{N} = - (8 π^{2})^{- 1} \int {tr}_{N} (F_{A} \land F_{A})$ . Zero-FI regular ADHM and framed twistor data, Trace-normalized instanton number and self-dual sectors

[F2]

The trace curvature obeys $D_{A} F_{A} = 0$ , and trace annihilates commutators, so $d tr F_{A} = 0$ . Principal curvature, Cartan structure equation, and Bianchi identity

[F3]

$H_{d R}^{2} (S^{4}) = 0$ under countable choice, and $S^{4}$ is simply connected. De rham cohomology of spheres, $S^{n}$ is simply connected for every $n \geq 2$

[F4]

A flat connection has local parallel frames and path-independent parallel transport on sufficiently small coordinate balls. A flat connection admits local parallel frames, Flat connections have locally path-independent parallel transport on a coordinate ball

[F5]

The total Chern form is $\det (I - F_{A} / (2 π i))$ , and its real de Rham class is the realification of the topological Chern class. Chern, Pontryagin, and Euler characteristic forms, Chern classes from the projective-bundle relation, Characteristic forms represent topological characteristic classes over the reals, The de Rham theorem

Proof

1.1givenF1F2algebra

In a local unitary frame let $F_{\det A}$ be the curvature of the induced connection on $\det E$ . The induced connection on the top exterior power has local connection matrix $tr A$ , hence $F_{\det A} = tr F_{A}$ . Put $η = - i tr F_{A}$ , a real two-form because $F_{A}$ is anti-Hermitian. By [F2], $d η = 0$ , and the linearity of trace in $F_{A}^{+} = 0$ gives $*_{g} η = - η$ .

2.1A1F3step 1.1algebra

By [A1] and [F3], $η$ is exact, so choose a smooth real one-form $α$ with $η = d α$ . The ASD equation gives $\int_{S^{4}} η \land *_{g} η = - \int_{S^{4}} d α \land η = - \int_{S^{4}} d (α \land η) = 0$ , since $d η = 0$ and $S^{4}$ is compact without boundary. The integrand $η \land *_{g} η$ is the nonnegative squared norm density of the real form $η$ ; zero integral forces $η = 0$ pointwise. Hence $F_{\det A} = 0$ and the determinant connection is flat.

3.1F3F4step 2.1construct

Choose the unit vector $v_{\infty}$ in $(\det E)_{\infty}$ specified by the determinant of the framing. For any $x \in S^{4}$ , parallel transport $v_{\infty}$ along a piecewise smooth path from $\infty$ to $x$ . If two paths are chosen, their concatenation is a loop. By [F3] the loop is null-homotopic; subdivide a null-homotopy square into finitely many rectangles mapped into coordinate balls, which is possible by compactness and a Lebesgue number for the pulled-back coordinate-ball cover. On each rectangle [F4] makes boundary transport the identity. Multiplying around the subdivision cancels the interior edges, so the two path transports agree. The resulting section is smooth by local parallel frames, is parallel by construction, and agrees with $v_{\infty}$ ; uniqueness follows from uniqueness of parallel sections along each path. Its unit norm is preserved by unitary parallel transport, so the determinant framing reduces the structure group to $S U (N)$ .

4.1F1F5step 2.1algebra∎

Writing $C = - F_{A} / (2 π i)$ , the degree-four term of $\det (I + C)$ is $c_{2} (A) = \frac{1}{2} ((tr C) \land (tr C) - tr (C \land C))$ . Step 2.1 gives $tr F_{A} = 0$ , so this reduces to $c_{2} (A) = (8 π^{2})^{- 1} {tr}_{N} (F_{A} \land F_{A})$ . Therefore $q_{N} = - \int_{S^{4}} c_{2} (A)$ . By [F5] and the de Rham integration pairing, $\int_{S^{4}} c_{2} (A) = ⟨ c_{2} (E), [S^{4}] ⟩$ , an integer. Finally $q_{N} = (8 π^{2})^{- 1} \int B (F_{A} \land F_{A})$ for the positive invariant form $B (X, Y) = - {tr}_{N} (X Y)$ ; since $F_{A} = - *_{g} F_{A}$ and $B$ is positive definite, $q_{N} = - (8 π^{2})^{- 1} \int B (F_{A} \land *_{g} F_{A}) \leq 0$ . This proves the stated sign and nonnegativity of $k = - q_{N}$ .

Determinant flatness and charge normalization for framed U(N) ASD data

Statement

Statement references

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources