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Zero-charge positive real monad

Example

Let $N \geq 2$ , $k = 0$ , $V = C^{0}$ , and let $W = C^{N}$ with its standard Hermitian form and fixed unitary framing. The unique maps $B_{1}, B_{2} \in End (V)$ , $I : W \to V$ , and $J : V \to W$ are zero. Their ADHM moments vanish, the stability and costability conditions are vacuous, and the raw monad is $0 ⟶ W \otimes O ⟶ 0.$ Its middle cohomology is the framed trivial bundle $W \otimes O$ , positive on every real-line section space. The normalized gauge group $U (0)$ has one element, so no gauge or normal-form parameters remain. Its algebraic and topological second Chern classes vanish and its Euler characteristic is $N$ .

Example references

Positive real linear monad normal forms

ADHM linear data, stability, costability and FI levels

Chern character and charge normalization of a linear twistor monad

Donaldson, Sections 3–4 and 6

Facts & Assumptions

Given: $V = 0$ , $W = C^{N}$ with $N \geq 2$ , and its prescribed unitary framing.

[F1]

On the zero vector space there is exactly one endomorphism and exactly one map to or from it; stability and costability hold vacuously and $U (0)$ is the trivial group. ADHM linear data, stability, costability and FI levels

[F2]

The finite monad resolution has algebraic $ch (F) = N - k H_{a l g}^{2}$ , topological $c_{2}^{t o p} (F) = k H_{t o p}^{2}$ , and $χ (F) = N - 2 k$ . The charge supplier defines the twistor projection by $π ([v]) = {span}_{C} (v, ȷ v)$ , with fibers the real incidence lines, and proves $π^{*} u = H_{t o p}^{2}$ in the positive orientation; its conclusion $c_{2} (E) = k u$ is conditional on the topological pullback $F ≅ π^{*} E$ . Chern character and charge normalization of a linear twistor monad

Proof

Given: The zero-dimensional ADHM data and the standard Hermitian space $W$ .

1.1givenF1algebra

Every map into or out of $V = 0$ is the unique zero map. Thus both moments are zero. Every word vector $B_{i_{1}} \dots B_{i_{m}} I (w)$ is zero, including the $m = 0$ vectors $I (w)$ for $w \in W$ , so their span is $0 = V$ . No nonzero invariant subspace exists; hence stability and costability hold. Since $U (0)$ has one element, the data have no gauge parameters.

2.1givenstep 1.1algebra

Substituting $V = 0$ into the monad leaves $0 \to W \otimes O \to 0$ . Its middle cohomology is canonically $W \otimes O$ , and the identity on $W$ is the unitary framing at infinity. The standard form on $W$ induces the same positive form on every real-line section space. There are no end maps or hidden Euler summands.

3.1F2step 2.1algebra∎

Setting $k = 0$ in [F2] gives $ch (F) = N$ , $c_{2} (F) = 0$ , and $χ (F) = N$ . Directly, $F = W \otimes O$ has these same classes; if it is written as $π^{*} E$ , naturality gives $c_{2} (E) = 0$ under the charge lemma's stated pullback condition. Thus the example has zero charge and retains all $N$ framing directions.

Depends on

Dependency tree · 2 levels

Sources

Simon K. Donaldson, The ADHM construction of Yang–Mills instantons