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Chern character and charge of the normalized twistor monad

Statement

Let $S^{4} = H P^{1}$ have the unit round metric and the orientation $d x^{0} \land d x^{1} \land d x^{2} \land d x^{3}$ in the stereographic coordinates $z_{1} = x^{0} + i x^{1}$ , $z_{2} = x^{2} + i x^{3}$ . Write $Z = P_{C} (H^{2}) ≅ C P^{3}$ , fix the right quaternionic structure $j$ with $j^{2} = - 1$ , and define $π : Z \to S^{4}$ by $π ([v]) = {span}_{C} (v, j v)$ . Its fibers are the real twistor lines. The complex orientation on $Z$ is the one for which vertical complex orientation followed by the displayed base orientation is the complex orientation. Let $O_{Z} (1)$ be the dual of the tautological line bundle and set $H = c_{1} (O_{Z} (1)) \in H^{2} (Z; Z)$ , viewed in rational cohomology for $ch$ . Let $u \in H^{4} (S^{4}; Z)$ satisfy $⟨ u, [S^{4}] ⟩ = 1$ .

Fix integers $k \geq 0$ and $N \geq 2$ . Let $E \to S^{4}$ be a framed Hermitian rank- $N$ bundle, with a smooth unitary connection $A$ and Ward bundle $E$ whose underlying smooth bundle is $π^{*} E$ . Write $F_{A}$ for the anti-Hermitian curvature and $F_{A}^{+} = (F_{A} + *_{g} F_{A}) / 2$ , where $*_{g}$ is the Hodge star for the stated metric and orientation; assume $F_{A}^{+} = 0$ . Define

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

Suppose $E$ is the middle cohomology of a holomorphic complex exact at its end terms,

$O_{Z} (- 1)^{\oplus k} ⟶ O_{Z}^{\oplus (2 k + N)} ⟶ O_{Z} (1)^{\oplus k} .$

Assume this middle cohomology is locally free. Then $ch (E) = N - k H^{2}$ , $c_{1} (E) = 0$ , and $c_{2} (E) = k H^{2}$ . For the stated positive twistor orientation,

$π^{*} u = H^{2} .$

Consequently $k = ⟨ c_{2} (E), [S^{4}] ⟩ = - q_{N} (E, A)$ .

Statement references

Zero-FI regular ADHM and framed twistor data

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

Donaldson, §§4–6, printed pp. 6–11

Nakajima–Yoshioka, §3.1, printed pp. 17–19

Facts & Assumptions

Given: The normalized monad, the smooth Ward pullback identification, the specified orientation, and a framed unitary ASD connection on $S^{4}$ .

[F1]

The total Chern character is additive on exact complexes of vector bundles, $ch (O (1)) = e^{H}$ , $ch (O (- 1)) = e^{- H}$ , and on $C P^{3}$ all classes of degree greater than six vanish. The Chern character and the Todd class

[F2]

For the oriented submersion $π$ , the pullback of the positive point class is Poincaré dual to the twistor fibre. Each fibre is a complex projective line, and $H$ integrates to $1$ on it. Zero-FI regular ADHM and framed twistor data, The normal Thom class realizes the Poincare dual of a closed submanifold, Integral cohomology ring of complex projective space

[F3]

For a framed unitary ASD $U (N)$ bundle on the round $S^{4}$ , the determinant connection is flat and the specified trace normalization gives $- q_{N} = ⟨ c_{2} (E), [S^{4}] ⟩ \geq 0$ . Determinant flatness and charge normalization for framed U(N) ASD data

[F4]

Chern classes are natural under pullback, the second Chern class of a bundle with $c_{1} = 0$ is represented by $- {ch}_{2}$ , and Chern–Weil forms represent the corresponding real characteristic classes. Characteristic forms represent topological characteristic classes over the reals, The Chern character and the Todd class

Proof

Given: The normalized monad and the Ward pullback identification in the Statement.

1.1F1F4algebra

In the Grothendieck group, exactness of the monad gives $[E] = (2 k + N) [O] - k [O (- 1)] - k [O (1)]$ . By [F1], its Chern character is $(2 k + N) - k e^{- H} - k e^{H} = N - k H^{2}$ , because the constant terms cancel to $N$ , the linear terms cancel, the quadratic contribution is $- k H^{2}$ , and terms of degree at least four vanish on $Z$ . Therefore $c_{1} (E) = 0$ and $c_{2} (E) = - {ch}_{2} (E) = k H^{2}$ .

1.2F2algebra

Let $x \in S^{4}$ . In the explicit incidence chart of the Definition, $μ_{1} = z_{1}$ , $μ_{2} = z_{2}$ at $λ_{2} = 0$ , so the vertical complex orientation followed by the positive stereographic orientation is the complex orientation on $Z$ . The submersion pulls the oriented normal disk at $x$ back to the normal disk of its fibre; fibrewise normalization and uniqueness of the Thom class, followed by its relative-to-absolute map, therefore pull the positive point dual to the positive fibre dual. The class $u$ is Poincaré dual to the positively oriented point $x$ , so $π^{*} u$ is Poincaré dual to the fibre $L_{x}$ . By [F2] that fibre has its complex line orientation; as a linear $C P^{1} \subset C P^{3}$ its Poincaré dual is $H^{2}$ , since $\int_{L_{x}} H = 1$ and $H^{2}$ generates degree-four cohomology. The twistor orientation clause in the Definition makes this the positive pullback class, hence $π^{*} u = H^{2}$ .

2.1F3step 1.1step 1.2algebra∎

Write $c_{2} (E) = m u$ for the integer $m = ⟨ c_{2} (E), [S^{4}] ⟩$ . Naturality and the Ward pullback in the Statement give $m H^{2} = π^{*} c_{2} (E) = c_{2} (E) = k H^{2}$ by step 1.1 and step 1.2. Since $H^{2}$ is nonzero, $m = k$ . Finally [F3] gives $- q_{N} = m = k$ , with the nonnegative sign fixed by the ASD equation.

Chern character and charge of the normalized twistor monad

Statement

Statement references

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources