Conditional zero-FI regular ADHM and framed ASD correspondence

By [A1], every regular complex orbit meets the zero real level in exactly one $U(k)$ orbit. Thus inclusion of the zero-level regular locus induces the stated diffeomorphic identification with the regular complex quotient. The compact action [F1] commutes with gauge and preserves both zero moments, so the identification is $T$-equivariant.

To construct the forward map from an arbitrary regular zero-moment datum, use [F4]'s global raw monad. Write $F=\ker \beta \mathrm{/}\mathrm{im}\alpha$ and $K=V\oplus V\oplus W$. The identity $\beta (v)=\alpha (\sigma v{)}^{†}$ gives $\ker \beta (v)=(\mathrm{im}\alpha (\sigma v){)}^{\perp }$. The standard first-slot-linear Hermitian form on $K$ therefore descends to a pairing between $F_v$ and $F_{\sigma v}$: changing either representative by its $\alpha$-image pairs to zero. The annihilator identities show that the induced pairing is nondegenerate; the Hermitian symmetry follows from that on $K$. Conjugate pullback makes it holomorphic. On a finite real line, [F4]'s Euler decomposition identifies its section space with $\ker D_z$ and the pairing with the restricted positive form; at infinity these are exactly $W$ and its fixed frame. The two locally split monad sequences also split as smooth bundles. Their total Chern classes give $c(F)=((1-H)(1+H){)}^{-k}$, hence $c_1(F)=0$ and $c_2(F)=k{H}^2$. Thus every hypothesis of the inverse Ward construction [A4] is met. It produces a smooth global framed unitary ASD connection, and [F2]–[F3] give $q_N=-k$ with the specified orientation and trace.

We now prove the additional Ward bridge over $D=\mathbb{C}[ϵ]\mathrm{/}({ϵ}^2)$; it is not part of [A4]'s ordinary Statement. A first-order ASD object means a locally free rank-$N$ module over $C_{S^4}^{\mathrm{\infty }}\otimes D$, with a nondegenerate $D$-Hermitian form positive on reduction, unitary framing and form-preserving connection ${\mathrm{\nabla }}_D$, with $F_{{\mathrm{\nabla }}_D}^{+}=0$ coefficientwise. Morphisms are framed unitary $D$-linear connection isomorphisms. Pull back and define the vertical/horizontal operator using [A4]'s incidence. Its square vanishes by the same curvature identity coefficientwise. Viewed over $\mathbb{C}$, this is an integrable Dolbeault operator on a smooth rank-$2N$ bundle, so [F5] supplies holomorphic frames. Multiplication by $ϵ$ is an intertwining map, hence holomorphic, with square zero and image equal to kernel of complex rank $N$. At a point choose $N$ holomorphic sections $s_a$ whose $ϵs_a$ form a basis of that image. The $2N$ sections $s_a,ϵs_a$ remain a complex frame nearby by a nonzero determinant. Thus the $s_a$ are a holomorphic $D$-basis. The pairing is holomorphic by metric compatibility as in [A4], and every real-line restriction is the constant free module of its base fibre. This gives the forward first-order Ward object and preserves its framing.

By [A2], regular complex orbits correspond exactly to locally free framed sheaves on ${\mathbb{P}}^2$: stability gives torsion-freeness and costability gives local freeness. At positive level, $k=1$, $B_1=B_2=J=0$ and $I{I}^{†}=\zeta >0$ satisfy both moment equations and are stable, whereas $V\subset \ker J$ violates costability. Thus the positive-FI quotient includes a non-locally-free point and is not identified here with the ordinary ASD sector. This proves the boundary assertion without using any Ward-family claim.

This Ward connection is the raw projection connection on the finite chart. On the complexified line-parameter chart $X$, let $a_X={\alpha }_{{\lambda }_1}$, $b_X={\alpha }_{{\lambda }_2}$ and $S_X=\ker {\beta }_{{\lambda }_1}\cap \ker {\beta }_{{\lambda }_2}$. Along the real locus these are $a_z,b_z,\ker D_z$; the complement and invertible Gram persist near each real line. Along a parameter direction $v$ in $\mathcal{D}=\ker (de)$ at $[\lambda ]$, the point of $Z$ is fixed, so differentiating $\beta (\lambda )s=0$ for a section $s(X)\in S_X$ gives $\beta (\lambda )ds(v)=0$. At the real locus write the component of $ds(v)$ perpendicular to $\ker D_z$ as $a_{z}u+b_{z}w$. Since $\beta (\lambda )=f_z(-{\lambda }_2,{\lambda }_1)$ on these two summands, its kernel condition is $-{\lambda }_{2}u+{\lambda }_{1}w=0$. Hence $(u,w)=({\lambda }_{1}r,{\lambda }_{2}r)$ and that component is $\alpha (\lambda )r$, zero in monad cohomology. Evaluation therefore transports the canonical relative derivative to $P_{z}ds(v)$. The cotangent uniqueness in [A4]'s incidence construction identifies the full connection with $Pd$. The global Ward bundle supplies its smooth extension at infinity.

Conversely let $F_D$ be a holomorphic locally free $D$-bundle in the stated monad range. Its underlying complex bundle $G$ has rank $2N$ and is trivial on every real line; no positive complex form on $G$ is assumed. Here is the needed section-bundle argument without that assumption. Algebraize $G$ by [A3] and form the universal proper flat line family $p:\mathcal{L}\to \mathrm{Gr}(2,4)$, $e:\mathcal{L}\to Z$, with its graph charts as in [F5]. The locally free $e^{\ast }G$ is base-flat. At a real line its fibre has $H^1=0$ and $H^0={\mathbb{C}}^{2N}$. The degree-one base-change map is therefore surjective. By [F5] it is an isomorphism; finiteness from the projective cohomology complex and Nakayama kill the degree-one stalk near that line. The degree-one local-freeness criterion then makes degree-zero base change surjective, hence an isomorphism with locally free section bundle of rank $2N$. Evaluation is an isomorphism on the real line. Removing the proper image of its determinant-zero locus makes it an isomorphism on an entire neighboring line family. Analytifying evaluation and using that holomorphic functions on a product with ${\mathbb{P}}^1$ are constant along its fibres identifies the actual analytic section bundle, exactly as in [F5]. Multiplication by $ϵ$ commutes with evaluation and has image equal to kernel of rank $N$. The local frame argument of step 1.3 now makes this section bundle free of rank $N$ over $\mathcal{O}\otimes D$ and evaluation $D$-linear. Restriction to the real locus is a smooth free $D$-bundle.

Conversely, start with a framed ASD connection. By [A4] it has its positive Ward bundle, and the reconstruction supplier produces a monad of ranks $k,2k+N,k$, including flat summands. Apply [A5] to this monad to extract its regular zero-moment orbit. Applying step 1.2 to that orbit returns the original framed Ward bundle; the inverse identity in [A4] returns the original connection. Starting with raw data, the cohomology isomorphism in [A4] followed by the chain lifting and inverse extraction in [A6] returns that same normalized monad up to exactly $U(k)$. Thus both composites on framed classes are identities. This uses [A5] only in its stated ASD-to-data direction.

On the graph-chart universal family, $\mathcal{D}=\ker (de)$ consists of variations $v$ with $v\lambda =0$. The canonical flat derivative along $\mathcal{D}$ on $e^{\ast }{F}_D$ transfers by evaluation to the pulled-back section bundle. The restriction of cotangent forms gives $p_{\ast }{\mathcal{D}}^{\ast }\cong {\mathrm{\Omega }}_{\mathrm{Gr}(2,4)}^1$: on each line its kernel is $\mathcal{O}(-1{)}^{\oplus 2}$ with $H^0=H^1=0$, and the map is equivalently the restriction of linear covectors on $\mathrm{Hom}({\mathbb{C}}^2,{\mathbb{C}}^2)$ to all planes $v\lambda =0$. These planes span that four-dimensional space; both spaces of covectors have dimension four. The same isomorphism tensored with $D$ therefore pushes the partial derivative to a unique full $D$-connection, with the Leibniz rule. It is flat on every alpha-plane, hence ASD on the real locus by the incidence curvature polynomial. Duals, conjugate pullback, pairing and morphisms commute with evaluation and this unique construction. Consequently it preserves the induced $D$-Hermitian form and unitary framing. Evaluation intertwines the reconstructed vertical/horizontal operator with the original one, and starting from a connection recovers all its alpha-plane derivatives, which determine the full connection. These prove inverse identities on framed $D$-objects and morphisms. All constructions commute with reduction and $ϵ\mapsto rϵ$, $r\in \mathbb{R}$; this also proves the first-order Ward naturality missing from the ordinary supplier.

The action here is the compact $T$ of [F1]. Since ${\bar{t}}_i=t_i^{-1}$ and $h^{†}=h^{-1}$, [F4]'s chain factors are $(t_1{t}_2{)}^{-1}$, $\mathrm{diag}(t_2^{-1},t_1^{-1},h)$ and $1$; the induced map at infinity is $h$. Ward reconstruction and inverse extraction commute with these maps by [A4] and [A6]. Thus the bijection of step 3.1 is $T$-equivariant. No assertion for nonunitary complex-torus parameters is needed.

For precision, a tangent vector at a fixed ordinary framed object is a first-order object with identified reduction, modulo isomorphisms inducing identity on that reduction. On the raw side write $x_D=x+ϵ\dot{x}$; the tangent quotient is the real-linear kernel of $(d{\mu }_{\mathbb{C}},d{\mu }_{\mathbb{R}}{)}_x$ modulo $([\xi ,B_1],[\xi ,B_2],\xi I,-J\xi )$, ${\xi }^{†}=-\xi$, restricted to the stated monad/base-change range. On the ASD side smooth first-order bundle transitions differ from the reduced ones by an additive $\mathrm{End}(E)$ cocycle. A partition of unity [F6] splits it: in common bundle notation, for $c_{ij}+c_{jk}=c_{ik}$, $u_i={\sum }_j{\rho }_j{c}_{ij}$ gives $u_i-u_j=c_{ij}$. A change $1-ϵh_0^{-1}{h}_1\mathrm{/}2$ normalizes a metric $h_0+ϵh_1$, and a smooth adjustment near infinity fixes the framing. Thus these tangents are represented by $A+ϵa$, $a\in {\mathrm{\Omega }}^1(S^4,\mathfrak{u}(E))$, with $(d_{A}a{)}^{+}=0$, modulo $d_A\xi$ for smooth $\xi \in \mathrm{\Gamma }(\mathfrak{u}(E))$ with $\xi (\mathrm{\infty })=0$. Here $\mathfrak{u}(E)$ is the bundle of skew-Hermitian endomorphisms and $d_A$ is the covariant exterior derivative. Restrict to those Ward objects admitting the exact monad, section, coefficient-image and base-change conditions already imposed in the Statement; nothing asserts them for all infinitesimal ASD objects.

Apply [A6] over $D$ to precisely those monads. Its chain lifting also extends over $D$: the two locally split kernel/image sequences give the same Hom restriction maps, whose kernels and cokernels vanish by the degree-zero, degree-one and degree-two groups of $\mathcal{O}(-1)$ and $\mathcal{O}(-2)$. The projective twisting-sheaf calculation in [F5] supplies those vanishings over $D$, and $H^0(\mathcal{O})=D$, so the lift is uniquely a constant $D$-linear chain map. Uniqueness makes it preserve the lifted pairing whenever the cohomology map does. The raw construction remains valid because every required inverse matrix has invertible reduction, with $(M_0+ϵM_1{)}^{-1}=M_0^{-1}-ϵM_0^{-1}{M}_1{M}_0^{-1}$, and its real-line Euler decomposition holds over $D$. Together with steps 1.3, 2.3 and 3.2 and [A3]'s exact comparison this gives inverse maps on the restricted first-order classes. If a holomorphic $D$-object needs algebraization, apply absolute GAGA to its coherent underlying complex bundle and to its holomorphic nilpotent endomorphism; full faithfulness transports the $D$-action and exactness transports kernel and image. Its locally free $D$-structure follows from the same rank-$N$ frame argument, so no general relative algebraization theorem is being assumed. The comparison with framed-sheaf families uses [A2] only with its stipulated relative cohomology ranks and base-change isomorphisms. On fixed reductions all first-order equations, derivatives, changes of bases and gauge actions are real-linear in the $ϵ$ coefficients; evaluation and the inverse uniquely determined connection respect their sums. The induced inverse tangent maps are therefore real-linear. This proves the stated first-order compatibility in exactly its prescribed range, including $k=0$ and flat summands, and completes every clause.